statistique, théorie et gestion de portefeuille - Docs at ISFA

UNIVERSITE DE NICE-SOPHIA ANTIPOLIS - UFR SCIENCES
Ecole Doctorale de Sciences Fondamentales et Appliquées
THESE
Présentée pour obtenir le titre de
Docteur en SCIENCES
de l’Université de Nice - Sophia Antipolis
Spécialité : Physique
par
Yannick MALEVERGNE
Risques extrêmes en finance :
statistique, théorie et gestion de portefeuille
Soutenue publiquement le 20 décembre 2002 devant le jury composé de :
Jean-Claude AUGROS Professeur, Université Lyon I (Président)
Jean-Paul LAURENT Professeur, Université Lyon I (Co-directeur de Thèse)
Jean-François MUZY Chargé de recherche au CNRS
Michael ROCKINGER Professeur, HEC Lausanne
Bertrand ROEHNER Professeur, Université Paris VII (Rapporteur)
Didier SORNETTE Directeur de recherche au CNRS (Directeur de Thèse)
à l’Institut de Science Financière et d’Assurances - Université Claude Bernard - Lyon I

Table des matières
Avant-propos 11
Introduction 13
I Etude et modélisation des propriétés des rentabilités des actifs financiers 19
1 Faits stylisés des rentabilités boursières 21
1.1 Rappel des faits stylisés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.1.1 La distribution des rendements . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.1.2 Propriétés de dépendances temporelles . . . . . . . . . . . . . . . . . . . . . . 23
1.1.3 Autres faits stylisés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2 De la difficulté de représenter la distribution des rendements . . . . . . . . . . . . . . . 27
1.3 Modélisation des propriétés de dépendance des rendements . . . . . . . . . . . . . . . . 28
1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Modèles phénoménologiques de cours 31
2.1 Bulles rationnelles multi-dimensionnelles et queues épaisses . . . . . . . . . . . . . . . 32
2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.2 Hypothèses du modèle de Blanchard et Watson . . . . . . . . . . . . . . . . . . 34
2.1.3 Généralisation des bulles rationnelles en dimensions quelconques . . . . . . . . 36
2.1.4 Théorie du renouvellement des produits de matrices aléatoires . . . . . . . . . . 37
2.1.5 Conséquences pour les bulles rationnelles . . . . . . . . . . . . . . . . . . . . . 39
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Des bulles rationnelles aux krachs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.1 Les modèles de bulles rationnelles . . . . . . . . . . . . . . . . . . . . . . . . . 43
3

4 Table des matières
2.2.2 Bulles rationnelles pour un actif isolé . . . . . . . . . . . . . . . . . . . . . . . 46
2.2.3 Généralisation des bulles rationnelles à un nombre arbitraire de dimensions . . . 49
2.2.4 Modèle de krach avec taux de hasard . . . . . . . . . . . . . . . . . . . . . . . 54
2.2.5 Modèle de croissance non stationnaire . . . . . . . . . . . . . . . . . . . . . . 56
2.2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Distributions exponentielles étirées contre distributions régulièrement variables 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2 Statistiques descriptives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.1 Les données . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2.2 Existence de dépendance temporelle . . . . . . . . . . . . . . . . . . . . . . . 70
3.3 Propriétés extrêmes d’un processus à mémoire longue . . . . . . . . . . . . . . . . . . 70
3.3.1 Résultats théoriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3.2 Exemples de convergence lente vers les distributions des valeurs extrêmes et de
Pareto généralisées . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.3.3 Génération d’un processus à mémoire longue avec des marginales prescrites . . 73
3.3.4 Simulations numériques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3.5 Estimateurs des paramètres des GEV et GPD pour les données réelles . . . . . . 77
3.4 Estimation paramétrique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.4.1 Définition d’une famille de distributions à trois paramètres . . . . . . . . . . . . 78
3.4.2 Méthodologie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.3 Résultats empiriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.4.4 Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5 Comparaison du pouvoir descriptif des différentes familles . . . . . . . . . . . . . . . . 84
3.5.1 Comparaison entre les quatre distributions paramétriques et la distribution globale 85
3.5.2 Comparaison directe des modèles de Pareto et Exponentiel-Etiré . . . . . . . . 86
3.6 Discussion et conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4 Relaxation de la volatilité 135
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Table des matières 5
4.2 Mémoire longue et distinction entre chocs endogènes et exogènes . . . . . . . . . . . . 139
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5 Approche comportementale des marchés financiers : l’apport des modèles d’agents 149
5.1 Prix d’un actif et excès de demande . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.2 Modèles d’opinion contre modèles de marché . . . . . . . . . . . . . . . . . . . . . . . 152
5.3 Modèles à agents adaptatifs contre modèles à agents non adaptatifs . . . . . . . . . . . . 154
5.4 Conséquences des phénomènes d’imitation et d’antagonisme... . . . . . . . . . . . . . . 155
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6 Comportements mimétiques et antagonistes : bulles hyperboliques, krachs et chaos 159
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.2 Le modèle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.3 Analyse qualitative des propriétés dynamiques . . . . . . . . . . . . . . . . . . . . . . 165
6.4 Analyse quantitative des bulles spéculatives en régime chaotique dans le cas symétrique 167
6.5 Propriétés statistiques des rendements dans le cas symétrique . . . . . . . . . . . . . . 169
6.6 Cas asymétriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.7 Effets de taille finie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Appendice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
II Etude des propriétés de dépendances entre actifs financiers 179
7 Etude de la dépendance à l’aide des copules 181
7.1 Les copules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.2 Quelques familles de copules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.2.1 Les copules gaussiennes et copules de Student . . . . . . . . . . . . . . . . . . 184
7.2.2 Les copules archimédiennes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.3 Tests empiriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.3.1 Tests paramétriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6 Table des matières
7.3.2 Tests non paramétriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8 Tests de copule gaussienne 191
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.2 Généralités sur les copules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.2.1 Définitions et résultats importants concernant les copules . . . . . . . . . . . . 195
8.2.2 Dépendance entre deux variables aléatoires . . . . . . . . . . . . . . . . . . . . 196
8.2.3 La copule gaussienne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.2.1 La copule de Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.3 Tester l’hypothèse de copule gaussienne . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.3.1 statistique de test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.3.2 Procédure de test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8.3.3 Sensibilité de la méthode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
8.4 Résultats empiriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.4.1 Les devises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.4.2 Les matières premières : les métaux . . . . . . . . . . . . . . . . . . . . . . . . 209
8.4.3 Les actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
9 Mesure de la dépendance extrême entre deux actifs financiers 237
9.1 Les différentes mesures de dépendances extrêmes . . . . . . . . . . . . . . . . . . . . . 238
9.1.1 Coefficients de corrélation conditionnels . . . . . . . . . . . . . . . . . . . . . 243
9.1.2 Mesures de dépendances conditionnelles . . . . . . . . . . . . . . . . . . . . . 249
9.1.3 Coefficient de dépendance de queue . . . . . . . . . . . . . . . . . . . . . . . . 252
9.1.4 Synthèse et discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.2 Estimation du coefficient de dépendance de queue . . . . . . . . . . . . . . . . . . . . . 299
9.2.1 Mesure intrinsèque de dépendance extrême . . . . . . . . . . . . . . . . . . . . 303
9.2.2 Coefficient de dépendance de queue pour un modèle à facteurs . . . . . . . . . 305
9.2.3 Etude empirique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Table des matières 7
9.2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
9.3 Synthèse de la description de la dépendance entre actifs financiers . . . . . . . . . . . . 340
III Mesures des risques extrêmes et application à la gestion de portefeuille 343
10 La mesure du risque 345
10.1 La théorie de l’utilité . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
10.1.1 Théorie de l’utilité en environnement certain . . . . . . . . . . . . . . . . . . . 346
10.1.2 Théorie de la décision face au risque . . . . . . . . . . . . . . . . . . . . . . . . 347
10.1.3 Théorie de la décision face à l’incertain . . . . . . . . . . . . . . . . . . . . . . 350
10.2 Les mesures de risque cohérentes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
10.2.1 Définition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
10.2.2 Quelques exemples de mesures de risque cohérentes . . . . . . . . . . . . . . . 354
10.2.3 Représentation des mesures de risque cohérentes . . . . . . . . . . . . . . . . . 355
10.2.4 Conséquences sur l’allocation de capital . . . . . . . . . . . . . . . . . . . . . . 356
10.2.5 Critique des mesures cohérentes de risque . . . . . . . . . . . . . . . . . . . . . 357
10.3 Les mesures de fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
10.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
11 Portefeuilles optimaux et équilibre de marché 363
11.1 Les limites de l’approche moyenne - variance . . . . . . . . . . . . . . . . . . . . . . . 364
11.2 Prise en compte des grands risques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
11.2.1 Optimisation sous contrainte de capital économique . . . . . . . . . . . . . . . . 365
11.2.2 Optimisation sous contrainte de fluctuations autour du rendement espéré . . . . . 366
11.2.3 Optimisation sous d’autres contraintes . . . . . . . . . . . . . . . . . . . . . . . 366
11.3 Equilibre de marché . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
11.5 Annexe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
12 Gestion des risques grands et extrêmes 373
12.1 Comprendre et gérer les risques grands et extrêmes . . . . . . . . . . . . . . . . . . . . 374

8 Table des matières
12.1.1 Distributions des rendements à queues épaisses . . . . . . . . . . . . . . . . . . 376
12.1.2 Dépendance temporelle intermittente à l’origine des grandes pertes . . . . . . . 376
12.1.3 Dépendance de queue et contagion . . . . . . . . . . . . . . . . . . . . . . . . 377
12.1.4 Nature multidimensionnelle des risques . . . . . . . . . . . . . . . . . . . . . . 378
12.1.5 Petits risques, grands risques et rendement . . . . . . . . . . . . . . . . . . . . 378
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
12.2 Minimiser l’impact des grands co-mouvements . . . . . . . . . . . . . . . . . . . . . . 381
12.2.1 Quantifier les grands co-mouvements . . . . . . . . . . . . . . . . . . . . . . . 382
12.2.2 Dépendance de queue génerée par un modèle à facteur . . . . . . . . . . . . . . 383
12.2.3 implémentation pratique et conséquences . . . . . . . . . . . . . . . . . . . . . 384
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
13 Gestion de portefeuille sous contraintes de capital économique 387
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
13.1 Définition et concepts importants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
13.1.1 Distributions de Weibull modifiées . . . . . . . . . . . . . . . . . . . . . . . . 392
13.1.2 Equivalence de queue pour les fonctions de distribution . . . . . . . . . . . . . 393
13.1.3 La copule gaussienne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
13.2 Distribution de richesse d’un portefeuille pour différentes structures de dépendance . . . 395
13.2.1 Distribution de richesse pour des actifs indépendants . . . . . . . . . . . . . . . 395
13.2.2 Distribution de richesse pour des actifs comonotones . . . . . . . . . . . . . . . 396
13.2.3 Distribution de richesse sous hypothèse de copule gaussienne . . . . . . . . . . 398
13.3 Value-at-Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
13.4 Portefeuilles Optimaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
13.4.1 Portefeuilles à risque minimum . . . . . . . . . . . . . . . . . . . . . . . . . . 401
13.4.2 Portefeuilles VaR-efficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
13.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
14 Gestion de Portefeuilles multimoments et équilibre de marché 423
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
14.2 Mesure des grands risques d’un portefeuille . . . . . . . . . . . . . . . . . . . . . . . . 428

Table des matières 9
14.2.1 Pourquoi les moments d’ordres élevés permettent-ils de quantifier de plus grands
risques ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
14.2.2 Quantification des fluctuations d’un actif . . . . . . . . . . . . . . . . . . . . . 429
14.2.3 Exemples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
14.3 La frontière efficiente généralisée . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
14.3.1 Frontière en l’absence d’actif sans risque . . . . . . . . . . . . . . . . . . . . . 432
14.3.2 Frontière efficiente en présence d’actif sans risque . . . . . . . . . . . . . . . . 433
14.3.3 Théorème de séparation en deux fonds . . . . . . . . . . . . . . . . . . . . . . 433
14.3.4 Influence du taux d’intérêt sans risque . . . . . . . . . . . . . . . . . . . . . . . 434
14.4 Classifications des actifs et des portefeuilles . . . . . . . . . . . . . . . . . . . . . . . . 434
14.4.1 Méthode d’ajustement du risque . . . . . . . . . . . . . . . . . . . . . . . . . . 435
14.4.2 Risque marginal d’un actif au sein d’un portefeuille . . . . . . . . . . . . . . . 436
14.5 Un nouveau modèle d’équilibre de marché . . . . . . . . . . . . . . . . . . . . . . . . 437
14.5.1 Equilibre en marché homogène . . . . . . . . . . . . . . . . . . . . . . . . . . 438
14.5.2 Equilibre en marché hétérogène . . . . . . . . . . . . . . . . . . . . . . . . . . 438
14.6 Estimation de la distribution de probabilité jointe du rendement de plusieurs actifs . . . 439
14.6.1 Brève exposition et justification de la méthode . . . . . . . . . . . . . . . . . . 440
14.6.2 Transformation d’une variable aléatoire en une variable gaussienne . . . . . . . 441
14.6.3 Determination de la distribution jointe : maximum d’entropie et copule gaussienne442
14.6.4 Test empirique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
14.7 Choix d’une famille exponentielle pour paramétrer les distributions marginales . . . . . 444
14.7.1 Les distributions de Weibull modifiées . . . . . . . . . . . . . . . . . . . . . . 444
14.7.2 Transformation des lois de Weibull en gaussiennes . . . . . . . . . . . . . . . . 445
14.7.3 Test empirique et estimation des paramètres . . . . . . . . . . . . . . . . . . . 445
14.8 Développement en cumulants de la distribution du portefeuille . . . . . . . . . . . . . . 446
14.8.1 Lien entre moments et cumulants . . . . . . . . . . . . . . . . . . . . . . . . . 446
14.8.2 Cas de distributions symétriques . . . . . . . . . . . . . . . . . . . . . . . . . 447
14.8.3 Cas de distributions asymétriques . . . . . . . . . . . . . . . . . . . . . . . . . 449
14.8.4 Tests empiriques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
14.9 Peut-on avoir le beurre et l’argent du beurre ? . . . . . . . . . . . . . . . . . . . . . . . 450
14.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453

10 Table des matières
Références . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
Conclusions et Perspectives 491
A Evaluation de la conduite du projet de thèse 495
A.1 Bref résumé du sujet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
A.2 Eléments de contexte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
A.2.1 Choix du sujet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
A.2.2 Choix de l’encadrement et du laboratoire d’accueil . . . . . . . . . . . . . . . . 496
A.2.3 Financement du projet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
A.3 Evolution du projet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
A.3.1 Elaboration du projet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
A.3.2 Conduite du projet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
A.3.3 Retombées scientifiques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
A.4 Compétences acquises et enseignements personnels . . . . . . . . . . . . . . . . . . . . 499
A.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
Bibliographie 503

Avant-propos
La rédaction d’une thèse est un travail de longue haleine, exigeant, quelque déroutant voire décourageant,
tout particulièrement lorsque les difficultés s’accumulent et semblent insurmontables. Mais au final, combien
est captivante, passionnante et enthousiasmante cette expérience, première réelle occasion de mettre
à profit la somme de connaissances patiemment engrangée au cours de longues années d’études.
C’est lorsque vient le moment de la rédaction finale que l’on mesure l’ensemble du chemin parcouru.
Les hésitations et difficultés rencontrées tout au long de la route s’évanouissent et seuls les succès et
satisfactions reviennent en mémoire. Ressurgissent aussi les noms de ceux qui vous ont accompagné sur
cette route, et c’est à eux que je souhaiterai, avant toute chose, adresser ces quelques remerciements.
Le travail que j’ai mené durant ces deux années de thèse doit énormément à Didier Sornette et Jean-Paul
Laurent qui ont accepté de le diriger. Je tiens avant tout à leur exprimer ma gratitude pour ces années
passées à leur contact et voudrais dire le grand plaisir et l’immense satisfaction que j’ai eu à travailler
avec eux. Ils ont su à la fois me guider judicieusement et me laisser une grande latitude quant aux choix
des thèmes de recherche que je souhaitais poursuivre, ce dont je leur suis infiniment reconnaissant.
Je tiens aussi à remercier chaleureusement les professeurs Gouriéroux et Roehner qui m’ont fait l’honneur
d’accepter d’être les rapporteurs de cette thèse. Les questions et remarques qu’ils ont formulées,
ainsi que les discussions que nous avons eues ont été pour moi du plus grand intérêt car elles ont
été génératrices d’idées nouvelles et m’ont ainsi permis d’entrevoir d’autres directions de recherche,
complémentaires de celle suivies jusqu’ici. Je souhaite également exprimer ma reconnaissance aux professeurs
Augros et Rockinger ainsi qu’à Jean-François Muzy – avec qui j’ai eu en outre le grand plaisir
de collaborer – qui ont bien voulu consacrer une part de leur temps à juger mon travail et accepté de
participer à mon jury de thèse.
J’aimerais également remercier les nombreuses personnes avec qui j’ai été amené à travailler sur l’un
ou l’autre des divers thèmes de recherche que j’ai abordé au cours de ma thèse. En premier lieu, je
souhaiterais dire toute mon estime et ma reconnaissance à Vladilen Pisarenko, pour l’aide précieuse
qu’il m’a apporté tout au long de ces deux années. Je voudrais aussi remercier Jorgen Andersen avec
qui j’ai passé les trois mois qui ont précédé le début de ma thèse durant mon stage de DEA ainsi qu’Ali
Chabaane et Françoise Turpin du groupe BNP-Paribas avec qui j’ai entamé une collaboration fructueuse.
Je souhaiterais aussi remercier Anne Sornette pour son soutien et ses encouragements constants mais
également pour m’avoir incité à participer au projet de “Nouveau chapitre de la thèse”, expérience qui
s’est avérée être très intéressante et enrichissante. Incidemment, je remercie Nadjia Hohweiller, consultante
en bilan de compétence à la chambre de commerce et d’industrie de Nice, qui a participé à la
rédaction de ce chapitre et Emmanuel Tric de l’Association Bernard Gregory qui soutient ce projet.
Je voudrais également remercier Bernard Gaypara et Yann Ageon pour l’aide précieuse qu’ils m’ont
apportée dans le domaine informatique, Bernard pour ce qui a été de résoudre les (nombreuses) pannes
de mon ordinateur et autres problèmes d’impression et Yann pour ses compétences de programmeur qui
11

12
m’ont souvent été d’un grand secours.
Enfin, je tiens à remercier ma famille, mes ami(e)s, mes proches et tous ceux qui m’ont encouragé tout
au long de ces deux années. Qu’ils voient dans l’accomplissement de ce travail un témoignage de ma
reconnaissance pour le soutien et la confiance qu’ils m’ont toujours manifestés.
Nice, Janvier 2003

Introduction
La multiplication des risques majeurs est l’une des caractéristiques des sociétés modernes, à tel point que,
reprenant le titre d’un des plus célèbres ouvrages du sociologue allemand Ulrich Beck, on n’hésite plus
à les qualifier de sociétés du risque. En outre, notre conception même du risque a évolué de sorte que les
événements catastrophiques ne semblent plus désormais perçus par nos concitoyens comme s’imposant
à nos sociétés du fait d’un destin injuste et implacable mais comme résultant essentiellement de notre
propre développement technologique dont la complexité atteint un tel degré qu’elle engendre “naturellement”
des situations de crises pour le présent (Tchernobyl, AZF...) comme pour l’avenir (réchauffement
planétaire et changement climatique, par exemple). Cette prolifération des sources de risques impose
alors aux organisations publiques, mais aussi aux institutions privées telles que les compagnies d’assurances
ou les banques, de nouvelles obligations quant à la détermination, au contrôle et à la gestion de ces
risques extrêmes afin d’en rendre les conséquences supportables pour la communauté sans pour autant
mettre en péril les organismes qui ont en charge de les assurer.
Dans le secteur strictement financier, les krachs représentent peut-être les événements les plus frappants
de toute une catégorie d’événements extrêmes, et de plus en plus fréquemment l’activité financière doit
en subir les effets néfastes. Que l’on songe, pour fixer les idées, que le krach d’octobre 1987 a englouti en
quelques jours plus de mille milliards de dollars ou encore que le récent krach de la nouvelle économie
a conduit à un effondrement de près d’un tiers de la capitalisation boursière mondiale par rapport à
son niveau de 1999. Or, si le rôle de l’argent est de constituer une réserve de valeur, encore faut-il être
capable d’en maîtriser les fluctuations afin notamment de ne pas voir, en un instant, englouti l’épargne
de toute une vie, ruiné les projets d’expansion d’une entreprise ou anéanti l’économie d’une nation. Il
est donc absolument nécessaire de se doter d’outils et de normes permettant de mieux appréhender les
risques extrêmes sur les marchés financiers. Les instances bancaires mondiales, pleinement conscientes
de cette nécessité, ont émis des instructions en se sens, au travers des recommandations du comité de
Bâle (1996, 2001), en proposant des modèles de gestion interne des risques et en imposant des montants
minimaux de fonds propres en adéquation avec les expositions aux risques. Cependant, quelques critiques
se sont élevées contre ces recommandations (Szergö 1999, Danielsson, Embrechts, Goodhart, Keating,
Muennich, Renault et Shin 2001) jugées inadaptées et pouvant même conduire à une déstabilisation des
marchés. Cette controverse fait ressortir toute l’importance d’une meilleure compréhension des risques
extrêmes, de leurs conséquences et des moyens de s’en prémunir.
Ce challenge comporte à notre avis deux volets. D’une part il est indispensable d’être capable de mieux
quantifier les risques extrêmes, et cela passe par le développement d’outils statistiques permettant de
dépasser le cadre gaussien dans lequel s’ancre la théorie financière classique héritée de Bachelier (1900),
Markovitz (1959), et Black et Scholes (1973) notamment. D’autre part, il convient de s’interroger sur
la façon dont peut s’intégrer à la gestion de portefeuille la prise en compte des risques extrêmes. En
effet, il est fondamental de savoir si, à l’instar de ce que nous apprend la théorie financière classique
basée sur l’approche moyenne-variance, les risques extrêmes demeurent diversifiables ? Si tel n’est pas
le cas, il conviendra alors d’envisager l’utilisation d’autres moyens que la constitution de portefeuilles
13

14 Introduction
pour espérer se couvrir contre ce type de risques en ayant recours notamment aux produits dérivés, pour
autant qu’ils soient capables de fournir une réelle assurance contre les grandes variations de cours - ce
qui fût loin d’être le cas lors du krach de 1987 - ou peut-être avoir recours à la mutualisation comme en
assurance par exemple.
Le problème du contrôle des risques extrêmes en finance et plus particulièrement son application à la gestion
de portefeuille peuvent sembler totalement étrangers au domaine de la physique. Cela n’est cependant
qu’une apparence puisque dès ses premiers pas, la finance a partagé avec la physique, la devançant
même parfois, de nombreuses méthodes et outils telle la marche aléatoire brownienne (Bachelier 1900,
Einstein 1905), l’utilisation de la notion de probabilité subjective qui permet de rendre compte du comportement
des agents économiques dans l’incertain (Savage 1954) mais aussi d’interpréter des expériences
de mécanique quantique (Caves, Fluchs et Schack 2002), ou encore la notion de compromis moyennevariance
en théorie du portefeuille Markovitz (1959) comme en algorithmique quantique (Maurer 2001),
ainsi que l’équation de diffusion de la chaleur qui sert également à décrire l’évolution du prix d’une
option dans l’univers de Black et Scholes (1973), et plus récemment l’utilisation de modèles d’agents en
interactions (variante du modèle d’Ising par exemple) permettant de dépasser le cadre standard de l’agent
économique représentatif et ainsi de mieux percevoir les mécanismes fondamentaux à l’oeuvre sur les
marchés financiers.
C’est donc dans les méthodes, mais aussi dans les concepts, qu’il faut chercher un lien entre des matières
en apparence si différentes que la finance et la physique. En effet, la mesure du risque fait appel à des
notions de théorie de l’information, de la décision et du contrôle dynamique qui ont une longue tradition
en physique, quant à la gestion de portefeuille, de manière un peu schématique, on peut affirmer que
ce n’est rien d’autre qu’un exemple particulier de problème d’optimisation sous contraintes, comme
on en rencontre dans de nombreux domaines. En cela, d’un point de vue purement mathématique, la
recherche de portefeuilles optimaux n’est guère différente de la recherche des états d’équilibres d’un
système thermodynamique. En effet, dans le premier cas, on cherche à obtenir une valeur minimale du
risque associé aux variations de richesse du portefeuille pour une valeur fixée du rendement moyen et
un montant de la richesse initiale donnée. Dans le second cas, on cherche - pour ce qui est de l’étude de
l’ensemble micro-canonique - à maximiser l’entropie (ou minimiser la néguentropie) du système sous
la contrainte d’une énergie totale et d’un nombre de particules fixé. On peut donc, formellement, faire
un parallèle (certes un peu rapide) entre risque et néguentropie, rendement espéré et énergie totale et
richesse initiale et nombre de particules.
Ceci étant, le parallèle s’arrête malheureusement là et l’on ne peut espérer d’une application simple
et directe de certains résultats généraux de la thermodynamique statistique ou de mécanique quantique
qu’elle nous permette d’obtenir d’intéressants enseignements pour la gestion de portefeuille. La distinction
essentielle entre ces deux problèmes d’optimisation que sont la gestion de portefeuille et la physique
statistique vient en fait de la différence entre les ordres de grandeur en jeu dans ces deux situations : les
plus gros portefeuilles contiennent au maximum quelques milliers d’actifs alors que les systèmes thermodynamiques
auxquels nous sommes usuellement confrontés sont constitués de quelques cent milles
milliards de milliards (NA 10 23 ) de particules. Pour un tel nombre de particules, le théorème de la
limite centrale s’applique pleinement, y compris pour des événements de très faibles probabilités, et fournit
une simplification fort utile. De plus, vu le nombre gigantesque d’entités considérées, les fluctuations
relatives typiques des grandeurs mesurées par rapport à leurs valeurs moyennes (espérées) sont de l’ordre
de 1/ √ NA ∼ 10 −11 − 10 −12 , ce qui les rend tout à fait imperceptibles.
Les choses sont toutes autres pour ce qui est de la gestion de portefeuille, où il convient de caractériser
avec rigueur la distribution de rendement dudit portefeuille. En effet, du fait du nombre le plus souvent
modéré d’actifs qui constituent un portefeuille et de par la structure sous-jacente des distributions de
chacun des actifs, la distribution de rendement du portefeuille s’avère très éloignée de la distribution

Introduction 15
gaussienne, contrairement à ce que pourrait conduire à penser le théorème de la limite centrale et à ce
que l’on observe le plus souvent en physique statistique. Il est donc nécessaire d’estimer la distribution
de rendement du portefeuille, ce qui peut être réalisé de manière directe pour une allocation de capital
donnée ou bien de manière plus générale en se penchant sur le problème de l’estimation de la distribution
jointe de tous les actifs à inclure dans le portefeuille. La première approche est certes beaucoup plus
simple et rapide puisqu’elle ne consiste qu’en l’estimation d’une distribution monovariée, mais il est
aussi très clair qu’elle n’est en fait guère satisfaisante car elle néglige une grande partie de l’information
observable et extractible à partir des données recueillies sur les marchés financiers et dont seule est
capable de rendre compte la distribution multivariée des actifs. Cependant, on peut garder à l’esprit que
ces deux approches se rejoignent dans la mesure où l’on sait que la connaissance des distributions de
rendements de tous les portefeuilles (pour toutes les allocations de capital possibles) est équivalente à
la connaissance de la distribution multivariée. Au final, la seconde méthode semble préférable, et parait
être celle qui mobilise aujourd’hui le plus d’énergie aussi bien dans la recherche académique que privée.
L’attaque frontale du problème de la détermination de la distribution multivariée des actifs est ardue et à
notre avis beaucoup moins instructive que l’étude séparée du comportement des distributions marginales
de chaque actif d’une part et de la structure de dépendance entre ces actifs d’autre part. C’est pourquoi
nous avons privilégié cette démarche-ci, avec pour objectif principal de rendre compte de manière la plus
précise possible des diverses sources de risques : risques associés individuellement à chaque actif d’une
part et risques collectivement liés à l’ensemble des actifs d’autre part.
Ceci nous a conduit dans un premier temps (partie I) à nous intéresser au comportement marginal des
actifs dans le but de faire ressortir les points essentiels qui nécessitaient une compréhension approfondie
afin de mieux cerner l’origine des risques inhérents aux actifs individuels. Dans notre optique, se focalisant
sur l’impact des risques extrêmes, les krachs - succédant aux bulles spéculatives - présentent un
attrait tout particulier, au même titre que les quelques plus grands mouvements observés sur les marchés.
En effet, conformément à la célèbre loi des 80-20 de V. Pareto, une toute petite fraction d’événements (ici
les krachs et autres grands mouvements) suffit à rendre compte de la plus grande partie des conséquences
(ici le rendement à long terme des actifs, par exemple) résultant de l’ensemble de tous les événements. En
outre, de par leurs répétitions qui semblent de plus en plus fréquentes, on ne peut plus faire l’économie de
négliger ces événements extrêmes, en laissant au seul hasard / destin le soin de décider de leurs effets tout
autant macro-économiques que sur chacun d’entre nous. Il est donc souhaitable de mieux comprendre
comment et pourquoi ces événements se produisent.
Pour cela, nous avons mené notre étude selon trois directions complémentaires. Nous avons tout d’abord
commencé par une étude empirique conduisant à une description statistique et statique des grands risques.
Nous avons ensuite poursuivi par une étude phénoménologique prenant en compte le caractère dynamique
de l’occurrence des extrêmes, et dont le but était de décrire de manière ad hoc les phénomènes
observés. Nous avons enfin terminé par une étude “microscopique” ou micro-structurelle nous permettant
d’aborder de manière fondamentale les mécanismes à l’œuvre sur les marchés et de relier les propriétés
statistiques statiques et dynamiques des cours à des comportements individuels et des organisations de
marchés.
D’un point de vue statistique, il est nécessaire d’estimer précisément la probabilité d’occurrence des
événements extrêmes, qui se traduisent par le fait désormais bien connu que les distributions de rendement
des actifs ont des “queues épaisses”. Cependant, encore faut-il être capable de caractériser le plus
parfaitement possible ces “queues épaisses”, c’est-à-dire sans négliger ni surestimer ces événements
extrêmes. De plus, il faut garder à l’esprit que toute description paramétrique est sujette à l’erreur de
modèle et qu’ainsi plusieurs représentations doivent être considérées afin de cerner correctement les effets
d’une telle source d’erreur. C’est pour cela que nous nous sommes, tout au long de notre étude,
intéressés à deux classes de distributions à queues épaisses : les distributions régulièrement variables et

16 Introduction
les distributions exponentielles étirées dont nous avons discuté la pertinence et les conséquences vis-à-vis
de la sous / surestimation des risques (chapitres 1, 2 et 3).
Nous nous sommes ensuite intéressés à la manière dont la volatilité retourne à un niveau d’“équilibre”
après une longue période de forte variabilité des cours (chapitre 4). On sait en effet que la volatilité
présente un phénomène de persistance très marqué, et qu’après une période anormalement haute (ou
basse), elle relaxe vers un niveau moyen, que l’on peut en quelque sorte associer à un niveau d’équilibre.
L’étude de cette relaxation est très importante du point de vue de la gestion des risques, car elle permet
d’une part d’estimer la durée typique d’une période anormalement turbulente ou calme et d’autre part
elle apporte un moyen de prédiction relativement fiable de la valeur future de la volatilité, dont on connaît
l’intérêt pour tout ce qui concerne le “pricing” de produits dérivés notamment. De plus, et c’est le point
clé de notre développement, cela nous a permis de proposer un mécanisme expliquant l’impact du flux
d’information sur la dynamique de la volatilité. Ceci ouvre la voie à une étude systématique de l’effet
de l’arrivée de telle ou telle information sur les marchés aussi bien ex post, pour tout ce qui concerne
l’analyse des causes des grands mouvements de cours, qu’ex ante pour tout ce qui touche à l’étude de
scénarii et l’anticipation des effets des grands chocs.
Enfin, nous avons voulu explorer un peu plus avant les causes microscopiques responsables des phénomènes
observés sur les marchés (chapitres 5 et 6). Pour cela, nous avons construit un modèle d’agents en
interaction, relativement parcimonieux, visant à rendre compte de manière plus réaliste que ne le font
les modèles actuels de la croissance super-exponentielle des cours lors des phases de bulles spéculatives
qui conduisent à d’importantes surestimations du prix des actifs et finalement à de fortes corrections.
Les résultats obtenus sont intéressants et ont confirmé l’importance de certains types de comportements
des agents conduisant à des “emballées” des marchés et par suite aux périodes de grandes volatilités et
aux fluctuations extrêmes. Cependant, on ne peut, en l’état, espérer intégrer ces modèles, encore trop
rudimentaires, à une chaîne de décision concernant la politique de gestion des risques d’une institution.
Néanmoins, nous pensons que ce genre d’outil présente un attrait particulier et devrait, à terme, fournir
d’utiles informations permettant une meilleure estimation / prévision des risques à venir. En particulier,
ces modèles devraient permettre d’obtenir, à l’aide de la génération de scénarii, des estimations des
probabilités associées aux événements rares dans certaines phases de marchés, estimation beaucoup plus
fiable que ne le sont les actuelles estimations subjectives (Johnson, Lamper, Jefferies, Hart et Howison
2001).
Le deuxième maillon de la chaîne de reconstruction de la distribution multivariée des rendements d’actifs
consiste en l’étude de leur structure de dépendance, problème que nous aborderons dans la partie II. En
effet, les risques ne sont pas uniquement dus au comportement marginal de chaque actif mais également
à leur comportement collectif. Celui-ci peut être étudié à l’aide d’objets mathématiques nommés copules
qui permettent de capturer complètement la dépendance entre les actifs. En fait, nous verrons aux
chapitres 7 et 8 que la détermination de la copule est une affaire délicate et que là encore le risque
d’erreur de modèle est important, expressément pour ce qui est des risques extrêmes. Ainsi, nous verrons
qu’il s’avère nécessaire de mener une étude spécifique de la dépendance entre les extrêmes, ce que
nous ferrons au chapitre 9, en développant une méthode d’estimation du coefficient de dépendance de
queue, c’est-à-dire de la probabilité qu’un actif subisse une très grande perte sachant qu’un autre actif
par exemple ou que le marché dans son ensemble a lui aussi subi une très grande perte. Cette quantité est
à notre avis absolument cruciale car elle quantifie de manière très simple le fait que les risques extrêmes
puissent être ou non diversifiés par agrégation au sein de portefeuilles. En effet, soit le coefficient de
dépendance de queue est nul et les extrêmes se produisent de manière asymptotiquement indépendante
ce qui permet alors d’envisager de les diversifier, soit ils demeurent asymptotiquement dépendants et on
ne peut qu’espérer minimiser la probabilité d’occurrence des mouvements extrêmes concomitants.
Synthétisant les résultats de ces deux premières parties, nous pourrons alors nous poser la question pro-

Introduction 17
prement dite de la gestion de portefeuille, ce qui sera l’objet de la partie III. Avant d’en arriver au
problème de la recherche des portefeuilles optimaux, qui n’est en fait qu’un problème mathématique,
nous nous interrogerons sur la manière de choisir un portefeuille optimal, ce qui nous amènera à discuter,
au chapitre 10, de la théorie de la décision et de divers moyens de quantifier les risques associés à
la distribution de rendement d’un portefeuille. La question n’est en fait pas triviale car elle revient à se
demander comment synthétiser le mieux possible en un seul nombre l’information contenue dans une
fonction (de distribution) toute entière et donc une infinité de nombres. Ce résumé de l’information ne
pouvant qu’être partiel, il convient, afin d’en conserver la fraction la plus pertinente, de bien cerner nos
objectifs et pour cela de bien comprendre les buts que l’on cherche à atteindre en définissant ce qu’est la
notion de grand risque ou de risque extrême dans le contexte de la sélection de portefeuilles.
A notre avis, la gestion de portefeuille doit répondre à deux objectifs. Premièrement, l’allocation des
actifs au sein du portefeuille doit satisfaire à des contraintes portant sur le capital économique, c’està-dire
sur la somme qui doit être investie dans un actif sans risque pour permettre au gestionnaire de
faire face à ses obligations en dépit des fluctuations de la valeur de marché du portefeuille. L’objectif
primordial étant, bien évidemment d’éviter la ruine. Deuxièmement, le portefeuille doit répondre à un
objectif de rentabilité fixé. Il convient donc d’être capable de quantifier la propension du portefeuille à
réaliser l’objectif qui a été défini. Plus clairement, l’objectif de rentabilité est généralement déterminé
par la valeur du rendement moyen (ou espéré) à atteindre et dont on souhaite ne pas voir trop s’écarter
les rendements réalisés. Il est alors nécessaire d’être capable d’estimer les fluctuations que le portefeuille
subit typiquement. Donc, la notion de risque présente au minimum un double sens, dans la mesure où
elle concerne d’une part le capital économique et d’autre part l’amplitude des fluctuations statistiques du
rendement autour du rendement à atteindre.
Moyennant cela, nous verrons ensuite comment mettre à profit ces mesures de risques grands et extrêmes
afin de construire des portefeuilles optimaux supportant le mieux possible les grandes fluctuations des
marchés financiers (chapitre 11), et en particulier comment minimiser l’impact des grands co-mouvements
entre actifs qui ne peuvent être totalement diversifiés (chapitre 12), puis comment satisfaire la contrainte
portant sur le capital économique au travers de mesures telles que la Value-at-Risk ou l’Expected-shortall
(chapitre 13) et enfin, comment construire des portefeuilles dont les rendements réalisés s’écartent le
moins possible de l’objectif de rentabilité initialement fixé sous l’effet des grandes fluctuations (chapitre
14), ce qui nous amènera aussi à discuter des conséquences du choix de portefeuilles minimisant l’impact
des risques grands et extrêmes sur l’équilibre des marchés.
Enfin, nous synthétiserons l’ensemble des résultats obtenus, discuterons de leurs conséquences vis-à-vis
de la gestion des risques extrêmes et conclurons par quelques perspectives de recherches futures.

18 Introduction

Première partie
Etude et modélisation des propriétés des
rentabilités des actifs financiers
19

Chapitre 1
Faits stylisés des rentabilités boursières
L’objet de ce chapitre est de présenter les principaux faits stylisés des rendements boursiers, d’en discuter
la pertinence vis-à-vis des grands principes de la finance ou de certains modèles phénoménologiques,
mais aussi de remettre en cause ou généraliser certains résultats considérés comme acquis et bien établis
alors même que d’autres caractéristiques toutes aussi réalistes semblent devoir être mises à jour.
Par fait stylisé nous désignons toute caractéristique commune des séries temporelles issues des marchés
financiers. Ainsi donc, nous ne nous intéressons pas à une étude événementielle des marchés, étude
où l’on cherche à expliquer le mouvement des cours par l’arrivée de telle ou telle information, mais
plutôt à une approche statistique dont le but est d’identifier et d’isoler les traits communs des séries
temporelles issues des cours des actifs financiers. En écrivant cela, nous ne cherchons pas à opposer ces
deux approches, qui n’ont rien d’antinomiques, et peuvent même se révéler complémentaires comme
nous le verrons à la fin de ce chapitre.
L’étude des faits stylisés repose sur les données fournies par les séries temporelles des prix d’actifs
financiers et de leurs rendements. Dans la suite, nous définirons par Pt le prix d’un actif à l’instant t et
Pt+1 le prix de cet actif à l’instant t + ∆t, où ∆t désigne l’unité de temps considérée, qui peut aussi bien
être la minute que la journée ou le mois. Le rendement de l’actif à l’instant t sera quant à lui noté rt et
défini par
rt = ln Pt
Pt−1
Pt − Pt−1
, (1.1)
Pt−1
qui est en toute rigueur le rendement logarithmique mais diffère peu du rendement traditionnel tant que
celui-ci demeure très petit devant un. La différence essentielle vient de ce que le rendement traditionnel
est borné inférieurement par moins un, tandis que le rendement logarithmique est définie sur la droite
réelle toute entière, ce qui constitue une distinction importante pour la détermination de la distribution
marginale des rendements. De plus, dans ces conditions le logarithme du prix ln Pt suit un procesus dont
les incréments sont donnés par les rendements rt.
Une question cruciale se pose alors, à savoir celle de la stationnarité des données sur lesquelles se basent
les études statistiques. L’hypothèse de la stationnarité des prix des actifs est évidemment à rejeter, puisque
ces derniers dérivent d’un processus intégré, comme le traduit l’équation (1.1). En revanche, hypothèse de
stationnarité semble beaucoup plus acceptable pour les séries de rendements et est très souvent faite, car
nécessaire à l’analyse statistique des données 1 , même s’il nous faut reconnaitre qu’elle est parfois sujette
à caution. En effet, certaines études auxquelles nous feront référence dans la suite ont été menées sur des
1 Dans le cas de séries non stationnaires, l’étude reste dans une certaine mesure possible, mais fait appel à des méthodes plus
complexes et dont l’emploi n’est valable que dans le cadre de modèles plus ou moins spécifiques (Gouriéroux et Jasiak 2001)
21

22 1. Faits stylisés des rentabilités boursières
séries longues d’un siècle. Or, sur de telles durées, l’évolution du contexte économique, réglementaire
mais aussi l’introduction de nouveaux produits financiers semblent laisser à penser que les modes de
fonctionnement des marchés se sont considérablement modifiés au fil du temps ce qui devrait se refléter
dans les cours par des non-stationnarités. De plus, même sur de brèves périodes, des effets saisonniers
apparaissent et l’on note par exemple des comportements anormaux en début et fin de journée, de semaine
ou d’année. Cependant, ces effets étant identifiés, il est possible de les corriger. En résumé, la
stationnarité pose potentiellement un problème important, mais on peut espérer en limiter les effets par
la dessaisonalisation des données et la considération d’échantillons de taille raisonnable (de cinq à dix
ans tout au plus) sélectionnant des séquences de marchés relativement homogènes.
Dans ce qui suit, nous allons essentiellement présenter les propriétés des rendements journaliers ou intradays,
et sauf indication contraire, le terme rendement fera implicitement référence aux rendements calculés
à une échelle de temps inférieure ou égale à la journée. Lorsque nous considérerons des rendements
mensuels par exemple, il en sera explicitement fait mention.
Enfin, nous voulons souligner que le rappel des faits stylisés que nous donnons dans ce chapitre ne
prétend pas à l’exhaustivité, mais se concentre sur la description de ceux qui nous seront les plus utiles
pour la suite de notre exposé, et nous renvoyons le lecteur aux nombreux articles de revue sur le sujet
comme par exemple Pagan (1996), Cont (2001) ou encore Engle et Patton (2001) ainsi qu’aux ouvrages
de Campbell, Lo et MacKinlay (1997) ou Gouriéroux et Jasiak (2001) pour une approche économétrique
et Mantegna et Stanley (1999), Bouchaud et Potters (2000) ou Roehner (2001, 2002) pour une vision de
physiciens.
1.1 Rappel des faits stylisés
Nous allons maintenant exposer les principaux faits stylisés en commençant par la distribution des rendements
dont il semble aujourd’hui certain qu’elle possède des “queues épaisses”, puis nous présenterons
les propriétés de dépendances temporelles des séries financières essentiellement caractérisées par l’absence
de corrélation entre les rendements mais une forte persistance de la volatilité, après quoi nous
terminerons par quelques propriétés complémentaires de ces séries financières.
1.1.1 La distribution des rendements
La toute première particularité des rendements financiers est de suivre des lois dites “à queues épaisses”.
Par loi à queues épaisses nous désignons l’ensemble des lois régulièrement variables à l’infini, c’està-dire
pour simplifier, les lois équivalentes à l’infini à une loi de puissance (pour une définition exacte
de la notion de variation régulière voir Bingham, Goldie et Teugel (1987)). Ce comportement est radicalement
différent de celui généralement admis durant la première moitié du XXème siècle. En effet,
depuis des travaux pionniers de Bachelier (1900) repris et étendus par Samuelson (1965, 1973), tout
le monde s’accordait sur le fait que les distributions de rendements suivaient des distributions gaussiennes.
Il faudra attendre Mandelbrot (1963) et son étude des prix pratiqués sur le marché du coton, puis
Fama (1963, 1965a) pour clairement rejeter cette hypothèse et en venir à considérer des distributions en
lois de puissance, et plus particulièrement les lois stables de Lévy, dont l’une des caractéristiques est
de ne pas admettre de second moment et donc d’avoir une variance infinie. Or, des tests directs sur la
plupart des séries financières permettent clairement de conclure à l’existence de ce second moment. En
effet, par agrégation temporelle des rendements, c’est-à-dire lorsque l’on passe des rendements journaliers
aux rendements mensuels ou à des rendements calculés à des échelles encore plus importantes, on
note une (lente) convergence de la distribution vers la gaussienne (Bouchaud et Potters 2000, Campbell

1.1. Rappel des faits stylisés 23
et al. 1997, Mantegna et Stanley 1999). Donc, les lois stables de Lévy ne sauraient elles non plus convenir
à la description des distributions de rentabilités boursières, du moins dans leur globalité.
Durant les années 80-90, de gigantesques bases de données contenant les cours des actifs financiers sur
de longues périodes et enregistrées à de très hautes fréquences voient le jours. Dans le même temps,
l’économétrie et plus particulièrement l’économétrie financière développe de nouveaux outils, si bien
qu’à la fois sur le plan méthodologique que sur le plan de la quantité de données accessibles, une petite
révolution se produit. C’est alors que l’on en vient à cerner de manière beaucoup plus précise le comportement
asymptotique des distributions de rendements des actifs financiers. De nombreuses études
montrent qu’effectivement les distributions de rendements se comportent comme des lois régulièrement
variables dont l’exposant de queue est compris entre deux et demi et quatre de sorte que les deux premiers
moments de ces distributions existent et peut-être aussi les troisième et quatrième. Ces études ont
été menées sur diverses sortes d’actifs mais conduisent toutes à des conclusions similaires. On pourra
notamment consulter Longin (1996), Lux (1996), Pagan (1996), Gopikrishnan, Meyer, Amaral et Stanley
(1998) pour des études concernant les marchés d’actions ainsi que Dacorogna, Müller, Pictet et de Vries
(1992), de Vries (1994) ou encore Guillaume, Dacorogna, Davé, Müller, Olsen et Pictet (1997) pour ce
qui est des taux de change.
Cependant, quelques voix se sont récemment élevées pour suggérer qu’il serait peut-être sage de tempérer
l’enthousiasme général à l’égard des lois de puissances. En effet, selon Mantegna et Stanley (1995) les
résultats de Mandelbrot (1963) décrivent certes de façon correcte une grande partie de la distribution
des rendements mais, de manière ultime, le comportement en loi stable de Lévy est erroné et doit être
tronqué par une décroissance exponentielle. On sait d’ailleurs que ce type de distribution converge - par
convolution - de manière extraordinairement lente vers la gaussienne (Mantegna et Stanley 1994), ce qui
est conforme aux observations empiriques. Dans le même esprit, Gouriéroux et Jasiak (1998) concluent
que la densité de probabilité du titre Alcatel décroît plus vite que toute loi de puissance. Enfin, Laherrère
et Sornette (1999) affirment de manière plus précise que les distributions dites exponentielles étirées
semblent fournir une meilleure description des distributions de rendements que les lois de puissance,
non seulement dans les queues extrêmes mais aussi sur une large gamme de rendements. Aussi à des fins
de généralité, il semble judicieux d’étendre la notion de distributions à queues épaisses aux distributions
sous-exponentielles, c’est-à-dire qui ne décroissent pas plus vite qu’une exponentielle à l’infini. Pour une
définition rigoureuse ainsi qu’une synthèse des propriétés de ces distributions, nous renvoyons le lecteur
à Embrechts, Klüppelberg et Mikosh (1997) et Goldie et Klüppelberg (1998).
Au vu du flou qui demeure quant au comportement exact des distributions de rendements, un réexamen
de leur comportement asymptotique semble nécessaire. En particulier, il parait indispensable de comparer
le pouvoir descriptif des deux familles de distributions que nous venons d’évoquer, et donc nous
reviendrons en détail sur ce problème dans un prochain paragraphe. Nous délaissons temporairement
ce point pour nous tourner maintenant vers le deuxième trait caractéristique des séries financières, à savoir
l’existence de structures de dépendances temporelles non triviales. Ceci est d’ailleurs un élément
fondamental de la compréhension de la difficulté à déterminer précisément la distribution marginale des
rendements.
1.1.2 Propriétés de dépendances temporelles
Les propriétés de dépendances temporelles que nous allons résumer dans ce paragraphe dérivent toutes
de l’un des grands principes fondamentaux de la théorie financière, à savoir le principe de non arbitrage.
Sans rentrer ici dans le détail, nous nous bornerons à dire de manière sommaire que ce principe postule
l’impossibilité d’obtenir un gain certain sur les marchés financiers. La logique d’un tel postulat tient dans
le fait que si un agent détecte une telle opportunité, il va immédiatement en tirer profit et celle-ci va donc

24 1. Faits stylisés des rentabilités boursières
Volatility
0.01
0.009
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
Jan 70 Dec 85
FIG. 1.1 – Volatilité réalisée (carrés des rendements) de l’indice Standard & Poor’s 500 entre janvier
1970 et décembre 1985.
disparaître. Ainsi, hormis de manière fugitive, une telle situation ne peut se produire. Cela implique en
particulier que les rendements futurs ne peuvent être prédits.
Une conséquence immédiate et très communément observée est l’absence d’auto-corrélation entre les
rendements 2 (Fama 1971, Pagan 1996, Gouriéroux et Jasiak 2001). Cette absence de corrélations temporelles
est aisément justifiable par l’absence d’opportunité d’arbitrage (statistique). En effet, s’il existe
des corrélations significatives, il devient possible de prédire les prix futurs, ce qui offre - du moins statistiquement
- une possibilité de gagner de l’argent à coup sûr. Ainsi, selon Mandelbrot (1971) l’absence
d’opportunité d’arbitrage conduit à blanchir le spectre des changements de prix et donc à faire disparaître
les corrélations temporelles. Ceci a longtemps était l’un des supports de la théorie de l’efficience
des marchés financiers. En effet, l’absence de corrélation justifie notamment l’hypothèse remontant à
Bachelier (1900) selon laquelle les prix suivent une marche aléatoire (mouvement Brownien). Cela dit, à
partir du moment où l’on admet que le processus stochastique décrivant l’évolution au cours du temps des
rendements n’est pas gaussien, sa seule fonction d’auto-corrélation ne peut suffire à le définir. En particulier,
l’absence de corrélation ne peut permettre de conclure que le prix (plus précisément le logarithme
du prix) suit un processus à incréments indépendants, ce qui serait faux.
Ceci se vérifie d’ailleurs simplement lorsque l’on observe une série de rendements. En effet, il apparaît
que les périodes de grande volatilité alternent avec les périodes de volatilité plus faible montrant ainsi
que la volatilité 3 présente un phénomène de persistance (voir figure 1.1). Ce phénomène est très similaire
2 En toute rigueur, il existe deux limites à cette assertion. Premièrement, en dessous d’un intervalle de temps de l’ordre de
quelques secondes à quelques minutes, les effets de microstructure des marchés deviennent importants et justifient que l’on
observe l’existence de corrélations significatives (Campbell et al. 1997). Deuxièmement, pour les rendements calculés à des
échelles de temps de l’ordre du mois ou plus, il semble là aussi que des corrélations significatives apparaissent. Cela peut
s’expliquer par le fait que sur des échelles de temps de cet ordre, le rendement d’un actif possède un contenu économique
beaucoup plus marqué qu’un rendement journalier essentiellement dominé par des activités de (noise) trading pouvant être
totalement découplées de toutes réalités économiques.
3 Nous parlons ici de volatilité dans un sens relativement vague, qui peut recouvrir à la fois l’écart type des rendements
ou simplement leur amplitude. On peut toutefois noter qu’une telle persistance s’observe aussi sur la sknewness et la kurtosis
(Jondeau et Rockinger 2000, par exemple).

1.1. Rappel des faits stylisés 25
au phénomène d’intermittence observé en turbulence (Frisch 1995). Il caractérise clairement le fait qu’il
existe une certaine dépendance dans les séries financières : la séquence des rendements n’apparaît pas de
manière indépendante. En fait, on constate que la valeur absolue des rendements ou de leurs carrés sont
fortement corrélés et que cette corrélation persiste : la fonction d’auto-corrélation des valeurs absolues
des rendements décroît typiquement comme une loi de puissance dont l’exposant est compris entre 0.2
et 0.4 (Cont, Potters et Bouchaud 1997, Liu, Cizeau, Meyer, Peng et Stanley 1997). Une fois encore,
cela est conforme au principe de non arbitrage, car la volatilité n’étant pas une grandeur signée, elle
ne permet en rien de déduire les rendements futurs. De manière générale, toute fonction de la volatilité
(ou de l’amplitude du rendement) est autorisée à présenter des corrélations temporelles significatives.
Pour quelques exemples voir notamment Cont (2001) mais aussi Muzy, Delour et Bacry (2000), où il est
montré que la corrélation du logarithme de la valeur absolue des rendements décroît comme le logarithme
du décalage temporel entre les rendements considérés et possède donc une mémoire longue.
Ce phénomène de persistance (ou mémoire longue) est caractéristique de la volatilité mais sa mise en
évidence est quelques fois délicate et sujette à caution comme l’ont montré Granger et Teräsvirta (1999),
qui ont construit un processus auto-régressif du premier ordre non-linéaire passant les tests standards de
mémoire longue ou Andersson, Eklund et Lyhagen (1999) qui ont prouvé le contraire, c’est-à-dire qu’un
processus à mémoire longue pouvait passer les tests de linéarité.
Pour en terminer avec les exemples sur les corrélations temporelles, il convient de citer l’effet de levier
mis en évidence par Black (1976) puis Christie (1982) et Bouchaud, Matacz et Potters (2001) notamment,
et que l’on rencontre pour les rendements d’actions et les taux d’intérêt mais pas les taux de change.
Cet effet de levier se traduit par une corrélation négative entre la volatilité à un instant donné et les
rendements antérieurs, ce qui signifie que les mouvements de prix à la baisse ont tendance à entraîner
une augmentation de la volatilité. Il semble d’ailleurs que cet effet de levier soit à l’origine de la cascade
causale observée par Arnéodo, Muzy et Sornette (1998) sur les marchés d’actions 4 .
En résumé, de par la contrainte de non arbitrage, il est impossible d’observer des corrélations significatives
entre des fonctions des variables passées et les rendements futurs qui permettraient de prédire ces
derniers. En revanche, il est tout à fait possible d’observer - et c’est effectivement le cas - d’importantes
corrélations entre des fonctions des variables passées et la volatilité future, ceci ne permettant pas de
réaliser de prédiction sur les rendements.
Enfin, outre l’existence de fortes dépendances entre volatilité future d’une part, et volatilité et rendements
passés d’autre part, il convient de remarquer que globalement, celle-ci a tendance à retourner à un niveau
moyen, que l’on peut interpréter comme son niveau “normal”. Ce phénomène de retour à la moyenne
est bien observé lorsque l’on s’intéresse à la volatilité prédite par la plupart des modèles usuels : lorsque
l’horizon de prévision augmente, on constate que la volatilité tend vers une même valeur moyenne (voir
Engle et Patton (2001) par exemple). Il est alors très intéressant d’étudier le mode de relaxation de la
volatilité vers son niveau moyen, ce qui a été réalisé par Sornette, Malevergne et Muzy (2002) et dont
nous parlerons un peu plus en détail dans un paragraphe ultérieur.
1.1.3 Autres faits stylisés
Les faits stylisés précédemment évoqués sont les faits stylisés principaux universellement admis. Ce sont
ceux dont doit au moins rendre compte tout modèle de cours réaliste. Cela peut sembler peu, mais en fait,
il est déjà assez difficile de trouver de manière ad hoc des processus satisfaisant ces quelques restrictions.
Nous en donnerons quelques exemples dans la suite de ce chapitre. A coté de ces grands faits stylisés,
existent plusieurs autres caractéristiques importantes dont notamment la dissymétrie gains/pertes pour
4 Cette interprétation nous a été suggérée par J.F. Muzy lors d’une conversation privée.

26 1. Faits stylisés des rentabilités boursières
Hang-Seng
10000
1000
↓
↓
↓
↓
100
70 75 80 85 90 95 100
FIG. 1.2 – Indice Hang Seng (bourse de Hong Kong) entre janvier 1970 et Decembre 2000.
les actions, les accélérations super-exponentielles des cours lors des phases de bulles spéculatives ou
encore la multifractalité et la corrélation entre volatilité et volume de transaction, ce dernier point étant
intuitivement évident (Gopikrishnan, Plerou, Gabiax et Stanley 2000).
Concernant la dissymétrie gains/pertes, il a été clairement établi par Johansen et Sornette (1998) et Johansen
et Sornette (2002) que les marchés d’actions sont sujets à des pertes cumulées de très grandes
amplitudes, alors que l’on n’observe pas systématiquement de phénomènes de telle ampleur pour les
gains. De plus, ce phénomène ne semble pas toucher les marchés de change, où une plus grande symétrie
parait de mise. Ceci est en fait naturel, car la hausse ou la baisse d’un taux de change n’est que relative à
la position que l’on adopte : une hausse du taux euro/dollar par exemple correspond évidemment à une
baisse du taux dollar/euro.
Pour ce qui est de la croissance super-exponentielle du prix des actifs, celle-ci est illustrée sur la figure
1.2. On y observe que le prix (représenté en échelle logarithmique) croit en moyenne de manière linéaire
en fonction du temps, ce qui caractérise une croissance exponentielle et donc un taux de croissance
constant. Mais, un examen plus approfondi montre qu’en fait se succèdent une série de phases de croissance
accélérée - et donc plus rapides que la moyenne - interrompues par de fortes corrections du prix.
Ce phénomène a été qualifié par Roehner et Sornette (1998) de “sharp peak, flat trough” et semble tirer
son origine des phénomènes d’imitation et effets de foule observés sur les marchés. Nous reviendrons
beaucoup plus en détail sur ce point précis aux chapitres 5 et 6.
Enfin le dernier point que nous aborderons concernera la quantification de la régularité de la trajectoire
des prix d’actifs financiers. Arnéodo et al. (1998) et Fisher, Calvet et Mandelbrot (1998) ont conclu au
caractère multifractal de ces trajectoires. En effet, ils ont montré que quel que soit l’actif considéré, son
spectre multifractal est compris strictement entre zéro et un, ce qui assure que sa trajectoire est presque
partout continue mais non dérivable, et a la forme d’une parabole dont le maximum est voisin de 0.6 5 .
Par comparaison, le mouvement brownien étant auto similaire - et donc (mono) fractal - son spectre vaut
simplement un demi. Ceci conduit à de très fortes contraintes sur les processus permettant d’obtenir ce
type de trajectoires, mais un bémol s’impose quant aux résultats de ces études car de nombreux auteurs,
5 Il est intéressant de remarquer que ces observations conduisent à rejeter un grand nombre de modèles en temps continu tels
que les processus de diffusion, les processus de Lévy ou les processus à saut.
Date
↓
↓
↓
↓

1.2. De la difficulté de représenter la distribution des rendements 27
dont Veneziano, Moglen et Bras (1995), Avnir, Biham, Lidar et Malcai (1998), Bouchaud, Potters et
Meyer (2000), LeBaron (2001) ou encore Sornette et Andersen (2002), ont montré qu’il était très facile
de mettre en évidence un caractère multifractal pour des processus dont il est connu qu’ils ne possèdent
aucune propriété de fractalité, simplement du fait de la taille finie des séries temporelles considérées ou
d’autres mécanismes faisant intervenir des non-linéarités.
1.2 De la difficulté de représenter la distribution des rendements
Nous venons de voir dans la première partie du paragraphe précédent que la distribution des rendements
semblait pouvoir être décrite par des distributions régulièrement variables d’indice de queue de l’ordre
de trois ou quatre. Cependant, une autre hypothèse a récemment vu le jour, hypothèse selon laquelle les
distributions de rendements suivraient des lois exponentielles étirées. Or, si ces deux types de distributions
possèdent beaucoup de points communs, elles présentent néanmoins une différence majeure dont
l’impact théorique est très important.
Pour ce qui est des points communs, ces deux familles de distributions appartiennent à la classe des
distributions sous-exponentielles et sont donc à même de produire des événements extrêmes avec une
probabilité élevée. En particulier, pour cette classe de variables aléatoires, les grandes déviations de la
somme d’un grand nombre de ces variables sont dominées par les grandes déviations d’une seule d’entre
elles : la somme est grande si l’une des variables est grande (Embrechts et al. 1997, Sornette 2000), ce
qui contraste énormément avec les distributions dites super-exponentielles où chaque variable contribue
de manière significative (et globalement équivalente) à la somme (Frisch et Sornette 1997).
Cela dit, il existe une différence majeure : les distributions sous-exponentielles admettent des moments
de tous ordres alors que les distributions régulièrement variables n’admettent de moments que d’ordre
inférieur à leur indice de queue. Cette remarque montre combien il est crucial de pouvoir trancher entre
ces deux types de distributions, car nous verrons en partie III que ces moments vont jouer un rôle essentiel
dans la représentation du comportement qu’adoptent les agents économiques face au risque ainsi que
dans la modélisation des décisions qu’ils prennent. Il est donc absolument fondamental d’être sûr de leur
existence.
L’incertitude qui semble naître quant à la nature exacte de la distribution des rendements dans les
extrêmes et les difficultés à la déterminer précisément est à notre avis en grande partie liée aux dépendances
temporelles présentent dans les séries financières. En effet, lorsque les observations sont supposées
indépendantes et identiquement distribuées, la théorie montre que les estimateurs usuels - tel l’estimateur
de Hill (1975), qui permet de déterminer l’exposant de queue d’une distribution régulièrement variable,
mais aussi l’estimateur de Pickands (1975) - sont asymptotiquement consistants et normalement distribués
(Embrechts et al. 1997), ce qui permet, par exemple, d’estimer de façon précise l’incertitude
sur la mesure d’un indice de queue et éventuellement d’effectuer des tests d’égalité entre les indices de
queues de différents actifs ou entre la queue positive et négative d’un même actif afin de mettre, ou non,
à jour d’éventuelles dissymétries (Jondeau et Rockinger 2001).
Mais, lorsque les données présentent une certaine dépendance temporelle, on ne peut plus guère espérer
que la consistance asymptotique de ces estimateurs (Rootzèn, Leadbetter et de Haan 1998), l’incertitude
de la mesure étant quant à elle beaucoup plus importante que celle fournit sous hypothèse d’asymptotique
normalité. En effet, Kearns et Pagan (1997) ont montré que pour des processus de type (G)ARCH,
qui permettent de rendre raisonnablement compte des dépendances des séries financières (voir paragraphe
suivant), la déviation standard de l’estimateur de Hill peut être sept à huit fois plus grande que
celle estimée pour des séries dont les données sont indentiquement et indépendamment distribuées. Ces

28 1. Faits stylisés des rentabilités boursières
résultats sont encore pires pour l’estimateur de Pickand, dont on sait qu’il joue un rôle important dans
la détermination empirique du domaine d’attraction des lois extrêmes auquel appartient une distribution
donnée.
Cette dernière remarque est cruciale et conduit à remettre potentiellement en cause bon nombre d’études
sur le comportement asymptotique des distributions de rendements. En effet, négliger les dépendances
temporelles revient à minorer l’incertitude des estimateurs comme dans le cas des études se fondant sur la
théorie des valeurs extrêmes menées par Longin (1996) ou Lux (1997) notamment, où l’on peut dire que
l’hypothèse selon laquelle les distributions de rendements sont régulièrement variables à l’infini semble
avoir été un peu rapidement et peut-être inconsidérément acceptée.
De plus, sur le plan théorique, il a été démontré que certaines classes de modèles supportant l’hypothèse
de distributions régulièrement variables devaient être abandonnées. En effet Lux et Sornette (2002) et
Malevergne et Sornette (2001a) ont prouvé que les modèles phénoménologiques à bruit multiplicatif du
type Blanchard (1979) et Blanchard et Watson (1982), et qui permettent de justifier la notion de bulles
rationnelles (Colletaz et Gourlaouen 1989, Broze, Gouriéroux et Szafarz 1990), conduisent, de par la
condition de non arbitrage, à des distributions de rendement régulièrement variables dont l’exposant de
queue est nécessairement inférieur à un (voir chapitre 2), ce que toutes les études empiriques menées
jusqu’à ce jour ont réfuté. Enfin, parmi les divers modèles de prix d’actifs passés en revue dans Sornette
et Malevergne (2001), il ressort notamment que les modèles de Johansen, Sornette et Ledoit (1999) et
Johansen, Ledoit et Sornette (2000) - dans lesquels est introduit un taux de krach qui en assure la stationnarité
- admettent des dynamiques de prix compatibles aussi bien avec des distributions de rendements
régulièrement variables qu’exponentielles étirées.
Au vu de ces incertitudes aussi bien empiriques que théoriques, nous avons jugé nécessaire de mener
une étude comparative du pouvoir descriptif des deux représentations possibles des distributions de rendements
(voir Malevergne, Pisarenko et Sornette (2002) présenté au chapitre 3). Il ressort de cette étude
que, d’un point de vue statistique, les exponentielles étirées rendent au moins aussi bien compte des distributions
de rendements que les lois de puissances et autres lois à variations régulières, mais que du fait
de la dépendance temporelle, il ne nous est pas possible de dire que l’une est réellement meilleure que
l’autre. Nous devons donc nous borner à dire que les exponentielles étirées fournissent une alternative
parfaitement crédible aux distributions régulièrement variables, mais nous sommes malheureusement
incapable de trancher la question de l’existence ou non des moments au-delà d’un certain ordre.
1.3 Modélisation des propriétés de dépendance des rendements
Le paragraphe précédent vient de nous permettre de comprendre que d’un point de vue statistique, il n’est
pas vraiment justifié de privilégier les distributions régulièrement variables au profit des distributions exponentielles
étirées ou vice-versa. En fait, comme nous allons le voir maintenant, l’étude de l’évolution
dynamique des cours fournit un moyen de spécifier un peu mieux ce choix. En effet, la prise en compte
des faits stylisés concernant la dépendance temporelle de la volatilité impose des contraintes suffisamment
fortes pour permettre de définir de façon assez satisfaisante le processus stochastique décrivant
l’évolution des cours, ce qui, en retour, fixe la distribution marginale des rendements compatible avec le
processus spécifié. L’intérêt de trouver un processus stochastique est donc double : décrire la dynamique
des cours en tenant compte de la dépendance exhibée par la volatilité et fournir des jalons quant à la
distribution marginale des rendements.
Les premières descriptions des processus de prix en terme de marches aléatoires à incréments indépendants
ou la modélisation de la dynamique des rendements à l’aide de processus ARMA ont rapidement

1.3. Modélisation des propriétés de dépendance des rendements 29
été abandonnées en raison de leur incapacité à rendre compte de la mémoire longue de la volatilité et
notamment des bouffées de volatilité. En effet, par construction, les marches aléatoires à incréments
indépendants ne présentent aucune dépendance de volatilité tandis que la fonction d’auto-corrélation des
processus ARMA décroît de manière exponentielle (voir Gouriéroux et Jasiak (2001) par exemple) et
non de manière algébrique. Une alternative aux processus ARMA est fournie par les processus ARIMA
fractionnaires (Granger et Joyeux 1980, Hosking 1981) qui présentent des corrélations à longues portées
mais dont l’usage n’est pas bien adapté à la modélisation de la volatilité.
La première réelle ébauche de solution a en fait été proposée par Engle (1982) et les modèles autorégressifs
conditionnellement hétéroskédastiques (ARCH) qui furent ensuite généralisés par Bollerslev
(1986) avec les modèles GARCH. Dans cette famille de modèles, les rendements sont décomposés
comme un produit :
rt = σt · εt, (1.2)
où la volatilité σt suit un processus auto-régressif et εt est un bruit blanc indépendant de σt, ce qui assure
l’indépendance des rendements à des dates successives. Dans un modèle ARCH, la volatilité présente σt
ne dépend que des réalisations passées des rendements rt−1, rt−2, · · · alors que pour un modèle GARCH,
elle est aussi fonction des volatilités passées σt−1, σt−2, · · ·. En pratique, le modèle GARCH possède un
indéniable avantage sur le modèle ARCH car il est beaucoup plus parcimonieux : prendre en compte la
dernière volatilité et les deux derniers rendements (et donc trois paramètres) est en général suffisant alors
que pour un modèle ARCH, il n’est pas rare de devoir considérer les dix ou quinze derniers rendements
réalisés.
Du point de vue de l’adéquation vis-à-vis des faits stylisés, ces processus permettent de rendre compte
de la lente décroissance de l’auto-corrélation de la volatilité, ce qui était leur premier objectif et du retour
à la moyenne. De plus, on peut monter facilement (voir Embrechts et al. (1997) par exemple) à l’aide de
la théorie des équations de renouvellement stochastique (Kesten 1973, Goldie 1991) que la distribution
marginale des rendements est à variation régulière.
Une des limites de cette modélisation est de ne pas rendre compte convenablement de la lente décroissance
de la corrélation du logarithme de la volatilité, de la multifractalité 6 ni de l’effet de levier. Cependant,
ce dernier point peut être corrigé en considérant le modèle EGARCH de Nelson (1991) qui n’est
rien d’autre qu’un processus GARCH sur le logarithme de la volatilité et non la volatilité elle-même ou
encore les modèles TARCH de Glosten, Jagannanthan et Runkle (1993) et Zakoian (1994) qui sont de
modèles GARCH à seuil. Nous n’irons pas plus avant dans la description de la vaste famille des modèles
ARCH et de leur généralisation et nous renvoyons le lecteur aux nombreux articles de revues et livres
sur le sujet dont notamment Bollerslev, Chou et Kroner (1992), Bollerslev, Engle et Nelson (1994) et
Gouriéroux (1997).
La seconde alternative a été présentée plus récemment par Bacry, Delour et Muzy (2001) qui ont développé
un processus de marche aléatoire multifractale (MRW), processus en temps continu infiniment logdivisible
(Bacry et Muzy 2002, Muzy et Bacry 2002). Comme dans un processus EGARCH, c’est ici le
logarithme de la volatilité qui est modélisé. Ce processus est en fait un processus gaussien stationnaire
dont l’auto-corrélation du logarithme de la volatilité est spécifiée de sorte qu’elle décroisse proportionnellement
au logarithme du décalage entre les volatilités, ce qui est un des faits stylisés présentés plus
haut et dont ne rendait pas compte les modèles (G)ARCH. C’est en fait l’avancée majeure proposée
par ce type de modèle par rapport aux autres processus déjà existant, d’autant qu’il rend compte, par
construction même, des corrélations à longue portée de la volatilité et est l’un des rares processus connus
à satisfaire de manière théorique, et non artificiellement, les contraintes liées à la multifractalité.
6 Sur des échantillons de taille finie, Baviera, Biferale, Mantegna et Vulpiani (1998) ont cependant noté la présence - pure-
ment artificielle - d’une apparente multifractalité.

30 1. Faits stylisés des rentabilités boursières
Pour ce qui est de la distribution marginale des rendements, celle-ci est aussi à queues épaisses et
régulièrement variables mais avec des exposants de queue beaucoup plus élevés que ceux précédemment
rencontrés. En effet, lorsque le modèle est calibré sur les données réelles, il indique nettement que les
moments existent jusqu’à un ordre supérieur à vingt, ce qui est bien plus important que ce qui est habituellement
admis. Cela montre une fois de plus combien il est difficile de décider du comportement
asymptotique de la distribution marginale des rendements.
Enfin, ce modèle a permis à Sornette et al. (2002) de prédire d’un point de vue théorique le mode de
relaxation de la volatilité, c’est-à-dire comment décroît (ou croit) la volatilité lorsqu’elle retourne vers sa
valeur moyenne. Lillo et Mantegna (2002) ont récemment observé qu’après un grand choc, la volatilité
relaxait selon une loi de puissance dont l’exposant était étrangement proche de celui de la fonction d’autocorrélation
de la volatilité, ce qui pouvait conduire à penser à une relation de type fluctuation-dissipation
entre les corrélations des fluctuations de la volatilité “ à l’équilibre” et la dissipation entraînant le retour à
l’équilibre après un choc. En fait il n’en n’est rien, et nous avons montré que le mode de relaxation (plus
précisément l’exposant de la loi de puissance) dépendait de la nature du choc. Si une importante nouvelle
est suffisante par elle-même pour faire bouger le marché (comme le coup d’état contre Gorbatchev en
1991, par exemple) la volatilité relaxe avec un exposant -1/2, indépendamment de l’amplitude du choc.
Au contraire, si le choc est dû à une accumulation de “petites” mauvaises nouvelles, la relaxation se fait
avec un exposant en général plus faible dont la valeur dépend de l’amplitude du choc.
Cette approche, vérifiée expérimentalement, a apporté une nouvelle confirmation de l’intérêt du modèle
MRW, puisqu’il semble le seul modèle à même de décrire ce changement de mode de relaxation selon
la nature du choc. En particulier, la relaxation observée sur un processus GARCH n’en dépend pas.
Enfin, cette approche permet d’entrevoir une méthode de classement systématique des chocs de volatilité
selon leur nature “endogène” ou “exogène”. Cependant, une limite actuelle de ce modèle est de traiter
de manière totalement équivalente les gains et les pertes, négligeant de ce fait l’effet de levier. Une
solution envisageable serait de considérer un processus MRW asymétrique tel celui proposé par Pochart
et Bouchaud (2002).
1.4 Conclusion
L’objet de ce chapitre était d’exposer les principaux faits stylisés des rentabilités boursières, que l’on
peut résumer comme suit :
– distribution des rendements à queues épaisses,
– dépendance temporelle complexe : absence de corrélation temporelle entre les rendements, mais corrélation
persistante de la volatilité,
– croissance super-exponentielle transitoire mettant en évidence l’existence de différentes phases de
marché,
– et caractère multifractal des trajectoires des prix des actifs financiers.
Certaines justifications théoriques peuvent être apportées à ces faits stylisés en faisant appel aussi bien
à des principes fondamentaux de la théorie financière qu’à de simples modèles phénoménologiques. En
retour, cela permet de limiter le champ des processus acceptables pour la modélisation des prix des actifs
financiers. Pour autant, cette démarche reste superficielle en ce qu’elle ne permet pas une meilleure
compréhension des mécanismes de fonctionnement des marchés financiers et des comportements des
agents prenant position sur ces marchés. C’est pourquoi il sera nécessaire de nous intéresser à l’aspect
“microscopique” ou microstructurel des marchés financiers, ce qui sera l’objet des chapitres 5 et 6. Mais
avant cela, nous allons, dans les chapitres qui suivent, présenter certains des résultats que nous avons
obtenus et qui permettent de justifier les assertions que nous avons formulées tout au long de ce chapitre.

Chapitre 2
Modèles phénoménologiques de cours
Nous présentons dans ce chapitre quelques modèles phénoménologiques de cours d’actifs afin d’illustrer
notre affirmation du chapitre 1, section 1.2 selon laquelle il était tout aussi possible de construire des
modèles permettant de justifier que la distribution stationnaire des rendements est régulièrement variable
ou exponentielle étirée.
Pour cela, nous commençons par montrer que les modèles simples de bulles rationnelles 1 à la Blanchard
et Watson (1982) ne sont en fait pas compatibles avec les données empiriques, car s’ils conduisent effectivement
à justifier l’existence de distributions de rendements régulièrement variables, l’indice de queue
auquel conduisent ces modèles est beaucoup plus faible que celui réellement estimé sous cette hypothèse.
Nous passons ensuite en revue deux modèles qui satisfont aux contraintes empiriques en vérifiant la plupart
des faits stylisés exposés au chapitre 1, et qui permettent de justifier tout autant l’hypothèse de
distribution hyperbolique que l’hypothèse de distribution exponentielle étirée.
1 Le lecteur est invité à se référer à Blanchard et Watson (1982), Colletaz et Gourlaouen (1989), Broze et al. (1990) ou
Adams et Szarfarz (1992) notamment pour une revue sur le sujet.
31

32 2. Modèles phénoménologiques de cours
2.1 Bulles rationnelles multi-dimensionnelles et queues épaisses
Lux et Sornette (2002) ont démontré que les queues des distributions inconditionnelles des incréments de
prix et de rendements associées au modèle de bulles rationnelles de Blanchard et Watson (1982) suivent
des lois de puissance (décroissent de manière hyperbolique), dont l’exposant de queue µ est inférieur à
un sur un large domaine. Bien que les queues en lois de puissance soient une caractéristique marquante
relevée sur les données empiriques, la valeur numérique µ < 1 est en désaccord avec les estimations habituelles
qui, selon ce type de modèle, donnent µ 3. Parmi les quatre hypothèses soutenant le modèle
de bulles rationnelles de Blanchard et Watson (rationalité des agents, condition de non arbitrage, dynamique
multiplicative, bulles indépendantes pour chaque actif), nous démontrons que le même résultat
µ < 1 reste valable lorsque que l’on relaxe la dernière de ces hypothèses, i.e en permettant le couplage
entre les différentes bulles des divers actifs. En conséquence, des extensions non linéaires de la dynamique
des bulles ou une relaxation partielle du principe d’évaluation rationnel des prix sont nécessaires
si l’on souhaite rendre compte des observations empiriques.
Reprint from : Y. Malevergne and D. Sornette (2001), “Multi-dimensional bubbles and fat tails”, Quantitative
Finance 1, 533-541.

2.1. Bulles rationnelles multi-dimensionnelles et queues épaisses 33
Q UANTITATIVE F INANCE V OLUME 1 (2001) 533–541 RESEARCH PAPER
I NSTITUTE OF P HYSICS P UBLISHING quant.iop.org
Multi-dimensional rational bubbles
and fat tails
Y Malevergne 1,2 and D Sornette 1,3
1 Laboratoire de Physique de la Matière Condensée CNRS UMR 6622,
Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France
2 Institut de Science Financière et d’Assurances, Université Lyon I, 43 Bd du
11 Novembre 1918, 69622 Villeurbanne Cedex, France
3 Institute of Geophysics and Planetary Physics and Department of Earth and
Space Science, University of California, Los Angeles, CA 90095, USA
E-mail: Yannick.Malevergne@unice.fr and sornette@unice.fr
Received 20 July 2001, in final form 20 August 2001
Published 14 September 2001
Online at stacks.iop.org/Quant/1/533
Abstract
Lux and Sornette have demonstrated that the tails of the unconditional
distributions of price differences and of returns associated with the model of
rational bubbles of Blanchard and Watson follow power laws (i.e. exhibit
hyperbolic decline), with an asymptotic tail exponent µ

34 2. Modèles phénoménologiques de cours
Y Malevergne and D Sornette Q UANTITATIVE F INANCE
on the process Xt. There is a huge literature on theoretical
refinements of this model and on the empirical detectability of
RE bubbles in financial data (see Camerer (1989) and Adam
and Szafarz (1992) for surveys of this literature).
Recently, Lux and Sornette (1999) studied the implications
of the RE bubble models for the unconditional distribution
of prices, price changes and returns resulting from a
more general discrete-time formulation extending (2) by allowing
the multiplicative factor at to take arbitrary values and
be i.i.d. random variables drawn from some non-degenerate
probability density function (pdf) Pa(a). The model can also
be generalized by considering non-normal realizations of bt
with distribution Pb(b) with E[bt] = 0, where E[·] is the expectation
operator. Since in (2) the bubble Xt denotes the difference
between the observed price and the fundamental price,
the ‘bubble’ regimes refer to the cases when Xt explodes exponentially
under the action of successive multiplications by
factor at,at+1,...with a majority of them larger than 1 in absolute
value but different, thus adding a stochastic component
to the standard model of Blanchard and Watson (1982).
For this large class of stochastic processes, Lux and
Sornette (1999) have shown that the distribution of returns
is a power law whose exponent µ is enforced by the nofree-lunch
condition to remain lower than one. Although
power-law tails are a pervasive feature of empirical data,
these characterizations are in strong disagreement with the
usual empirical estimates which find µ ≈ 3 (de Vries 1994,
Lux 1996, Pagan 1996, Guillaume et al 1997, Gopikrishnan
et al 1998). Thus, Lux and Sornette (1999) concluded that
exogenous rational bubbles are hardly compatible with the
empirical distribution data.
At this stage, one could argue that there is a logical trap
in the finding µ

2.1. Bulles rationnelles multi-dimensionnelles et queues épaisses 35
Q UANTITATIVE F INANCE Multi-dimensional rational bubbles
for instance Hommes (2001) and Challet et al (2001) and
references therein). Following many others before us, we
conclude that the hypotheses of rational expectations and of noarbitrage
condition are useful approximation or starting points
for model constructions.
Within this framework, the price Pt of an asset at time t
should obey the equation
Pt = δ · EQ[Pt+1|Ft]+dt ∀{Pt}t0, (4)
where dt is an exogeneous ‘dividend’, δ is the discount factor
and EQ[Pt+1|Ft] is the expectation of Pt+1 conditioned upon the
knowledge of the filtration up to time t under the risk-neutral
probability Q.
It is important to stress that the rationality of both
expectations and behaviour does not necessarily imply that
the price of an asset be equal to its fundamental value. In
other words, there can be rational deviations of the price from
this value, called rational bubbles. A rational bubble can arise
when the actual market price depends positively on its own
expected rate of change. This is thought to sometimes occur in
asset markets and constitutes the very mechanism underlying
the models of Blanchard (1979) and Blanchard and Watson
(1982). Indeed, the ‘forward’ solution of (4) is well-known to
be
+∞
Ft =
i=0
δ i · EQ[dt+i|Ft]. (5)
It is straightforward to check by replacement that the general
solution of (4) is the sum of the forward solution (5) and of an
arbitrary component Xt
where Xt has to obey the single condition:
Pt = Ft + Xt, (6)
Xt = δ · EQ[Xt+1|Ft]. (7)
With only two fundamental and quite reasonable assumptions,
it is thus possible to derive an equation (6) (with (7)) justifying
the existence of price fluctuations and deviations from the
fundamental value Ft, for equilibrium markets with rational
agents. However, the dynamics of the bubbles remains
unknown and is a priori completely arbitrary apart from the
no-arbitrage constraint (7).
2.2. Bubble dynamics
One of the main contributions of the model of Blanchard
and Watson (1982), and its possible generalizations (8), is
to propose a bubble dynamics which is both compatible with
most of the empirical stylized facts of price time series and
sufficiently simple to allow for a tractable analytical treatment.
Indeed, the stochastic autoregressive process
Xt+1 = atXt + bt, (8)
where {at} and {bt} are i.i.d. random variables, is well-known
to lead to fat-tailed distributions and volatility clustering.
Kesten (1973) (see Goldie (1991) for a modern extension)
has shown that, if E[ln |a|] < 0, the stochastic process {Xt}
admits a stationary solution (i.e. with a stationary distribution
function), whose distribution density P(X) is a power law
P(X) ∼ X −1−µ with exponent µ satisfying
E[|a| µ ] = 1, (9)
provided that E[|b| µ ] < ∞. It is easy to show that
without other constraints every exponent µ > 0 can be
reached. Consider, for example, a sequence of variables {at}
independently and identically distributed according to a lognormal
law with localization parameter a0 < 1 and scale
ln a0
parameter σ . Equation (9) simply yields µ =−2 σ 2 , which
shows that, varying σ , µ can range over the entire positive real
line.
The second interesting point is that the process (8) allows
volatility clustering. It is easy to see that a large value of Xt will
be followed by a large value of Xt+1 with a large probability.
The change of variables Xt = Y 2
t with at = vZ2 t and bt = uZ2 t
maps exactly the process (8) onto an ARCH(1) process
u + vYt 2 , (10)
Yt+1 = Zt
where Zt is a Gaussian random variable. The ARCH processes
are known to account for volatility clustering. Therefore, the
process (8) also exhibits volatility clustering.
This class (8) of stochastic processes thus provides
several interesting features of real price series (Roman et al
2001). There is however an objection related to the fact that,
without additional assumptions, the bubble price can become
arbitrarily negative and can then lead to a negative price:
Pt < 0. In fact, a negative price is not as meaningless as
often taken for granted, as shown in Sornette (2000). But,
even without allowing for negative prices, it is reasonable to
argue that near Pt = 0, other mechanisms come into play and
modify equation (8) in the neighbourhood of a vanishing price.
For instance, when a market undergoes a too strong and abrupt
loss, quotations are interrupted.
2.3. The independent bubbles assumption
The Blanchard and Watson model assumes that there is only
one asset and one bubble. In other words, the evolution of
each asset does not depend on the dynamics of the others.
But, in reality, there is no such thing as an isolated asset.
Stock markets exhibit a variety of inter-dependences, based
in part on the mutual influences between the USA, European
and Japanese markets. In addition, individual stocks may be
sensitive to the behaviour of the specific industry as a whole
to which they belong and to a few other indicators, such as
the main indices, interest rates and so on. Mantegna (1999)
and Bonanno et al (2001) have indeed shown the existence
of a hierarchical organization of stock interdependences.
Furthermore, bubbles often appear to be not isolated features
of a set of markets. For instance, Flood et al (1984) tested
whether a bubble simultaneously existed across the nations,
such as Germany, Poland, and Hungary, that experienced
hyperinflation in the early 1920s. Coordinated corrections
to what may be considered to be correlated bubbles can
sometimes be detected. One of the most prominent examples
535

36 2. Modèles phénoménologiques de cours
Y Malevergne and D Sornette Q UANTITATIVE F INANCE
is found in the market appreciations observed in many of the
world markets prior to the world market crash in October 1987
(Barro et al 1989). The collective growth of most of the
markets worldwide was interrupted by a worldwide market
crash: from the opening on 14 October 1987 through the
market close on 19 October major indices of market valuation
in the United States declined by 30% or more. Furthermore,
all major markets in the world declined substantially in the
month: out of 23 major industrial countries, 19 had a decline
greater than 20%. This is one of the most striking evidences
of the existence of correlations between corrections to bubbles
across the world markets. Similar intermittent coordination
of bubbles have been detected among the significant bubbles
followed by large crashes or severe corrections in Latin-
American and Asian stock markets (Johansen and Sornette
2000).
These empirical facts suggest an improvement on the onedimensional
bubble model by introducing a multi-dimensional
generalization.
2.4. Position of the problem
As shown in Lux and Sornette (1999), the one-asset model
(8) suffers from a deficiency: the power law tail exponents
predicted by the Blanchard and Watson model are not
compatible with the empirical facts. The proof relies on
the following ingredients: the no-free-lunch condition and
the one-dimensional dynamics of the bubble. The onedimensional
dynamics of the bubble implies that the equation
(9) holds, while the no-free-lunch condition imposes
E[a] > 1. (11)
Putting these two equations together allows Lux and Sornette
(1999) to conclude that necessarily µ

2.1. Bulles rationnelles multi-dimensionnelles et queues épaisses 37
Q UANTITATIVE F INANCE Multi-dimensional rational bubbles
A generalization of the two-dimensional case to arbitrary
dimensions leads to the following stochastic random equation
(SRE)
Xt = AtXt−1 + Bt
(16)
where (Xt, Bt) are d-dimensional vectors. Each component
of Xt can be thought of as the difference between the price
of the asset and its fundamental price. The matrices (At)
are identically independent distributed d × d-dimensional
stochastic matrices. We assume that Bt are identically
independent distributed random vectors and that (Xt) is a
causal stationary solution of (16). Generalizations introducing
additional arbitrary linear terms at larger time lags such as
Xt−2,... can be treated with slight modifications of our
approach and yield the same conclusions. We shall thus
confine our demonstration on the SRE of order 1, keeping in
mind that our results apply analogously to arbitrary orders of
regressions.
To formalize the SRE in a rigorous manner, we introduce
in a standard way the probability space (, F, P) and a
filtration (Ft). Here P represents the product measure P =
PX ⊗ PA ⊗ PB, where PX, PA and PB are the probability
measures associated with {Xt}, {At} and {Bt}. We further
assume as is customory that the stochastic process (Xt) is
adapted to the filtration (Ft).
In the following, we denote by |·|the Euclidean norm and
by ||·||the corresponding norm for any d × d-matrix A
||A|| = sup |Ax|. (17)
|x|=1
Now, we will formalize the ‘no-free-lunch’ condition for
the SRE (16) and show that it entails in particular that the
spectral radius (largest eigenvalue) of E[At] must be equal to
the inverse of the discount factor, hence it must be larger than
1.
3.3. The no-free-lunch condition
3.3.1. No-free-lunch condition under the risk-neutral
probability measure The ‘no-free-lunch’ condition is
equivalent to the existence of a probability measure Q
equivalent to P such that, for all self-financing portfolios t,
t
S0,t is a Q-martingale, where S0,t = t−1 i=0 δi −1 , δi = (1+ri) −1
is the discount factor for period i and ri is the corresponding
risk-free interest rate.
It is natural to assume that, for a given period i, the
discount rates ri are the same for all assets. In frictionless
markets, a deviation for this hypothesis would lead to arbitrage
opportunities. Furthermore, since the sequence of matrices
{At} is i.i.d. and therefore stationary, this implies that δt or rt
must be constant and equal respectively to δ and r.
Under those conditions, we have the following
proposition:
Proposition 1. The stochastic process
Xt = AtXt−1 + Bt
satisfies the no-arbitrage condition if and only if
(18)
EQ[A] = 1
δ Id. (19)
The proof is given in appendix A.
The condition (19) imposes some stringent constraints on
admissible matrices At. Indeed, while At are not diagonal in
general, their expectation must be diagonal. This implies that
the off-diagonal terms of the matrices At must take negative
values, sufficiently often so that their averages vanish. The offdiagonal
coefficients quantify the influence of other bubbles on
a given one. The condition (19) thus means that the average
effect of other bubbles on any given one must vanish. It is
straightforward to check that, in this linear framework, this
implies an absence of correlation (but not of dependence)
between the different bubble components E[X (k) X (ℓ) ] = 0
for any k = ℓ, where X (j) denotes the jth component of the
bubble X.
In contrast, the diagonal elements of At must be mostly
positive in order for EP[Aii] = δ−1 , for all i, to hold true. In
fact, on economic grounds, we can exclude the cases where
the diagonal elements take negative values. Indeed, a negative
value of Aii at a given time t would imply that X (i)
t might
abruptly change sign between t − 1 and t, what does not seem
to be a reasonable financial process.
3.3.2. Consequence for the no-free-lunch condition under
historic probability measure The historical P and riskneutral
Q probability measures are equivalent. This means
that there exists a non-negative matrix h(θ) = hij (θij ) such
that, for each element indexed by i, j, wehave
EP[Aij ] = EQ[hij · Aij ] (20)
= hij (θ 0 ij ) · EQ[Aij ] for some θ 0 ij ∈ R. (21)
The second equation comes from the well-known result:
f(θ)· g(θ)dµ(θ) = g(θ0) · f(θ)dµ(θ) for some θ0 ∈ R.
(22)
We thus get
EP[Aij ] = 0 if i = j (23)
EP[Aij ] = 1
δ (i)
if i = j, (24)
where the δ (i) can be different. We can thus write
EP[A] = δ −1 where δ −1 = diag[δ (1)−1 ,...,δ (d)−1 ]. (25)
Appendix B gives a proof showing that δ (i) is indeed the
genuine discount factor for the ith bubble component.
4. Renewal theory for products of
random matrices
In the following, we will consider that the random d × d
matrices At are invertible matrices with real entries. We will
denote by GLd(R) the group of these matrices.
537

38 2. Modèles phénoménologiques de cours
Y Malevergne and D Sornette Q UANTITATIVE F INANCE
4.1. Definitions
Definition 1 (Feasible matrix). A matrix M ∈ GLd(R) is
P-feasible if there exists an n ∈ N and M1,...,Mn ∈
supp(P) such that M = M1 ...Mn and if M has a simple real
eigenvalue q(M) which, in modulus, exceeds all other
eigenvalues of M.
Definition 2. For any matrix M ∈ GLd(R) and M ′ its
transpose, MM ′ is a symmetric positive definite matrix. We
define λ(M) as the square root of the smallest eigenvalue of
MM ′ .
4.2. Theorem
We extend the theorem 2.7 of Davis et al (1999), which
synthesized Kesten’s theorems 3 and 4 in Kesten (1973), to the
case of real valued matrices. The proof of this theorem is given
in Le Page (1983). We stress that the conditions listed below do
not require the matrices (An) to be non-negative. Actually, we
have seen that, in order for the rational expectation condition
not to lead to trivial results, the off-diagonal coefficients of
(An) have to be negative with sufficiently large probability
such that their means vanish.
Theorem 1. Let (An) be an i.i.d. sequence of matrices in
GLd(R) satisfying the following set of conditions:
H1: for some ɛ>0, EPA [||A||ɛ ] < 1,
H2: for every open U ⊂ Sd−1 (the unit sphere in Rd ) and for
all x ∈ Sd−1 there exists an n such that
xA1 ...An
Pr
∈ U > 0. (26)
||xA1 ...An||
H3: the group {ln |q(M)|,M is PA-feasible} is dense in R .
H4: for all r ∈ R d , Pr{A1r + B1 = r} < 1.
H5: there exists a κ0 > 0 such that
EPA ([λ(A1)] κ0 ) 1. (27)
H6: with the same κ0 > 0 as for the previous condition, there
exists a real number u>0 such that
sup{||A1||, EPA
||B1||} κ0+u
< ∞ ,
||A1|| −u (28)
< ∞.
EPA
Provided that these conditions hold,
• there exists a unique solution κ1 ∈ (0,κ0] to the equation
1
lim ln EPA ||A1 ...An||
n→∞ n κ1 = 0, (29)
• if (Xn) is the stationary solution to the SRE in (16) then
X is regularly varying with index κ1. In other words, the
tail of the marginal distribution of each of the
components of the vector X is asymptotically a power
law with exponent κ1.
538
4.3. Comments on the theorem
4.3.1. Intuitive meaning of the hypotheses A suitable
property for an economic model is the existence of a stationary
solution, i.e. the solution Xt of the SRE (16) should not
blow up. This condition is ensured by the hypothesis (H1).
Indeed, EPA [ln ||A||] < 0 implies that the Lyapunov exponent
of the sequence {An} of i.i.d. matrices is negative (Davis
et al 1999). And it is well known that the negativity of the
Lyapunov exponent is a sufficient condition for the existence
of a stationary solution Xt, provided that E[ln + ||B||] < ∞.
However, this condition can lead to too fast a decay of the
tail of the distibution of {X}. This phenomenon is prevented
by (H5) which means intuitively that the multiplicative
factors given by the elements of At sometimes produce an
amplification of Xt. In the one-dimensional bubble case, this
condition reduces to the simple rule that at must sometimes
be larger than 1 so that a bubble can develop. Otherwise, the
power law tail will be replaced by an exponential tail.
So, (H1) and (H5) keep the balance between two opposite
objectives: to obtain a stationary solution and to observe a
fat-tailed distribution for the process (Xt).
Another desirable property for the model is ergodicity: we
expect the price process (Xt) to explore the entire space Rd .
This is ensured by hypotheses (H2) and (H4): hypothesis (H2)
allows Xt to visit the neighbourhood of any point in Rd , and
(H4) forbids the trajectory to be trapped at some points.
Hypotheses (H3) and (H6) are more technical ones. The
hypothesis (H6) simply ensures that the tails of the distributions
of At and Bt are thinner than the tail created by the SRE (16),
so that the observed tail index is really due to the dynamics of
the system and not to the heavy-tail nature of the distributions
of At or Bt. The hypothesis (H3) expresses some kind of
aperiodicity condition.
4.3.2. Intuitive meaning of (29) The equation (29)
determining the tail exponent κ1 reduces to (9) in the onedimensional
case, which is simple to handle. In the multidimensional
case, the novel feature is the non-diagonal nature
of the multiplication of matrices which does not allow in
general for an explicit equation similar to (9).
4.4. Constraint on the tail index
The first conclusion of the theorem above shows that the tail
index κ1 of the process (Xt) is driven by the behaviour of the
matrices (At). We will then state a proposition in which we
give a majoration of the tail index.
Proposition 2. A necessary condition for the process (16) to
have a tail exponent κ1 > 1 is that the spectral radius of
[A] be smaller than 1:
EPA
κ1 > 1 ⇒ ρ(EPA [A])

2.1. Bulles rationnelles multi-dimensionnelles et queues épaisses 39
Q UANTITATIVE F INANCE Multi-dimensional rational bubbles
5. Consequences for rational expectation
bubbles
We have seen in section 3.3 from proposition 1 that, as a
result of the no-arbitrage condition, the spectral radius of the
matrix EP[A] = 1
δ Id is greater than 1. As a consequence, by
application of the converse of proposition 2, this proves that
the tail index κ1 of the distribution of (X) is smaller than 1.
Using the same arguments as in Lux and Sornette (1999),
that we do not recall here, it can be shown that the distribution
of price differences and price returns follows, at least over
an extended range of large returns, a power law distribution
whose exponent remains lower than 1. This result generalizes
to arbitrary d-dimensional processes the result of Lux and
Sornette (1999). As a consequence, d-dimensional rational
expectation bubbles linking several assets suffer from the
same discrepancy compared to empirical data as the onedimensional
bubbles. It therefore appears that accounting for
possible dependences between bubbles is not sufficient to cure
the Blanchard and Watson model: a linear multi-dimensional
bubble dynamics such as (16) is hardly reconcilable with some
of the most fundamental stylized facts of financial data at a very
elementary level.
This result does not rely on the diagonal property of the
matrices E[At] but only on the value of its spectral radius
imposed by the no-arbitrage condition. In other words, the fact
that the introduction of dependences between asset bubbles is
not sufficient to cure the model can be traced to the constraint
introduced by the no-arbitrage condition, which imposes that
the averages of the off-diagonal terms of the matrices At
vanish. As we indicated before, this implies zero correlations
(but not the absence of dependence) between asset bubbles.
It thus seems that the multi-dimensional generalization is
constrained so much by the no-arbitrage condition that the
multi-dimensional bubble model almost reduces to an average
one-dimensional model. With this insight, our present result
generalizing that of Lux and Sornette (1999) is natural.
To our knowledge, there are two possible remedies. The
first one is based on the rational bubble and crash model
of Johansen et al (1999, 2000) which abandons the linear
stochastic dynamics (8) in favour of an essentially arbitrary
and nonlinear dynamics controlled by a crash hazard rate.
A jump process for crashes is added to the process, with a
crash hazard rate evolving with time such that the rational
expectation condition is ensured. This model is squarely
based on the rational expectation framework and shows that
changing the dynamics of the Blanchard and Watson model
allows satisfying results to be obtained, as the corresponding
return distributions can be made to exhibit reasonable fat tails.
The second solution (Sornette 2001) requires the existence
of an average exponential growth of the fundamental price at
some return rate rf > 0 larger than the discount rate. With
the condition that the price fluctuations associated with bubbles
must on average grow with the mean market return rf , it can be
shown that the exponent of the power law tail of the returns is
no more bounded by 1 as soon as rf is larger than the discount
rate r and can take essentially arbitrary values. This second
approach amounts to abandoning the rational pricing theory
(6) with (4) and keeping only the no-arbitrage constraint (7)
on the bubble component. This hypothesis may be true in the
case of a firm, sometimes in the case of an industry (railways in
the 19th century, oil and computer in the 20th for instance), but
is hard to defend in the case of an economy as a whole. The
real long run interest rate in the US is approximately 3.5%,
and the real rate of growth of profits since World War II is
about 2.1%. Thus, for the economy as a whole, the discounted
sum is always finite. It would be interesting to investigate the
interplay between inter-dependence between several bubbles
and this exponential growth model.
As a final positive remark balancing our negative result,
we stress that the stochastic multiplicative multi-asset rational
bubble model presented here provides a natural mechanism for
the existence of a ‘universal’ tail exponent µ across different
markets.
Acknowledgments
We acknowledge helpful discussions and exchanges with
J P Laurent, T Lux, V Pisarenko and M Taqqu. We also
thank T Mikosch for providing access to Le Page (1983) and
V Pisarenko for a critical reading of the manuscript. Any
remaining error is ours.
Appendix A. Proof of proposition 1 on
the no-arbitrage condition
Let t be the value at time t of any self-financing portfolio:
t = W ′
t Xt, (A.1)
where W ′
t = (W1,...,Wd) is the vector whose components
are the weights of the different assets and the prime denotes
the transpose. The no-free-lunch condition reads:
t = δ · EQ[t+1|Ft] ∀{t}t0. (A.2)
Therefore, for all self-financing strategies (Wt), wehave
W ′
t+1 EQ[A] − 1
δ Id
Xt = 0 ∀ Xt ∈ R d , (A.3)
where we have used the fact that (Wt+1) is (Ft)-measurable
and that the sequence of matrices {At} is i.i.d.
The strategy W ′
t = (0,...,0, 1, 0,...,0) (1 in ith
position), for all t, is self-financing and implies
(ai1,ai2,...,aii − 1
δ ,...,aid)
·(X (1)
t ,X (2)
t ,...,X (i)
t ,...,X (d)
t ) ′ = 0, (A.4)
for all Xt ∈ Rd . We have called aij the (i, j)th coefficient of
the matrix EQ[A]. As a consequence,
(ai1,ai2,...,aii − 1
δ ,...,aid) = 0 ∀i, (A.5)
and
EQ[A] = 1
δ Id. (A.6)
539

40 2. Modèles phénoménologiques de cours
Y Malevergne and D Sornette Q UANTITATIVE F INANCE
The no-arbitrage condition thus implies: EQ[A] = 1
δ Id.
We now show that the converse is true, namely that if
EQ[A] = 1
δ Id is true, then the no-arbitrage condition is
verified. Let us thus assume that EQ[A] = 1
δ Id holds. Then
W ′
EQ[Pt+1|Ft] = EQ[W ′
t+1 · Xt+1|Ft] (A.7)
= W ′
t+1 · EQ[Xt+1|Ft] (A.8)
= W ′
t+1 · EQ[At+1Xt + Bt+1|Ft] (A.9)
= W ′
t+1 · EQ[At+1|Ft] · Xt (A.10)
= W ′
t+1 · EQ[A] · Xt
(A.11)
= 1 ′
W t+1
δ · Xt. (A.12)
The condition that the portfolio is self-financing is
t+1 Xt = W ′
t Xt, which means that the weights can be
rebalanced a priori arbitrarily between the assets with the
constraint that the total wealth at the same time remains
constant. We can thus write
EQ[t+1|Ft] = 1 ′
W t+1 · Xt (A.13)
δ
= 1
δ t. (A.14)
Therefore, the discounted process {t} is a Q-martingale.
Appendix B. Proof that δ (i) is the
discount factor for the ith bubble
component in the historical space
Here, we express the no-free-lunch condition in the historical
space (or real space). The condition we will obtain is the socalled
‘rational expectation condition’, which is a little bit less
general than the condition detailed in the previous appendix A.
Given the prices {X (i)
k }kt of an asset, labelled by i, until
the date t, the best estimation of its price at t +1isEP[X (i)
t+1 |Ft].
So, the RE condition leads to
EP[X (i)
t+1 |Ft] − X (i)
t
X (i) = r
t
(i)
t , (A.15)
where r (i)
t is the return of the asset i between t and t +1. As
previously, we will assume in what follows that r (i)
t = r (i) is
time independent. Thus, the rational expectation condition for
the assets i reads
X (i)
t = δ (i) · EPX
= δ (i) · EP
X (i)
t+1 |Ft
X (i)
t+1 |Ft
, (A.16)
, (A.17)
where δ (i) is the discount factor.
A priori, each asset has a different return. Thus,
introducing the vector ˜Xt whose ith component is X(i) t
δ (i) ,we
can summarize the rational expectation condition as
540
˜Xt = EP [Xt+1|Ft] . (A.18)
Again, we evaluate the conditional expectation of (16),
and using the fact that {At} are i.i.d., we have
= EP[A]Xt−1. (A.19)
EP Xt|Ft−1
This equation together with (A.18), leads to
˜Xt−1 = EP[A]Xt−1, (A.20)
which can be rewritten
EP[A] − δ−1
Xt−1 = 0, (A.21)
where δ−1 = diag[δ (1)−1 ...δ (d)−1 ] is the matrix whose ith
diagonal component is δ (i)−1 and 0 elsewhere.
The equation (A.21) must be true for every Xt−1 ∈ Rd ,
thus
EP[A] = δ−1 , (A.22)
which is the result announced in section (3.3.2).
Appendix C. Proof of proposition 2 on
the condition κ1 < 1
First step. Behaviour of the function
1
f(κ)= lim ln EPA ||An ...A1||
n→∞ n κ1
in the interval [0,κ0].
(A.23)
In the fourth step of the proof of theorem 3, Kesten (1973)
shows that the function f has the following properties:
• f is continuous on [0,κ0],
• f(0) = 0 and f(κ0) >0,
• f ′ (0) 1, it is necessary that
f(1)

2.1. Bulles rationnelles multi-dimensionnelles et queues épaisses 41
Q UANTITATIVE F INANCE Multi-dimensional rational bubbles
f(k)
f(1)
0
Figure A1. Schematic shape of the function f(κ)defined in (A.23).
where ρ(EPA [A]) is the spectral radius of A. By definition,
|| EPA [A] n
|| | EPA [A] n
xmax| =λ n max . (A.29)
Then
1
lim
n→∞ n ln EPA ||An
1
...A1|| lim ln ρ(EPA [A])n
n→∞ n
= ln ρ(EPA [A]) . (A.30)
Now, suppose that ρ(EPA [A]) 1. We obtain
1
f(1) = lim
n→∞ n ln EPA ||An ...A1|| 0, (A.31)
which is in contradiction with the necessary condition (A.24).
Thus,
References
f(1)

42 2. Modèles phénoménologiques de cours
2.2 Des bulles rationnelles aux krachs
Nous étudions et généralisons de diverses manières le modèle de bulles rationnelles introduit dans la
littérature économique par Blanchard et Watson (1982). Les bulles sont présentées comme équivalentes
aux modes de Goldstone du prix fondamental de l’équation de “pricing” rationnel, associés à la brisure
de symétrie liée à l’existence d’un dividende non nul. Généralisant les bulles en terme de processus stochastique
multiplicatif, nous résumons le résultat de Lux et Sornette (2002) selon lequel la condition
de non arbitrage impose que la queue des distributions de rendement est régulièrement variable, d’indice
de queue µ < 1. Nous rappelons ensuite le principal résultat de Malevergne et Sornette (2001a),
qui étend le modèle de bulles rationnelles à un nombre quelconque de dimensions d : un nombre d
de séries financières sont rendues linéairement interdépendantes via une matrice de couplage aléatoire.
Nous dérivons la condition de non arbitrage dans ce contexte et, à l’aide de la théorie de renouvellement
des matrices aléatoires, nous étendons le théorème de Lux et Sornette (2002) et démontrons que les
queues des distributions inconditionnelles des rendements de chaque actif sont régulièrement variables
et de même indice de queue inférieur à un. Bien que le comportement asymptotique en loi de puissance
des queues de distributions soit une caractéristique marquante ressortant de l’étude des données empiriques,
l’exposant de queue inférieur à un est en contradiction flagrante avec l’estimation empirique
généralement admise qui est de l’ordre de trois. Nous discutons alors deux extensions du modèle de
bulles rationnelles en accord avec les faits stylisés.
Reprint from : D. Sornette et Y. Malevergne (2001),“From rational bubbles to crashes”, Physica A 299,
40-59.

2.2. Des bulles rationnelles aux krachs 43
Physica A 299 (2001) 40–59
From rational bubbles to crashes
www.elsevier.com/locate/physa
D. Sornette a;b; ∗ , Y. Malevergne a
a Laboratoire de Physique de la Matiere Condensee, CNRS UMR 6622 and Universite de Nice-Sophia
Antipolis, 06108 Nice Cedex 2, France
b Institute of Geophysics and Planetary Physics and Department of Earth and Space Science,
University of California, Los Angeles, CA 90095, USA
Abstract
We study and generalize in various ways the model of rational expectation (RE) bubbles
introduced by Blanchard and Watson in the economic literature. Bubbles are argued to be the
equivalent of Goldstone modes of the fundamental rational pricing equation, associated with
the symmetry-breaking introduced by non-vanishing dividends. Generalizing bubbles in terms of
multiplicative stochastic maps, we summarize the result of Lux and Sornette that the no-arbitrage
condition imposes that the tail of the return distribution is hyperbolic with an exponent ¡1.
We then outline the main results of Malevergne and Sornette, who extend the RE bubble model
to arbitrary dimensions d: a number d of market time series are made linearly interdependent
via d × d stochastic coupling coe cients. We derive the no-arbitrage condition in this context
and, with the renewal theory for products of random matrices applied to stochastic recurrence
equations, we extend the theorem of Lux and Sornette to demonstrate that the tails of the
unconditional distributions associated with such d-dimensional bubble processes follow power
laws, with the same asymptotic tail exponent ¡1 for all assets. The distribution of price
di erences and of returns is dominated by the same power-law over an extended range of
large returns. Although power-law tails are a pervasive feature of empirical data, the numerical
value ¡1 is in disagreement with the usual empirical estimates ≈ 3. We then discuss two
extensions (the crash hazard rate model and the non-stationary growth rate model) of the RE
bubble model that provide two ways of reconciliation with the stylized facts of nancial data.
c○ 2001 Elsevier Science B.V. All rights reserved.
1. The model of rational bubbles
Blanchard [1] and Blanchard and Watson [2] originally introduced the model of
rational expectations (RE) bubbles to account for the possibility, often discussed in
∗ Corresponding author. Laboratoire de Physique de la Matiere Condensee, CNRS UMR 6622 and Universite
de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France. Fax: +33-4-92-07-67-54.
E-mail addresses: sornette@unice.fr (D. Sornette), yannick.malevergne@unice.fr (Y. Malevergne).
0378-4371/01/$ - see front matter c○ 2001 Elsevier Science B.V. All rights reserved.
PII: S 0378-4371(01)00281-3

44 2. Modèles phénoménologiques de cours
D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59 41
the empirical literature and by practitioners, that observed prices may deviate signi -
cantly and over extended time intervals from fundamental prices. While allowing for
deviations from fundamental prices, rational bubbles keep a fundamental anchor point
of economic modelling, namely that bubbles must obey the condition of rational expectations.
In contrast, recent works stress that investors are not fully rational, or have
at most bounded rationality, and that behavioral and psychological mechanisms, such
as herding, may be important in the shaping of market prices [3–5]. However, for
uid assets, dynamic investment strategies rarely perform over simple buy-and-hold
strategies [6], in other words, the market is not far from being e cient and little arbitrage
opportunities exist as a result of the constant search for gains by sophisticated
investors. Here, we shall work within the conditions of rational expectations and of
no-arbitrage condition, taken as useful approximations. Indeed, the rationality of both
expectations and behavior often does not imply that the price of an asset be equal to its
fundamental value. In other words, there can be rational deviations of the price from
this value, called rational bubbles. A rational bubble can arise when the actual market
price depends positively on its own expected rate of change, as sometimes occurs in
asset markets, which is the mechanism underlying the models of Refs. [1,2].
In order to avoid the unrealistic picture of ever-increasing deviations from fundamental
values, Blanchard [2] proposed a model with periodically collapsing bubbles in
which the bubble component of the price follows an exponential explosive path (the
price being multiplied by at =a¿1) with probability and collapses to zero (the price
being multiplied by at = 0) with probability 1− . It is clear that, in this model, a bubble
has an exponential distribution of lifetimes with a nite average lifetime =(1 − ).
Bubbles are thus transient phenomena. The condition of rational expectations imposes
that a =1=( ), where is the discount factor. In order to allow for the start of new
bubbles after the collapse, a stochastic zero mean normally distributed component bt
is added to the systematic part of Xt. This leads to the following dynamical equation
Xt+1 = atXt + bt ; (1)
where, as we said, at =a with probability and at = 0 with probability 1 − . Both
variables at and bt do not depend on the process Xt. There is a huge literature on theoretical
re nements of this model and on the empirical detectability of RE bubbles in
nancial data (see Refs. [7,8], for surveys of this literature). Model (1) has also been
explored in a large variety of contexts, for instance in ARCH processes in econometry
[9], 1D random- eld Ising models [10] using Mellin transforms, and more recently
using extremal properties of the G-harmonic functions on non-compact groups [11]
and the Wiener–Hopf technique [12]. See also Ref. [13] for a short review of other
domains of applications including population dynamics with external sources, epidemics,
immigration and investment portfolios, the internet, directed polymers in
random media ::: :
Large |Xk| are generated by intermittent ampli cations resulting from the multiplication
by several successive values of |a| larger than one. We now o er a simple
“mean- eld” type argument that clari es the origin of the power law fat tail. Let us

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42 D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59
call p¿ the probability that the absolute value of the multiplicative factor a is found
larger than 1. The probability to observe n successive multiplicative factors |a| larger
than 1 is thus pn ¿. Let us call |a¿| the average of |a| conditioned on being larger than
1:|a¿| is thus the typical absolute value of the ampli cation factor. When n successive
multiplicative factors occur with absolute values larger than 1, they typically lead to
an ampli cation of the amplitude of X by |a¿| n . Using the fact that the additive term
bk ensures that the amplitude of Xk remains of the order of the standard deviation or
of other measures of typical scales b of the distribution Pb(b) when the multiplicative
factors |a| are less than 1, this shows that a value of Xk of the order of |X |≈ b|a¿| n
occurs with probability
p n
ln |X |= b 1
¿ = exp (n ln p¿) ≈ exp ln p¿ =
(2)
ln |a¿| (|X |= b)
with =lnp¿=ln |a¿|, which can be rewritten as p¿|a¿| = 1. Note the similarity
between this last “mean- eld” equation and the exact solution (8) given below. The
power law distribution is thus the result of an exponentially small probability of creating
an exponentially large value [14]. Expression (2) does not provide a precise
determination of the exponent , only an approximate one since we have used a kind
of mean- eld argument in the de nition of |a¿|.
In the next section, we recall how bubbles appear as possible solutions of the fundamental
pricing equation and play the role of Goldstone modes of a price-symmetry
broken by the dividend ow. We then describe the Kesten generalization of rational
bubbles in terms of random multiplicative maps and present the fundamental result [15]
that the no-arbitrage condition leads to the constraint that the exponent of the power
law tail is less than 1. We then present an extension to arbitrary multidimensional
random multiplicative maps: a number d of market time series are made linearly interdependent
via d × d stochastic coupling coe cients. We show that the no-arbitrage
condition imposes that the non-diagonal impacts of any asset i on any other asset j = i
has to vanish on average, i.e., must exhibit random alternative regimes of reinforcement
and contrarian feedbacks. In contrast, the diagonal terms must be positive and
equal on average to the inverse of the discount factor. Applying the results of renewal
theory for products of random matrices to stochastic recurrence equations (SRE), we
extend the theorem of Ref. [15] and demonstrate that the tails of the unconditional
distributions associated with such d-dimensional bubble processes follow power laws
(i.e., exhibit hyperbolic decline), with the same asymptotic tail exponent ¡1 for all
assets. The distribution of price di erences and of returns is dominated by the same
power-law over an extended range of large returns. In order to unlock the paradox,
we brie y discuss the crash hazard rate model [16,17] and the non-stationary growth
model [18]. We conclude by proposing a link with the theory of speculative pricing
through a spontaneous symmetry-breaking [19].
We should stress that, due to the no-arbitrage condition that forms the backbone of
our theoretical approach, correlations of returns are vanishing. In addition, the multiplicative
stochastic structure of the models ensures the phenomenon of volatility

46 2. Modèles phénoménologiques de cours
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clustering. These two stylized facts, taken for granted in our present approach, will
not be discussed further.
2. Rational bubbles of an isolated asset [15]
2.1. Rational expectation bubble model and Goldstone modes
We rst brie y recall that pricing of an asset under rational expectations theory is
based on the two following hypothesis: the rationality of the agents and the “no-free
lunch” condition.
Under the rational expectation condition, the best estimation of the price pt+1 of
an asset at time t + 1 viewed from time t is given by the expectation of pt+1 conditioned
upon the knowledge of the ltration {Ft} (i.e., sum of all available information
accumulated) up to time t: E[pt+1|Ft].
The “no-free lunch” condition imposes that the expected returns of every assets are
all equal under a given probability measure Q equivalent to the historical probability
measure P. In particular, the expected return of each asset is equal to the return r of
the risk-free asset (which is assumed to exist), and thus the probability measure Q is
named the risk neutral probability measure.
Putting together these two conditions, we are led to the following valuation formula
for the price pt:
pt = EQ[pt+1|Ft]+dt ∀{pt}t¿0 ; (3)
where dt is an exogeneous “dividend”, and =(1+r) −1 is the discount factor. The rst
term in the r.h.s. quanti es the usual fact that something tomorrow is less valuable than
today by a factor called the discount factor. Intuitively, the second term, the dividend,
is added to express the fact that the expected price tomorrow has to be decreased by
the dividend since the value before giving the dividend incorporates it in the pricing.
The “forward” solution of (3) is well-known to be the fundamental price
p f
t =
+∞
i=0
i EQ[dt+i|Ft] : (4)
It is straightforward to check by replacement that the sum of the forward solution (4)
and of an arbitrary component Xt
pt = p f
t + Xt ; (5)
where Xt has to obey the single condition of being an arbitrary martingale
Xt = EQ[Xt+1|Ft] (6)
is also the solution of (3). In fact, it can be shown [20] that (5) is the general solution
of (3).
Here, it is important to note that, in the framework of the Blanchard and Watson
model, the speculative bubbles appear as a natural consequence of the valuation formula

2.2. Des bulles rationnelles aux krachs 47
44 D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59
(3), i.e., of the no free-lunch condition and the rationality of the agents. Thus, the
concept of bubbles is not an addition to the theory, as sometimes believed, but is
entirely embedded in it.
Notice also that the component Xt in (5) plays a role analogous to the Goldstone
mode in nuclear, particle and condensed-matter physics [21,22]. Goldstone modes are
the zero-wavenumber zero-energy modal uctuations that attempt to restore a broken
symmetry. For instance, consider a Bloch wall between two semi-in nite magnetic
domains of opposite spin directions selected by opposite magnetic eld at boundaries
far away. At non-zero temperature, capillary waves are excited by thermal uctuations.
The limit of very long-wavelength capillary modes correspond to Goldstone modes that
tend to restore the translational symmetry broken by the presence of the Bloch wall
[23].
In the present context, as shown in Ref. [19], the
p →−p parity symmetry (7)
is broken by the “external” eld embodied in the dividend ow dt. Indeed, as can be
seen from (3) and its forward solution (4), the fundamental price is identically zero in
absence of dividends. Ref. [19] has stressed the fact that it makes perfect sense to think
of negative prices. For instance, we are ready to pay a (positive) price for a commodity
that we need or like. However, we will not pay a positive price to get something we
dislike or which disturb us, such as garbage, waste, broken and useless car, chemical
and industrial hazards, etc. Consider a chunk of waste. We will be ready to buy it
for a negative price, in other words, we are ready to take the unwanted commodity if
it comes with cash. Positive dividends imply positive prices, negative dividends lead
to negative prices. Negative dividends correspond to the premium to pay to keep an
asset for instance. From an economic view point, what makes a share of a company
desirable is its earnings, that provide dividends, and its potential appreciation that give
rise to capital gains. As a consequence, in absence of dividends and of speculation,
the price of share must be nil and the symmetry (7) holds. The earnings leading to
dividends d thus act as a symmetry-breaking “ eld”, since a positive d makes the share
desirable and thus develop a positive price.
It is now clear that the addition of the bubble Xt, which can be anything but for
the martingale condition (6), is playing the role of the Goldstone modes restoring the
broken symmetry: the bubble price can wander up or down and, in the limit where it
becomes very large in absolute value, dominate over the fundamental price, restoring
the independence with respect to dividend. Moreover, as in condensed-matter physics
where the Goldstone mode appears spontaneously since it has no energy cost, the
rational bubble itself can appear spontaneously with no dividend.
The “bubble” Goldstone mode turns out to be intimately related to the “money” Goldstone
mode introduced by Bak et al. [24]. Ref. [24] introduces a dynamical many-body
theory of money, in which the value of money in equilibrium is not xed by the
equations, and thus obeys a continuous symmetry. The dynamics breaks this continuous
symmetry by xating the value of money at a level which depends on initial

48 2. Modèles phénoménologiques de cours
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conditions. The uctuations around the equilibrium, for instance in the presence of
noise, are governed by the Goldstone modes associated with the broken symmetry. In
apparent contrast, a bubble represents the arbitrary deviation from fundamental valuation.
Introducing money, a given valuation or price is equivalent to a certain amount
of money. A growing bubble thus corresponds to the same asset becoming equivalent
to more and more cash. Equivalently, from the point of view of the asset, this can be
seen as cash devaluation, i.e., in ation. The “bubble” Goldstone mode and the “money”
Goldstone mode are thus two facets of the same fundamental phenomenon: they both
are left unconstrained by the valuation equations.
2.2. The no-arbitrage condition and fat tails
Following Ref. [15], we study the implications of the RE bubble models for the
unconditional distribution of prices, price changes and returns resulting from a general
discrete-time formulation extending (1) by allowing the multiplicative factor at to take
arbitrary values and be i.i.d. random variables drawn from some non-degenerate probability
density function (pdf) Pa(a). The model can also be generalized by considering
non-normal realizations of bt with distribution Pb(b) with EP[bt] = 0, where EP[ · ]is
the unconditional expectation with respect to the probability measure P.
Provided EP[ln a] ¡ 0 (stationarity condition) and if there is a number such that
0 ¡EP[|b| ] ¡ + ∞, such that
EP[|a| ]=1 (8)
and such that EP[|a| ln |a|] ¡ + ∞, then the tail of the distribution of X is asymptotically
(for large X ’s) a power law [25,26]
PX (X )dX ≈ C
dX ; (9)
|X | 1+
with an exponent given by the real positive solution of (8).
Rational expectations require in addition that the bubble component of asset prices
obeys the “no free-lunch” condition
EQ[Xt+1|Ft]=Xt
(10)
where ¡1 is the discount factor and the expectation is taken conditional on the
knowledge of the
rst
ltration (information) until time t. Condition (10) with (1) imposes
and then
EQ[a]=1= ¿ 1 ; (11)
EP[a] ¿ 1 ; (12)
on the distribution of the multiplicative factors at.
Consider the function
M( )=EP[a ] : (13)

2.2. Des bulles rationnelles aux krachs 49
46 D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59
Fig. 1. Convexity of M( ). This enforces that the exponent solution of (8) is ¡1.
It has the following properties:
(1) M(0) = 1 by de nition,
(2) M ′ (0) = EP[ln a] ¡ 0 from the stationarity condition,
(3) M ′′ ( )=EP[(ln a) 2 |a| ] ¿ 0, by the positivity of the square,
(4) M(1)=1= ¿ 1 by the no-arbitrage result (12).
M( ) is thus convex and is shown in Fig. 1. This demonstrate that ¡1 automatically
(see Ref. [15] for a detailed demonstration). It is easy to show [15] that the
distribution of price di erences has the same power law tail with the exponent ¡1
and the distribution of returns is dominated by the same power-law over an extended
range of large returns [15]. Although power-law tails are a pervasive feature of empirical
data, these characterizations are in strong disagreement with the usual empirical
estimates which nd ≈ 3 [27–31]. Lux and Sornette [15] concluded that exogenous
rational bubbles are thus hardly reconcilable with some of the stylized facts of nancial
data at a very elementary level.
3. Generalization of rational bubbles to arbitrary dimensions [32]
3.1. Generalization to several coupled assets
In reality, there is no such thing as an isolated asset. Stock markets exhibit a variety
of inter-dependences, based in part on the mutual in uences between the USA,
European and Japanese markets. In addition, individual stocks may be sensitive to
the behavior of the speci c industry as a whole to which they belong and to a few
other indicators, such as the main indices, interest rates and so on. Mantegna et al.
[33,34] have indeed shown the existence of a hierarchical organization of stock interdependences.
Furthermore, bubbles often appear to be not isolated features of a set of
markets. For instance, Ref. [35] tested whether a bubble simultaneously existed across

50 2. Modèles phénoménologiques de cours
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the nations, such as Germany, Poland, and Hungary, that experienced hyperin ation
in the early 1920s. Coordinated bubbles can sometimes be detected. One of the most
prominent example is found in the market appreciations observed in many of the world
markets prior to the world market crash in October 1987 [36]. Similar intermittent
coordination of bubbles have been detected among the signi cant bubbles followed
by large crashes or severe corrections in Latin American and Asian stock markets
[37]. It is therefore desirable to generalize the one-dimensional RE bubble model (1)
to the multi-dimensional case. One could also hope a priori that this generalization
would modify the result ¡1 obtained in the one-dimensional case and allow for a
better adequation with empirical results. Indeed, 1d-systems are well-known to exhibit
anamalous properties often not shared by higher dimensional systems. Here however,
it turns out that the same result ¡1 holds, as we shall see.
In the case of several assets, rational pricing theory again dictates that the fundamental
price of each individual asset is given by a formula like (3), where the speci c
dividend ow of each asset is used, with the same discount factor. The corresponding
forward solution (4) is again valid for each asset. The general solution for each asset
is (5) with a bubble component Xt di erent from an asset to the next. The di erent
bubble components can be coupled, as we shall see, but they must each obey the martingale
condition (6), component by component. This imposes speci c conditions on
the coupling terms, as we shall see.
Following this reasoning, we can therefore propose the simplest generalization of a
bubble into a “two-dimensional” bubble for two assets X and Y with bubble prices, respectively,
equal to Xt and Yt at time t. We express the generalization of the Blanchard–
Watson model as follows:
Xt+1 = atXt + btYt + t ; (14)
Yt+1 = ctXt + dtYk + t ; (15)
where at, bt, ct and dt are drawn from some multivariate probability density function.
The two additive noises t and t are also drawn from some distribution function with
zero mean. The diagonal case bt = ct = 0 for all t recovers the previous one-dimensional
case with two uncoupled bubbles, provided t and t are independent.
Rational expectations require that Xt and Yt obey both the “no-free lunch”
condition (10), i.e., · EQ[Xt+1|Ft]=Xt and · EQ[Yt+1|Ft]=Yt. With (14; 15); this
gives
(EQ[at] − −1 )Xt + EQ[bt]Yt =0; (16)
EQ[ct]Xt +(EQ[dt] − −1 )Yt =0; (17)
where we have used that t and t are centered. The two equations (16; 17) must
be true for all times, i.e., for all values of Xt and Yt visited by the dynamics. This

2.2. Des bulles rationnelles aux krachs 51
48 D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59
imposes EQ[bt]=EQ[ct]=0 and EQ[at]=EQ[dt]= −1 . We are going to retrieve this
result more formally in the general case.
3.2. General formulation
A generalization to arbitrary dimensions leads to the following stochastic random
equation (SRE):
Xt = AtXt−1 + Bt ; (18)
where (Xt; Bt) are d-dimensional vectors. Each component of Xt can be thought of
as the price of an asset above its fundamental price. The matrices (At) are identically
independent distributed d × d-dimensional stochastic matrices. We assume that Bt are
identically independent distributed random vectors and that (Xt) is a causal stationary
solution of (18). Generalizations introducing additional arbitrary linear terms at larger
time lags such as Xt−2;::: can be treated with slight modi cations of our approach
and yield the same conclusions. We shall thus con ne our demonstration on the SRE
of order 1, keeping in mind that our results apply analogously to arbitrary orders of
regressions.
In the following, we denote by |·| the Euclidean norm and by · the corresponding
norm for any d × d-matrix A
A =sup|Ax|
: (19)
|x|=1
Technical details are given in [32].
3.3. The no-free lunch condition
The valuation formula (3) and the martingale condition (6) given for a single asset
easily extends to a basket of assets. It is natural to assume that, for a given period t, the
discount rate rt(i), associated with asset i, are all the same. In frictionless markets, a
deviation for this hypothesis would lead to arbitrage opportunities. Furthermore, since
the sequence of matrices {At} is assumed to be i.i.d. and therefore stationary, this
implies that t or rt must be constant and equal, respectively, to and r.
Under those conditions, we have the following proposition.
Proposition 1. The stochastic process
Xt = AtXt−1 + Bt
satis es the no-arbitrage condition if and only if
(20)
EQ[A]= 1 Id : (21)
The proof is given in Ref. [32] in which this condition (21) is also shown to hold true
under the historical probability measure P.

52 2. Modèles phénoménologiques de cours
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The condition (21) imposes some stringent constraints on admissible matrices At.
Indeed, while At are not diagonal in general, their average must be diagonal. This
implies that the o -diagonal terms of the matrices At must take negative values, sufciently
often so that their averages vanish. The o -diagonal coe cients quantify the
in uence of other bubbles on a given one. The condition (21) thus means that the
average e ect of other bubbles on any given one must vanish. It is straightforward to
check that, in this linear framework, this implies an absence of correlation (but not an
absence of dependence) between the di erent bubble components E[X (k) X (‘) ]=0 for
any k = ‘.
In contrast, the diagonal elements of At must be positive in majority in order for
EP[Aii]= (i)−1,
for all i’s, to hold true. In fact, on economic grounds, we can exclude
the cases where the diagonal elements take negative values. Indeed, a negative value
of Aii at a given time t would imply that X (i)
t abruptly change sign between t − 1 and
t, what does not seem to be a reasonable nancial process.
3.4. Renewal theory for products of random matrices
In the following, we will consider that the random d × d matrices At are invertible
matrices with real entries. We use Theorem 2:7 of Davis et al. [38], which synthesized
Kesten’s Theorems 3 and 4 in Ref. [25], to the case of real valued matrices. The proof
of this theorem is given in Ref. [39]. We stress that the conditions listed below do
not require the matrices (An) to be non-negative. Actually, we have seen that, in order
for the rational expectation condition not to lead to trivial results, the o -diagonal
coe cients of (An) have to be negative with su ciently large probability such that
their means vanish.
Theorem 1. Let (An) be an i.i.d. sequence of matrices in GLd(R) satisfying the following
set of conditions that we state in a heuristic manner (see Ref. [32] for technical
details). Provided that the following conditions hold;
H1: stationarity condition;
H2: ergodicity;
H3: intermittent ampli cation of the random matrices;
H4: the fattailness of the distribution is not controlled by that of the additive
part Bt;
then;
• there exists a unique solution ∈ (0; 0] to the equation
1
lim
n→∞ n ln EP[A1 ···An ]=0; (22)
• If (Xn) is the stationary solution to the stochastic recurrence equation in (18)
then X is regularly varying with index . In other words; the tail of the marginal

2.2. Des bulles rationnelles aux krachs 53
50 D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59
distribution of each of the components of the vector X is asymptotically a power
law with exponent .
Eq. (22) determining the tail exponent reduces to (8) in the one-dimensional case,
which is simple to handle. In the multi-dimensional case, the novel feature is the
non-diagonal nature of the multiplication of matrices which does not allow in general
for an explicit equation similar to (8).
It is important to stress that the tails of the distribution of returns for all the components
of the bubble decrease with the same tail index . This model thus provides
a natural setting for rationalizing the well-documented empirical observation that the
exponent is found to be approximately the same for most assets. The constraint on
its value discussed in the next paragraph does not diminish the value of this remark,
as explained in Section 5.
3.5. Constraint on the tail index
The rst conclusion of the theorem above shows that the tail index of the process
(Xt) is driven by the behavior of the matrices (At). We will then state a proposition
in which we give a majoration of the tail index.
Proposition 2. A necessary condition to have ¿1 is that the spectral radius (largest
eigenvalue) of EP[A] be smaller than 1:
¿1 ⇒ (EP[A]) ¡ 1 : (23)
The proof is given in Ref. [32]. This proposition, put together with Proposition 1 above,
allows us to derive our main result. We have seen in Section 3.3 from Proposition 1
that, as a result of the no-arbitrage condition, the spectral radius of the matrix EP[A]
is greater than 1. As a consequence, by application of the converse of Proposition 2,
we nd that the tail index of the distribution of (X) is smaller than 1. This result
does not rely on the diagonal property of the matrices EP[At] but only on the value
of the spectral radius imposed by the no-arbitrage condition.
This result generalizes to arbitrary d-dimensional processes the result of Ref. [15]. As
a consequence, d-dimensional rational expectation bubbles linking several assets su er
from the same discrepancy compared to empirical data as the one-dimensional bubbles.
It would therefore appear that exogenous rational bubbles are hardly reconcilable with
some of the most fundamental stylized facts of nancial data at a very elementary level.
At this stage, we have to ask the question: what is wrong with the model of rational
bubbles? Two alternative answers are explored below: either we believe in the standard
valuation formula and we are led to extend the restricted framework described by the
Blanchard and Watson’s model; or we believe in the existence of the bubbles within
their framework and we have to generalize the valuation formula. In the next two
sections, we will discuss these two points of view.

54 2. Modèles phénoménologiques de cours
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4. The crash hazard rate model [16,17]
In the stylised framework of a purely speculative asset that pays no dividends—i.e.,
with zero fundamental price—and in which we ignore information asymmetry and the
market-clearing condition, the price of the asset equals the price of the bubble and the
valuation formula (3) leads to the familiar martingale hypothesis for the bubble price:
for all t ′ ¿t t→t ′EQ[X (t ′ )|Ft]=X (t) : (24)
This equation is nothing but a generalisation of Eq. (6) to a continuous time formulation,
in which t→t ′ denotes the discount factor from time t to time t′ .
We consider a general bubble dynamics given by
dX = m(t) X (t)dt − X (t)dj; (25)
where m(t) can be any non-linear causal function of X itself. We add a jump process
j to capture the possibility that the bubble exhibits a crash. j is thus zero before the
crash and one afterwards. The random nature of the crash occurrence is modeled by the
cumulative distribution function Q(t) of the time of the crash, the probability density
function q(t)=dQ=dt and the hazard rate h(t)=q(t)=[1 − Q(t)]. The hazard rate is the
probability per unit of time that the crash will happen in the next instant provided it
has not happened yet, i.e.:
EQ[dj|Ft]=h(t)dt: (26)
Expression (25) assumes that, during a crash, the bubble drops by a xed percentage
∈ (0; 1), say between 20% and 30% of the bubble price.
Using EQ[X (t +dt)|Ft]=(1+r dt)X (t), where r is the riskless discount rate, taking
the expectation of (25) conditioned on the ltration up to time t and using Eq. (26),
we get
EQ[dX |Ft]=m(t)X (t)dt − X (t)h(t)dt = rX (t)dt; (27)
which yields
m(t) − r = h(t) : (28)
If the crash hazard rate h(t) increases, the return m(t)−r above the riskless interest rate
increases to compensate the traders for the increasing risk. Reciprocally, if the dynamics
of the bubble shoots up, the rational expectation condition imposes an increasing crash
risk in order to ensure the absence of arbitrage opportunities: the risk-adjusted return
remains constant equal to the risk-free rate. The corresponding equation for the bubble
price, conditioned on the crash not to have occurred, is
t
X (t)
log = rt + h(t
X (t0)
t0
′ )dt ′
before the crash : (29)
The integral t
t0 h(t′ )dt ′ is the cumulative probability of a crash until time t. This gives
the logarithm of the bubble price as the relevant observable. It has successfully been

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52 D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59
applied to the 1929 and 1987 Wall Street crashes up to about 7:5 years prior to the
crash [40,17].
The higher the probability of a crash, the faster the bubble must increase (conditional
on having no crash) in order to satisfy the martingale condition. Reciprocally,
the higher the bubble, the more dangerous is the probability of a looming crash. Intuitively,
investors must be compensated by a higher return in order to be induced to
hold an asset that might crash. This is the only e ect that this model captures. Note
that the bubble dynamics can be anything and the bubble can in particular be such
that the distribution of returns are fat tails with an exponent ≈ 3 without loss of
generality [41].
Ilinski [42] raised the concern that the martingale condition (24) leads to a model
which “assumes a zero return as the best prediction for the market”. He continues:
“No need to say that this is not what one expects from a perfect model of market
bubble! Buying shares, traders expect the price to rise and it is re ected (or caused)
by their prediction model. They support the bubble and the bubble support them!”.
In other words, Ilinski [42] criticises a key economic element of the model [16,17]:
market rationality. This point is captured by assuming that the market level is expected
to stay constant (up to the riskless discount rate) as written in equation (24). Ilinski
claims that this equation (24) is wrong because the market level does not stay constant
in a bubble: it rises, almost by de nition.
This misunderstanding addresses a rather subtle point of the model and stems from
the di erence between two di erent types of returns:
(1) The unconditional return is indeed zero as seen from (24) and re ects the fair
game condition.
(2) The conditional return, conditioned upon no crash occurring between time t and
time t ′ , is non-zero and is given by Eq. (28). If the crash hazard rate is increasing with
time, the conditional return will be accelerating precisely because the crash becomes
more probable and the investors need to be remunerated for their higher risk.
Thus, the expectation which remains constant in Eq. (24) takes into account the
probability that the market may crash. Therefore, conditionally on staying in the bubble
(no crash yet), the market must rationally rise to compensate buyers for having taken
the risk that the market could have crashed.
The market price re ects the equilibrium between the greed of buyers who hope the
bubble will in ate and the fear of sellers that it may crash. A bubble that goes up is
just one that could have crashed but did not. The model [16,17] is well summarised
by borrowing the words of another economist: “(...) the higher probability of a crash
leads to an acceleration of [the market price] while the bubble lasts”. Interestingly, this
citation is culled from the very same article by Blanchard [1] that Ilinski [42] cites as
an alternative model more realistic than the model [16,17]. We see that this is in fact
more of an endorsement than an alternative.
A simple way to incorporate a di erent level of risk aversion into the model [16,17]
is to say that the probability of a crash in the next instant is perceived by traders as
being K times bigger than it objectively is. This amounts to multiplying our hazard

56 2. Modèles phénoménologiques de cours
D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59 53
rate h(t) byK, and once again this makes no substantive di erence as long as K is
bounded away from zero and in nity. Risk aversion is a central feature of economic
theory, and it is generally thought to be stable within a reasonable range, associated
with slow-moving secular trends such as changes in education, social structures and
technology. Ilinski [42] rightfully points out that risk perceptions are constantly changing
in the course of real-life bubbles, but wrongfully claims that the model [16,17]
violates this intuition. In this model, risk perceptions do oscillate dramatically throughout
the bubble, even though subjective aversion to risk remains stable, simply because
it is the objective degree of risk that the bubble may burst that goes through wild
swings. For these reasons, the criticisms put forth by Ilinski, far from making a dent
in the economic model [16,17], serve instead to show that it is robust, exible and
intuitive.
To summarize, the crash hazard rate model is such that the price dynamics can be
essentially arbitrary, and in particular such that the corresponding returns exhibit a
reasonable fat tail. A jump process for crashes is added, with a crash hazard rate such
that the rational expectation condition is ensured.
5. The non-stationary growth model [18]
In the previous section, we have presented a model which assumes that the fundamental
valuation formula remains valid and have generalized Blanchard and Watson’s
framework by reformulating the rational expectation condition with a jump crash process.
We now consider the second view point which consists in rejecting the validity
of the valuation formula while keeping the decomposition of the price of an asset into
the sum of a fundamental price and a bubble term. In this aim, we present a possible
modi cation of the rational bubble model of Blanchard and Watson, recently proposed
in Ref. [18], which involves an average exponential growth of the fundamental price
at some return rate rf ¿ 0 larger than the discount rate.
5.1. Exponentially growing economy
Recall that (5) shows that the observable market price is the sum of the bubble
component Xt and of a “fundamental” price p f
t
pt = p f
t + Xt : (30)
Thus, waiving o the valuation formula (3), let us assume that the fundamental price
p f
t is growing exponentially as
p f
t = p0e rft
(31)
at the rate rf and the bubble price is following (1).
Note that this formulation is compatible with the standard valuation formula as long
as rf ¡r, provided the cash- ow dt at time t also grows with the same exponential

2.2. Des bulles rationnelles aux krachs 57
54 D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59
rate rf, i.e.: dt = d0erft . Indeed, the standard valuation formula then applies and leads
to
p f
∞ dte
t =
k=0
krf
ekr
d0
e
r − rf
rft
(32)
up to the rst order in rf − r. The discussion of the case rf ¿r, for which (32) loses
its meaning, is the subject of the sequel in which we follow [19].
Putting (1) and (31) together with (30), we obtain
pt+1 = p f
t+1 + atXt + bt = atpt +(e rf − at)p f
t + bt :
Replacing p
(33)
f
t in (30) by p0erft given in (31) leads to
pt =e rft (p0 +ât) ; (34)
where we have de ned the “reduced” bubble price following:
ât+1 = ate −rf ât +e −rf e −rft bt : (35)
Thus, if we allow the additive term bt in (1) to also grow exponentially as
bt =e rft ˆbt ; (36)
where ˆbt is a stationary stochastic white noise process, we obtain
ât+1 = ate −rf ât +e −rf ˆbt ; (37)
which is of the usual form. Intuitively, the additive term represents the background of
“normal” uctuations around the fundamental price (31). Their “normal” uctuations
have thus to grow with the same growth rate in order to remain stationary in relative
value.
In addition, replacing p f
t in (33) again by p0e rft leads to
pt+1 = atpt +e rft [p0(e rf − at)+ ˆbt] : (38)
The expression (38) has the same form as (1) with a di erent additive term [p0(e r −
at)+ ˆbt] replacing bt. The structure of this new additive term makes clear the origin of
the factor e rft : as we said, it re ects nothing but the average exponential growth of the
underlying economy. The contributions bt =e rft ˆbt are then nothing but the uctuations
around this average growth.
5.2. The value of the tail exponent
The condition E[ln a] ¡rf ensures that E[ln(ae −rf )] ¡ 0 which is now the stationarity
condition for the process ât de ned by (37). The conditions 0 ¡E[|e rf ˆbt| ] ¡ + ∞
(which is the same condition 0 ¡E[|bt| ] ¡ + ∞ as before) and the solution of
E[|ae −rf | ] = 1 (39)
together with the constraint E[|ae −rf | ln |ae −rf |] ¡ + ∞ (which is the same as
E[|a| ln |a|] ¡ + ∞) leads to an asymptotic power law distribution for the reduced

58 2. Modèles phénoménologiques de cours
D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59 55
price variable ât of the form Pâ(â) ≈ Câ=|â| 1+ , where is the real positive solution
of (39). Note that the condition E[ln(ae −rf )] ¡ 0 which is E[ln(a)] ¡rf now allows
for positive average growth rate of the product atat−1at−2 ···a2a1a0.
Consider the illustrative case where the multiplicative factors at are distributed according
to a log-normal distribution such that E[ln a]=lna0 (where a0 is thus the most
probable value taken by at) and of variance 2 . Then
E[|ae −rf
| ] = exp −rf + ln a0 + 2
2
2
: (40)
Equating (40) to 1 to get according to Eq. (39) gives
=2 rf − ln a0
2
= rf − ln a0
r − ln a0
=1+ rf − r
r − ln a0
: (41)
We have used the notation 1= =1+r for the discount rate r de ned in terms of the
discount factor . The second equality in (41) uses E[a]=a0e 2 =2 .
First, we retrieve the result [15] that ¡1 for the initial RE model (1) for which
rf = 0 and ln a0 ¡ 0. However, as soon as rf ¿r −ln ; we get
¿1 ; (42)
and can take arbitrary values. Technically, this results fundamentally from the structure
of the process in which the additive noise grows exponentially to mimick the
growth of the bubble which alleviates the bound ¡1. Note that rf does not need
to be large for the result (42) to hold. Take for instance an annualized discount rate
ry =2%; an annualized return r y
f = 4% and a0 =1:0004. Expression (41) predicts
which is compatible with empirical data.
=3;
5.3. Price returns
The observable return is
where
Rt = pt+1 − pt
pt
= t
f
pt+1 − pf t
p f
t
= pf
t+1 − pf t + Xt+1 − Xt
p f
t + Xt
+ Xt+1 − Xt
p f
t
= t rf + ât+1 − ât
p0
; (43)
t = pf t
p f
1
=
: (44)
t + Xt 1+(ât=p0)
In order to derive the last equality in the right-hand-side of (43), we have used the
de nition of the return of the fundamental price (neglecting the small second order
di erence between e rf − 1 and rf). Expression (43) shows that the distribution of

2.2. Des bulles rationnelles aux krachs 59
56 D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59
returns Rt of the observable prices is the same as that of the product of the random
variable t by rf+(ât+1−ât)=p0. Now, the tail of the distribution of rf+(ât+1−ât)=p0 is
the same as the tail of the distribution of ât+1 −ât; which is a power law with exponent
solution of (39), as shown rigorously in Ref. [15].
It remains to show that the product of this variable rf +(ât+1 − ât)=p0 by t has
the same tail behavior as rf +(ât+1 − ât)=p0 itself. If rf +(ât+1 − ât)=p0 and t were
independent, this would follow from results in Ref. [43] who demonstrates that for two
independent random variables and with Proba(| | ¿x) ≈ cx − and E[ + ] ¡ ∞
for some ¿0; the random product obeys Proba(| | ¿x) ≈ E[ ]x − .
However, rf+(ât+1−ât)=p0 and t are not independent as both contain a contribution
from the same term ât. However, when âtp0; t is close to 1 and the previous result
should hold. The impact of ât in becomes important when ât becomes comparable
to p0.
It is then convenient to rewrite (43) using (44) as
Rt =
rf
1+(ât=p0) + ât+1 − At
=
p0 +ât
We can thus distinguish two regimes:
rf
1+(ât=p0) + (ate −rf − 1)ât +e −rf bt
p0 +ât
: (45)
• for not too large values of the reduced bubble term ât; speci cally for ât ¡p0, the
denominator p0 +ât changes more slowly than the numerator of the second term, so
that the distribution of returns will be dominated by the variations of this numerator
(ate −rf − 1)ât +e −rf bt and, hence, will follow approximately the same power-law
as for ât; according to the results of Ref. [43].
• For large bubbles, ât of the order of or greater than p0; the situation changes,
however: from (45), we see that when the reduced bubble term ât increases without
bound, the rst term rf=(1+(ât=p0)) goes to 0 while the second term becomes
asymptotically ate −r − 1. This leads to the existence of an absolute upper bound for
the absolute value of the returns.
To summarize, we expect that the distribution of returns will therefore follow a
power-law with the same exponent as for ât, but with a nite cut-o (see [18]
for details). This is validated by numerical simulations shown in Fig. 2 taken from
Ref. [18].
Thus, when the price uctuations associated with bubbles on average grow with the
mean market return rf; we nd that the exponent of the power law tail of the returns
is no more bounded by 1 as soon as rf is larger than the discount rate r and can take
essentially arbitrary values. It is remarkable that this condition rf ¿r corresponds to
the paradoxical and unsolved regime in fundamental valuation theory where the forward
valuation solution (4) loses its meaning, as discussed in Ref. [19]. In analogy
with the theory of bifurcations and their normal forms, Ref. [19] proposed that this
regime might be associated with a spontaneous symmetry breaking phase corresponding

60 2. Modèles phénoménologiques de cours
D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59 57
Fig. 2. Double logarithmic scale representation of the complementary cumulative distribution of the “monthly”
returns Rt de ned in (43) of the synthetic total price sum of the exponential growing fundamental price and
the bubble price. The continuous (resp. dashed) line corresponds to the positive (resp. negative) returns. The
distribution is well-described by an asymptotic power law with an exponent in agreement with the prediction
≈ 3:3 given by the equations (39) and (41) and shown as the straight line. The small di erences between
the predicted slope and the numerically generated ones are within the error bar of ±0:3 obtained from a
standard maximum likelihood Hill estimation. From Ref. [18].
to a spontaneous valuation in absence of dividends by pure speculative imitative
processes.
6. Conclusion
Despite its elegant formulation of the bubble phenomenon, the Blanchard and Watson’s
model su ers from a lethal discrepancy: it does not seem to comply with the
empirical data, i.e., it cannot generate power law tails whose exponent is greater than
1, in disagreement with the empirical tail index found around 3. We have summarized
the demonstration that this result holds true both for a bubble de ned for a single asset
as well as for bubbles on any set of coupled assets, as long as the rational expectation
condition holds.
In order to reconcile the theory with the empirical facts on the tails of the distributions
of returns, two alternative models have been presented. The “crash hazard rate”
model extends the formulation of Blanchard and Watson by replacing the linear stochastic
bubble price equation by an arbitrary dynamics solely constrained by the no-arbitrage
condition made to hold with the introduction of a jump process. The “growth rate
model” departs more audaciously from standard economic models since it discards
one of the pillars of the standard valuation theory, but putting itself rmly in the
regime rf ¿r for which the fundamental valuation formula breaks down. In addition to
allowing for correct values of the tail exponent, it provides a generalization of the

2.2. Des bulles rationnelles aux krachs 61
58 D. Sornette, Y. Malevergne / Physica A 299 (2001) 40–59
fundamental valuation formula by providing an understanding of its breakdown as
deeply associated with a spontaneous breaking of the price symmetry. Its implementation
for multi-dimensional bubbles is straightforward and the results obtained in
Section 5 carry over naturally in this case. This provides an explanation for why the
tail index seems to be the same for any group of assets as observed empirically. This
work begs for the introduction of a generalized eld theory which would be able to
capture the spontaneous breaking of symmetry, recover the fundamental valuation formula
in the normal economic case rf ¡r and extend it to the still unexplored regime
rf ¿r.
Acknowledgements
We acknowledge helpful discussions and exchanges with J.V. Andersen, A. Johansen,
J.P. Laurent, O. Ledoit, T. Lux, V. Pisarenko and M. Taqqu and thank T. Mikosch for
providing access to Ref. [39].
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Chapitre 3
Distributions exponentielles étirées contre
distributions régulièrement variables
Un large consensus semble s’être développé en faveur de l’hypothèse selon laquelle les distributions
empiriques de rendement des séries financières sont régulièrement variables, avec un indice de queue de
l’ordre de trois. Nous montrons tout d’abord, à l’aide de tests synthétiques réalisés sur des séries temporelles
dont la volatilité présente de longues dépendances et dont les distributions marginales sont soit
des lois de puissance soit des exponentielles étirées, que les estimateurs basés sur les distributions des
extrêmes généralisés ou les distributions de Pareto généralisées ne sont pas à même de distinguer ces deux
classes de distributions, en contradiction avec de nombreuses études antérieures. Ensuite, nous utilisons
une représentation paramétrique de la distribution de rendements qui englobe à la fois des distributions
régulièrement variables ainsi que rapidement variables comme notamment les exponentielles-étirées.
Utilisant le théorème de Wilk, nous comparons la validité de ces différentes hypothèses et concluons que
les distributions exponentielles-étirées et les distributions de Pareto fournissent une description convenable
des données et ne peuvent être distinguées lorsque l’on s’intéresse à des quantiles suffisamment
élevés. Nous introduisons ensuite un nouveau test basé sur le fait que dans une certaine limite, les distributions
exponentielles-étirées tendent vers les distributions de Pareto. La conclusion de notre ensemble
de tests est que probablement les distributions de rendements d’actifs financiers décroissent moins vite
que toute exponentielle-étirée mais plus vite que toute loi de puissance d’exposant raisonnable. Pour terminer,
nous discutons les implications de nos résultats sur le problème d’existence des moments et pour
l’estimation du risque.
63

64 3. Distributions exponentielles étirées contre distributions régulièrement variables

Empirical Distributions of Log-Returns:
Exponential or Power-like?
Y. Malevergne 1,2 , V. Pisarenko 3 , and D. Sornette 1,4
1 Laboratoire de Physique de la Matière Condensée
CNRS UMR6622 and Université de Nice-Sophia Antipolis
Parc Valrose, 06108 Nice Cedex 2, France
2 Institut de Science Financière et d’Assurances - Université Lyon I
43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
3 International Institute of Earthquake Prediction Theory and Mathematical Geophysics
Russian Ac. Sci. Warshavskoye sh., 79, kor. 2, Moscow 113556, Russia
4 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science
University of California, Los Angeles, California 90095
e-mails: Yannick.Malevergne@unice.fr, Vlad@sirus.mitp.ru and sornette@unice.fr
Abstract
A large consensus now seems to take for granted that the distributions of empirical returns of financial
time series are regularly varying, with a tail exponent close to 3. First, we show by synthetic tests performed
on time series with long-range time dependence in the volatility with both Pareto and Stretched-
Exponential distributions that standard generalized extreme value (GEV) and Generalized Pareto Distribution
(GPD) estimators are quite inefficient and cannot distinguish reliably between the two classes
of distributions, in contradiction with previous results. Then, we use a parametric representation of the
distribution of returns of 100 years of daily return of the Dow Jones Industrial Average and over 1 years
of 5-minutes returns of the Nasdaq Composite index, encompassing both a regularly varying distribution
in one limit of the parameters and rapidly varying distributions of the class of the Stretched-Exponential
(SE) distributions in other limits. Using the method of nested hypothesis testing (Wilk theorem), we
conclude that both the SE distributions and Pareto distributions provide reliable descriptions of the data
and cannot be distinguished for sufficiently high thresholds. Then, we introduced a novel encompassing
test based on the discovery that the SE contains the Pareto law in a certain limit that confirms that the SE
encompasses any power law tail distribution. However, our best estimation of the parameters of the SE
model suggests that the tails decay probably more slowly that a pure SE. Summing up all the evidence
provided by our battery of tests, it seems that the tails ultimately decay slower than any SE but probably
faster than power laws with reasonable exponents. We discuss the implications of our results on the
“moment condition failure” and for risk estimation and management.
1
65

66 3. Distributions exponentielles étirées contre distributions régulièrement variables
1 Motivation of the study
The determination of the precise shape of the tail of the distribution of returns is a major issue both from
a practical and from an academic point of view. For practitioners, it is crucial to accurately estimate the
low value quantiles of the distribution of returns (profit and loss) because they are involved in almost all
the modern risk management methods. From an academic perspective, many economic and financial theories
rely on a specific parameterization of the distributions whose parameters are intended to represent the
“macroscopic” variables the agents are sensitive to.
The distribution of returns is one of the most basic characteristics of the markets and many papers have been
devoted to it. Contrarily to the average or expected return, for which economic theory provides guidelines
to assess them in relation with risk premium, firm size or book-to-market equity (see for instance Fama
and French (1996)), the functional form of the distribution of returns, and especially of extreme returns, is
much less constrained and still a topic of active debate. Naively, the central limit theorem would lead to
a Gaussian distribution for sufficiently large time intervals over which the return is estimated. Taking the
continuous time limit such that any finite time interval is seen as the sum of an infinite number of increments
thus leads to the paradigm of log-normal distributions of prices and equivalently of Gaussian distributions
of returns, based on the pioneering work of Bachelier (1900) later improved by Samuelson (1965). The lognormal
paradigm has been the starting point of many financial theories such as Markovitz (1959)’s portfolio
selection method, Sharpe (1964)’s market equilibrium model or Black and Scholes (1973)’s rational option
pricing theory. However, for real financial data, the convergence in distribution to a Gaussian law is very
slow (Campbell et al. 1997, Bouchaud and Potters 2000, for instance), much slower that predicted for
independent returns. As table 1 shows, the excess kurtosis (which is zero for a normal distribution) remains
large even for monthly returns, testifying of significant deviations from normality, of the heavy tail behavior
of the distributions of returns and of significant dependences between returns (Campbell et al. 1997).
Another idea rooted in economic theory consists in invoking the “Gibrat principle” (Simon 1957) initially
used to account for the growth of cities and of wealth through a mechanism combining stochastic multiplicative
and additive noises (Levy et al. 1996, Sornette and Cont 1997, Biham et al 1998, Sornette 1998)
leading to a Pareto distribution of sizes (Champenowne 1953, Gabaix 1999). Rational bubble models a
la Blanchard and Watson (1982) can also be cast in this mathematical framework of stochastic recurrence
equations and leads to distribution with power law tails, albeit with a strong constraint on the tail exponent
(Lux and Sornette 2002, Malevergne and Sornette 2001). These frameworks suggest that an alternative and
natural way to capture the heavy tail character of the distributions of returns is to use distributions with
power-like tails (Pareto, Generalized Pareto, stable laws) or more generally, regularly-varying distributions
(Bingham et al 1987) 1 , the later encompassing all the former.
In the early 1960s, Mandelbrot (1963) and Fama (1965) presented evidence that distributions of returns can
be well approximated by a symmetric Levy stable law with tail index b about 1.7. These estimates of the
power tail index have recently been confirmed by Mittnik et al. (1998), and slightly different indices of the
stable law (b = 1.4) were suggested by Mantegna and Stanley (1995, 2000).
On the other hand, there are numerous evidences of a larger value of the tail index b ∼ = 3 (Longin 1996,
Guillaume et al. 1997, Gopikrishnan et al. 1998, Müller et al. 1998, Farmer 1999, Lux 2000). See also the
various alternative parameterizations in term of the Student distribution (Blattberg and Gonnedes 1974, Kon
1984), hyperbolic distributions (Eberlein et al. 1998, Prause 1998), normal inverse Gaussian distributions
(Barndorff-Nielsen 1997), and Pearson type-VII distributions (Nagahara and Kitagawa 1999), which all have
an asymptotic power law tail and are regularly varying. Thus, a general conclusion of this group of authors
1 The general representation of a regularly varying distribution is given by ¯F(x) = L(x) · x −α , where L(·) is a slowly varying
function, that is, limx→∞L(tx)/L(x) = 1 for any finite t.
2

concerning tail fatness can be formulated as follows: tails of the distribution of returns are heavier than a
Gaussian tail and heavier than an exponential tail; they certainly admit the existence of a finite variance
(b > 2), whereas the existence of the third (skewness) and the fourth (kurtosis) moments is questionable.
These apparent contradictory results actually do not apply to the same quantiles of the distributions of
returns. Indeed, Mantegna and Stanley (1995) have shown that the distribution of returns can be described
accurately by a Lévy law only within a range of approximately nine standard deviations, while a faster decay
of the distribution is observed beyond. This almost-but-not-quite Lévy stable description explains (in part)
the slow convergence of the returns distribution to the Gaussian law under time aggregation (Sornette 2000).
And it is precisely outside this range where the Lévy law applies that a tail index of about three have been
estimated. This can be seen from the fact that most authors who have reported a tail index b ∼ = 3 have used
some optimality criteria for choosing the sample fractions (i.e., the largest values) for the estimation of the
tail index. Thus, unlike the authors supporting stable laws, they have used only a fraction of the largest
(positive tail) and smallest (negative tail) sample values.
It would thus seem that all has been said on the distributions of returns. However, there are dissenting
views in the literature. Indeed, the class of regularly varying distributions is not the sole one able to account
for the large kurtosis and fat-tailness of the distributions of returns. Some recent works suggest alternative
descriptions for the distributions of returns. For instance, Gouriéroux and Jasiak (1998) claim that the
distribution of returns on the French stock market decays faster than any power law. Cont et al. (1997)
have proposed to use exponentially truncated stable distributions (Cont et al. 1997) while Laherrère and
Sornette (1999) suggest to fit the distributions of stock returns by the Stretched-Exponential (SE) law. These
results, challenging the traditional hypothesis of power-like tail, offer a new representation of the returns
distributions and need to be tested rigorously on a statistical ground.
A priori, one could assert that Longin (1996)’s results should rule out the exponential and Stretched-
Exponential hypotheses. Indeed, his results, based on extreme value theory, show that the distributions
of log-returns belong to the maximum domain of attraction of the Fréchet distribution, so that they are necessarily
regularly varying power-like laws. However, his study, like almost all others on this subject, has
been performed under the assumption that (1) financial time series are made of independent and identically
distributed returns and (2) the corresponding distributions of returns belong to one of only three possible
maximum domains of attraction. However, these assumptions are not fulfilled in general. While Smith
(1985)’s results indicate that the dependence of the data does not constitute a major problem in the limit
of large samples, we shall see that it can significantly bias standard statistical methods for samples of size
commonly used in extreme tails studies. Moreover, Longin’s conclusions are essentially based on an aggregation
procedure which stresses the central part of the distribution while smoothing the characteristics of
the tail, which are essential in characterizing the tail behavior.
In addition, real financial time series exhibit GARCH effects (Bollerslev 1986, Bollerslev et al. 1994) leading
to heteroskedasticity and to clustering of high threshold exceedances due to a long memory of the
volatility. These rather complex dependent structures make difficult if not questionable the blind application
of standard statistical tools for data analysis. In particular, the existence of significant dependence in the
return volatility leads to dramatic underestimates of the true standard deviation of the statistical estimators
of tail indices. Indeed, there are now many examples showing that dependences and long memories as well
as nonlinearities mislead standard statistical tests (Andersson et al. 1999, Granger and Teräsvirta 1999, for
instance). Consider the Hill’s and Pickand’s estimators, which play an important role in the study of the
tails of distributions. It is often overlooked that, for dependent time series, Hill’s estimator remains only
consistent but not asymptotically efficient (Rootzen et al. 1998). Moreover, for financial time series with a
dependence structure described by a GARCH process, Kearns and Pagan (1997) have shown that the standard
deviation of Hill’s estimator obtained by a bootstrap method can be seven to eight time larger than the
standard deviation derived under the asymptotic normality assumption. These figures are even worse for
3
67

68 3. Distributions exponentielles étirées contre distributions régulièrement variables
Pickand’s estimator.
The question then arises whether the many results and seemingly almost consensus obtained by ignoring
the limitations of usual statistical tools could have led to erroneous conclusions about the tail behavior of
the distributions of returns. Here, we propose to investigate once more this delicate problem of the tail
behavior of distributions of returns in order to shed new lights. To this aim, we investigate two time series:
the daily returns of the Dow Jones Industrial Average (DJ) Index over a century and the five-minutes returns
of the Nasdaq Composite index (ND) over one year from April 1997 to May 1998. These two sets of data
have been chosen since they are typical of the data sets used in most previous studies. Their size (about
20,000 data points), while significant compared with those used in investment and portfolio analysis, is
however much smaller than recent data-intensive studies using ten of millions of data points (Gopikrishnan
et al. 1998, Matia et al. 2002, Mizuno et al. 2002).
Our first conclusion is that none of the standard parametric family distributions (Pareto, exponential, stretchedexponential
and incomplete Gamma) fits satisfactorily the DJ and ND data on the whole range of either
positive or negative returns. While this is also true for the family of stretched exponential distribution, this
family appears to be the best among the four considered parametric families, in so far as it is able to fit the
data over the largest interval. Our second and main conclusion comes from the discovery that the Pareto
distribution is a limit case of the Stretched-Exponential distribution. This allows us to test the encompassing
of these two models with respect to the true data, from which it seems that the Stretched-Exponential distribution
is the most relevant model of returns distributions. The regular decay of the fractional exponent of
the SE model together with the regular increase of the tail index of the Pareto model lead us to think that the
extreme tail of true distribution of returns is fatter that any stretched-exponential, strictly speaking -i.e., with
a strickly positive fractional exponent- but thinner than any power law. Notwithstanding our best efforts, we
cannot conclude on the exact nature of the far-tail of distributions of returns. As already mentioned, other
works have proposed the so-called inverse-cubic law (b = 3) based on the analysis of distributions of returns
of high-frequency data aggregated over hundreds up to thousands of stocks. This aggregating procedure
leads to novel problems of interpretation. We think that the relevant question for most practical applications
is not to determine what is the true asymptotic tail but what is the best effective description of the tails in
the domain of useful applications. As we shall show below, it may be that the extreme asymptotic tail is
a regularly varying function with tail index b = 3 for daily returns, but this is not very useful if this tail
describes events whose recurrence time is a century or more. Our present work must thus be gauged as
an attempt to provide a simple efficient effective description of the tails of distribution of returns covering
most of the range of interest for practical applications. We feel that the efforts requested to go beyond the
tails analyzed here, while of great interest from a scientific point of view to potentially help unravel market
mechanisms, may be too artificial and unreachable to have significant applications.
The paper is organized as follows.
The next section is devoted to the presentation of our two data sets and to some of their basic statistical
properties, emphasizing their fat tailed behavior. We discuss, in particular, the importance of the so-called
“lunch effect” for the tail properties of intraday returns. We then demonstrate the presence of a significant
temporal dependence structure and study the possible non-stationary character of these time series.
Section 3 attempts to account for the temporal dependence of our time series and investigates its effect on
the determination of the extreme behavior of the tails of the distribution of returns. In this goal, we build
two simple long memory stochastic volatility processes whose stationary distributions are by construction
either asymptotically regularly varying or exponential. We show that, due to the long range dependence
on the volatility, the estimation with standard statistical estimators is severely biased. This leads to a very
unreliable estimation of the parameters. These results justify our re-examination of previous claims of
regularly varying tails.
4

To fit our two data sets, section 4 proposes a general parametric representation of the distribution of returns
encompassing both a regularly varying distribution in one limit of the parameters and rapidly varying distributions
of the class of stretched exponential distributions in another limit. The use of regularly varying
distributions have been justified above. From a theoretical view point, the class of stretched exponentials is
motivated in part by the fact that the large deviations of multiplicative processes are generically distributed
with stretched exponential distributions (Frisch and Sornette 1997). Stretched exponential distributions are
also parsimonious examples of the important subset of sub-exponentials, that is, of the general class of
distributions decaying slower than an exponential. This class of sub-exponentials share several important
properties of heavy-tailed distributions (Embrechts et al. 1997), not shared by exponentials or distributions
decreasing faster than exponentials.
The descriptive power of these different hypotheses are compared in section 5. We first consider nested
hypotheses and use Wilk’s test to this aim. It appears that both the stretched-exponential and the Pareto
distributions are the most parsimonous models compatible with the data with a slight advantage in favor
of the stretched exponential model. Then, in order to directly compare the descriptive power of these two
models, we perform encompassing tests, which prove the validity of the two representations, but for different
quantile ranges. Finally we show that these two distributions can be set within a single model.
Section 7 summarizes our results and concludes.
2 Some basic statistical features
2.1 The data
We use two sets of data. The first sample consists in the daily returns 2 of the Dow Jones Industrial Average
Index (DJ) over the time interval from May 27, 1896 to May 31, 2000, which represents a sample size
n = 28415. The second data set contains the high-frequency (5 minutes) returns of Nasdaq Composite (ND)
index for the period from April 8, 1997 to May 29, 1998 which represents n=22123 data points. The choice
of these two data sets is justified by their similarity with (1) the data set of daily returns used by Longin
(1996) particularly and (2) the high frequency data used by Guillaume et al. (1997), Lux (2000), Müller et
al. (1998) among others.
For the intra-day Nasdaq data, there are two caveats that must be addressed. First, in order to remove the
effect of overnight price jumps, we have determined the returns separately for each of 289 days contained in
the Nasdaq data and have taken the union of all these 289 return data sets to obtain a global return data set.
Second, the volatility of intra-day data are known to exhibit a U-shape, also called “lunch-effect”, that is, an
abnormally high volatility at the begining and the end of the trading day compared with a low volatility at
the approximate time of lunch. Such effect is present in our data, as depicted on figure 1, where the average
absolute returns are shown as a function of the time within a trading day. It is desirable to correct the data
from this systematic effect. This has been performed by renormalizing the 5 minutes-returns at a given
moment of the trading day by the corresponding average absolute return at the same moment. We shall refer
to this time series as the corrected Nasdaq returns in constrast with the raw (incorrect) Nasdaq returns and
we shall examine both data sets for comparison.
Although the distributions of positive and negative returns are known to be very similar (Jondeau and
Rockinger 2001, for instance), we have chosen to treat them separately. For the Dow Jones, this gives
us 14949 positive and 13464 negative data points while, for the Nasdaq, we have 11241 positive and 10751
negative data points.
2 Throughout the paper, we will use compound returns, i.e., log-returns.
5
69

70 3. Distributions exponentielles étirées contre distributions régulièrement variables
Table 1 summarizes the main statistical properties of these two time series (both for the raw and for the
corrected Nasdaq returns) in terms of the average returns, their standard deviations, the skewness and the
excess kurtosis for four time scales of five minutes, an hour, one day and one month. The Dow Jones
exhibits a significaltly negative skewness, which can be ascribed to the impact of the market crashes. The
raw Nasdaq returns are significantly positively skewed while the returns corrected for the “lunch effect” are
negatively skewed, showing that the lunch effect plays an important role in the shaping of the distribution
of the intra-day returns. Note also the important decrease of the kurtosis after correction of the Nasdaq
returns for lunch effect, confirming the strong impact of the lunch effect. In all cases, the excess-kurtosis are
high and remains significant even after a time aggregation of one month. Jarque-Bera’s test (Cromwell et
al. 1994), a joint statistic using skewness and kurtosis coefficients, is used to reject the normality assumption
for these time series.
2.2 Existence of time dependence
It is well-known that financial time series exhibit complex dependence structures like heteroskedasticity
or non-linearities. These properties are clearly observed in our two times series. For instance, we have
estimated the statistical characteristic V (for positive random variables) called coefficient of variation
V = Std(X)
E(X)
, (1)
which is often used as a testing statistic of the randomness property of a time series. It can be applied to
a sequence of points (or, intervals generated by these points on the line). If these points are “absolutely
random,” that is, generated by a Poissonian flow, then the intervals between them are distributed according
to an exponential distribution for which V = 1. If V > 1 are associated with a clustering phenomenon. We estimated V = V (u) for extrema X > u
and X < −u as function of threshold u (both for positive and for negative extrema). The results are shown
in figure 2 for the Dow Jones daily returns. As the results are essentially the same for the Nasdaq, we do not
show them. Figure 2 shows that, in the main range |X| < 0.02, containing ∼ 95% of the sample, V increases
with u, indicating that the “clustering” property becomes stronger as the threshold u increases.
We have then applied several formal statistical tests of independence. We have first performed the Lagrange
multiplier test proposed by Engle (1984) which leads to the T · R 2 test statistic, where T denotes the sample
size and R 2 is the determination coefficient of the regression of the squared centered returns xt on a constant
and on q of their lags xt−1,xt−2,··· ,xt−q. Under the null hypothesis of homoskedastic time series, T · R 2
follows a χ 2 -statistic with q degrees of freedom. The test have been performed up to q = 10 and, in every
case, the null hypothesis is strongly rejected, at any usual significance level. Thus, the time series are
heteroskedastics and exhibit volatility clustering. We have also performed a BDS test (Brock et al. 1987)
which allows us to detect not only volatility clustering, like in the previous test, but also departure from
iid-ness due to non-linearities. Again, we strongly rejects the null-hypothesis of iid data, at any usual
significance level, confirming the Lagrange multiplier test.
3 Can long memory processes lead to misleading measures of extreme properties?
Since the descriptive statistics given in the previous section have clearly shown the existence of a significant
temporal dependence structure, it is important to consider the possibility that it can lead to erroneous conclusions
on estimated parameters. We first briefly recall the standard procedures used to investigate extremal
6

properties, stressing the problems and drawbacks arising from the existence of temporal dependence. We
then perform a numerical simulation to study the behavior of the estimators in presence of dependence. We
put particular emphasis on the possible appearance of significant biases due to dependence in the data set.
Finally, we present the results on the extremal properties of our two DJ and ND data sets in the light of the
bootstrap results.
3.1 Some theoretical results
Two limit theorems allow one to study the extremal properties and to determine the maximum domain of
attraction (MDA) of a distribution function in two forms.
First, consider a sample of N iid realizations X1,X2,··· ,XN. Let X ∧ denotes the maximum of this sample.
Then, the Gnedenko theorem states that if, after an adequate centering and normalization, the distribution of
X ∧ converges to a non-degenerate distribution as N goes to infinity, this limit distribution is then necessarily
the Generalized Extreme Value (GEV) distribution defined by
H ξ(x) = exp
When ξ = 0, H ξ(x) should be understood as
Thus, for N large enough
71
−(1 + ξ · x) −1/ξ
. (2)
H ξ=0(x) = exp[−exp(−x)]. (3)
Pr X ∧ < x = H ξ
x − µ
ψ
, (4)
for some value of the centering parameter µ, scale factor ψ and tail index ξ. It should be noted that the
existence of non-degenerate limit distribution of properly centered and normalized X ∧ is a rather strong limitation.
There are a lot of distribution functions that do not satisfy this limitation, e.g., infinitely alternating
functions between a power-like and an exponential behavior.
The second limit theorem is called after Gnedenko-Pickands-Balkema-de Haan (GPBH) and its formulation
is as follows. In order to state the GPBH theorem, we define the right endpoint xF of a distribution function
F(x) as xF = sup{x : F(x) < 1}. Let us call the function
Pr{X − u ≥ x | X > u} ≡ ¯Fu(x) (5)
the excess distribution function (DF). Then, this DF ¯Fu(x) belongs to the Maximum Domain of Attraction of
H ξ(x) defined by eq.(2) if and only if there exists a positive scale-function s(u), depending on the threshold
u, such that
limu→xF sup 0≤x≤xF −u | ¯Fu(x) − ¯G(x/ξ,s(u))| = 0 , (6)
where
G(x/ξ,s) = 1 + ln(H ξ(x/s)) = 1 − (1 + ξ · x/s) −1/ξ . (7)
By taking the limit ξ → 0, expression (7) leads to the exponential distribution. The support of the distribution
function (7) is defined as follows:
0 x < ∞,
0 x −d/ξ,
if ξ 0
if ξ < 0.
(8)
Thus, the Generalized Pareto Distribution has a finite support for ξ < 0.
7

72 3. Distributions exponentielles étirées contre distributions régulièrement variables
The form parameter ξ is of paramount importance for the form of the limiting distribution. Its sign determines
three possible limiting forms of the distribution of maximua: If ξ > 0, then the limit distribution is
the Fréchet power-like distribution; If ξ = 0, then the limit distribution is the Gumbel (double-exponential)
distribution; If ξ < 0, then the limit distribution has a support bounded from above. All these three distributions
are united in eq.(2) by this parameterization. The determination of the parameter ξ is the central
problem of extreme value analysis. Indeed, it allows one to determine the maximum domain of attraction
of the underling distribution. When ξ > 0, the underlying distribution belongs to the Fréchet maximum
domain of attraction and is regularly varying (power-like tail). When ξ = 0, it belongs to the Gumbel MDA
and is rapidly varying (exponential tail), while if ξ < 0 it belongs to the Weibull MDA and has a finite right
endpoint.
3.2 Examples of slow convergence to limit GEV and GPD distributions
There exist two ways of estimating ξ. First, if there is a sample of maxima (taken from sub-samples of
sufficiently large size), then one can fit to this sample the GEV distribution, thus estimating the parameters
by Maximum Likelihood method. Alternatively, one can prefer the distribution of exceedance over a large
threshold given by the GPD (7), whose tail index can be estimated with Pickands’ estimator or by Maximum
Likelihood, as previously. Hill’s estimator cannot be used since it assumes ξ > 0, while the essence
of extreme value analysis is, as we said, to test for the class of limit distributions without excluding any
possibility, and not only to determine the quantitative value of an exponent. Each of these methods has its
advantages and drawbacks, especially when one has to study dependent data, as we show below.
Given a sample of size N, one consider the q-maxima drawn from q sub-samples of size p (such that p · q =
N) to estimate the parameters (µ,ψ,ξ) in (4) by Maximum Likelihood. This procedure yields consistent and
asymptotically Gaussian estimators, provided that ξ > −1/2 (Smith 1985). The properties of the estimators
still hold approximately for dependent data, provided that the interdependence of data is weak. However, it
is difficult to choose an optimal value of q of the sub-samples. It depends both on the size N of the entire
sample and on the underlying distribution: the maxima drawn from an Exponential distribution are known
to converge very quickly to Gumbel’s distribution (Hall and Wellnel 1979), while for the Gaussian law,
convergence is particularly slow (Hall 1979).
The second possibility is to estimate the parameter ξ from the distribution of exceedances (the GPD) or
Pickand’s estimator. For this, one can use either the Maximum Likelihood estimator or Pickands’ estimator.
Maximum Likelihood estimators are well-known to be the most efficient ones (at least for ξ > −1/2 and for
independent data) but, in this particular case, Pickands’ estimator works reasonably well. Given an ordered
sample x1 ≤ x2 ≤ ···xN of size N, Pickands’ estimator is given by
ˆξk,N = 1
ln2 ln xk − x2k
. (9)
x2k − x4k
For independent and identically distributed data, this estimator is consistent provided that k is chosen so that
k → ∞ and k/N → 0 as N → ∞. Morover, ˆ ξk,n is asymptotically normal with variance
σ( ˆ ξk,N) 2 · k = ξ2 (22ξ+1 + 1)
(2(2ξ . (10)
− 1)ln2) 2
In the presence of dependence between data, one can expect an increase of the standard deviation, as reported
by Kearns and Pagan (1997). For time dependence of the GARCH class, Kearns and Pagan (1997) have
indeed demonstrated a significant increase of the standard deviation of the tail index estimator, such as
Hill’s estimator, by a factor more than seven with respect to their asymptotic properties for iid samples. This
leads to very inaccurate index estimates for time series with this kind of temporal dependence.
8

Another problem lies in the determination of the optimal threshold u of the GPD, which is in fact related to
the optimal determination of the sub-samples size q in the case of the estimation of the parameters of the
distribution of maximum.
In sum, none of these methods seem really satisfying and each one presents severe drawbacks. The estimation
of the parameters of the GEV distribution and of the GPD may be less sensitive to the dependence of
the data, but this property is only asymptotic, thus it requires a bootstrap investigation to be able to compare
the real power of each estimation method.
As a first simple example illustrating the possibly very slow convergence to the limit distributions of extreme
value theory mentioned above, let us consider a simulated sample of iid Weibull random variables (we thus
fulfill the most basic assumption of extreme values theory, i.e, iid-ness). For such a distribution, one must
obtain in the limit N → ∞ the exponential distribution (ξ = 0). We took two values for the exponent of
the Weibull distribution: c = 0.7 and c = 0.3, with d = 1 (scale parameter). Such the estimation of ξ by
the distribution of the GPD of exceedance should give estimated values of ξ close to zero in the limit of
large N. In order to use the GPD, we have taken the conditional Weibull distribution under condition X >
Uk,k = 1...15, where the thresholds Uk are choosen as: U1 = 0.1; U2 = 0.3; U3 = 1; U4 = 3; U5 = 10; U6 =
30; U7 = 100; U8 = 300; U9 = 1000; U10 = 3000; U11 = 10 4 ; U12 = 3 · 10 4 ; U13 = 10 5 ; U14 = 3 · 10 5 and
U15 = 10 6 .
For each simulation, the size of the sample above the considered threshold Uk is choosen equal to 50,000
in order to get small standard deviations. The Maximum-Likelihood estimates of the GPD form parameter
ξ are shown in figure 3. For c = 0.7, the threshold U7 gives an estimate ξ = 0.0123 with standard deviation
equal to 0.0045, i.e., the estimate for ξ differs significantly from zero (which is the correct theoretical limit
value). This occurs notwithstanding the huge size of the implied data set; indeed, the probability PrX > U7
for c = 0.7 is about 10 −9 , so that in order to obtain a data set of conditional samples from an unconditional
data set of the size studied here (50,000 realizations above U7), the size of such unconditional sample should
be approximately 10 9 times larger than the number of “peaks over threshold”, i.e., it is practically impossible
to have such a sample. For c = 0.3 the convergence to the theoretical value zero is even slower. Indeed,
even the largest financial datasets for a single asset, drawn from high frequency data, are no larger than or
of the order of one million points 3 . The situation does not change even for data sets one or two orders of
magnitudes larger as considered in (Gopikrishnan et al. 1998), obtained by aggregating thousands of stocks
4 . Thus, although the GPD form parameter should be zero theoretically in the limit of large sample for the
Weibull distribution, this limit cannot be reached for any available sample sizes.
This is a clear illustration that a rapidly varying distribution, like the Weibull distribution with exponent
smaller than one, i.e., a Stretched-Exponential distribution, can be mistaken for a Pareto or any other regularly
varying distribution for any practical applications.
3.3 Generation of a long memory process with a well-defined stationary distribution
In order to study the performance of the various estimators of the tail index ξ and the influence of interdependence
of sample values, we have generated six samples with distinct properties. The first three samples
are made of iid realizations drawn respectively from a Pareto Distribution with tail index b = 3 and from
a Stretched-Exponential distribution with exponent c = 0.3 and c = 0.7. The three other samples contain
realizations exhibiting long-range memory with the same three distributions as for the first three samples:
a regularly varying distribution with tail index b = 3 and a Stretched-Exponential distribution with expo-
3 One year of data sampled at the 1 minute time scale gives approximately 1.2 · 10 5 data points
4 In this case, another issue arises concerning the fact that the aggregation of returns from differents assets may distord the
information and the very structure of the tails of the pdfs, if they exhibit some intrinsic variability (Matia et al. 2002).
9
73

74 3. Distributions exponentielles étirées contre distributions régulièrement variables
nent c = 0.3 and c = 0.7. Thus, the three first samples are the iid counterparts of the later ones. The
sample with regularly varying iid distributions converges to the Fréchet’s maximum domain of attraction
with ξ = 1/3 = 0.33, while the iid Stretched-Exponential distribution converges to Gumbel’s maximum domain
of attraction with ξ = 0. We now study how well can one distinguish between these two distributions
belonging to two different maximum domains of attraction.
For the stochastic processes with long memory, we use a simple stochastic volatility model. First, we
construct a Gaussian process {Xt}t≥1 with correlation function
C(t) =
1
(1+|t|) α
if |t| ≤ T ,
0 if |t| > T .
It should be noted that, in order for the stochastic process to be well-defined, the correlation function must
satisfy a positivity condition. More precisely, the spectral density (the Fourier transform of the correlation
function) must remain positive. This condition imposes that the duration of the memory T be larger than a
constant depending on α.
The next step consists in building the process {Ut}t≥1, defined by
(11)
Ut = Φ(Xt) , (12)
where Φ(·) is the Gaussian distribution function. The process {Ut}t≥1 exhibits exactly the same long range
dependence as the process {Xt}t≥1. This is ensured by the property of invariance of the copula under
strickly increasing change of variables. Let us recall that a copula is the mathematical embodiement of the
dependence structure between different random variables (Joe 1997, Nelsen 1998). The process {Ut}t≥1
thus possesses a Gaussian copula dependence structure with long memory and uniform marginals 5 .
In the last step, we define the volatility process
σt = σ0 ·U −1/b
t , (13)
which ensures that the stationary distribution of the volatility is a Pareto distribution with tail index b. Such
a distribution of the volatility is not realistic in the bulk which is found to be approximately a lognormal
distribution for not too large volatilities (Sornette et al. 2000), but is in agreement with the hypothesis of
an asymptotic regularly varying distribution. A change of variable more complicated than (13) can provide
a more realistic behavior of the volatility on the entire range of the distribution but our main goal is not to
provide a realistic stochastic volatility model but only to exhibit a long memory process with well-defined
prescribed marginals in order to test the influence of a long range dependence structure.
The return process is then given by
rt = σt · εt , (14)
where the εt are Gaussian random variables independent from σt. The construction (14) ensures the decorrelation
of the returns at every time lag. The stationary distribution of rt admits the density
p(r) = 2 b−1
b − 1 r2 b
2 · Γ , , (15)
2 2 rb+1
b−1 r2
which is regularly varying at infinity since Γ 2 , 2 goes to Γ
b−1
2 . This completes the construction and
characterization of our long memory process with regularly varying stationary distribution.
5 Of course, one can make the correlation as small as one wants under an adequate choice of a strickly increasing transformation
but this does not change the fact that the dependence remains unchanged. This is another illustration of the fact that the correlation
is not always a good and adapted measure of dependence (Malevergne and Sornette 2002).
10

In order to obtain a process with Stretched-Exponential distribution with long range dependence, we apply
to {rt}t≥1 the following increasing mapping G : r → y
⎧
⎪⎨ (x0 + ln
G(r) =
⎪⎩
r
r0 )1/c r > r0
sgn(r) · |r| 1/c |r| ≤ r0
−(r0 + ln|r/r0|) 1/c (16)
r < −r0 .
This transformation gives a stretched exponential of index c for all values of the return larger than the scale
factor r0. This derives from the fact that the process {rt}t≥1 admits a regularly varying distribution function,
characterized by ¯Fr(r) = 1 − Fr(r) = L(r)|r| −b , for some slowly varying function L. As a consequence, the
stationary distribution of {Yt}t≥1 is given by
¯FY (y) = L r0e −x0 exp(y c ) e br0
which is a Stretched-Exponential distribution.
x b 0
75
· e −b|y|c
, ∀|y| > r0, (17)
= L ′ (y) · e −b|y|c
, L ′ is slowly varying at infinity, (18)
To summarize, starting with a long memory Gaussian process, we have defined a long memory process
characterized by a stationary distribution function of our choice, thanks to the invariance of the temporal
dependence structure (the copula) under strictly increasing change of variable. In particular, this approach
gives long memory processes with a regularly varying marginal distribution and with a stretched-exponential
distribution. Notwithstanding the difference in their marginals, these two processes possess by construction
exactly the same time dependence. This allows us to compare the impact of the same dependence on these
two classes of marginals.
3.4 Results of numerical simulations
We have generated 1000 samples of each kind (iid Stretched-Exponential, iid Pareto, long memory process
with a Pareto distribution and with a Stretched-Exponential distribution). Each sample contains 10,000
realizations, which is approximately the number of points in each tail of our real samples. In order to
generate the Gaussian process with correlation function (11), we have used the algorithm based on Fast
Fourier Transform described in Beran (1994). The parameter T has been set to 250 and α to 0.5 (it can be
checked that for α = 0.5 the lower bound for T is equal to 23).
Panel (a) of table 2 presents the mean values and standard deviations of the Maximum Likelihood estimates
of ξ, using the Generalized Extreme Value distribution and the Generalized Pareto Distribution for the three
samples of iid data. To estimate the parameters of the GEV distribution and study the influence of the
sub-sample size, we have grouped the data in clusters of size q = 10,50,100 and 200. For the analysis in
terms of the GPD, we have considered four different large thresholds u, corresponding to the quantiles 90%,
95%, 99% and 99.5%. The estimates obtained from the distribution of maxima are significantly different
from the theoretical ones: 0.2 in average over the four different size of sub-samples intead of 0.0 for the
Stretched-Exponential distribution with c = 0.7, 1.0 instead of 0.0 for c = 0.3 for the Stretched-Exponential
distribution and 0.40 instead of 0.33 for the Pareto Distribution. At the same time, the standard deviation of
these estimator remains very low. This significant bias of the estimator is a clear sign that the distribution of
the maximum has not yet converged to the asymptotic GEV distribution, even for subsamples of size 200.
The results are better with smaller biases for the Maximum Likelihood estimates obtained from the GPD.
However, the standard deviations are significantly larger than in the previous case, which testifies of the high
variability of this estimator. Thus, for such sample sizes, the GEV and GPD Maximum Likelihood estimates
seem not very reliable due to an important bias for the former and large statistical fluctuations for the later.
11

76 3. Distributions exponentielles étirées contre distributions régulièrement variables
Nevertheless, an optimistic view point is that, discarding the largest quantile of 0.995, the GPD estimator is
compatible with a value ξ = 0.32 ± 0.1 for the iid data with Pareto distribution while it is compatible with
ξ = (−0.21) − (0.19) ± 0.6 − 0.8 for c = 0.3 and with ξ = (−0.13) − (0.11) ± 0.04 − 0.5 for c = 0.7. The
consistency of the GPD estimator in the case of the Pareto distribution in contrast with the wild variations for
the Stretched-Exponential distributions suggests that one could conclude that the first result qualifies (correctly)
a regularly varying function while the second one either (correctly) disqualifies a regularly varying
function or more conservatily is unable to conclude. In other words, when the sample data is truely Pareto,
the GPD estimator seems to be able to retrieve this information reliably, in constrast with the GEV estimator
which is quite unreliable in all cases.
Panel (b) of table 2 presents the same results for dependent data. The GEV estimates exhibit in each case a
significant bias, either positive or negative and a huge increase of the standard deviation in the case of the
Stretched-Exponential with exponent c = 0.7. Interestingly, the GEV estimator for the Pareto distribution is
utterly wrong. The situation is different for the GPD estimates which show a weak bias not really sensitive to
the quantile. In constrast, the standard deviations of the GPD estimators strongly increase with the quantile,
which is natural since the number of obervations decreases accordingly. The GPD behaves surprisingly well
and seems to be the only one able to perfom a reasonable estimation of the tail index.
To summarize, the Maximum Likelihood estimators derived form the GEV or GPD distributions are not very
efficient in the presence of dependence in the data and of non-asymptotic effects due to the slow convergence
toward the asymptotic GEV or GPD distributions. The only positive note is that the GPD estimator correctly
recovers the range of the index ξ with an uncertainty smaller than 20% for data with a pure Pareto distribution
while it is cannot reject the hypothesis that ξ = 0 when the data is generated with a Stretched-Exponential
distribution, albeit with a very large uncertainty, in other words with very little power.
Table 3 focuses on the results given by Pickands’ estimator for the tail index of the GPD. For each threshods
u, corresponding to the quantiles 90%, 95%, 99% and 99.5% respectively, the results of our simulations are
given for two particular values of k (defined in 9) corresponding to N/k = 4, which is the largest admissible
value, and N/k = 10 corresponding to be sufficiently far in the tail of the GPD. Table 3 provides the mean
value and the numerically estimated as well as the theoretical (given by (10)) standard deviation of ˆ ξk,N.
Panel (a) gives the result for iid data. The mean values do not exhibit a significant bias for the Pareto
distribution and the Stretched-Exponential with c = 0.7, but are utterly wrong in the case c = 0.3 since
the estimates are comparable with those given for the Pareto distribution. In each case, we note a very
good agreement between the empirical and theoretical standard deviations, even for the larger quantiles
(and thus the smaller samples). Panel (b) presents the results for dependent data. The estimated standard
deviations remains of the same order as the theoretical ones, contrarily to results reported by Kearns and
Pagan (1997) for GARCH processes. However, like these authors, we find that the bias, either positive or
negative, becomes very significant and leads on to misclassify a Stretched-Exponential distribution with
c = 0.3 for a Pareto distribution with b = 3. Thus, in presence of dependence, Pickands’ estimator is
unreliable.
To summarize, the impact of the dependence can add a severe scatter of estimators which increase their
standard deviation. The determination of the maximum domain of attraction with usual estimators does
not appear to be a very efficient way to study the extreme properties of dependent times series. Almost all
the previous studies which have investigated the tail behavior of asset returns distributions have focused on
these methods (see the influencial works of Longin (1996) for instance) and may thus have led to spurious
results on the determination of the tail behavior. In particular, our simulations show that rapidly varying
function may be mistaken for regularly varying functions. Thus, according to our simulations, this casts
doubts on the strength of the conclusion of previous works that the distributions of returns are regularly
varying as seems to have been the consensus until now and suggests to re-examine the possibility that the
distribution of returns may be rapidly varying as suggested by Gouriéroux and Jasiak (1998) or Laherrère
12

and Sornette (1999) for instance. We now turn to this question using the framework of GEV and GDP
estimators just described.
3.5 GEV and GPD estimators of the Dow Jones and Nasdaq data sets
We have applied the same analysis as in the previous section on the real samples of the Dow Jones and
Nasdaq (raw and corrected) returns. To this aim, we have randomly generated one thousand sub-samples,
each sub-sample being constituted of ten thousand data points in the positive or negative parts of the samples
respectively (without replacement). Obviously, among the one thousand sub-samples, many of them are
interdependent as they contain parts of the same observed values. With this database, we have estimated the
mean value and standard deviations of Pickands’ estimator for the GPD derived from the upper quantiles of
these distributions, and of ML-estimators for the distribution of maximum and for the GPD. The results are
given in tables 4 and 5.
These results confirm the confusion about the tail behavior of the returns distributions and it seems impossible
to exclude a rapidly varying behavior of their tails. Indeed, even the estimations performed by Maximum
Likelihood with the GPD tail index, which have appeared as the less unreliable estimator in our previous
tests, does not allow us to reject the hypothesis that the tails of the empirical distributions of returns are
rapidly varying.
For the Nasdaq dataset, accounting for the lunch effect does not yield a significant change in the estimations,
except for a very strong increase of the standard variation of the GPD Maximum Likelihood estimator. This
results from the fact that extremes are no more dominated by the few largest realizations of the returns at
the begining or the end of trading days. Indeed, panel (b) of table 4 shows that the sample variance of the
GPD maximum likelihood estimate vanishes for quantile 99% and 99.5%. It is due to the important overlap
of the sub-samples together with the impact of the extreme realizations of the returns at the open or close
trading days. In panel (c), which corresponds to the Nasdaq data corrected for the lunch effect, the sample
variance vanishes only in one case (instead of four), clearly showing that extremes are less dominated by
the large returns of the beginning and at the end of each day.
As a last non-parametric attempt to distinguish between a regularly varying tail and a rapidly varying tail
of the exponential or Stretched-Exponential families, we study the Mean Excess Function which is one of
the known methods that often can help in deciding what parametric family is appropriate for approximation
(see for details Embrechts et al. (1997)). The Mean Excess Function MEF(u) of a random value X (also
called “shortfall” when applied to negative returns in the context of financial risk management) is defined as
MEF(u) = E(X − u|X > u) . (19)
The Mean Excess Function MEF(u) is obviously related to the GPD for sufficiently large threshold u and
its behavior can be derived in this limit for the three maximum domains of attraction. In addition, more
precise results can be given for particular radom variables, even in a non-asymptotic regime. Indeed, for an
exponential random variable X, the MEF(u) is just a constant. For a Pareto random variable, the MEF(u)
is a straight increasing line, whereas for the Streched-Exponential and the Gauss distributions the MEF(u)
is a decreasing function. We evaluated the sample analogues of the MEF(u) (Embrechts et al. 1997, p.296)
which are shown in figure 4. All attempts to find a constant or a linearly increasing behavior of the MEF(u)
on the main central part of the range of returns were ineffective. In the central part of the range of negative
returns (|X| > 0.002; q ∼ = 98% for ND data, and |X| > 0.025 ; q ∼ = 96% for DJ data), the MEF(u) behaves
like a convex function which exclude both exponential and power (Pareto) distributions. Thus, the MEF(u)
tool does not support using any of these two distributions.
In view of the stalemate reached with the above non-parametric approaches and in particular with the stan-
13
77

78 3. Distributions exponentielles étirées contre distributions régulièrement variables
dard extreme value estimators, the sequel of this paper is devoted to the investigation of a parametric approach
in order to decide which class of extreme value distributions, rapidly versus regularly varying, accounts
best for the empirical distributions of returns.
4 Fitting distributions of returns with parametric densities
Since our previous results lead to doubt the validity of the rejection of the hypothesis that the distribution of
returns are rapidly varying, we now propose to pit a parametric champion for this class of functions against
the Pareto champion of regularly varying functions. To represent the class of rapidly varying functions, we
propose the family of Stretched-Exponentials. As discussed in the introduction, the class of stretched exponentials
is motivated in part from a theoretical view point by the fact that the large deviations of multiplicative
processes are generically distributed with stretched exponential distributions (Frisch and Sornette 1997).
Stretched exponential distributions are also parsimonious examples of sub-exponential distributions with fat
tails for instance in the sense of the asymptotic probability weight of the maximum compared with the sum
of large samples (Feller 1971). Notwithstanding their fat-tailness, Stretched Exponential distributions have
all their moments finite 6 , in constrast with regularly varying distributions for which moments of order equal
to or larger than the index b are not defined. This property may provide a substantial advantage to exploit in
generalizations of the mean-variance portfolio theory using higher-order moments (Rubinstein 1973, Fang
and Lai 1997, Hwang and Satchell 1999, Sornette et al. 2000, Andersen and Sornette 2001, Jurczenko and
Maillet 2002, Malevergne and Sornette 2002, for instance ). Moreover, the existence of all moments is
an important property allowing for an efficient estimation of any high-order moment, since it ensures that
the estimators are asymptotically Gaussian. In particular, for Stretched-Exponentially distributed random
variables, the variance, skewness and kurtosis can be well estimated, contrarily to random variables with
regularly varying distribution with tail index in the range 3 − 5.
4.1 Definition of a general 3-parameters family of distributions
We thus consider a general 3-parameters family of distributions and its particular restrictions corresponding
to some fixed value(s) of two (one) parameters. This family is defined by its density function given by:
A(b,c,d,u) x
fu(x|b,c,d) =
−(b+1) exp −
x c
d if x u > 0
(20)
0 if x < u.
Here, b,c,d are unknown parameters, u is a known lower threshold that will be varied for the purposes of
our analysis and A(b,c,d,u) is a normalizing constant given by the expression:
A(b,c,d,u) =
db c
Γ(−b/c,(u/d) c , (21)
)
where Γ(a,x) denotes the (non-normalized) incomplete Gamma function. The parameter b ranges from
minus infinity to infinity while c and d range from zero to infinity. In the particular case where c = 0,
the parameter b also needs to be positive to ensure the normalization of the probability density function
(pdf). The interval of definition of this family is the positive semi-axis. Negative log-returns will be studied
by taking their absolute values. The family (20) includes several well-known pdf’s often used in different
applications. We enumerate them.
6 However, they do not admit an exponential moment, which leads to problems in the reconstruction of the distribution from the
knowledge of their moments (Stuart and Ord 1994).
14

1. The Pareto distribution:
79
Fu(x) = 1 − (u/x) b , (22)
which corresponds to the set of parameters (b > 0,c = 0) with A(b,c,d,u) = b·u b . Several works have
attempted to derive or justified the existence of a power tail of the distribution of returns from agentbased
models (Challet and Marsili 2002), from optimal trading of large funds with sizes distributed
according to the Zipf law (Gabaix et al. 2002) or from stochastic processes (Biham et al 1998, 2002).
2. The Weibull distribution:
Fu(x) = 1 − exp −
x
c +
d
u
c , (23)
d
with parameter set (b = −c,c > 0,d > 0) and normalization constant A(b,c,d,u) = c
dc exp
u c
d .
This distribution is said to be a “Stretched-Exponential” distribution when the exponent c is smaller
than 1, namely when the distribution decays more slowly than an exponential distribution.
3. The exponential distribution:
Fu(x) = 1 − exp − x
d
u
+ , (24)
d
with parameter set (b = −1, c = 1, d > 0) and normalization constant A(b,c,d,u) = 1
d exp− u
d .
The exponential family can for instance derive from a simple model where stock price dynamics is
governed by a geometrical (multiplicative) Brownian motion with stochastic variance. Dragulescu
and Yakovenko (2002) have found an excellent fit of this model with the Dow-Jones index for time
lags from 1 to 250 trading days, within an asymptotic exponential tail of the distribution of log-returns
with a time-dependent exponent.
4. The incomplete Gamma distribution:
Fu(x) = 1 − Γ(−b,x/d)
Γ(−b,u/d)
with parameter set (b, c = 1, d > 0) and normalization A(b,c,d,u) =
d b
Γ(−b,u/d) .
Thus, the Pareto distribution (PD) and exponential distribution (ED) are one-parameter families, whereas
the stretched exponential (SE) and the incomplete Gamma distribution (IG) are two-parameter families. The
comprehensive distribution (CD) given by equation (20) contains three unknown parameters.
Interesting links between these different models reveal themselves under specific asymptotic conditions.
For instance, in the limit b → +∞, the Pareto model becomes the Exponential model (Bouchaud and Potters
2000). Indeed, provided that the scale parameter u of the power law is simultaneously scaled as u b = (b/α) b ,
we can write the tail of the cumulative distribution function of the PD as u b /(u + x) b which is indeed of the
form u b /x b for large x. Then, u b /(u + x) b = (1 + αx/b) −b → exp(−αx) for b → +∞. This shows that the
Exponential model can be approximated with any desired accuracy on an arbitrary interval (u > 0,U) by
the (PD) model with parameters (β,u) satisfying u b = (b/α) b . Although the value b → +∞ does not give
strickly speaking a Exponential distribution, the limit b → +∞ provides any desired approximation to the
Exponential distribution, uniformly on any finite interval (u,U).
More interesting for our present study is the behavior of the (SE) model when c → 0. In this limit, and
provided that
u
c c · → β,
d
as c → 0 . (26)
15
(25)

80 3. Distributions exponentielles étirées contre distributions régulièrement variables
the (SE) model goes to the Pareto model. Indeed, we can write
c
dc · xc−1
· exp − xc − uc dc
u
c = c ·
d
xc−1
u
c
x
c
exp − · − 1 ,
uc d u
β · x −1
u
c exp −c · ln
d
x
, as c → 0
u
β · x −1
exp −β · ln x
,
u
β uβ
, (27)
xβ+1 which is the pdf of the (PD) model with tail index β. The condition (26) comes naturally from the properties
of the maximum-likelihood estimator of the scale parameter d given by equation (47) and underlined by
equation (90) in the appendices at the end of the paper. It implies that, as c → 0, the characteristic scale d
of the (SE) model must also go to zero with c to ensure the convergence of the (SE) model towards the (PD)
model.
This shows that the Pareto model can be approximated with any desired accuracy on an arbitrary interval
(u > 0,U) by the (SE) model with paramters (c,d) satisfying equation (26) where the arrow is replaced by an
equality. Although the value c = 0 does not give strickly speaking a Stretched-Exponential distribution, the
limit c → 0 provides any desired approximation to the Pareto distribution, uniformly on any finite interval
(u,U). This deep relationship between the SE and PD models allows us to understand why it can be very
difficult to decide, on a statistical basis, which of these models fits the data best.
4.2 Methodology
We start with fitting our two data sets (DJ and ND) by the five distributions enumerated above (20) and (22-
25). Our first goal is to show that no single parametric representation among any of the cited pdf’s fits the
whole range of the data sets. Recall that we analyze separately positive and negative returns (the later being
converted to the positive semi-axis). We shall use in our analysis a movable lower threshold u, restricting by
this threshold our sample to observations satisfying to x > u.
In addition to estimating the parameters involved in each representation (20,22-25) by maximum likelihood
for each particular threshold u 7 , we need a characterization of the goodness-of-fit. For this, we propose
to use a distance measure between the estimated distribution and the sample distribution. Many distances
can be used: mean-squared error, Kullback-Liebler distance 8 , Kolmogorov distance, Spearman distance (as
in Longin (1996)) or Anderson-Darling distance, to cite a few. We can also use one of these distances
to determine the parameters of each pdf according to the criterion of minimizing the distance between the
estimated distribution and the sample distribution. The chosen distance is thus useful both for characterizing
and for estimating the parametric pdf. In the later case, once an estimation of the parameters of particular
distribution family has been obtained according to the selected distance, we need to quantify the statistical
significance of the fit. This requires to derive the statistics associated with the chosen distance. These
statistics are known for most of the distances cited above, in the limit of large sample.
We have chosen the Anderson-Darling distance to derive our estimated parameters and perform our tests
of goodness of fit. The Anderson-Darling distance between a theoretical distribution function F(x) and its
7 The estimators and their asymptotic properties are derived in appendix A.
8 This distance (or divergence, strictly speaking) is the natural distance associated with maximum-likelihood estimation since
it is for these values of the estimated parameters that the distance between the true model and the assumed model reaches its
minimum.
16

empirical analog FN(x), estimated from
· a sample of N realizations, is evaluated as follows:
[FN(x) − F(x)]
ADS = N 2
dF(x) (28)
F(x)(1 − F(x))
= −N − 2
N
∑
1
{wk log(F(yk)) + (1 − wk)log(1 − F(yk))}, (29)
where wk = 2k/(2N + 1), k = 1...N and y1 ... yN is its ordered sample. If the sample is drawn
from a population with distribution function F(x), the Anderson-Darling statistics (ADS) has a standard
AD-distribution free of the theoretical df F(x) (Anderson and Darling 1952), similarly to the χ 2 for the
χ 2 -statistic, or the Kolmogorov distribution for the Kolmogorov statistic. It should be noted that the ADS
weights the squared difference in eq.(28) by 1/F(x)(1 − F(x)) which is nothing but the inverse of the variance
of the difference in square brackets. The AD distance thus emphasizes more the tails of the distribution
than, say, the Kolmogorov distance which is determined by the maximum absolute deviation of Fn(x) from
F(x) or the mean-squared error, which is mostly controlled by the middle of range of the distribution. Since
we have to insert the estimated parameters into the ADS, this statistic does not obey any more the standard
AD-distribution: the ADS decreases because the use of the fitting parameters ensures a better fit to the sample
distribution. However, we can still use the standard quantiles of the AD-distribution as upper boundaries
of the ADS. If the observed ADS is larger than the standard quantile with a high significance level (1 − ε),
we can then conclude that the null hypothesis F(x) is rejected with significance level larger than (1 − ε). If
we wish to estimate the real significance level of the ADS in the case where it does not exceed the standard
quantile of a high significance level, we are forced to use some other method of estimation of the significance
level of the ADS, such as the bootstrap method.
In the following, the estimates minimizing the Anderson-Darling distance will be refered to as AD-estimates.
The maximum likelihood estimates (ML-estimates) are asymptotically more efficient than AD-estimates for
independent data and under the condition that the null hypothesis (given by one of the four distributions (22-
25), for instance) corresponds to the true data generating model. When this is not the case, the AD-estimates
provide a better practical tool for approximating sample distributions compared with the ML-estimates.
We have determined the AD-estimates for 18 standard significance levels q1 ...q18 given in table 6. The
corresponding sample quantiles corresponding to these significance levels or thresholds u1 ...u18 for our
samples are also shown in table 6. Despite the fact that thresholds uk vary from sample to sample, they
always corresponded to the same fixed set of significance levels qk throughout the paper and allows us to
compare the goodness-of-fit for samples of different sizes.
4.3 Empirical results
The Anderson-Darling statistics (ADS) for five parametric distributions (Weibull or Stretched-Exponential,
Generalized Pareto, Gamma, exponential and Pareto) are shown in table 7 for two quantile ranges, the first
top half of the table corresponding to the 90% lowest thresholds while the second bottom half corresponds
to the 10% highest ones. For the lowest thresholds, the ADS rejects all distributions, except the Stretched-
Exponential for the Nasdaq. Thus, none of the considered distributions is really adequate to model the data
over such large ranges. For the 10% highest quantiles, only the exponential model is rejected at the 95%
confidence level. The Stretched-Exponential distribution is the best, just before the Pareto distribution and
the Incomplete Gamma that cannot be rejected. We now present an analysis of each case in more details.
17
81

82 3. Distributions exponentielles étirées contre distributions régulièrement variables
4.3.1 Pareto distribution
Figure 5a shows the cumulative sample distribution function 1 − F(x) for the Dow Jones Industrial Average
index, and in figure 5b the cumulative sample distribution function for the Nasdaq Composite index. The
mismatch between the Pareto distribution and the data can be seen with the naked eye: if samples were
taken from a Pareto population, the graph in double log-scale should be a straight line. Even in the tails,
this is doubtful. To formalize this impression, we calculate the Hill and AD estimators for each threshold u.
Denoting y1 ... ynu the ordered sub-sample of values exceeding u where Nu is size of this sub-sample,
the Hill maximum likelihood estimate of parameter b is (Hill 1975)
1
ˆbu =
Nu
The standard deviations of ˆbu can be estimated as
Nu
∑ 1
log(yk/u)
−1
. (30)
Std(ˆbu) = ˆbu/ √ Nu, (31)
under the assumption of iid data, but very severely underestimate the true standard deviation when samples
exhibit dependence, as reported by Kearns and Pagan (1997).
Figure 6a and 6b shows the Hill estimates ˆbu as a function of u for the Dow Jones and for the Nasdaq.
Instead of an approximately constant exponent (as would be the case for true Pareto samples), the tail index
estimator increases until u ∼ = 0.04, beyond which it seems to slow its growth and oscillates around a value
≈ 3 − 4 up to the threshold u ∼ = .08. It should be noted that interval [0,0.04] contains 99.12% of the sample
whereas interval [0.04,0.08] contains only 0.64% of the sample. The behavior of ˆbu for the ND shown
in figure 6b is similar: Hill’s estimates ˆbu seem to slow its growth already at u ∼ = 0.0013 corresponding
to the 95% quantile. Are these slowdowns of the growth of ˆbu genuine signatures of a possible constant
well-defined asymptotic value that would qualify a regularly varying function?
As a first answer to this question, table 8 compares the AD-estimates of the tail exponent b with the corresponding
maximum likelihood estimates for the 18 intervals u1 ...u18. Both maximum likelihood and
Anderson-Darling estimates of b steadily increase with the threshold u (excepted for the highest quantiles
of the positive tail of the Nasdaq). The corresponding figures for positive and negative returns are very
close each to other and almost never significantly different at the usual 95% confidence level. Some slight
non-monotonicity of the increase for the highest thresholds can be explained by small sample sizes. One can
observe that both MLE and ADS estimates continue increasing as the interval of estimation is contracting to
the extreme values. It seems that their growth potential has not been exhausted even for the largest quantile
u18, except for the positive tail of the Nasdaq sample. This statement might be however not very strong
as the standard deviations of the tail index estimator also grow when exploring the largest quantiles.
However, the non-exhausted growth is observed for three samples out of four. Moreover, this effect is
seen for several threshold values and we can add that random fluctuations would distort the b-curve
in a random manner, i.e, now up now down, whearas we note in three cases an increasing curve.
Assuming that the observation, that the sample distribution can be approximated by a Pareto distribution
with a growing index b, is correct, an important question arises: how far beyond the sample this growth will
continue? Judging from table 8, we can think this growth is still not exhausted. Figure 7 suggests a specific
form of this growth, by plotting the hill estimator ˆbu for all four data sets (positive and negative branches
of the distribution of returns for the DJ and for the ND) as a function of the index n = 1,...,18 of the 18
quantiles or standard significance levels q1 ...q18 given in table 6. Similar results are obtained with the AD
estimates. Apart from the positive branch of the ND data set, all other three branches suggest a continuous
growth of the Hill estimator ˆbu as a function of n = 1,...,18. Since the quantiles q1 ...q18 given in table 6
18

have been chosen to converge to 1 approximately exponentially as
1 − qn = 3.08 e −0.342n , (32)
the linear fit of ˆbu as a function of n shown as the dashed line in figure 7 corresponds to
83
ˆbu(qn) = 0.08 + 0.626ln 3.08
. (33)
1 − qn
Expression (33) suggests an unbound logarithmic growth of ˆbu as the quantile approaches 1. For instance,
for a quantile 1 −q = 0.1%, expression (33) predicts ˆbu(1 −q = 10 −3 ) = 5.1. For a quantile 1 −q = 0.01%,
expression (33) predicts ˆbu(1 − q = 10 −4 ) = 6.5, and so on. Each time the quantile 1 − q is divided by
a factor 10, the apparent exponent ˆbu(q) is increased by the additive constant ∼ = 1.45: ˆbu((1 − q)/10) =
ˆbu(1 − q) + 1.45. This very slow growth uncovered here may be an explanation for the belief and possibly
mistaken conclusion that the Hill and other estimators of the tail index tends to a constant for high quantiles.
Indeed, it is now clear that the slowdowns of the growth of ˆbu seen in figures 6 decorated by large fluctuations
due to small size effects is mostly the result of a dilatation of the data expressed in terms of threshold u.
When recast in the more natural logarithm scale of the quantiles q1 ...q18, this slowdown disappears. Of
course, it is impossible to know how long this growth given by (33) may go on as the quantile q tends to 1.
In other words, how can we escape from the sample range when estimating quantiles? How can we estimate
the so-called “high quantiles” at the level q > 1 − 1/T where T is the total number of sampled points.
Embrechts et al. (1997) have summarized the situation in this way: “there is no free lunch when it comes to
high quantiles estimation!” It is possible that ˆbu(q) will grow without limit as would be the case if the true
underlying distribution was rapidly varying. Alternatively, ˆbu(q) may saturate to a large value, as predicted
for instance by the traditional GARCH model which yields tails indices which can reach 10 − 20 (Engle
and Patton 2001, Starica and Pictet 1999) or by the recent multifractal random walk (MRW) model which
gives an asymptotic tail exponent in the range 20 − 50 (Muzy et al. 2000, Muzy et al. 2001). According to
(33), a value ˆbu ≈ 20 (respectively 50) would be attained for 1 − q ≈ 10 −13 (respectively 1 − q ≈ 10 −34 )! If
one believes in the prediction of the MRW model, the tail of the distribution of returns is regularly varying
but this insight is completely useless for all practical purposes due to the astronomically high statistics that
would be needed to sample this regime. In this context, we cannot hope to get access to the true nature of the
pdf of returns but only strive to define the best effective or apparent most parsimonious and robust model.
We do not discuss here the new class of estimation issues raised by the MRW model, which is interesting in
itself but requires a specific analysis of its own left for another work.
The question of the exhaustion of the growth of the tail index is really crucial. Indeed, if it is unboundedly
increasing, it is the signature that the tails of the distributions of returns decay faster than any power-law, and
thus cannot be regularly varying. We revisit this question of the growth of the apparent exponent b, using
the notion of local exponent, in Appendix C as an attempt to better constraint this growth. The analysis
developed in this Appendix C basically confirms the first indication shown in figure 7 .
4.3.2 Weibull distributions
Let us now fit our data with the Weibull (SE) distribution (23). The Anderson-Darling statistics (ADS) for
this case are shown in table 7. The ML-estimates and AD-estimates of the form parameter c are represented
in table 9. Table 7 shows that, for the higest quantiles, the ADS for the Stretched-Exponential is the smallest
of all ADS, suggesting that the SE is the best model of all. Moreover, for the lowest quantiles, it is the sole
model not systematically rejected at the 95% level.
The c-estimates are found to decrease when increasing the order q of the threshold uq beyond which the
estimations are performed. In addition, the c-estimate is identically zero for u18. However, this does not
19

84 3. Distributions exponentielles étirées contre distributions régulièrement variables
automatically imply that the SE model is not the correct model for the data even for these highest quantiles.
Indeed, numerical simulations show that, even for synthetic samples drawn from genuine Stretched-
Exponential distributions with exponent c smaller than 0.5 and whose size is comparable with that of our
data, in about one case out of three (depending on the exact value of c) the estimated value of c is zero. This
a priori surprising result comes from condition (51) in appendix A which is not fulfilled with certainty even
for samples drawn for SE distributions.
Notwithstanding this cautionary remark, note that the c-estimate of the positive tail of the Nasdaq data equal
zero for all quantiles higher than q14 = 0.97%. In fact, in every cases, the estimated c is not significantly
different from zero - at the 95% significance level - for quantiles higher than q12-q14. In addition, table 10
gives the values of the estimated scale paremeter d, which are found very small - particularly for the Nasdaq
- beyond q12 = 95%. In constrast, the Dow Jones keeps significant scale factors until q16 − q17.
These evidences taken all together provide a clear indication on the existence of a change of behavior of
the true pdf of these four distributions: while the bulks of the distributions seem rather well approximated
by a SE model, a fatter tailed distribution than that of the (SE) model is required for the highest quantiles.
Actually, the fact that both c and d are extremely small may be interpreted according to the asymptotic
correspondence given by (26) and (27) as the existence of a possible power law tail.
4.3.3 Exponential and incomplet Gamma distribution
Let us now fit our data with the exponential distribution (24). The average ADS for this case are shown in
table 7. The maximum likelihood- and Anderson-Darling estimates of the scale parameter d are given in
table 11. Note that they always decrease as the threshold uq increases. Comparing the mean ADS-values
of table 7 with the standard AD quantiles, we can conclude that, on the whole, the exponential distribution
(even with moving scale parameter d) does not fit our data: this model is systematically rejected at the 95%
confidence level for the lowest and highest quantiles - excepted for the negative tail of the Nasdaq.
Finally, we fit our data by the IG-distribution (25). The mean ADS for this class of functions are shown in
table 7. The Maximum likelihood and Anderson Darling estimates of the power index b are represented in
table 12. Comparing the mean ADS-values of table 7 with the standard AD quantiles, we can again conclude
that, on the whole, the IG-distribution does not fit our data. The model is rejected at the 95% confidence
level excepted for the negative tail of the Nasdaq for which it is not rejected marginally (significance level:
94.13%). However, for the largest quantiles, this model becomes again revelant since it cannot be rejected
at the 95% level.
4.4 Summary
At this stage, two conclusions can be drawn. First, it appears that none of the considered distributions fit the
data over the entire range, which is not a surprise. Second, for the highest quantiles, three models seem to
be able to represent to data, the Gamma model, the Pareto model and the Stretched-Exponential model. This
last one has the lowest Anderson-Darling statistic and thus seems to be the most reasonable model among
the three models compatible with the data.
20

5 Comparison of the descriptive power of the different families
As we have seen by comparing the Anderson-Darling statistics corresponding to the four parametric families
(22-25), the best model in the sense of minimizing the Anderson-Darling distance is the Stretched-
Exponential distribution.
We now compare these four distributions with the comprehensive distribution (20) using Wilks’ theorem
(Wilks 1938) of nested hypotheses to check whether or not some of the four distributions are sufficient
compared with the comprehensive distribution to describe the data. We then turn to the Wald encompassing
test for non-nested hypotheses which provides a pairwise comparison of the different models.
5.1 Comparison between the four parametric families and the comprehensive distribution
According to Wilk’s theorem, the doubled generalized log-likelihood ratio Λ:
85
Λ = 2 log maxL(CD,X,Θ)
, (34)
maxL(z,X,θ)
has asymptotically (as the size N of the sample X tends to infinity) the χ 2 -distribution. Here L denotes the
likelihood function, θ and Θ are parametric spaces corresponding to hypotheses z and CD correspondingly
(hypothesis z is one of the four hypotheses (22-25) that are particular cases of the CD under some parameter
relations). The statement of the theorem is valid under the condition that the sample X obeys hypothesis z
for some particular value of its parameter belonging to the space θ. The number of degrees of freedom of
the χ 2 -distribution equals to the difference of the dimensions of the two spaces Θ and θ. Since dim(Θ) = 3
and dim(θ) = 2 for the Stretched-Exponential and Incomplet Gamma distributions; dim(θ) = 1 for the
Pareto and the Exponential distributions, we have one degree of freedom for the formers and two degrees of
freedom for the laters. The maximum of the likelihood in the numerator of (34) is taken over the space Θ,
whereas the maximum of the likelihood in the denominator of (34) is taken over the space θ. Since we have
always θ ⊂ Θ, the likelihood ratio is always larger than 1, and the log-likelihood ratio is non-negative. If the
observed value of Λ does not exceed some high-confidence level (say, 99% confidence level) of the χ 2 , we
then reject the hypothesis CD in favour of the hypothesis z, considering the space Θ redundant. Otherwise,
we accept the hypothesis CD, considering the space θ insufficient.
The doubled log-likelihood ratios (34) are shown in figures 8 for the positive and negative branches of the
distribution of returns of the Nasdaq and in figures 9 for the Dow Jones. The 95% χ 2 confidence levels for
1 and 2 degrees of freedom are given by the horizontal lines.
For the Nasdaq data, figure 8 clearly shows that Exponential distribution is completely insufficient: for all
lower thresholds, the Wilks log-likelihood ratio exceeds the 95% χ2 1 level 3.84. The Pareto distribution is
insufficient for thresholds u1 − u11 (92.5% of the ordered sample) and becomes comparable with the Comprehensive
distribution in the tail u12 − u18 (7.5% of the tail probability). It is natural that two-parametric
families Incomplete Gamma and Stretched-Exponential have higher goodness-of-fit than the one-parametric
Exponential and Pareto distributions. The Incomplete Gamma distribution is comparable with the Comprehensive
distribution starting with u10 (90%), whereas the Stretched-Exponential is somewhat better (u9 or
u8 , i.e., 70%). For the tails representing 7.5% of the data, all parametric families except for the Exponential
distribution fit the sample distribution with almost the same efficiency. The results obtained for the Dow
Jones data shown in figure 9 are similar. The Stretched-Exponential is comparable with the Comprehensive
distribution starting with u8 (70%). On the whole, one can say that the Stretched-Exponential distribution
performs better than the three other parametric families.
21

86 3. Distributions exponentielles étirées contre distributions régulièrement variables
We should stress that each log-likelihood ratio represented in figures 8 and 9, so-to say “acts on its own
ground,” that is, the corresponding χ 2 -distribution is valid under the assumption of the validity of each
particular hypothesis whose likelihood stands in the numerator of the double log-likelihood (34). It would
be desirable to compare all combinations of pairs of hypotheses directly, in addition to comparing each
of them with the comprehensive distribution. Unfortunately, the Wilks theorem can not be used in the
case of pair-wise comparison because the problem is not more that of comparing nested hypothesis (that
is, one hypothesis is a particular case of the comprehensive model). As a consequence, our results on the
comparison of the relative merits of each of the four distributions using the generalized log-likelihood ratio
should be interpreted with a care, in particular, in a case of contradictory conclusions. Fortunately, the main
conclusion of the comparison (an advantage of the Stretched-Exponential distribution over the three other
distribution) does not contradict our earlier results discussed above.
5.2 Pair-wise comparison of the Pareto model with the Stretched-Exponential model
In order to compare formally the descriptive power of the Stretched-Exponential distribution (the best twoparameters
ditribution) with that of the Pareto distribution (the best one-parameter distribution), we need to
use the methods for testing non-nested hypotheses. There are in fact many ways to perform such a test (see
Gouriéroux and Monfort (1994) for a review). Concerning the log-likelihood ratio test used in the previous
section for nested-hypotheses testing, a direct generalization for non-nested hypotheses has been provided
by Cox’s test (1961, 1962). However, such a test requires that the true distribution of the sample be nested in
one of the two considered models. Our previous investigations have shown that it is not the case, so we need
to use another testing procedure. Indeed, when comparing the Pareto model with the Stretched-Exponential
model, we have found that using the methodology of non-nested hypothesis leads to inconsistencies such
as negative variances of estimators. This is another (indirect) confirmation that neither the Pareto nor the
Stretched-Exponential distributions are the true distribution.
In the case where none of the tested hypotheses contain the true distribution, it can be useful to consider the
encompassing principle introduced by Mizon and Richard (1986). A model, (SE) say, is said to encompass
another model, (PD) for instance, if the representative of (PD), which is the closest to the best representative
of (SE), is also the best representative of (PD) per se. Here, the best representative of a model is the
distribution which is the nearest to the true distribution for the considered model. The detailled testing
procedure is based on the Wald and Score encompassing tests (Gouriéroux and Monfort 1994), which are
detailed in appendix D.
Table 13 presents the results of the tests for the null hypothesis “(SE) encompasses (PD)”. In every cases,
the null hypothesis cannot be rejected at the 95% significance level for quantiles higher than q6 = 0.5 and at
the 99% significance level for quantiles higher than q10 = 0.9. The unfilled entries for the largest quantiles
correspond to MLE giving c → 0. In this case, as shown in Appendix D.1.2, b † has a non-trivial and welldefined
limit which is nothing but the true value ˆb. Thus, the Wald tests is automatically verified at any
confidence levels. Thus, in the tail, the Stretched-Eponential model encompasses the Pareto model. We can
then conclude that it provides a description of the data which is at least as good as that given by the later.
In order to test whether the (SE) model is superior to the Pareto model, we should perform the reverse
test, namely the encompassing of the former model into the later. This task is difficult since the pseudotrue
values of the parameters (c,d) are not always well-defined as exposed in appendix D. Thus, Wald
encompassing test cannot be performed as in the previous case. As a remedy and alternative, we propose
a test of the null hypothesis H0 that the Pareto distribution is the true underlying distribution. This test is
22

ased on the fact that the quantity
ˆηT = ĉ
ĉ
u
+ 1
dˆ
goes to b under the null hypothesis, whatever c being positive or negative. This can be seen from the
asymptotic correspondence given by (26) and (27). Moreover, it is proved in appendix D that the variable
ˆηT
ζT = T − 1 , (36)
ˆb
asymptoticaly follows a χ 2 -distribution with one degree of freedom.
The results of this test are given in table 14. They show that H0 is more often rejected for the Dow Jones
than for the Nasdaq. Indeed, beyond quantile q12 = 95%, H0 cannot be rejected at the 95% confidence level
for the Nasdaq data. For the Dow Jones, we must consider quantiles higher than q18 = 99% in order not to
reject H0 at the 95% significance level. These results are in agreement with the central limit theorem: the
power-law regime (if it really exists) is pushed back to higher quantiles due to time agregation (recall that
our Dow Jones data is at the daily scale while our Nasdaq data is at the 5 minutes time scale).
In summary, the (SE) model encompasses the Pareto model as soon as one considers quantiles higher than
q6 = 50%. On the other hand, the null hypothesis that the true distribution is the Pareto distribution is
strongly rejected until quantiles 90% − 95% or so. Thus, within this range, the (SE) model seems the best.
But, for the very highest quantiles (above 95% − 98%) we cannot any more reject the hypothesis that the
Pareto model is the right one. Thus, for these extreme quantiles, this last model seems to slightly outperform
the (SE) model.
However, recall that section 4.1 has shown that the Pareto model is retrieved as a limiting case of the (SE)
model when the fractional exponent c and the scale factor d go to zero at an appropriated rate. Thus, all the
results above are compatible with the (SE) model in a generalized version. Indeed, defining a generalized
Stretched-Exponential model as
¯Fu(x) = exp − xc−uc dc
, c > 0
¯Fu(x) =
u b
x , c = 0 and b = limc→0 c
u c (37)
d = b,
our tests show the relevance of this representation and its superiority over all the models considered here.
Indeed, we have shown that it is the best (i.e, the most parcimonious) representation of the data for all
quantiles above q9 = 80%.
6 Discussion and Conclusions
We have presented a statistical analysis of the tail behavior of the distributions of the daily log-returns of
the Dow Jones Industrial Average and of the 5-minutes log-returns of the Nasdaq Composite index. We
have emphasized practical aspects of the application of statistical methods to this problem. Although the
application of statistical methods to the study of empirical distributions of returns seems to be an obvious
approach, it is necessary to keep in mind the existence of necessary conditions that the empirical data must
obey for the conclusions of the statistical study to be valid. Maybe the most important condition in order to
speak meaningfully about distribution functions is the stationarity of the data, a difficult issue that we have
not considered here. In particular, the importance of regime switching is now well established (Ramcham
and Susmel 1998, Ang and Bekeart 2001) and should be accounted for.
23
87
(35)

88 3. Distributions exponentielles étirées contre distributions régulièrement variables
Our purpose here has been to revisit a generally accepted fact that the tails of the distributions of returns
present a power-like behavior. Although there are some disagreements concerning the exact value of the
power indices (the majority of previous workers accepts index values between 3 and 3.5, depending on the
particular asset and the investigated time interval), the power-like character of the tails of distributions of
returns is not subjected to doubts. Often, the conviction of the existence of a power-like tail is based on
the Gnedenko theorem stating the existence of only three possible types of limit distributions of normalized
maxima (a finite maximum value, an exponential tail, and a power-like tail) together with the exclusion of
the first two types by experimental evidence. The power-like character of the log-return tail ¯F(x) follows
then simply from the power-like distribution of maxima. However, in this chain of arguments, the conditions
needed for the fulfillment of the corresponding mathematical theorems are often omitted and not discussed
properly. In addition, widely used arguments in favor of power law tails invoke the self-similarity of the data
but are often assumptions rather than experimental evidence or consequences of economic and financial
laws.
Here, we have shown that standard statistical estimators of heavy tails are much less efficient that often
assumed and cannot in general clearly distinguish between a power law tail and a Stretched Exponential
tail even in the absence of long-range dependence in the volatility. In fact, this can be rationalized by our
discovery that, in a certain limit where the exponent c of the stretched exponential pdf goes to zero (together
with condition (26) as seen in the derivation (27)), the stretched exponential pdf tends to the Pareto distribution.
Thus, the Pareto (or power law) distribution can be approximated with any desired accuracy on an
arbitrary interval by a suitable adjustment of the pair (c,d) of the parameters of the stretched exponential
pdf. We have then turned to parametric tests which indicate that the class of Stretched Exponential distribution
provides a significantly better fit to empirical returns than the Pareto, the exponential or the incomplete
Gamma distributions. All our tests are consistent with the conclusion that the Stretched Exponential model
provides the best effective apparent and parsimonious model to account for the empirical data.
However, this does not mean that the stretched exponential (SE) model is the correct description of the tails
of empirical distributions of returns. Again, as already mentioned, the strength of the SE model comes
from the fact that it encompasses the Pareto model in the tail and offers a better description in the bulk of
the distribution. To see where the problem arises, we report in table 6 our best ML-estimates for the SE
parameters c (form parameter) and d (scale parameter) restricted to the quantile level q12 = 95%, which
offers a good compromise between a sufficiently large sample size and a restricted tail range leading to an
accurate approximation in this range.
Sample c d
ND positive returns 0.039 (0.138) 4.54 · 10 −52 (2.17· 10 −49 )
ND negative returns 0.273 (0.155) 1.90 · 10 −7 (1.38· 10 −6 )
DJ positive returns 0.274 (0.111) 4.81 · 10 −6 (2.49· 10 −5 )
DJ negative returns 0.362 (0.119) 1.02 · 10 −4 (2.87· 10 −4 )
One can see that c is very small (and all the more so for the scale parameter d) for the tail of positive returns
of the Nasdaq data suggesting a convergence to a power law tail. The exponents c for the three other tails
are an order of magnitude larger but our tests show that they are not incompatible with an asymptotic power
tail either.
Note also that the exponents c seem larger for the daily DJ data than for the 5-minutes ND data, in agreement
with an expected (slow) convergence to the Gaussian law according to the central limit theory (see Sornette
et al. (2000) and figures 3.6-3.8 pp. 68 of Sornette (2000) where it is shown that SE distributions are
approximately stable in family and the effect of aggregation can be seen to slowly increase the exponent c).
However, a t-test does not allow us to reject the hypotheses that the exponents c remains the same for a given
24

tail (positive or negative) of the Dow Jones data. Thus, we confirm previous results (Lux 1996, Jondeau and
Rockinger 2001, for instance) according to which the extreme tails can be considered as symmetric, at least
for the Dow Jones data.
These are the evidence in favor of the existence of an asymptotic power law tail. Balancing this, many
of our tests have shown that the power law model is not as powerful compared with the SE model, even
arbitrarily far in the tail (as far as the available data allows us to probe). In addition, our attempts for a
direct estimation of the exponent b of a possible power law tail has failed to confirm the existence of a
well-converged asymptotic value (except maybe for the positive tail of the Nasdaq). In constrast, we have
found that the exponent b of the power law model systematically increases when going deeper and deeper
in the tails, with no visible sign of exhausting this growth. We have proposed tentative parameterization of
this growth of the apparent power law exponent. We note again that this behavior is expected from models
such as the GARCH or the Multifractal Random Walk models which predict asymptotic power law tails but
with exponents of the order of 20 or larger, that would be sampled at unattainable quantiles.
Attempting to wrap up the different results obtained by the battery of tests presented here, we can offer
the following conservative conclusion: it seems that the four tails examined here are decaying faster than
any (reasonable) power law but slower than any stretched exponentials. Maybe log-normal or log-Weibull
distributions could offer a better effective description of the distribution of returns 9 . Such a model has
already been suggested by (Serva et al. 2002).
The correct description of the distribution of returns has important implications for the assessment of large
risks not yet sampled by historical time series. Indeed, the whole purpose of a characterization of the functional
form of the distribution of returns is to extrapolate currently available historical time series beyond
the range provided by the empirical reconstruction of the distributions. For risk management, the determination
of the tail of the distribution is crucial. Indeed, many risk measures, such as the Value-at-Risk
or the Expected-Shartfall, are based on the properties of the tail of the distributions of returns. In order to
assess risk at probability levels of 95% or more, non-parametric methods have merits. However, in order to
estimate risks at high probability level such as 99% or larger, non-parametric estimations fail by lack of data
and parametric models become unavoidable. This shift in strategy has a cost and replaces sampling errors
by model errors. The considered distribution can be too thin-tailed as when using normal laws, and risk will
be underestimated, or it is too fat-tailed and risk will be over estimated as with Lévy law and possibly with
Pareto tails according to the present study. In each case, large amounts of money are at stake and can be lost
due to a too conservative or too optimistic risk measurement.
Our present study suggests that the Paretian paradigm leads to an overestimation of the probability of large
events and therefore leads to the adoption of too conservative positions. Generalizing to larger time scales,
the overly pessimistic view of large risks deriving from the Paretian paradigm should be all the more revised,
due to the action of the central limit theorem. Finally, an additional note of caution is in order. This study has
focused on the marginal distributions of returns calculated at fixed time scales and thus neglects the possible
occurrence of runs of dependencies, such as in cumulative drawdowns. In the presence of dependencies
between returns, and especially if the dependence is non stationary and increases in time of stress, the
characterization of the marginal distributions of returns is not sufficient. As an example, Johansen and
Sornette (2002) have recently shown that the recurrence time of very large drawdowns cannot be predicted
from the sole knowledge of the distribution of returns and that transient dependence effets occurring in time
of stress make very large drawdowns more frequent, qualifying them as abnormal “outliers.”
9 Let us stress that we are speaking of a log-normal distribution of returns, not of price! Indeed, the standard Black and Scholes
model of a log-normal distribution of prices is equivalent to a Gaussian distribution of returns. Thus, a log-normal distribution of
returns is much more fat tailed, and in fact bracketed by power law tails and stretched exponential tails.
25
89

90 3. Distributions exponentielles étirées contre distributions régulièrement variables
A Maximum likelihood estimators
In this appendix, we give the expressions of the maximum likelihood estimators derived from the four
distributions (22-25)
A.1 The Pareto distribution
According to expression (22), the Pareto distribution is given by
u
b Fu(x) = 1 − ,
x
x ≥ u (38)
and its density is
Let us denote by
L PD
fu(x|b) = b ub
x b+1
T (ˆb) = max
b
T
∑
i=1
(39)
ln fu(xi|b) (40)
the maximum of log-likekihood function derived under hypothesis (PD). ˆb is the maximum likelihood estimator
of the tail index b under such hyptothesis.
The maximum of the likelihood function is solution of
which yields
ˆb =
1
T
T
∑
i=1
lnxi − lnu
1 1
+ lnu −
b T ∑lnxi = 0, (41)
−1
, and
1
T LPD T (ˆb) = ln ˆb
u −
1 + 1
. (42)
ˆb
Moreover, one easily shows that ˆb is asymptotically normally distributed:
√ N(ˆb − b) ∼ N (0,b). (43)
A.2 The Weibull distribution
The Weibull distribution is given by equation (23) and its density is
fu(x|c,d) = c
d c · e( u d ) c
x c−1 · exp
The maximum of the log-likelihood function is
L SE
T (ĉ, d) ˆ = max
T ∑Ti=1 T
∑
i=1
c,d
T
∑
i=1
−
x
d
c
, x ≥ u. (44)
ln fu(xi|c,d) (45)
Thus, the maximum likehood estimators (ĉ, d) ˆ are solution of
1
c =
1
T ∑T xi c xi
i=1 u ln
u
1 xi c −
u − 1 1 T
T ∑ ln
i=1
xi
d
,
u
(46)
c = uc
xi
c
− 1.
T u
(47)
26

Equation (46) depends on c only and must be solved numerically. Then, the resulting value of c can be
reinjected in (47) to get d. The maximum of the log-likelihood function is
1
T LSE T (ĉ, d) ˆ = ln ĉ ĉ − 1
+
dˆ ĉ T
T
∑
i=1
91
lnxi − 1. (48)
Since c > 0, the vector √ N(ĉ − c, ˆ
d − d) is asymptotically normal, with a covariance matrix whose
expression is given in appendix D by the inverse of K11.
It should be noted that maximum likelihood equations (46-47) admit a solution with positive c not for
all possible samples (x1,··· ,xN). Indeed, the function
h(c) = 1
c −
1
T ∑T xi c xi
i=1 u ln
u
c +
− 1 1 T
ln
T
xi
, (49)
u
1
T ∑T xi
i=1 u
which is the total derivative of LSE T (c, d(c)), ˆ is a decreasing function of c. It means, as one can expect,
that the likelihood function is concave. Thus, a necessary and sufficient condition for equation (46) to
admit a solution is that h(0) is positive. After some calculations, we find
which is positive if and only if
h(0) = 2 1
T
xi ∑ln u
2 xi
T ∑ln u
∑
i=1
2 1 xi − T ∑ln2 u
, (50)
2 1 xi
2
T
∑ln −
u
1 2 xi
T
∑ln > 0. (51)
u
However, the probability of occurring a sample providing a negative maximum-likelyhood estimate of
c tends to zero (under Hypothesis of SE with a positive c) as
Φ − c√
N σ
√ e
σ 2π Nc − c2N 2σ2 , (52)
i.e. exponentially with respect to N. Here σ2 is the variance of the limit Gaussian distribution of
maximum-likelihood c-estimator that can be derived explicitly. If h(0) is negative, LSE T reaches its
maximum at c = 0 and in such a case
1
T LSE
1 xi
T (c = 0) = −ln
T
∑ln −
u
1
T ∑lnxi − 1. (53)
Now, if we apply maximum likelihood estimation based on SE assumption to samples distributed
differently from SE, then we can get negative c-estimate with some positive probability not tending
to zero with N → ∞. If sample is distributed according to Pareto distribution, for instance, then
maximum-likelihood c-estimate converges in probability to a Gaussian random variable with zero
mean, and thus the probability of negative c-estimates converges to 0.5.
A.3 The Exponential distribution
The Exponential distribution function is given by equation (24), and its density is
fu(x|d) = exp
u
d exp −
d
x
,
d
x ≥ u. (54)
27

92 3. Distributions exponentielles étirées contre distributions régulièrement variables
The maximum of the log-likelihood function is reach at
and is given by
T
d ˆ = 1
T ∑ xi − u, (55)
i=1
1
T LED T ( d) ˆ = −(1 + lnd). ˆ
(56)
The random variable √ N( ˆ
d − d) is asymptotically normally distributed with zero mean and variance d 2 /N.
A.4 The Incomplete Gamma distribution
The expression of the Incomplete Gamma distribution function is given by (25) and its density is
fu(x|b,d) =
d b
Γ −b, u
d
· x −(b+1)
exp −
x
, x ≥ u. (57)
d
Let us introduce the partial derivative of the logarithm of the incomplete Gamma function:
Ψ(a,x) = ∂
∞ 1
lnΓ(a,x) = dt lnt t
∂a Γ(a,x) x
a−1 e −t . (58)
The maximum of the log-likelihood function is reached at the point (ˆb, ˆ
d) solution of
and is equal to
1
T
T
∑
i=1
1
T
1
T LIG
T (ˆb, d) ˆ = −lndˆ − lnΓ
ln xi
d
T
xi
∑
i=1 d =
−b, u
d
= Ψ
−b, u
d
1
Γ −b, u
d
+ (b + 1) · Ψ
28
, (59)
u
−b e
d
− u d − b, (60)
−b, u
d
+ b −
1
Γ −b, u
d
u
−b e
d
− u d . (61)

B Minimum Anderson-Darling Estimators
We derive in this appendix the expressions allowing the calculation of the parameters which minimize the
Anderson-Darling distance between the assumed distribution and the true distribution.
Given the ordered sample x1 ≤ x2 ≤ ··· ≤ xN, the AD-distance is given by
ADN = −N − 2
N
∑
k=1
[wk logF(xk|α) + (1 − wk)log(1 − F(xk|α))], (62)
where α represents the vector of paramaters and wk = 2k/(2N + 1). It is easy to show that the minimun is
reached at the point ˆα solution of
B.1 The Pareto distribution
N
∑
k=1
Applying equation (63) to the Pareto distribution yields
1 − wk
log(1 − F(xk|α)) = 0. (63)
F(xk|α)
N
∑
k=1
1 −
wk
u
xk
b ln u
xk
This equation always admits a unique solution, and can easily be solved numerically.
B.2 Stretched-Exponential distribution
In the Stretched-Exponential case, we obtain the two following equations
with
N
∑
k=1
1 − wk
ln
Fk
u
d
∑
k=1
=
N
∑
k=1
93
ln u
. (64)
xk
u
c − ln
d
xk
xk c
d d
= 0, (65)
N
1 − wk
(u
Fk
c − x c k ) = 0, (66)
Fk = 1 − exp − uc − xc k
dc
. (67)
After some simple algebraic manipulations, the first equation can be slightly simplified, to finally yields
N
∑
k=1
N
∑
k=1
1 − wk
ln
Fk
xk
u
1 − wk
Fk
xk
u
xk
c
u
c
− 1
= 0, (68)
= 0. (69)
However, these two equations remain coupled. Moreover, we have not yet been able to prove the unicity of
the solution.
29

94 3. Distributions exponentielles étirées contre distributions régulièrement variables
B.3 Exponential distribution
In the exponential case, equation(63) becomes
with
N
wk
∑
k=1
Fk
Here again, we can show that this equation admits a unique solution.
− 1 (u − xk) = 0, (70)
u − xk
Fk = 1 − exp − . (71)
d
30

C Local exponent
In this Appendix, we come back to the notion of a local Pareto exponent discussed above in section 4.3.1,
with figure 7 and expression (33). We call the local exponent β.
Generally speaking, any positive smooth function g(x) can be represented in the form
g(x) = 1/x β(x) , x 1, (72)
by defining β(x) = −ln(g(x))/lnx. Thus, the local index β(x) of a distribution F(x) will be defined as
β(x) = −
95
ln(1 − F(x))
. (73)
ln(x)
For example, the log-normal distribution can be mistaken for a power law over several decades with very
slowly (logarithmically) varying exponents if its variance is large (see figures 4.2 and 4.3 in section 4.1.3
of (Sornette 2000)). When we approximate a sample distribution by some parametric family with moving
index β(x), it is important that β(x) should have as small a variation as possible, i.e., that the representation
(72) be parsimonious. Given a sample x1,x2,··· ,xN, drawn from a distribution function F(x), x ≥ 1, the tail
index β(x) is consistently estimated by
ˆβ(x) = lnN − ln
∑i 1 {xi>x}
, (74)
ln(x)
where 1 {·} is the indicator function which equals one if its argument is true and zero otherwhise. The
asymptotic distribution of the estimator is easily derived and reads:
with
N 1/2 ·
e ln(x)·ˆ β(x) − e ln(x)·β(x) d
−→ N (0,σ 2 ), (75)
σ 2 =
1
. (76)
F(x) · (1 − F(x))
As an example, let us illustrate the properties of the local index β(x) for regularly varying distributions. The
general representation of any regularly varying distribution is given by ¯F(x) = L(x) · x −α , where L(·) is a
slowly varying function. In such a case, the local power index can be written as
β(x) = α − lnL(x)
, (77)
ln(x)
which goes to α as x goes to infinity, as expected from the build-in slow variation of L(x). The upper panel
of figure 10 shows the local index β(x) for a simulated Pareto distribution with power index b = 1.2 (X > 1).
The estimate β(x) oscillates very closely to the true value b = 1.2.
Let us now assume that we observe a regularly varying local exponent
β(x) = L(x) · x c , with c > 0, (78)
it would be clearly the charaterization of a Stretched-Exponential distribution
¯F(x) = exp −L ′ (x) · x c , (79)
31

96 3. Distributions exponentielles étirées contre distributions régulièrement variables
where L ′ (x) = ln(x) L(x). The lower panel of figure 10 shows the local index β(x) for a simulated Stretched-
Exponential distribution with c = 0.3. In this case, the local index β(x) continuously increases, which corresponds
to our intuition that an Stretched-Exponential behaves like a power-law with an always increasing
exponent.
Figure 11 shows the local index β(x) for a distribution constructed by joining two Pareto distributions with
exponents b1 = 0.70 and b2 = 1.5 at the cross-over point u1 = 10. In this case, the local index β(x) again
increases but not so quickly as in previous Stretched-Exponential case. Even for such large sample size
(n = 15000), the “final” β(x) is about 1.3 which is still less than the true b-value for the second Pareto part
(b = 1.5), showing the existence of strong cross-over effects. Such very strong cross-over effects occurring
in the presence of a transition from a power law to another power law have already been noticed in (Sornette
et al. 1996).
Similarly to the local index β(x) based on the Pareto distribution taken as a reference, we define the notion
of a local exponent c(x) taking Stretched-Exponential distributions as a reference. Indeed, given any
sufficiently smooth positive function g(x), one can always find a function c(x) such that
g(x) = exp 1 − x c(x)
, x ≥ 1, (80)
with c(x) = ln(1 − ln(g(x)))/ln(x). Obviously, for any Stretched-Exponential distribution with exponent c,
the local exponent c(x) converges to c as x goes to infinity. This property is the same as for the local index
β(x) which goes to the true tail index β for regularly varying distributions.
Figure 12 shows the sample tail (continuous line), the local index β(x) (dashed line) and the local exponent
c(x) (dash-dotted line) for the negative tail of the Nasdaq five-minutes returns. The local exponent c(x)
clearly reaches an asymptotic value ∼ = 0.32 for large enough values of the returns. In contrast, the local exponent
β(x) remains continuously increasing. The lower panel shows in double logarithmic scale the local
index β(x). Over a large range, β(x) increases approximately a power law of index 0.77 while beyond the
quantile 99% (see the inset) it behaves like a power law with smaller index equal to 0.54 implying a decelerating
growth. The goodness of fit of the regression of lnβ(x) on lnx has been qualified by a χ 2 test which
does not allow to reject this model at any usual confidence level. Note that a power law dependence of the
local Pareto exponent β(x) as a function of x qualifies a Stretched-Exponential distribution, according to (78)
and (79). The second regime fitted with the exponent c = 0.54 seems still perturbed by a cross-over effect
as it does not retrieve the value c ∼ = 0.32 which characterizes the tail of the distributions of returns according
to the Stretched-Exponential model. These fits quantifying the growth of the local Pareto exponent are not
in contradiction with figure 7 and expression (33). Indeed, a decreasing positive exponent is an alternative
description for a logarithmic growth and vice-versa. These fits provide an improved quantification of the
previously rough characterization of the growth of the exponent estimated per quantile shown with 7 and
expression (33) but using a better characterization of the “local” exponent.
The positive tail of the Nasdaq and both tails of the Dow Jones exhibit exactly the same continuously increasing
behaviour with the same characteristics and we are not showing them. Taken together, these obervations
suggest that the Stretched-Exponential representation provides a better model of the tail behavior of large
returns than does a regularly varying distribution.
32

D Testing non-nested hypotheses with the encompassing principle
D.1 Testing the Pareto model against the (SE) model
D.1.1 Pseudo-true value
Let us consider the two models, stretched exponential (SE) and Pareto (P). The pdf’s associated with these
two models are f1(x|c,d) and f2(x|b) respectively . Under the true distribution f0(x), we will determine the
pseudo-true values of the maximum likelihood estimators ˆb, ĉ and d, ˆ namely, the values of these parameters
which minimize the (Kullback-Leibler) distance between the considered model and the true distribution
(Gouriéroux and Monfort 1994). Thus, the pseudo-true values b∗ , c∗ and d∗ of ˆb, ĉ and dˆ appear as the
expected values of the estimators under f0. For instance
b ∗ = arginf
b E0
ln f0(x)
, (81)
f2(x|b)
where, in all what follows, E0[·] denotes the expectation under the probability measure associated with the
true distribution f0. Thus, b∗ is simply solution of
∂
∂b E0
ln f0(x)
= 0, (82)
f2(x|b)
which yields
b ∗ = (E0[lnx] − lnu) −1 , (83)
and is consistently estimated by the maximum likelihood estimator
ˆb
1
=
T ∑lnxi
−1 − lnu . (84)
In fact, the maximum likelihood estimator ˆb converges to its pseudo-true value, with (ˆb−b ∗ ) asymptotically
normally distributed with zero mean.
Similarly, the maximum likelihood estimators ĉ and ˆ
d converge to their pseudo-true values c ∗ and d ∗ .
D.1.2 Binding functions and encompassing
Let us now ask what is the value b † (c,d) of the parameter b for which f2(x|b) is the nearest to f1(x|c,d), for
a given (c,d). Such a value b † is the binding function and is solution of
b † (c,d) = arginf
b E1
ln f1(x|c,d)
, (85)
f2(x|b)
which only involves f1 and f2 but not the true distribution f0. The binding function is given by
b †
(c,d) = E1 ln x
− ln
d
u
−1 . (86)
d
After some calculations, we find
E1 ln x
d
97
= c
dc e( u d ) c
·∞
dx ln
u
x
d xc−1 e −( x d ) c
, (87)
= ln u
d + e( u d ) c
c Γ
u
c 0, ,
d
(88)
33

98 3. Distributions exponentielles étirées contre distributions régulièrement variables
so that the binding function can be expressed as
b † (c,d) = c · e −( u d ) c
−1 u
c Γ 0, , when c > 0 (89)
d
In the case when c goes to zero and c and d are related by equation (47), we can still calculate b † . Indeed,
we can first show that
u
c ∼
d
1
1
c T
T
ln xi
−1 =
u
1
c · ˆb. (90)
Now, using the asymptoptic relation
we conclude that
∑
i=1
e x · Γ(0,x) ∼ x −1 , as x → ∞, (91)
b † (ĉ, ˆ
d) = ˆb, as ĉ → 0. (92)
This result is in fact natural because the PD model can be seen formally as the limit of the SE model for
c → 0 under the condition
as previously exposed in section 4.1
u
c c · → β,
d
as c → 0 , (93)
Now, following Mizon and Richard (1986), the model (SE) with pdf f1 is said to encompass the model
(PD) with pdf f2 if the best representative of (PD) -with parameter b ∗ - is also the distribution nearest to the
best representative of (SE) -with parameters (c ∗ ,d ∗ ). Thus, (SE) is said to encompass (PD) if and only if
b ∗ = b † (c ∗ ,d ∗ ).
The reverse situation can be considered in order to study the encompassing of the model (SE) by the model
(PD). Such situation occurs if and only if
c ∗
D.1.3 Wald encompassing test
d ∗
c † (b∗ )
=
d † (b∗ )
. (94)
We first test the encompassing of (PD) into (SE), namely the null hypothesis H0 = {b ∗ = b † (c ∗ ,d ∗ )}.
Under this null hypothesis, it can be shown (Gouriéroux and Monfort 1994) that the random variable
√ T ˆb − b † (ĉ, ˆ
d) is asymptotically normally distributed with zero mean and variance V given by
V = K −1
C12]K −1
22 [C22 −C21C −1
11
+ K −1 −1
22
[C21C11 − ˜C21K −1
11
22 +
]C11[C −1
11 C12 − K −1
11
˜C12]K −1
22
where the expression of the coefficients involved in V will be given below. Thus, under H0, the random variable
ξT = T ˆb − b † (ĉ, d) ˆ
ˆV −1 ˆb − b † (ĉ, d) ˆ
, where ˆV −1 is a consistent estimator of V −1 , follows asymptotically
a χ2-distribution with one degrees of freedom.
The matrix K11 is given by
K11(i, j) = −E0
∂ 2 ln f1(x|c ∗ ,d ∗ )
∂αi∂α j
34
(95)
, i, j = 1,2, (96)

where α = (c ∗ ,d ∗ ). It can be consistently estimated by
The coefficient K22 is given by
ˆK11(1,1) = 1
ĉ u u
− ln2
+
ĉ2 dˆ
dˆ
1 T
T ∑ ln
i=1
2
ĉ xi xi
, (97)
dˆ
dˆ
ˆK11(1,2) = ˆK11(2,1) = c
ĉ u u
ln
−
d dˆ
dˆ
1 T
ĉ
xi xi
T ∑ ln
, (98)
ˆ
i=1 d dˆ
2 ˆK11(2,2)
ĉ
= . (99)
dˆ
K22 = −E0
which is consistently estimated by ˆK22 = ˆb −2 .
∂ 2 ln f (x|b ∗ )
∂b ∗2
99
= 1
, (100)
b∗2 Now, we have to calculate the two components of the vector ˜C12 = ˜C t 21 . Its first component is
˜C12(1)
∂ln f1(x|c
= E1
∗ ,d ∗ )
∂c∗ · ∂ln f2(x|b∗ )
∂b∗
, (101)
1 u
= E1 + ln
c∗ d∗
u
d∗ c∗
+ ln x x
− ln
d∗ d∗
x
d∗ c∗
1 u
· + ln
b∗ d∗
− ln x
d∗
,(102)
1 u
= + ln
c∗ d∗
u
d∗ c∗
1 u 1 u
· + ln − + ln
b∗ d c∗ d∗
u
d∗ c∗
· E1 ln x
d∗
1 u
+ + ln
b∗ d∗
· E1 ln x
d∗
− E1 ln x
d∗
x
d∗ c∗
2 x
− E1 ln
d∗
2 x
+ E1 ln
d∗
x
d∗ c∗ (103) .
Some simple calculations show that
E1
E1
ln x
d ·
x
c d
2 x
ln
d ·
x
c d
= 1 u
+ ln
c d ·
u
c
+ E1 ln
d
x
, (104)
d
2 u
= ln
d ·
u
c +
d
2
c E1
ln x
2 x
+ E1 ln , (105)
d d
which allows us to show that the first and third terms cancel out, and it remains
˜C12(1) = 1
E1 ln
c∗ x
d∗
− ln u
d∗
u
d∗ c∗
E1 ln x
d∗
− ln u
d∗
. (106)
The second component is
˜C12(2) =
∂ln f1(x|c
E1
∗ ,d ∗ )
∂d∗ · ∂ln f2(x|b∗ )
∂b∗ =
,
E1 −
(107)
c∗
d∗
u
1 +
d∗ c∗
x
−
d∗ c∗
1 u
· + ln
b∗ d∗
− ln x
d∗ =
,
−
(108)
c∗
d∗
u
1 +
d∗ c∗ 1 u
+ ln
b∗ d∗
u
− 1 +
d∗ c∗
E1 ln x
d∗ −
1 u
+ ln
b∗ d∗ x
E1
d∗ c∗
+ E1 ln x
d∗
x
d∗ c∗ . (109)
35

100 3. Distributions exponentielles étirées contre distributions régulièrement variables
Now, accounting for the relation E1[xc ] = uc +d c , the first and third term within the brackets cancel out, and
accounting for equation (104) yields
˜C12(2) = − c∗
d∗
1
u
+
c∗ d∗ c∗
· ln u
− E ln
d∗ x
d∗ . (110)
Finally, ˜C12 can be consistently estimated by replacing (c ∗ ,d ∗ ) by (ĉ, ˆ
d) and using the relation
Let us now define
g1(x|c ∗ ,d ∗ ) =
E
ln x
d
∂ln f1(x|c ∗ ,d ∗ )
∂c ∗
∂ln f1(x|c ∗ ,d ∗ )
∂d ∗
g2(x|b ∗ ) = ∂ln f2(x|b ∗ )
∂b ∗
The matrices Ci j are defined as
which can be consistently estimated by
= ln u 1
+
d c e( u d ) c
· Γ 0,
=
1
c∗ + ln u
d∗ − c∗
d∗
u
c . (111)
d
u
d∗ c∗
+ ln x
d∗ − ln x
d∗ 1 + u
d∗ c∗ − x
d∗ c∗
x c∗ d
(112)
= 1
+ lnu − lnx. (113)
b∗
Ci j = E0 gi(x) · g j(x) t , i, j = 1,2, (114)
Ĉi j = 1
T
T
∑
i=1
D.2 Testing the (SE) model against the Pareto model
gi(x) · g j(x) t , i, j = 1,2. (115)
Let us now assume that, beyond a given high threshold u, the true model is the Pareto model, that is, the true
returns distribution is a power law with pdf
This will be our null hypothesis H0.
f0(x|b) = b ub
, x ≥ u. (116)
xb+1 Now consider the maximum likelihood estimators (ĉ, d) ˆ of the model (SE). They are solution of equations
(46-47)
1
c =
d c = uc
T
1
T ∑T xi
i=1 u
1
T ∑T xi
i=1 u
T
xi
∑
i=1 u
Under H0, (ĉ, ˆ
d) converges to the pseudo-true values, solutions of
1
c
x
E0 u =
x
E0 u
c xi ln
u
c −
− 1 1 T
T ∑ ln
i=1
xi
,
u
(117)
c − 1. (118)
c
x ln
u
c − 1 − E0
ln x
, (119)
u
d c = E0 [x c ] − u c , (120)
36

where E0[·] denotes the expectation with respect to the power-law distribution f0. We have
x
c E0
u
E0 ln
=
b
, and c < b,
b − c
(121)
x
u
= 1
x
c E0 ln
u
,
b
(122)
x
u
=
b
.
(b − c) 2 (123)
Thus, we easily obtain that the unique solution of (119) is c = 0, and equation (120) does not make sense
any more. So, under H0, ĉ goes to zero and dˆ is not well defined. Thus, Wald test cannot be performed under
such a null hypothesis. We must find another way to test (SE) against H0.
In this goal, we remark that the quantity
ˆηT = ĉ
ĉ
u
+ 1
dˆ
101
(124)
is still well defined. Using (120), it is easy to show that, as T goes to infinity, this quantity goes to b,
whatever ĉ being positive or equal to zero. For positive c, this is obvious from (120) and (121), while for
c = 0, expanding (118) around ĉ = 0 yields
ĉ
ĉ u
dˆ
=
1
T
T
∑
i=1
ln xi
u
−1
(125)
= ˆb → b. (126)
In order to test the descriptive power of the (SE) model against the null hypothesis that the true model is the
Pareto model, we can consider the statistic
ˆηT
ζT = T − 1 , (127)
ˆb
which asymptoticaly follows a χ 2 -distribution with one degree of freedom. Indeed expanding the quantity
(xi/u) ĉ in power series around c = 0 gives
which allows us to get
where
xi
ĉ
∼= 1 + ĉ · log(
u
xi
u
) + ĉ2
2 · log2
xi
+
u
ĉ3
6 · log3
xi
+ ··· , as ĉ → 0, (128)
u
1
T ∑
xi
ĉ
∼= 1 + ĉ · S1 +
u
ĉ2
2 · S2 + ĉ3
3 S3, (129)
1
T ∑
xi
c
xi
log ∼= S1 + ĉ · S2 +
u u
ĉ2
2 S3, (130)
S1 = 1
T
S2 = 1
T
S3 = 1
T
T
∑
i=1
T
∑
i=1
T
∑
i=1
log
xi
u
log 2 xi
u
log 3 xi
u
37
, (131)
, (132)
. (133)
(134)

102 3. Distributions exponentielles étirées contre distributions régulièrement variables
Thus, we get from equation (117):
and from equations (124) and (118):
ĉ
1
2 S2 − S 2 1
1
2S1S2 − 1
3S3 ˆηT
ˆb − 1 = ĉ · S2 1
1 − 2S2 + ĉ
2 [S1S2 − 1
S1 + ĉ
2S2 + ĉ2
6 S3
, (135)
3 S3]
. (136)
Now, accounting for the fact that the variables ξ1 = S1 − b−1 , ξ2 = S2 − 2b−2 and ξ3 = S3 − 6b−3 are
asymptoticaly Gaussian random variables with zero mean and variance of order T −1/2 , we obtain, at the
lowest order in T −1/2 :
ĉ = b 2
2ξ1 − b
2 ξ2
, (137)
and
which shows that
S2 1
1 − 2S2 + ĉ
2 [S1S2 − 1
3S3] S1 + ĉ
2 S2 + ĉ2
6 S3
ˆηT
ˆb
− 1 = b2
= 2ξ1 − b
2 ξ2
, (138)
2ξ1 − b
2 ξ2
2
. (139)
We now use the fact that ξ1, ξ2 are asymptotically Gaussian random variables with zero mean. We find their
variances:
Var(ξ1) = 1
T b2 , Var(ξ2) = 20
T b4 , and Cov(ξ1,ξ2) = 4
.
T b3 (140)
Using equation (139), the random variable b(2ξ1 − bξ2/2) is in the limit of large T a Gaussian rv with zero
mean and variance 1/T , i.e., ζT is asymptoticaly distributed according to a χ 2 distribution with one degree
of freedom.
Thus, the test consists in accepting H0 if ζT ≤ χ2 1−ε (1) and rejecting it otherwise. (ε denotes the asymptotic
level of the test).
38

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43
107

108 3. Distributions exponentielles étirées contre distributions régulièrement variables
Mean St. Dev. Skewness Ex. Kurtosis Jarque-Bera
Nasdaq (5 minutes) † 1.80 ·10 −6 6.61 ·10 −4 0.0326 11.8535 1.30 ·10 5 (0.00)
Nasdaq (1 hour) † 2.40 ·10 −5 3.30 ·10 −3 1.3396 23.7946 4.40 ·10 4 (0.00)
Nasdaq (5 minutes) ‡ - 6.33 ·10 −9 3.85 ·10 −4 -0.0562 6.9641 4.50 ·10 4 (0.00)
Nasdaq (1 hour) ‡ 1.05 ·10 −6 1.90 ·10 −3 -0.0374 4.5250 1.58 ·10 3 (0.00)
Dow Jones (1 day) 8.96·10 −5 4.70 ·10 −3 -0.6101 22.5443 6.03 ·10 5 (0.00)
Dow jones (1 month) 1.80 ·10 −3 2.54 ·10 −2 -0.6998 5.3619 1.28 ·10 3 (0.00)
Table 1: Descriptive statistics for the Dow Jones returns calculated over one day and one month and for
the Nasdaq returns calculated over five minutes and one hour. The numbers within parenthesis represent
the p-value of Jarque-Bera’s normality test. (†) raw data, (‡) data corrected for the U-shape of the intra-day
volatility due to the lunch effect.
44

(a) Independent Data
Stretched-Exponential c=0.7 Stretched-Exponential c=0.3 Pareto Distribution b=3
Maximum Maximum Maximum
cluster 10 50 100 200 cluster 10 50 100 200 cluster 10 50 100 200
mean 0.1870 0.1734 0.2032 0.2290 mean 0.9313 0.9910 1.0617 1.1011 mean 0.3669 0.4007 0.4455 0.4769
Emp Std 0.0262 0.0340 0.0382 0.0415 Emp Std 0.0380 0.0464 0.0492 0.0508 Emp Std 0.0267 0.0353 0.0384 0.0407
GPD GPD GPD
quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995
mean 0.1108 0.0758 -0.1374 -0.0509 mean -0.1234 0.1947 -0.2147 -0.2681 mean 0.3336 0.3271 0.3193 0.2585
Emp Std 0.0346 0.1579 0.4648 0.4147 Emp Std 0.8139 0.6197 0.6625 0.597 Emp Std 0.0454 0.0604 0.1271 0.2991
(b) Dependent Data
Stretched-Exponential c=0.7 with long memory Stretched-Exponential c=0.3 with long memory Pareto with long memory b=3
Maximum Maximum Maximum
cluster 10 50 100 200 cluster 10 50 100 200 cluster 10 50 100 200
mean -0.2169 -0.2226 -0.2027 -0.1918 mean 0.1230 0.1143 0.1359 0.1569 mean 0.0566 0.0606 0.0794 0.0978
Emp Std 0.1553 0.1512 0.1553 0.1718 Emp Std 0.0328 0.0414 0.0450 0.0443 Emp Std 0.0422 0.0495 0.0533 0.0560
45
GPD GPD GPD
quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995
mean -0.1005 -0.0593 -0.0189 -0.0314 mean 0.0732 0.0740 0.0031 -0.0559 mean 0.2952 0.3203 0.3153 0.2570
Emp Std 0.0386 0.0503 0.1097 0.1700 Emp Std 0.0942 0.1804 0.3744 0.4822 Emp Std 0.0548 0.0694 0.1812 0.3382
Table 2: Mean values and standard deviations of the Maximum Likelihood estimates of the parameter ξ (inverse of the Pareto exponent) for the distribution
of maxima (cf. equation 4) when data are clustered in samples of size 10,50,100 and 200 and for the Generalized Pareto Distribution (7) for thresholds
u corresponding to quantiles 90%,95%,99% ans 99.5%. In panel (a), we have used iid samples of size 10000 drawn from a Stretched-Exponential
distribution with c = 0.7 and c = 0.3 and a Pareto distribution with tail index b = 3, while in panel (b) the samples are drawn from a long memory process
with Stretched-Exponential marginals and regularly-varying marginal as explained in the text.
109

110 3. Distributions exponentielles étirées contre distributions régulièrement variables
(a) Independent data
Stretched-Exponential c=0.7 Stretched-Exponential c=0.3 Pareto Distribution b=3
quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1
N/k=4 N/k=4 N/k=4
mean 0.0791 0.0689 0.1172 0.1432 mean 0.3843 0.4340 0.6301 0.7768 mean 0.3460 0.3496 0.3359 0.3334
emp. Std 0.5156 0.3614 0.1652 0.1154 emp. Std 0.5643 0.3925 0.1800 0.1340 emp. Std 0.5384 0.3631 0.1760 0.1175
th. Std 0.5148 0.3635 0.1636 0.1161 th. Std 0.5393 0.3847 0.1786 0.1302 th. Std 0.5358 0.3791 0.1691 0.1195
N/k=10 N/k=10 N/k=10
mean 0.0726 0.0652 0.0880 0.1083 mean 0.3613 0.3819 0.5028 0.5950 mean 0.3134 0.3379 0.3213 0.3382
emp. Std 0.8218 0.5925 0.2602 0.1839 emp. Std 0.8961 0.6141 0.2741 0.1963 emp. Std 0.9003 0.6056 0.2551 0.1933
th. Std 0.8133 0.5745 0.2577 0.1827 th. Std 0.8493 0.6027 0.2755 0.1983 th. Std 0.8425 0.5982 0.2668 0.1892
(b) Dependent data
Stretched-Exponential c=0.7 with long memory Stretched-Exponential c=0.3 with long memory Pareto b=3 with long memory
quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1
N/k=4 N/k=4 N/k=4
mean -0.0439 -0.0736 -0.2134 -0.3395 mean 0.1297 0.1215 0.0410 0.0235 mean 0.1458 0.1261 -0.0023 -0.0790
emp. Std 0.5396 0.3599 0.1643 0.1128 emp. Std 0.5557 0.3756 0.1633 0.1130 emp. Std 0.5572 0.3765 0.1633 0.1125
th. Std 0.5073 0.3576 0.1579 0.1108 th. Std 0.5183 0.3661 0.1620 0.1143 th. Std 0.5194 0.3663 0.1612 0.1130
46
N/k=10 N/k=10 N/k=10
mean -0.0371 -0.0307 -0.0928 -0.1781 mean 0.0961 0.1275 0.0682 0.0521 mean 0.1239 0.1461 0.0584 0.0196
emp. Std 0.8325 0.5754 0.2561 0.1821 emp. Std 0.8314 0.5796 0.2607 0.1801 emp. Std 0.8352 0.5811 0.2608 0.1801
th. Std 0.8028 0.5680 0.2524 0.1771 th. Std 0.8158 0.5793 0.2570 0.1814 th. Std 0.8188 0.5808 0.2567 0.1807
Table 3: Pickands estimates (9) of the parameter ξ for the Generalized Pareto Distribution (7) for thresholds u corresponding to quantiles 90%,95%,99%
ans 99.5% and two different values of the ratio N/k respectively equal to 4 and 10. In panel (a), we have used iid samples of size 10000 drawn from a
Stretched-Exponential distribution with c = 0.7 and c = 0.3 and a Pareto distribution with tail index b = 3, while in panel (b) the samples are drawn from
a long memory process with Stretched-Exponential marginals and regularly-varying marginal.

(a) Dow Jones
Positive Tail Negative Tail
Maximum Maximum
cluster 10 50 100 200 cluster 10 50 100 200
mean 0.2195 0.2150 0.2429 0.2884 mean 0.1346 0.1012 0.1222 0.1700
Emp Std 0.0850 0.1163 0.1342 0.1238 Emp Std 0.1318 0.1437 0.1549 0.1671
GPD GPD
quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995
mean 0.2479 0.2493 0.1808 0.3458 mean 0.2121 0.0244 0.2496 0.3118
Emp Std 0.0224 0.0436 0.0654 0.1209 Emp Std 0.0294 0.0344 0.0979 0.1452
(b) Nasdaq (Raw data)
Positive Tail Negative Tail
Maximum Maximum
cluster 10 50 100 200 cluster 10 50 100 200
mean 1.2961 1.1992 0.9858 0.7844 mean 1.2105 0.4707 0.4146 0.4200
Emp Std 0.2837 0.2955 0.4377 0.4679 Emp Std 0.2780 0.4625 0.4117 0.3980
GPD GPD
quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995
mean 0.2123 0.3025 0.5973 0.5973 mean 0.1477 0.2189 0.5973 0.5973
Emp Std 0.0693 0.0712 0 0 Emp Std 0.0468 0.0798 0 0
(c) Nasdaq (Correted data)
Positive Tail Negative Tail
Maximum Maximum
cluster 10 50 100 200 cluster 10 50 100 200
mean 0.2163 0.2652 0.2738 0.2771 mean 0.2530 0.2276 0.3173 0.3681
Emp Std 0.2461 0.2825 0.2364 0.2604 Emp Std 0.3345 0.3273 0.4303 0.4289
GPD GPD
quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995
mean 0.2569 0.4945 0.5943 0.5973 mean 0.2560 0.3185 0.5929 0.5847
Emp Std 0.1287 0.1683 0.0304 0.0000 Emp Std 0.1794 0.2137 0.0446 0.0883
Table 4: Mean values and standard deviations of the Maximum Likelihood estimates of the parameter ξ for
the distribution of maximum (cf. equation 4) when data are clustered in samples of size 10,50,100 and 200
and for the Generalized Pareto Distribution (7) for thresholds u corresponding to quantiles 90%,95%,99%
ans 99.5%. In panel (a), are presented the results for the Dow Jones, in panel (b) for the Nasdaq for raw data
and in panel (c) the Nasdaq corrected for the “lunch effect”.
47
111

112 3. Distributions exponentielles étirées contre distributions régulièrement variables
(a) Dow Jones
Negative Tail Positive Tail
quantile 0.995 0.99 0.95 0.90 quantile 0.995 0.99 0.95 0.90
N/k =4 N/k =4
mean 0.2491 -0.0366 0.2721 0.2268 mean 0.6427 -0.3035 0.4006 0.2338
emp. Std 0.3561 0.2382 0.0974 0.0639 emp. Std 0.3913 0.2233 0.0902 0.0695
th. Std 0.5274 0.3590 0.1674 0.1175 th. Std 0.5663 0.3510 0.1710 0.1177
N/k =10 N/k=10
mean 0.4449 -0.2539 0.0279 0.2905 mean -0.2228 0.8057 0.3622 0.3107
emp. Std 0.5354 0.5563 0.1314 0.0943 emp. Std 1.3601 0.5420 0.1301 0.1227
th. Std 0.8619 0.5568 0.2558 0.1877 th. Std 0.7891 0.6552 0.2686 0.1883
(b) Nasdaq (Raw data)
Negative Tail Positive Tail
quantile 0.995 0.99 0.95 0.90 quantile 0.995 0.99 0.95 0.90
N/k =4 N/k=4
mean 0.3189 0.0006 0.1055 0.0284 mean 1.0709 0.2538 0.1090 0.0386
emp. Std 0.2489 0.1295 0.0649 0.043 emp. Std 0.3461 0.2190 0.0626 0.0384
th. Std 0.5333 0.3605 0.1634 0.1144 th. Std 0.6222 0.3732 0.1634 0.1145
N/k =10 N/k =10
mean 0.2034 -0.1098 0.1634 0.2445 mean -0.8666 1.1228 0.1601 0.2846
emp. Std 0.6173 0.4596 0.0871 0.0571 emp. Std 0.6405 0.3499 0.1157 0.0762
th. Std 0.8281 0.5634 0.2604 0.1863 th. Std 0.7835 0.7042 0.2602 0.1875
(c) Nasdaq (Corrected data)
Negative Tail Positive Tail
quantile 0.995 0.99 0.95 0.90 quantile 0.995 0.99 0.95 0.90
N/k =4 N/k=4
mean 0.1910 0.3435 0.0616 0.2070 mean 1.4051 -0.1313 0.0255 0.2259
emp. Std 0.2238 0.1441 0.0487 0.0333 emp. Std 0.2299 0.1424 0.0739 0.0390
th. Std 0.5228 0.3787 0.1624 0.1172 th. Std 0.6742 0.3556 0.1617 0.1175
N/k =10 N/k =10
mean 0.0630 0.0543 0.4537 -0.0699 mean -0.3357 1.3961 0.4034 0.0770
emp. Std 0.4899 0.1004 0.0780 0.0614 emp. Std 0.6830 0.2481 0.0891 0.0643
th. Std 0.5137 0.3629 0.1727 0.1150 th. Std 0.7835 0.7521 0.2705 0.1820
Table 5: Pickands estimates (9) of the parameter ξ for the Generalized Pareto Distribution (7) for thresholds u
corresponding to quantiles 90%,95%,99% ans 99.5% and two different values of the ratio N/k respectiveley
equal to 4 and 10. In panel (a), are presented the results for the Dow Jones, in panel (b) for the Nasdaq for
raw data and in panel (c) the Nasdaq corrected for the “lunch effect”.
48

Nasdaq Dow Jones
Pos. Tail Neg. Tail Pos. Tail Neg. Tail
q 10 3 u nu 10 3 u nu 10 2 u nu 10 2 u nu
q1=0 0.0053 11241 0.0053 10751 0.0032 14949 0.0028 13464
q2=0.1 0.0573 10117 0.0571 9676 0.0976 13454 0.0862 12118
q3=0.2 0.1124 8993 0.1129 8601 0.1833 11959 0.1739 10771
q4=0.3 0.1729 7869 0.1723 7526 0.2783 10464 0.263 9425
q5=0.4 0.238 6745 0.2365 6451 0.3872 8969 0.3697 8078
q6=0.5 0.3157 5620 0.3147 5376 0.5055 7475 0.4963 6732
q7=0.6 0.406 4496 0.412 4300 0.6426 5980 0.6492 5386
q8=0.7 0.5211 3372 0.5374 3225 0.8225 4485 0.8376 4039
q9=0.8 0.6901 2248 0.7188 2150 1.0545 2990 1.1057 2693
q10=0.9 0.973 1124 1.0494 1075 1.4919 1495 1.6223 1346
q11=0.925 1.1016 843 1.1833 806 1.6956 1121 1.8637 1010
q12=0.95 1.2926 562 1.3888 538 1.9846 747 2.2285 673
q13=0.96 1.3859 450 1.4955 430 2.1734 598 2.4197 539
q14=0.97 1.53 337 1.639 323 2.413 448 2.7218 404
q15=0.98 1.713 225 1.8557 215 2.7949 299 3.1647 269
q16=0.99 2.1188 112 1.8855 108 3.5704 149 4.1025 135
q17=0.9925 2.3176 84 2.4451 81 3.9701 112 4.3781 101
q18=0.995 3.0508 56 2.7623 54 4.5746 75 5.0944 67
Table 6: Significance levels qk and their corresponding lower thresholds uk for the four different samples.
The number nu provides the size of the sub-sample beyond the threshold uk.
49
113

114 3. Distributions exponentielles étirées contre distributions régulièrement variables
Mean AD-statistic for u1 – u9
N-pos N-neg DJ-pos DJ-neg
Weibull 1.86 (89.34%) 1.34 (79.18%) 5.43 (99.82%) 3.85 (98.91%)
Gen. Pareto 4.45 (99.45%) 2.68 (96.01%) 12.47 (99.996%) 6.44 (99,99%)
Gamma 3.59 (98.59%) 2.82 (96.62%) 8.76 (99.996%) 7.23 (99.996%)
Exponential 3.64 (98.66%) 2.76 (96.36%) 13.96 (99.996%) 10.10 (99.996%)
Pareto 475.2 (99.996%) 441.7 (99.996%) 691.7 (99.996%) 607.9 (99.996%)
Mean AD-statistic for u10 – u18
Weibull 1.21 (74.29%) .988 (63.39%) .835 (53,52%) .849 (54.54%)
Gen. Pareto 2.29 (93.57%) 1.88 (89.52%) 1.95 (90.28%) 1.36 (79.67%)
Gamma 2.49 (95.00%) 1.90 (89.74%) 2.12 (92.01%) 1.63 (86.02%)
Exponential 3.26 (97.97%) 1.93 (90.02%) 4.52 (99.10%) 2.70 (96.11%)
Pareto 1.80 (88.60%) 1.77 (88.23%) 1.18 (73.01%) 1.65 (86.40%)
Table 7: Mean Anderson-Darling distances in the range of thresholds u1-u9 and in the range u10-u18. The
figures within parenthesis characterize the goodness of fit: they represent the significance levels with which
the considered model can be rejected.
50

Nasdaq Dow Jones
Pos. Tail Neg. Tail Pos. Tail Neg. Tail
MLE ADE MLE ADE MLE ADE MLE ADE
1 0.256 (0.002) 0.192 0.254 (0.002) 0.191 0.204 (0.002) 0.150 0.199 (0.002) 0.147
2 0.555 (0.006) 0.443 0.548 (0.006) 0.439 0.576 (0.005) 0.461 0.538 (0.005) 0.431
3 0.765 (0.008) 0.630 0.755 (0.008) 0.625 0.782 (0.007) 0.644 0.745 (0.007) 0.617
4 0.970 (0.011) 0.819 0.945 (0.011) 0.800 0.989 (0.010) 0.833 0.920 (0.009) 0.777
5 1.169 (0.014) 1.004 1.122 (0.014) 0.965 1.219 (0.013) 1.053 1.114 (0.012) 0.960
6 1.400 (0.019) 1.227 1.325 (0.018) 1.157 1.447 (0.017) 1.279 1.327 (0.016) 1.169
7 1.639 (0.024) 1.460 1.562 (0.024) 1.386 1.685 (0.022) 1.519 1.563 (0.021) 1.408
8 1.916 (0.033) 1.733 1.838 (0.032) 1.655 1.984 (0.030) 1.840 1.804 (0.028) 1.659
9 2.308 (0.049) 2.145 2.195 (0.047) 1.999 2.240 (0.041) 2.115 2.060 (0.040) 1.921
10 2.759 (0.082) 2.613 2.824 (0.086) 2.651 2.575 (0.067) 2.474 2.436 (0.066) 2.315
11 2.955 (0.102) 2.839 3.008 (0.106) 2.836 2.715 (0.081) 2.648 2.581 (0.081) 2.467
12 3.232 (0.136) 3.210 3.352 (0.145) 3.259 2.787 (0.102) 2.707 2.765 (0.107) 2.655
13 3.231 (0.152) 3.193 3.441 (0.166) 3.352 2.877 (0.118) 2.808 2.782 (0.120) 2.642
14 3.358 (0.183) 3.390 3.551 (0.198) 3.479 2.920 (0.138) 2.841 2.903 (0.144) 2.740
15 3.281 (0.219) 3.306 3.728 (0.254) 3.730 2.989 (0.173) 2.871 3.059 (0.186) 2.870
16 3.327 (0.313) 3.472 3.990 (0.384) 3.983 3.226 (0.263) 3.114 3.690 (0.318) 3.668
17 3.372 (0.366) 3.636 3.917 (0.435) 3.860 3.427 (0.322) 3.351 3.518 (0.350) 3.397
18 3.136 (0.415) 3.326 4.251 (0.578) 4.302 3.818 (0.441) 3.989 4.168 (0.506) 4.395
Table 8: Maximum Likelihood and Anderson-Darling estimates of the Pareto parameter b. Figures within
parentheses give the standard deviation of the Maximum Likelihood estimator.
51
115

116 3. Distributions exponentielles étirées contre distributions régulièrement variables
Nasdaq Dow Jones
Pos. Tail Neg. Tail Pos. Tail Neg. Tail
MLE ADE MLE ADE MLE ADE MLE ADE
1 1.007 (0.008) 1.053 0.987 (0.008) 1.017 1.040 (0.007) 1.104 0.975 (0.007) 1.026
2 0.983 (0.011) 1.051 0.953 (0.011) 0.993 0.973 (0.010) 1.075 0.910 (0.010) 0.989
3 0.944 (0.014) 1.031 0.912 (0.014) 0.955 0.931 (0.013) 1.064 0.856 (0.012) 0.948
4 0.896 (0.018) 0.995 0.876 (0.018) 0.916 0.878 (0.015) 1.038 0.821 (0.015) 0.933
5 0.857 (0.021) 0.978 0.861 (0.021) 0.912 0.792 (0.019) 0.955 0.767 (0.018) 0.889
6 0.790 (0.026) 0.916 0.833 (0.026) 0.891 0.708 (0.023) 0.873 0.698 (0.022) 0.819
7 0.732 (0.033) 0.882 0.796 (0.033) 0.859 0.622 (0.028) 0.788 0.612 (0.028) 0.713
8 0.661 (0.042) 0.846 0.756 (0.042) 0.834 0.480 (0.035) 0.586 0.531 (0.035) 0.597
9 0.509 (0.058) 0.676 0.715 (0.059) 0.865 0.394 (0.047) 0.461 0.478 (0.047) 0.527
10 0.359 (0.092) 0.631 0.522 (0.099) 0.688 0.304 (0.074) 0.346 0.403 (0.076) 0.387
11 0.252 (0.110) 0.515 0.481 (0.120) 0.697 0.231 (0.087) 0.158 0.379 (0.091) 0.337
12 0.039 (0.138) 0.177 0.273 (0.155) 0.275 0.269 (0.111) 0.207 0.357 (0.119) 0.288
13 0.057 (0.155) 0.233 0.255 (0.177) 0.274 0.247 (0.127) 0.147 0.428 (0.136) 0.465
14 0 0 0.215 (0.209) 0.194 0.283 (0.150) 0.174 0.448 (0.164) 0.641
15 0 0 0.091 (0.260) 0 0.374 (0.192) 0.407 0.451 (0.210) 0.863
16 0 0 0.064 (0.390) 0 0.372 (0.290) 0.382 0.022 (0.319) 0.110
17 0 0 0.158 (0.452) 0.224 0.281 (0.346) 0.255 0.178 (0.367) 0.703
18 0 0 0 0 0 0 0 0
Table 9: Maximum Likelihood and Anderson-Darling estimates of the form parameter c of the Weibull
(Stretched-Exponential) distribution.
52

Nasdaq Dow Jones
Pos. Tail Neg. Tail Pos. Tail Neg. Tail
MLE ADE MLE ADE MLE ADE MLE ADE
1 0.443 (0.004) 0.441 0.455 (0.005) 0.452 7.137 (0.060) 7.107 7.268 (0.068) 7.127
2 0.429 (0.006) 0.440 0.436 (0.006) 0.443 6.639 (0.082) 6.894 6.726 (0.094) 6.952
3 0.406 (0.008) 0.432 0.410 (0.009) 0.424 6.236 (0.113) 6.841 6.108 (0.131) 6.640
4 0.372 (0.011) 0.414 0.383 (0.012) 0.402 5.621 (0.155) 6.655 5.656 (0.175) 6.515
5 0.341 (0.015) 0.404 0.369 (0.016) 0.399 4.515 (0.215) 5.942 4.876 (0.235) 6.066
6 0.283 (0.020) 0.364 0.345 (0.021) 0.383 3.358 (0.277) 5.081 3.801 (0.305) 5.220
7 0.231 (0.026) 0.339 0.309 (0.028) 0.358 2.192 (0.326) 4.073 2.475 (0.366) 3.764
8 0.166 (0.034) 0.311 0.269 (0.039) 0.336 0.682 (0.256) 1.606 1.385 (0.389) 2.149
9 0.053 (0.030) 0.164 0.225 (0.057) 0.365 0.195 (0.163) 0.510 0.810 (0.417) 1.297
10 0.005 (0.010) 0.128 0.058 (0.057) 0.184 0.019 (0.048) 0.065 0.276 (0.361) 0.207
11 0.000 (0.001) 0.049 0.036 (0.053) 0.194 0.001 (0.003) 0.000 0.169 (0.316) 0.065
12 0.000 (0.000) 0.000 0.000 (0.001) 0.000 0.005 (0.025) 0.000 0.103 (0.291) 0.012
13 0.000 (0.000) 0.000 0.000 (0.001) 0.000 0.001 (0.010) 0.000 0.427 (0.912) 0.729
14 - - 0.000 (0.000) 0.000 0.009 (0.055) 0.000 0.577 (1.357) 3.509
15 - - 0.000 (0.000) - 0.149 (0.629) 0.282 0.613 (1.855) 9.640
16 - - 0.000 (0.000) - 0.145 (0.960) 0.179 0.000 (0.000) 0.000
17 - - 0.000 (0.000) 0.000 0.007 (0.109) 0.002 0.000 (0.000) 5.528
18 - - - - - - - -
Table 10: Maximum Likelihood and Anderson-Darling estimates of the form parameter d(×10 3 ) of the
Weibull (Stretched-Exponential) distribution.
53
117

118 3. Distributions exponentielles étirées contre distributions régulièrement variables
Nasdaq Dow Jones
Pos. Tail Neg. Tail Pos. Tail Neg. Tail
MLE ADE MLE ADE MLE ADE MLE ADE
1 0.441 (0.004) 0.441 0.458 (0.004) 0.451 7.012 (0.057) 7.055 7.358 (0.063) 7.135
2 0.435 (0.004) 0.431 0.454 (0.005) 0.444 6.793 (0.059) 6.701 7.292 (0.066) 6.982
3 0.431 (0.005) 0.424 0.452 (0.005) 0.438 6.731 (0.062) 6.575 7.275 (0.070) 6.890
4 0.428 (0.005) 0.416 0.453 (0.005) 0.437 6.675 (0.065) 6.444 7.358 (0.076) 6.938
5 0.429 (0.005) 0.415 0.458 (0.006) 0.443 6.607 (0.070) 6.264 7.429 (0.083) 6.941
6 0.429 (0.006) 0.411 0.464 (0.006) 0.447 6.630 (0.077) 6.186 7.529 (0.092) 6.951
7 0.436 (0.006) 0.413 0.472 (0.007) 0.453 6.750 (0.087) 6.207 7.700 (0.105) 7.005
8 0.447 (0.008) 0.421 0.483 (0.009) 0.463 6.920 (0.103) 6.199 8.071 (0.127) 7.264
9 0.462 (0.010) 0.425 0.503 (0.011) 0.482 7.513 (0.137) 6.662 8.797 (0.170) 7.908
10 0.517 (0.015) 0.468 0.529 (0.016) 0.496 8.792 (0.227) 7.745 10.205 (0.278) 9.175
11 0.540 (0.019) 0.479 0.551 (0.019) 0.514 9.349 (0.279) 8.148 10.835 (0.341) 9.751
12 0.574 (0.024) 0.489 0.570 (0.025) 0.516 10.487 (0.383) 9.265 11.796 (0.454) 10.657
13 0.615 (0.029) 0.526 0.594 (0.029) 0.537 11.017 (0.451) 9.722 12.598 (0.543) 11.581
14 0.653 (0.035) 0.543 0.627 (0.035) 0.564 11.920 (0.563) 10.626 13.349 (0.664) 12.386
15 0.750 (0.050) 0.625 0.671 (0.046) 0.594 13.251 (0.766) 12.062 14.462 (0.880) 13.521
16 0.917 (0.086) 0.741 0.760 (0.073) 0.674 15.264 (1.246) 13.943 15.294 (1.316) 13.285
17 0.991 (0.107) 0.783 0.827 (0.092) 0.744 15.766 (1.483) 14.210 17.140 (1.705) 15.327
18 1.178 (0.156) 0.978 0.857 (0.117) 0.742 16.207 (1.871) 13.697 16.883 (2.047) 13.476
Table 11: Maximum Likelihood- and Anderson-Darling estimates of the scale parameter d = 10 −3 d ′ of the
Exponential distribution.Figures within parentheses give the standard deviation of the Maximum Likelihood
estimator.
54

Nasdaq Dow Jones
Pos. Tail Neg. Tail Pos. Tail Neg. Tail
MLE ADE MLE ADE MLE ADE MLE ADE
MLE ADE MLE ADE MLE ADE MLE ADE
1 -1.03 -1.09 -1.00 -1.03 -1.12 -1.18 -0.100 -1.05
2 -1.02 -1.13 -0.934 -1.01 -1.01 -1.19 -0.862 -1.01
3 -0.931 -1.13 -0.821 -0.955 -0.921 -1.23 -0.710 -0.943
4 -0.787 -1.09 -0.701 -0.887 -0.766 -1.24 -0.594 -0.944
5 -0.655 -1.12 -0.636 -0.914 -0.458 -1.09 -0.397 -0.870
6 -0.395 -1.01 -0.518 -0.911 -0.119 -0.929 -0.118 -0.715
7 -0.142 -1.03 -0.351 -0.906 0.261 -0.763 0.251 -0.462
8 0.206 -1.09 -0.149 -0.97 0.881 -0.202 0.619 -0.160
9 0.971 -0.754 0.101 -1.35 1.31 0.127 0.930 -0.018
10 1.83 -1.04 1.17 -1.33 1.82 0.408 1.40 0.435
11 2.34 -0.441 1.45 -1.53 2.10 0.949 1.59 0.420
12 3.12 -0.445 2.52 -0.435 2.04 0.733 1.78 0.403
13 3.10 -0.444 2.63 -0.402 2.16 0.886 1.57 -0.375
14 3.35 1.43 2.89 -0.419 2.07 0.786 1.58 -0.425
15 3.27 1.57 3.36 1.35 1.82 -0.282 1.64 -2.75
16 3.30 2.97 3.80 -0.411 1.88 -0.129 3.60 -0.428
17 3.34 3.19 3.46 -0.412 2.35 -0.317 3.19 -0.433
18 2.74 2.90 4.22 -0.408 3.73 3.27 4.11 0.374
Table 12: Maximum Likelihood- and Anderson-Darling estimates of the form parameter b of the Incomplete
Gamma distribution.
55
119

120 3. Distributions exponentielles étirées contre distributions régulièrement variables
Dow Jones Positive Tail Dow Jones Negative Tail
ˆb b † Wald Test Score Test ˆb b † Wald Test Score Test
1 0.204 0.205 79.009% 78.793% 0.199 0.200 77.394% 77.166%
2 0.576 0.580 52.580% 52.300% 0.538 0.541 45.609% 45.385%
3 0.782 0.787 40.708% 40.474% 0.745 0.749 28.921% 28.792%
4 0.989 0.995 29.416% 29.251% 0.920 0.924 21.447% 21.354%
5 1.219 1.224 15.222% 15.159% 1.114 1.118 13.074% 13.027%
6 1.447 1.452 7.202% 7.181% 1.327 1.330 6.456% 6.439%
7 1.685 1.689 2.984% 2.978% 1.563 1.565 2.224% 2.221%
8 1.984 1.986 0.449% 0.448% 1.804 1.805 0.559% 0.558%
9 2.240 2.241 0.078% 0.078% 2.060 2.060 0.179% 0.179%
10 2.575 2.575 0.005% 0.005% 2.436 2.437 0.020% 0.020%
11 2.715 2.715 0.004% 0.004% 2.581 2.582 0.010% 0.010%
12 2.787 2.787 0.004% 0.004% 2.765 2.765 0.009% 0.009%
13 2.877 2.877 0.004% 0.004% 2.782 2.783 0.048% 0.048%
14 2.920 2.920 0.005% 0.005% 2.903 2.905 0.082% 0.082%
15 2.989 2.989 0.002% 0.002% 3.059 3.063 0.099% 0.099%
16 3.226 3.226 0.001% 0.001% 3.690 3.690 0.000% 0.000%
17 3.427 3.427 0.001% 0.001% 3.518 3.519 0.002% 0.002%
18 3.818 - - - 4.168 - - -
Nasdaq Positive Tail Nasdaq Negative Tail
ˆb b † Wald Test Score Test ˆb b † Wald Test Score Test
1 0.256 0.256 38.328% 38.238% 0.254 0.255 39.906% 39.812%
2 0.555 0.557 38.525% 38.352% 0.548 0.550 26.450% 26.369%
3 0.765 0.769 27.629% 27.505% 0.755 0.757 14.636% 14.601%
4 0.970 0.974 17.914% 17.844% 0.945 0.946 7.777% 7.763%
5 1.169 1.174 13.144% 13.094% 1.122 1.124 6.269% 6.258%
6 1.400 1.404 7.080% 7.060% 1.325 1.327 4.096% 4.090%
7 1.639 1.643 4.221% 4.211% 1.562 1.564 2.468% 2.464%
8 1.916 1.920 2.167% 2.162% 1.838 1.840 1.481% 1.480%
9 2.308 2.311 0.420% 0.419% 2.195 2.199 1.061% 1.059%
10 2.759 2.761 0.071% 0.071% 2.824 2.826 0.137% 0.136%
11 2.955 2.956 0.013% 0.013% 3.008 3.010 0.089% 0.089%
12 3.232 3.232 0.000% 0.000% 3.352 3.353 0.002% 0.002%
13 3.231 3.231 0.000% 0.000% 3.441 3.441 0.001% 0.001%
14 3.358 - - - 3.551 3.551 0.000% 0.000%
15 3.281 - - - 3.728 3.728 0.000% 0.000%
16 3.327 - - - 3.990 3.990 0.000% 0.000%
17 3.372 - - - 3.917 3.917 0.000% 0.000%
18 3.136 - - - 4.251 - - -
Table 13: Wald and Score encompassing Test for non-nested hypotheses. The p-value gives the significance
with which one can reject the null hypothesis: (SE) encompasses (PD). b † is the estimator of the Pareto
model under the hypothesis for the SE distribution. ˆb is the estimator under the true distribution for the
Pareto model.
56

Dow Jones positive tail Dow Jones negative tail
ˆηT ˆb p-value ˆηT ˆb p-value
1 1.044 0.204 100.000% 0.979 0.199 100.000%
2 1.123 0.576 100.000% 1.050 0.538 100.000%
3 1.229 0.782 100.000% 1.148 0.745 100.000%
4 1.351 0.989 100.000% 1.259 0.920 100.000%
5 1.493 1.219 100.000% 1.388 1.114 100.000%
6 1.653 1.447 100.000% 1.538 1.327 100.000%
7 1.835 1.685 100.000% 1.715 1.563 100.000%
8 2.069 1.984 100.000% 1.912 1.804 100.000%
9 2.294 2.240 100.000% 2.141 2.060 100.000%
10 2.605 2.575 99.997% 2.489 2.436 100.000%
11 2.732 2.715 99.165% 2.626 2.581 99.997%
12 2.809 2.787 98.487% 2.803 2.765 99.766%
13 2.895 2.877 94.723% 2.835 2.782 99.869%
14 2.943 2.920 94.010% 2.960 2.903 99.514%
15 3.027 2.989 95.032% 3.116 3.059 97.480%
16 3.261 3.226 80.157% 3.690 3.690 5.634%
17 3.447 3.427 58.002% 3.527 3.518 38.929%
18 3.818 3.818 - 4.168 4.168 -
Nasdaq positive tail Nasdaq negative tail
ˆηT ˆb p-value ˆηT ˆb p-value
1 1.019 0.256 100.000% 1.000 0.254 100.000%
2 1.118 0.555 100.000% 1.090 0.548 100.000%
3 1.225 0.765 100.000% 1.194 0.755 100.000%
4 1.347 0.970 100.000% 1.312 0.945 100.000%
5 1.486 1.169 100.000% 1.447 1.122 100.000%
6 1.650 1.400 100.000% 1.605 1.325 100.000%
7 1.840 1.639 100.000% 1.796 1.562 100.000%
8 2.069 1.916 100.000% 2.032 1.838 100.000%
9 2.393 2.308 100.000% 2.353 2.195 100.000%
10 2.799 2.759 99.994% 2.900 2.824 100.000%
11 2.974 2.955 98.132% 3.070 3.008 99.996%
12 3.232 3.232 22.570% 3.372 3.352 92.329%
13 3.232 3.231 28.945% 3.457 3.441 85.305%
14 3.358 3.358 - 3.563 3.551 69.909%
15 3.281 3.281 - 3.730 3.728 27.179%
16 3.327 3.327 - 3.991 3.990 13.117%
17 3.372 3.372 - 3.923 3.917 27.655%
18 3.136 3.136 - 4.251 4.251 -
Table 14: Test of the (SE) model against the null hypothesis that the true model is the Pareto model. The
p-value gives the significance with which one can reject the null hypothesis.
57
121

122 3. Distributions exponentielles étirées contre distributions régulièrement variables
mean absolute return |r|
x 10−4
10
9
8
7
6
5
4
3
2
0 10 20 30 40
time (by five minutes)
50 60 70 80
Figure 1: Average absolute return, as a function of time within a trading day. The U-shape characterizes the
so-called lunch effect.
58

Variation coefficient
Variation coefficient
4
3.5
3
2.5
2
1.5
1
0.5
Variation coeff.=std/mean of time intervals between pos. extremums of DJ
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Threshold u, (for extremums X > u)
4
3.5
3
2.5
2
1.5
1
0.5
Variation coeff.=std/mean of time intervals between neg. extremums, DJ
0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Threshold, u (for extremums X < −u)
Figure 2: Coefficient of variation V for the Dow Jones daily returns. An increase of V characterizes the
increase of “clustering”.
59
123

124 3. Distributions exponentielles étirées contre distributions régulièrement variables
MLE of ξ
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
MLE of GPD form parameter ξ from SE samples, n=50000
c=0.7
c=0.3
−0.2
0 2 4 6 8 10 12 14 16
Number of lower threshold h
Figure 3: Maximum Likelihood estimates of the GPD form parameter for Stretched-Exponentail samples of
size 50,000.
60

Mean excess function
Mean excess function
0.06
0.05
0.04
0.03
0.02
0.01
0.5
Mean excess functions for DJ−daily pos.(line) and neg.(pointwise)
0
0 0.05 0.1 0.15
Lower threshold, u
x 10−3
3.5
3
2.5
2
1.5
1
Mean excess functions for ND−5min pos.(line) and neg.(pointwise)
0
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01
Lower threshold, u
Figure 4: Mean excess functions for the Dow Jones daily returns (upper panel) and the Nasdaq five minutes
returns (lower panel). The plain line represents the positive returns and the dotted line the negative ones
61
125

126 3. Distributions exponentielles étirées contre distributions régulièrement variables
Complementary sample DF
Complementary sample DF
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
10 0
10 −1
10 −2
10 −3
10 −4
10 −5
10 −5
10 −6
Complementary DF of DJ−daily pos.(line),n=14949 and neg.(pointwise),n=13464
10 −4
10 −5
10 −3
10 −2
Absolute log−return, x
10 −4
10 −3
10 −1
Complementary DF of ND−5min pos.(line),n=11241 and neg.(pointwise) n=10751
Absolute log−return, x
Figure 5: Cumulative sample distributions for the Dow Jones (a) and for the Nasdaq (b) data sets.
62
10 0
10 −2

Hill‘s estimate b u
Hill‘s estimate b u
4.5
3.5
2.5
1.5
1
0.5
6
5
4
3
2
1
0
5
4
3
2
0
10 −5
10 −5
Hill‘s estimates of b u for DJ−daily pos.(line), n=14949, and neg.(pointwise),n=13464
10 −4
10 −3
Lower threshold, u
Hill‘s estimates of b u for ND−5min pos.(line),n=11241, and neg.(pointwise),n=10751
10 −4
Lower threshold, u
Figure 6: Hill estimates ˆbu as a function of the threshold u for the Dow Jones (a) and for the Nasdaq (b).
63
10 −2
10 −3
127

128 3. Distributions exponentielles étirées contre distributions régulièrement variables
b
5
4
3
2
1
ND<
ND>
DJ>
DJ<
0
0 5 10 15 20
Index n of the quantile q n
Figure 7: Hill estimator ˆbu for all four data sets (positive and negative branches of the distribution of returns
for the DJ and for the ND) as a function of the index n = 1,...,18 of the 18 quantiles or standard significance
levels q1 ...q18 given in table 6. The dashed line is expression (33) with 1 − qn = 3.08 e −0.342n given by
(32).
64

Wilks statistic (doubled log−likelihood ratio)
Wilks statistic (doubled log−likelihood ratio)
100
90
80
70
60
50
40
30
20
10
SE
IG
Wilks statistics for CD vs 4 parametric families, Npos, n=11241
PD
0 0.5 1 1.5 2 2.5 3
x 10 −3
0
Lower threshold, u
80
70
60
50
40
30
20
10
SE
ED
2
χ (.95)
2
Wilks statistics for CD vs 4 parametric families, Nneg, n=10751
ED
IG
PD
2
χ (.95)
1
0 0.5 1 1.5 2 2.5 3
x 10 −3
0
Lower threshold, u
Figure 8: Wilks statistic for the comprehensive distribution versus the four parametric distributions : Pareto
(PD), Weibull (SE), Exponential (ED) and Incomplete Gamma (IG) for the Nasdaq five minutes returns.
The upper panel refers to the positive returns and lower panel to the negative ones.
65
2
χ (.95)
2
2
χ (.95)
1
129

130 3. Distributions exponentielles étirées contre distributions régulièrement variables
Wilks statistic (doubled log−likelihood ratio)
Wilks statistic (doubled log−likelihood ratio)
80
70
60
50
40
30
20
10
SE
Wilks statistics for CD vs 4 parametric families, DJpos, n=14949
IG
PD
0
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
Lower threshold, u
80
70
60
50
40
30
20
10
SE
IG
ED
2
χ (.95)
2
Wilks statistics for CD vs 4 parametric families, DJneg, n=13464
PD
ED
2
χ (.95)
2
2
χ (.95)
1
2
χ (.95)
1
0
0 0.01 0.02 0.03
Lower threshold, u
0.04 0.05 0.06
Figure 9: Wilks statistic for the comprehensive distribution versus the four parametric distributions : Pareto
(PD), Weibull (SE), Exponential (ED) and Incomplete Gamma (IG) for the Dow Jones daily returns. The
upper panel refers to the positive returns and the lower panel to the negative ones.
66

Tail 1−F(x) and parameter b
Tail 1−F(x) and parameter b
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Sample tail 1−F(x)(thick) and "local" Pareto−b(thin), simulated Pareto−1.2,n=15000
10 20 30 40 50
x
60 70 80 90 100
Sample tail 1−F(x)(thick) and "local" Pareto−b(thin), simulated SE−0.3,n=15000
100 200 300 400 500
x
600 700 800 900 1000
Figure 10: Local index β(x) estimated for a Pareto distribution with tail index = 1.2 (upper panel) and a
Stretched exponential with exponent c = 0.3 (lower panel).
67
131

132 3. Distributions exponentielles étirées contre distributions régulièrement variables
Tail and parameter b
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Sample tail 1−F(x)(tick) and Pareto−b(thin), simulated crossover
Simulated 2 Pareto sample
b 1 =0.70; b 2 =1.5;
Lower threshold u 0 =1;
Crossover point u 1 =10;
PX>10≅0.1;
( )
20 40 60 80 100
x
120 140 160 180 200
Figure 11: Local index β(x) for a distribution constructed by joining two Pareto distributions with exponents
b1 = 0.70 and b2 = 1.5 at the cross-over point u1 = 10.
68

1.4
1.2
1
0.8
0.6
0.4
0.2
0 1 2 3 4 5 6 7
x 10 −3
0
log−return x
10 1
10 0
10 −1
10 −2
10 −5
10 1
10 0
10 −1
10 −3
y ∼ x 0.54
10 −4
10 −2
log−return x
y ∼ x 0.77
Figure 12: Upper panel: Sample tail (continuous line), local index β(x) (dashed line) and local exponent c(x)
(dash-dotted line) for the negative tail of the Nasdaq five minutes returns. Lower panel: doubled logarithmic
plot of the local index β(x). Over most of the data set, β(x) increases with the log-return x as a power law
of index 0.77 while beyond the quantile 99% (see the inset) it behaves like another power law of a smaller
index equal to 0.54. The goodness of fit with these two power laws has been checked by a χ 2 test.
69
10 −3
10 −2
133

134 3. Distributions exponentielles étirées contre distributions régulièrement variables

Chapitre 4
Relaxation de la volatilité
L’un des enjeux de la théorie financière est de comprendre comment le flux incessant d’information
arrivant sur les marchés financiers est incorporé dans le prix des actifs. Toutes les nouvelles n’ayant
pas le même impact, la question se pose de savoir si l’on peut distinguer les effets des attentats du 11
septembre 2001 ou du coup d’état contre Gorbatchev le 19 aout 1991 du crash de 1987 ou d’autre chocs
de volatilité de plus faible amplitude ? Utilisant un processus autorégressif à mémoire longue défini sur
le logarithme de la volatilité - le processus de marche aléatoire multifractale (MRW) - nous prédisons des
fonctions de réponse de la volatilité des prix totalement différentes aux grands chocs externes comparées
à celles que nous prédisons pour les chocs endogènes, i.e, qui résultent d’une accumulation constructive
(ou cohérente) d’un grand nombre de petits chocs.
Ces prédictions sont remarquablement bien confirmées empiriquement sur divers chocs de volatilité
d’amplitudes variées. Notre théorie permet de distinguer deux types d’évèments (endogènes et exogènes)
avec des signatures spécifiques et des précurseurs caractéristiques pour la classe des chocs endogènes.
Cela explique aussi l’origine de ce type de chocs par l’accumulation cohérente de petites mauvaises nouvelles,
et ainsi permet d’unifier les précédentes explications des grands krachs, incluant celui d’octobre
1987.
135

136 4. Relaxation de la volatilité

Volatility Fingerprints of Large Shocks:
Endogeneous Versus Exogeneous ∗
D. Sornette 1,2 , Y. Malevergne 1,3 and J.-F. Muzy 4
1 Laboratoire de Physique de la Matière Condensée CNRS UMR 6622
Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France
2 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science
University of California, Los Angeles, California 90095, USA
3 Institut de Science Financière et d’Assurances - Université Lyon I
43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
4 Laboratoire Systèmes Physiques de l’Environemment, CNRS UMR 6134
Université de Corse, Quartier Grossetti, 20250 Corte, France
email: sornette@unice.fr, Yannick.Malevergne@unice.fr and muzy@univ-corse.fr
fax: (310) 206 30 51
Forthcoming Risk
Abstract
Finance is about how the continuous stream of news gets incorporated into prices. But not all news
have the same impact. Can one distinguish the effects of the Sept. 11, 2001 attack or of the coup
against Gorbachev on Aug., 19, 1991 from financial crashes such as Oct. 1987 as well as smaller
volatility bursts? Using a parsimonious autoregressive process with long-range memory defined on the
logarithm of the volatility, we predict strikingly different response functions of the price volatility to great
external shocks compared to what we term endogeneous shocks, i.e., which result from the cooperative
accumulation of many small shocks. These predictions are remarkably well-confirmed empirically on
a hierarchy of volatility shocks. Our theory allows us to classify two classes of events (endogeneous
and exogeneous) with specific signatures and characteristic precursors for the endogeneous class. It also
explains the origin of endogeneous shocks as the coherent accumulations of tiny bad news, and thus
unify all previous explanations of large crashes including Oct. 1987.
1 Introduction
A market crash occurring simultaneously on most of the stock markets of the world as witnessed in Oct.
1987 would amount to the quasi-instantaneous evaporation of trillions of dollars. Market crashes are the
extreme end members of a hierarchy of market shocks, which shake stock markets repeatedly. Among
∗ We acknowledge helpful discussions and exchanges with E. Bacry and V. Pisarenko. This work was partially supported by the
James S. Mc Donnell Foundation 21st century scientist award/studying complex system.
1
137

138 4. Relaxation de la volatilité
recent events still fresh in memories are the Hong-Kong crash and the turmoil on US markets on oct. 1997,
the Russian default in Aug. 1998 and the ensuing market turbulence in western stock markets and the
collapse of the “new economy” bubble with the crash of the Nasdaq index in March 2000.
In each case, a lot of work has been carried out to unravel the origin(s) of the crash, so as to understand
its causes and develop possible remedies. However, no clear cause can usually be singled out. A case in
point is the Oct. 1987 crash, for which many explanations have been proposed but none has been widely
accepted unambiguously. These proposed causes include computer trading, derivative securities, illiquidity,
trade and budget deficits, over-inflated prices generated by speculative bubble during the earlier period, the
auction system itself, the presence or absence of limits on price movements, regulated margin requirements,
off-market and off-hours trading, the presence or absence of floor brokers, the extent of trading in the
cash market versus the forward market, the identity of traders (i.e. institutions such as banks or specialized
trading firms), the significance of transaction taxes, etc. More rigorous and systematic analyses on univariate
associations and multiple regressions of these various factors conclude that it is not at all clear what caused
the crash (Barro et al. 1989). The most precise statement, albeit somewhat self-referencial, is that the most
statistically significant explanatory variable in the October crash can be ascribed to the normal response of
each country’s stock market to a worldwide market motion (Barro et al. 1989).
In view of the stalemate reached by the approaches attempting to find a proximal cause of a market shock,
several researchers have looked for more fundamental origins and have proposed that a crash may be the climax
of an endogeneous instability associated with the (rational or irrational) imitative behavior of agents (see
for instance (Orléan 1989, Orléan 1995, Johansen and Sornette 1999, Shiller 2000)). Are there qualifying
signatures of such a mechanism? According to (Johansen and Sornette 1999, Sornette and Johansen 2001)
for which a crash is a stochastic event associated with the end of a bubble, the detection of such bubble would
provide a fingerprint. A large literature has emerged on the empirical detectability of bubbles in financial
data and in particular on rational expectation bubbles (see (Camerer 1989, Adam and Szafarz 1992) for a survey).
Unfortunately, the present evidence for speculative bubbles is fuzzy and unresolved at best, according
to the standard economic and econometric literature. Other than the still controversial (Feigenbaum 2001)
suggestion that super-exponential price acceleration (Sornette and Andersen 2002) and log-periodicity may
qualify a speculative bubble (Johansen and Sornette 1999, Sornette and Johansen 2001), there are no unambiguous
signatures that would allow one to qualify a market shock or a crash as specifically endogeneous.
On the other end, standard economic theory holds that the complex trajectory of stock market prices is the
faithful reflection of the continuous flow of news that are interpreted and digested by an army of analysts
and traders (Cutler et al. 1989). Accordingly, large shocks should result from really bad surprises. It is
a fact that exogeneous shocks exist, as epitomized by the recent events of Sept. 11, 2001 and the coup
against Gorbachev on Aug., 19, 1991, and there is no doubt about the existence of utterly exogeneous bad
news that move stock market prices and create strong bursts of volatility. However, some could argue that
precursory fingerprints of these events were known to some elites, suggesting the possibility the action of
these informed agents may have been reflected in part in stock markets prices. Even more difficult is the
classification (endogeneous versus exogeneous) of the hierarchy of volatility bursts that continuously shake
stock markets. While it is a common practice to associate the large market moves and strong bursts of
volatility with external economic, political or natural events (White 1996), there is not convincing evidence
supporting it.
Here, we provide a clear and novel signature allowing us to distinguish between an endogeneous and an
exogeneous origin to a volatility shock. Tests on the Oct. 1987 crash, on a hierarchy of volatility shocks
and on a few of the obvious exogeneous shocks validate the concept. Our theoretical framework combines
a rather novel but really powerful and parsimonious so-called multifractal random walk with conditional
probability calculations.
2

2 Long-range memory and distinction between endogeneous and exogeneous
shocks
While returns do not exhibit discernable correlations beyond a time scale of a few minutes in liquid arbitraged
markets, the historical volatility (measured as the standard deviation of price returns or more
generally as a positive power of the absolute value of centered price returns) exhibits a long-range dependence
characterized by a power law decaying two-point correlation function (Ding et al. 1993, Ding
and Granger 1996, Arneodo et al. 1998) approximately following a (t/T ) −ν decay rate with an exponent
ν ≈ 0.2. A variety of models have been proposed to account for these long-range correlations (Granger and
Ding 1996, Baillie 1996, Müller et al. 1997, Muzy et al. 2000, Muzy et al. 2001, Müller et al. 1997).
In addition, not only are returns clustered in bursts of volatility exhibiting long-range dependence, but they
also exhibit the property of multifractal scale invariance (or multifractality), according to which moments
mq ≡ 〈|rτ | q 〉 of the returns at time scale τ are found to scale as mq ∝ τ ζq , with the exponent ζq being a
non-linear function of the moment order q (Mandelbrot 1997, Muzy et al. 2000).
To make quantitative predictions, we use a flexible and parsimonious model, the so-called multifractal random
walk (MRW) (see Appendix A and (Muzy et al. 2000, Bacry et al. 2001)), which unifies these two
empirical observations by deriving naturally the multifractal scale invariance from the volatility long range
dependence.
The long-range nature of the volatility correlation function can be seen as the direct consequence of a
slow power law decay of the response function K∆(t) of the market volatility measured a time t after the
occurrence of an external perturbation of the volatility at scale ∆t. We find that the distinct difference
between exogeneous and endogeneous shocks is found in the way the volatility relaxes to its unconditional
average value.
The prediction of the MRW model (see Appendix B for the technical derivation) is that the excess volatility
Eexo[σ 2 (t) | ω0] − σ 2 (t), at scale ∆t, due to an external shock of amplitude ω0 relaxes to zero according to
the universal response
139
Eexo[σ 2 (t) | ω0] − σ2 2K0t−1/2 (t) ∝ e − 1 ≈ 1 √ , (1)
t
for not too small times, where σ 2 (t) = σ 2 ∆t is the unconditional average volatility. This prediction is
nothing but the response function K∆(t) of the MRW model to a single piece of very bad news that is
sufficient by itself to move the market significantly. This prediction is well-verified by the empirical data
shown in figure 1.
On the other hand, an “endogeneous” shock is the result of the cumulative effect of many small bad news,
each one looking relatively benign taken alone, but when taken all together collectively along the full path
of news can add up coherently due to the long-range memory of the volatility dynamics to create a large
“endogeneous” shock. This term “endogeneous” is thus not exactly adequate since prices and volatilities
are always moved by external news. The difference is that an endogeneous shock in the present sense is the
sum of the contribution of many “small” news adding up according to a specific most probable trajectory.
It is this set of small bad news prior to the large shock that not only led to it but also continues to influence
the dynamics of the volatility time series and creates an anomalously slow relaxation. Appendix C gives the
derivation of the specific relaxation (21) associated with endogeneous shocks.
Figure 2 reports empirical estimates of the conditional volatility relaxation after local maxima of the S&P100
intradaily series made of 5 minute close prices during the period from 04/08/1997 to 12/24/2001 (figure
1(a)). The original intraday squared returns have been preprocessed in order to remove the U-shaped volatility
modulation associated with the intraday variations of market activity. Figure 2(b) shows that the MRW
3

140 4. Relaxation de la volatilité
∫ dt E[σ(t) | ω 0 ] − E[σ(t)]
10 −1
10 −2
10 0
Nikkei 250 : Aug 19, 1991
S&P 500 : Aug 19, 1991
FT−SE 100 : Aug 19, 1991
CAC 40 : Sep 11, 2001
S&P 500 : Oct 19, 1987
10 1
Time (trading days)
Slope α = 1/2
August 19, 1991 : Putsh against President Gorbachev
September 11, 2001: Attack against the WTC
Figure 1: Cumulative excess volatility at scale ∆t, that is, integral over time of Eexo[σ 2 (t) | ω0] − σ 2 (t),
due to the volatility shock induced by the coup against President Gorbachev observed in three British,
Japanese and USA indices and the shock induced by the attack of September 11, 2002 against the World
Trade Center. The dashed line is the theoretical prediction obtained by integrating (1), which gives a ∝ √ t
time-dependence. The cumulative excess volatility following the crash of October 1987 is also shown with
circles. Notice that the slope of the non-constant curve for the October 1987 crash is very different from the
value 1/2 expected and observed for exogeneous shocks. This crash and the resulting volatility relaxation
can be interpreted as an endogeneous event.
4
10 2

Figure 2: Measuring the conditional volatility response exponent α(s) for S&P 100 intradaily time series
as a function of the endogeneous shock amplitude parameterized by s, defined by (19). (a) The original
5 minute intradaily time series from 04/08/1997 to 12/24/2001. The 5 minute de-seasonalized squared
returns are aggregated in order to estimate the 40 minutes and daily volatilities. (b) 40 minute log-volatility
covariance C40(τ) as a function of the logarithm of the lag τ. The MRW theoretical curve with λ 2 = 0.018
and T = 1 year (dashed line) provides an excellent fit of the data up to lags of one month. (c) Conditional
volatility response ln(Eendo[σ 2 (t) | s]) as a function of ln(t) for three shocks with amplitudes given by
s = −1, 0, 1. (d) Estimated exponent α(s) for ∆t = 40 minutes (•) as a function of s. The solid line
is the prediction corresponding to Eq. (22). The dashed line corresponds to the empirical MRW estimate
obtained by averaging over 500 Monte-Carlo trials. It fits more accurately for negative s (volatility lower
than normal) due to the fact that the estimations of the variance by aggregation over smaller scales is very
noisy for small variance values. The error bars give the 95 % confidence intervals estimated by Monte-Carlo
trials of the MRW process. In the inset, α(s) is compared for ∆t = 40 minutes (•) and ∆t = 1 day (×).
5
141

142 4. Relaxation de la volatilité
model provides a very good fit of the empirical volatility covariance in a range of time scales from 5 minutes
to one month. Fig. 2(c) plots in a double logarithmic representation, for the time scale ∆t = 40 minutes,
the estimated conditional volatility responses for s = 1, 0, −1, where the endogeneous shocks are parameterized
by e 2s σ 2 (t). A value s > 0 (resp. s < 0) corresponds to a positive bump (resp. negative dip) of the
volatility above (resp. below) the average level σ 2 (t). The straight lines are the predictions (Eqs. (24,22))
of the MRW model and qualify power law responses whose exponents α(s) are continuous function of the
shock amplitude s. Figure 1(d) plots the conditional response exponent α(s) as a function of s for the two
time scales ∆t = 40 minutes and ∆t = 1 day (inset). For ∆t = 40 minutes, we observe that α varies
between −0.2 for the largest positive shocks to +0.2 for the largest negative shocks, in excellent agreement
with MRW estimates (dashed line) and, for α ≥ 0, with Eq. (22) obtained without any adjustable parameters.
1 The error bars represent the 95 % confidence intervals estimated using 500 trials of synthetic MRW
with the same parameters as observed for the S&P 100 series. By comparing α(s) for different ∆t (inset),
we can see the the MRW model is thus able to recover not only the s-dependence of the exponent α(s) of
the conditional response function to endogeneous shocks but also its time scale ∆t variations: this exponent
increases as one goes from fine to coarse scales. Similar results are obtained for other intradaily time series
(Nasdaq, FX-rates, etc.). We also obtain the same results for 17 years of daily return times series of various
indices (French, German, canada, Japan, etc.).
In summary, the most remarkable result is the qualitatively different functional dependence of the response
(1) to an exogeneous compared to the response (24,22) to an endogeneous shock. The former gives a decay
of the burst of volatility ∝ 1/t 1/2 compared to 1/t α(s) for endogeneous shocks with amplitude e 2s σ 2 (t),
with an exponent α(s) being a linear function of s.
3 Discussion
What is the source of endogeneous shocks characterized by the response function (21)? Appendix D and
equation (29) predict that the expected path of the continuous information flow prior to the endogeneous
shock grows proportionally to the response function K(tc − t) measured in backward time to the shock
occuring at tc. In other words, conditioned on the observation of a large endogeneous shock, there is
specific set of trajectories of the news flow that led to it. This specific flow has an expectation given by (29).
This result allows us to understand the distinctive features of an endogeneous shock compared to an external
shock. The later is a single piece of very bad news that is sufficient by itself to move the market significantly
according to (1). In contrast, an “endogeneous” shock is the result of the cumulative effect of many small
bad news, each one looking relatively benign taken alone, but when taken all together collectively along
the full path of news can add up coherently due to the long-range memory of the log-volatility dynamics to
create a large “endogeneous” shock. This term “endogeneous” is thus not exactly adequate since prices and
volatilities are always moved by external news. The difference is that an endogeneous shock in the present
sense is the sum of the contribution of many “small” news adding up according to a specific most probable
trajectory. It is this set of small bad news prior to the large shock that not only led to it but also continues to
influence the dynamics of the volatility time series and creates the anomalously slow relaxation (21).
In this respect, this result allows us to rationalize and unify the many explanations proposed to account for
the Oct. 1987 crash: according to the present theory, each of the explanations is insufficient to explain the
crash; however, our theory suggests that it is the cumulative effect of many such effects that led to the crash.
In a sense, the different commentators and analysts were all right in attributing the origin of the Oct. 1987
crash to many different factors but they missed the main point that the crash was the extreme response of
1 The deviation of α(s) from expression (22) for negative s, originates from the error in the volatility estimation using a sum of
squared returns. The smaller the sum of squared returns, the larger the error is. As ∆t increases, this error becomes negligible
6

the system to the accumulation of many tiny bad news contributions. To test this idea, we note that the
decay of the volatility response after the Oct. 1987 crash has been described by a power law 1/t 0.3 (Lillo
and Mantegna 2001), which is in line with the prediction of our MRW theory with equation (22) for such
a large shock (see also figure 2 panel d). This value of the exponent is still significantly smaller than 0.5.
Figure 1 demonstrates further the difference between the relaxation of the volatility after this event shown
with circle and those following the exogenous coup against Gorbachev and the September 11 attack. There
is clearly a strong constrast which qualifies the Oct. 1987 crash as endogeneous, in the sense of our theory
of “conditional response.” This provides an independent confirmation of the concept advanced before in
(Johansen and Sornette 1999, Sornette and Johansen 2001).
It is also interesting to compare the prediction (21) with those obtained with a linear autoregressive model
of the type (5), in which ω(t) is replaced by σ(t). FIGARCH models fall in this general class. It is easy
to show in this case that this linear (in volatility) model predicts the same exponent for the response of the
volatility to endogeneous shocks, independently of their magnitude. This prediction is in stark constrast
with the prediction (21) of the log-volatility MRW model. The later model is thus strongly validated by our
empirical tests.
Appendix A: The Multifractal Randow Walk (MRW) model
The multifractal random walk model is the continuous time limit of a stochastic volatility model where
log-volatility 2 correlations decay logarithmically. It possesses a nice “stability” property related to its scale
invariance property: For each time scale ∆t ≤ T , the returns at scale ∆t, r∆t(t) ≡ ln[p(t)/p(t − ∆t)], can
be described as a stochastic volatility model:
143
r∆t(t) = ɛ(t) · σ∆t(t) = ɛ(t) · e ω∆t(t) , (2)
where ɛ(t) is a standardized Gaussian white noise independent of ω∆t(t) and ω∆t(t) is a nearly Gaussian
process with mean and covariance:
µ∆t = 1
2 ln(σ2∆t) − C∆t(0) (3)
C∆t(τ) = Cov[ω∆t(t), ω∆t(t + τ)] = λ 2
T
ln
|τ| + e−3/2
. (4)
∆t
σ 2 ∆t is the return variance at scale ∆t and T represents an “integral” (correlation) time scale. Such logarithmic
decay of log-volatility covariance at different time scales has been demonstrated empirically in
(Arneodo et al. 1998, Muzy et al. 2000). Typical values for T and λ 2 are respectively 1 year and 0.02.
According to the MRW model, the volatility correlation exponent ν is related to λ 2 by ν = 4λ 2 .
The MRW model can be expressed in a more familiar form, in which the log-volatility ω∆t(t) obeys an
auto-regressive equation whose solution reads
t
ω∆t(t) = µ∆t + dτ η(τ) K∆t(t − τ) , (5)
−∞
where η(t) denotes a standardized Gaussian white noise and the memory kernel K∆t(·) is a causal function,
ensuring that the system is not anticipative. The process η(t) can be seen as the information flow. Thus ω(t)
represents the response of the market to incoming information up to the date t. At time t, the distribution
2 The log-volatilty is the natural quantity used in canonical stochatic volatility models (see (Kim et al. 1998, and references
therein)).
7

144 4. Relaxation de la volatilité
of ω∆t(t) is Gaussian with mean µ∆t and variance V∆t = ∞
0 dτ K2 ∆t (τ) = λ2 ln
which entirely specifies the random process, is given by
∞
C∆t(τ) =
0
T e 3/2
∆t
. Its covariance,
dt K∆t(t)K∆t(t + |τ|) . (6)
Performing a Fourier tranform, we obtain ˆ K∆t(f) 2 = Ĉ∆t(f) = 2λ2 f −1
T f
0
which shows that for τ small enough
K∆t(τ) ∼ K0
λ 2 T
τ
sin(t)
t dt + O (f∆t ln(f∆t)) ,
for ∆t

Appendix C: “Conditional response” to an endogeneous shock
Let us consider the natural evolution of the system, without any large external shock, which nevertheless
exhibits a large volatility burst ω(t = 0) = ω0 at t = 0. From the definition (2) with (5) and (6), it is
clear that a large “endogeneous” shock requires a special set of realization of the “small news” {η(t)}. To
quantifies the response in such case, we can evaluate Eendo[σ 2 (t) | ω0] = Eendo[e 2ω(t) | ω0]. Since ω(t) is a
Gaussian process, the new process ω(t) conditional on ω0 remains Gaussian, so that
145
Eendo[σ 2 (t) | ω0] = Eendo[e 2ω(t) | ω0] (13)
= exp (2E[w(t) | ω0] + 2Var[ω(t) | ω0]) . (14)
Due to the still Gaussian nature of the condition log-volatility ω(t), we easily obtain using (3) and (4),
and
Let us set:
Eendo[ω(t) | ω0] = E[ω(t)] +
Cov[ω(t), ω0]
Var[ω0]
· (ω0 − E[ω0]) , (15)
= (ω0 − µ) · C(t)
+ µ , (16)
C(0)
Cov[ω(t), ω0] 2
Varendo[ω(t) | ω0] = Var[ω(t)] − , (17)
Var[ω0]
= C(0) 1 − C2 (t)
C2
. (18)
(0)
e 2ω0 = e 2s σ 2 (t) ⇒ ω0 − µ = s + C(0) (19)
By subsitution in (14), we obtain thanks to (3) and (4),
Eendo[σ 2 (t) | ω0] = σ2
(t) exp 2(ω0 − µ) · C(t)
C(0) − 2C2 =
(t)
,
C(0)
σ
(20)
2 α(s)+β(t) T
(t)
t
(21)
where
α(s) =
ln(
2s
, (22)
T e3/2
∆t )
β(t) = 2λ 2 ln(t/∆t)
ln(T e3/2 . (23)
/∆t)
Within the range ∆t < t

146 4. Relaxation de la volatilité
where
It is thus easy to obtain the estimate:
β ′ (t) = 2λ 2 ln(T/t)
ln(T e3/2 /∆t)
Varendo [σ2 (t) | ω0] ≤ Eendo[σ 2
(t) | ω0] ( T
∆t )2β(t) 1/2
− 1 6λ2 ln(te3/2 /∆t) Eendo[σ 2 (t) | ω0]
We thus conclude, that, for s large enough (i.e., α(s) large enough):
Eendo[σ 2 (t) | ω0] − σ 2 (t)
Varendo [σ 2 (t) | ω0]
1
6λ 2 ln(te 3/2 /∆t)
Over the first decade ∆t ≤ t ≤ 10∆t, the deviation of the conditional mean volatility from the unconditional
volatility σ 2 (t) is greater than the conditional variance, which ensures the existence of a strong deterministic
component of the conditional response above the stochastic components.
Expressions (24,22) are our two main predictions. These equations predict that the conditional response
function Eendo[σ2 (t) | ω0] of the volatility decays as a power law ∼ 1/tα
of the time since the endogeneous
shock, with an exponent α ≈ 2ω0 − ln(σ2
λ2 (t))
C(0) which depends linearly upon the amplitude ω0 of the
shock. Note in particular, that α changes sign: it is positive for w0 > 1
2 ln(σ2 (t)) and negative otherwise.
Appendix D: Determination of the sources of endogeneous shocks
What is the source of endogeneous shocks characterized by the response function (21)? To answer, let
us consider the process W (t) ≡ t
−∞ dτ η(τ), where η(t) is a standardized Gaussian white noise which
captures the information flow impacting on the volatility, as defined in (5). Extending the property (16), we
find that
t
Cov[W (t), ω0]
Eendo[W (t) | ω0] = · (ω0 − E[ω0]) ∝ (ω0 − E[ω0]) dτ K(−τ) . (29)
Var[ω0]
−∞
Expression (29) predicts that the expected path of the continuous information flow prior to the endogeneous
shock (i.e., for t < 0) grows like ∆W (t) = η(t)∆t ∼ K(−t)∆t ∼ ∆t/ √ −t for t < 0 upon the approach
to the time t = 0 of the large endogeneous shock. In other words, conditioned on the observation of a large
endogeneous shock, there is specific set of trajectories of the news flow η(t) that led to it. These conditional
news flows have an expectation given by (29).
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12

Chapitre 5
Approche comportementale des marchés
financiers : l’apport des modèles d’agents
Depuis quelques années, les sciences économiques et financières abandonnent progressivement le modèle
standard de l’agent économique représentatif dont on sait qu’il soutient notamment l’hypothèse d’efficience
des marchés, et ce dans le but de prendre en considération le fait que les marchés financiers
reflètent les actions et les émotions d’êtres réels. Ainsi s’est développé ce qu’il convient désormais d’appeler
l’économie ou la finance cognitive (voir par exemple Thaller (1993), Sheffrin (2000), Shleiffer
(2000) ou encore Goldberg et von Nitzsch (2001)). Le but de cette nouvelle approche est d’intégrer
la rationalité nécessairement limitée des agents, qui peuvent certes tenter au mieux de leurs capacités
d’être rationnels et d’agir de manière optimale, il n’en reste pas moins qu’ils seront toujours sujet à
des émotions ou des influences extérieures qu’ils ne contrôlent pas. Ainsi, il est normal que les dimensions
psychologique, sociologique et cognitive au sens large ayant une influence sur l’agent économique
soient prises en compte. Pour autant et en dépit de cette rationalité limitée des agents, les marchés sont le
plus souvent très proches de l’efficience. Ainsi il convient de se demander comment l’auto-organisation
des agents permet d’atteindre cette efficience et pourquoi / comment le système s’en écarte parfois de
manière spectaculaire.
C’est en fait tout l’enjeu de l’étude des modèles d’agents en interaction que de tenter d’apporter des
réponses à ces questions. L’approche théorique nouvelle offerte par ces modèles permet d’intégrer beaucoup
des limitations des agents réels et à partir de la description - même très sommaire - du comportement
des acteurs agissant sur les marchés, il est possible d’en déduire certains modes de fonctionnement. Cela
permet notamment de mieux comprendre comment, partant d’agents individuels non rationnels au sens
strict, peut émerger un comportement collectif conforme aux modèles rationnels, puis dans un deuxième
temps, d’étudier les déviations possibles et les relaxations vers l’état d’efficience du marché prédit par
ces modèles rationnels. Réciproquement, l’adéquation ou non des prévisions de ces modèles aux faits
stylisés, que nous avons décrits au chapitre 1, permet si ce n’est d’accepter ou de rejeter définitivement la
pertinence de certains types de comportements des agents vis-à-vis de la génération de l’organisation des
marchés, du moins de cerner les mécanismes les plus importants quant à la description des faits stylisés.
De plus, à la différence notoire de la physique et des sciences naturelles en général, il est quasiment
impossible de se livrer à des expériences sur les marchés financiers afin de tester directement telle ou
telle hypothèse ou l’impact d’une nouvelle réglementation. Certes, quelques simulations de marché ont
été conduites en laboratoire (Smith 1994, Smith 1998), mais le nombre restreint d’acteurs qui y prennent
part et le fait qu’ils sachent qu’ils participent à une expérience, en limite quelque peu la validité. Donc,
le développement de modèles de marchés réalistes est particulièrement intéressant dans la mesure où
149

150 5. Approche comportementale des marchés financiers : l’apport des modèles d’agents
ils pourraient servir de substitut de laboratoire. En fait cette approche est d’ores et déjà en train de se
développer, tant sur le plan académique dans le but d’étudier l’effet de l’introduction de la taxe Tobin
ou l’influence des ordres limites sur la volatilité du marché (Westerhoff 2001, Westerhoff 2002, par
exemple), que sur le plan pratique pour tester divers scénarii et mettre au point certaines stratégies.
Comme nous venons de le dire, l’émergence des modèles d’agents en interaction est en grande partie
due à l’échec des modèles standards dont les principales conséquences sont les propriétés d’efficience
des marchés et le fait que les prix des actifs financiers suivent des marches aléatoires. Le défaut majeur,
aujourd’hui bien établi, des modèles économiques standards est de considérer un agent dit représentatif,
ce qui revient à faire une approximation de “champ moyen”, alors qu’il apparaît très clairement que
l’hétérogénéité est un concept clé de la compréhension des marchés financiers et que malgrè cette
hétérogénéité, une auto-organisation voit le jour.
Il existe désormais un grand nombre de modèles d’agents permettant de rendre compte de tels ou tels
faits stylisés. Notre objectif n’est pas de les décrire un par un de manière exhaustive, ce qui n’aurait
d’ailleurs pas grand intérêt. Nous préférons nous attacher à présenter quelques grandes classes de
modèles d’agents parmi lesquelles nous distinguons d’une part les modèles d’opinion et les modèles
de marchés et d’autre part les modèles à agents adaptatifs et non adaptatifs. D’autres catégories peuvent
évidemment être établies, conduisant notamment à l’importante distinction entre modèles à hétérogénéité
forte ou faible, mais nous n’irons pas aussi loin, d’autant que nous souhaitons mettre en exergue les
quelques mécanismes récurrents se retrouvant dans la plupart des modèles.
Avant d’entrer dans le vif du sujet, il convient de spécifier un dernier point, à savoir, étant donnée une
population d’agents économiques s’échangeant un actif financier, comment se forme le prix de cet actif.
Un grand nombre de modèles continuent de supporter l’idée que la formation des prix sur les marchés
financiers peut résulter d’un équilibre entre offre et demande. Ce sont les modèles dits d’équilibres rationnels
adaptatifs, parmi lesquels on peut citer Brock et LeBaron (1996), Brock et Hommes (1997) ou
Grandmont (1998) par exemple. Cependant, au vu de résultats empiriques de plus en plus nombreux
(Brown, Walsh et Yuen 1997, Chordia, Roll et Subrahmanyam 2002), il parait extrêmement difficile de
continuer à admettre cette hypothèse et il devient nécessaire de reconnaître que les prix observés sur les
marchés financiers se forment en dehors de tout équilibre. Donc, avant de nous intéresser aux différentes
classes de modèles d’agents, nous allons montrer comment définir un pris hors équilibre.
5.1 Prix d’un actif et excès de demande
L’un des principaux enjeux de tout modèle d’agents est de rendre compte et d’expliquer l’évolution du
prix des actifs sur les marchés financiers. Dans le modèle du tâtonnement Walrasien, les agents signalent
le prix auquel ils sont prêts à acheter ou vendre, et après ajustement, la transaction est effectuée lorsqu’un
prix d’équilibre est atteint (Samuelson 1941, par exemple). Si ce type d’approche semblait convenir à la
modélisation des marchés financiers au début du siècle lorsque les prix étaient fixés une fois par jour
et non en continu, elle parait désormais très loin de refléter le fonctionnement des places financières
modernes, où l’on peut penser que les prix se forment en dehors de tout équilibre. Il est donc nécessaire de
savoir comment relier l’évolution des prix, hors de l’équilibre, à la répartition des agents entre acheteurs
et vendeurs. Une simple application de la loi de l’offre et de la demande nous indique que les prix doivent
augmenter lorsque le nombre d’acheteurs est supérieur au nombre de vendeurs et réciproquement, celuici
doit diminuer lorsque les vendeurs sont en plus grand nombre que les acheteurs.
Donc, il apparaît qu’une quantité naturelle dont dépend l’évolution des prix est l’excès de demande, que
nous noterons ED dans la suite, et qui représente la différence entre le nombre d’acheteurs et le nombre

5.1. Prix d’un actif et excès de demande 151
de vendeurs. Plus précisément, l’excès de demande est la différence entre la demande des acheteurs et
l’offre des vendeurs. Ainsi, la variation de prix dP (t), entre les instants t et t+dt, est fonction de l’excès
de demande ED(t), mais aussi a priori du prix lui-même P (t) :
dP (t)
dt
= f(P (t), ED(t)), (5.1)
où il suffit de choisir f(P (t), ED(t)) = P (t) · g(ED(t)), avec g(·) croissante et g(0) = 0 pour assurer
la positivité du prix P (t) à tout instant :
On peut noter que la quantité 1
P (t)
1
P (t)
dP (t)
dt
associé au prix P (t), si bien que r(t) est égal à g(ED(t)).
= g(ED(t)). (5.2)
· dP (t)
dt n’est en fait rien d’autre que le rendement instantané r(t)
Pour aller plus avant, il est nécessaire de spécifier un peu plus l’expression de la fonction g(·). En l’absence
de toute autre information, l’approche la plus simple consiste à remarquer que pour les faibles
excès de demande, un développement au premier ordre de g(·), pourvu que g ′ (0) existe et soit non nulle,
est déjà intéressant. En effet, il vient alors
1
P (t)
dP (t)
dt
ED(t)
= , (5.3)
λ
où l’on a noté g ′ (0) = λ −1 . Ce facteur λ mesure la liquidité ou profondeur du marché (Hausman, Lo et
MacKinlay 1992, Kempf et Korn 1999). Il représente l’excès de demande nécessaire pour faire bouger
le rendement instantané d’une unité et mesure donc la sensibilité du prix aux fluctuations de l’excès de
demande. La plupart des modèles que nous rencontrerons dans la suite utilisent cette représentation.
Farmer (1998) a montré que cette expression qui n’apparaît ici que comme une approximation (linéaire)
est en fait la relation exacte entre le prix et l’excès de demande si l’on admet que les ordres traités
simultanément le sont au même prix. Ceci est réaliste pour des marchés où le carnet d’ordre est traité en
continu, comme sur EURONEXT par exemple, mais devient peu crédible lorsque les ordres sont traités
de manière décentralisée par différents market-makers comme sur le NYSE.
En conclusion, la relation (5.3), même si elle admet certaines justifications théoriques, n’est généralement
qu’approximative, il est donc important de se demander quel est son domaine de validité. Les études
empiriques de Campbell et al. (1997), Chan et Fong (2000) ou Plerou, Gopikrishnan, Gabaix et Stanley
(2001) ont montré que lorsque l’excès de demande devient trop important, des non-linéarités - traduisant
un phénomène de saturation - deviennent appréciables. Ainsi, selon Plerou et al. (2001) la fonction g(·)
peut être raisonnablement approchée par la fonction tangente hyperbolique. Notons, que dans ce cas,
l’approximation (5.3) reste valable pour les excès de demande modérés. Ceci n’est plus vrai dans la
représentation de Zhang (1999) où la fonction g(·) est la fonction racine carrée, qui n’admet pas de
dérivée en zéro. Cette représentation en terme de racine carrée apparaît lorsque l’on admet qu’un ordre
peut ne pas être exécuté instantanément, mais requière d’autant plus de temps que sa taille est importante.
Son intérêt a récemment été confirmé par l’étude empirique de Lillo, Farmer et Mantegna (2002).
Cette dernière remarque met en lumière l’influence du temps d’exécution d’un ordre sur l’évolution du
prix, ce qui constitue en fait une autre des limitations de (5.3) (et de 5.1). En effet, il est incorrect de
considérer que l’évolution du prix à l’instant t ne dépend que de l’excès de demande au même instant :
Chordia et al. (2002) notamment ont montré que les grandes fluctuations de prix étaient aussi affectées
par les excès de demande passés. Plus précisément ceci semble là encore dépendre de l’organisation du
marché, puisque selon Brown et al. (1997) cet effet est plus important pour les marchés organisés autour

152 5. Approche comportementale des marchés financiers : l’apport des modèles d’agents
de market-makers comme sur le NYSE que pour les marchés organisés autour d’un carnet d’ordres
gérés informatiquement comme sur EURONEXT ou sur le marché australien ASX.
Malgré ces quelques limitations, la modélisation (5.3) est la plus souvent retenue du fait de sa simplicité,
ce qui est pleinement justifiable lorsque cette simplification permet d’obtenir des solutions analytiques
pour le modèle considéré. Malheureusement, la plupart des modèles d’agents que l’on rencontre dans
la littérature doivent, du fait de leur complexité, être étudiés numériquement. Du coup, cette approche
simplificatrice, n’a de raison d’être que dans la mesure où elle facilite l’interprétation et la compréhension
des résultats numériques en permettant de réduire l’ensemble des paramètres du système à son minimum.
5.2 Modèles d’opinion contre modèles de marché
Outre la modélisation du prix des actifs financiers, un des objectifs de l’étude des marchés en terme de
modèle d’agents est de s’intéresser à la dynamique de répartition de ces agents en différents groupes :
acheteurs/vendeurs, fondamentalistes/chartistes ou encore imitateurs/antagonistes C’est pour cela, qu’une
fois fixées les règles de formation des groupes, l’étude des modèles d’agents reprend les principes directeurs
de l’étude de n’importe quel système dynamique.
Un point important dans la distinction entre les différents types de modèles d’agents est de savoir si la
variable prix est endogène au modèle ou exogène. Par exogène, nous entendons que le prix est fixé par
l’excès de demande et donc par la composition des différents groupes d’agents, mais qu’il n’influe pas sur
la formation de ces groupes. En ce sens, de tels modèles d’agents ne sont rien d’autres que des modèles
d’opinion, où le déséquilibre entre les différents groupes est traduit par un prix, à l’aide de l’équation
(5.3). Au contraire, lorsque le prix devient une variable endogène, la formation des différents groupes
d’agents dépend de celui-ci, notamment par l’intermédiaire de règles de transition entre groupes. Il y
a donc, d’une part une action de la composition des groupes sur le prix (via 5.3), et d’autre part une
rétro-action du prix sur la composition des groupes (via les règles de formation/transition). On est alors
réellement en présence d’un modèle de marché.
Parmi les nombreux modèles d’opinion ayant vu le jour, le modèle de Cont et Bouchaud (2000) présente
l’avantage d’être d’une grande généralité. En effet, il ne spécifie nullement la nature des interactions
entre les agents - celle-ci étant extrêmement difficile à déterminer - et va même jusqu’à tirer profit de
cette indétermination en reprenant l’idée défendue par Kirman (1983) selon laquelle la communication
entre agents économiques peut être représentée comme un réseau de connexions aléatoires. Cont et Bouchaud
(2000) formalisent alors le problème de sorte qu’il apparaisse exactement comme un problème de
percolation, phénomène critique bien connu des mathématiciens et physiciens.
Plus précisément, Cont et Bouchaud (2000) considèrent un ensemble de N agents choisissant d’être
acheteurs, vendeurs ou de ne pas prendre position sur le marché, avec des probabilités respectivement
égales à a, a et 1−2a, et a ∈]0, 1/2[. Ces agents créent des coalitions au sein desquelles chaque individu
possède la même opinion, ce qui peut rendre compte par exemple de l’existence et de l’évolution de
groupes d’investisseurs associés au sein de fonds mutuels. Les coalitions se forment de manière aléatoire
par appariement entre agents. Ainsi, on peut considérer que les agents sont situés sur les nœuds d’un
graphe, et que deux agents en relation sont liés par une connexion entre les nœuds qu’ils occupent.
Un groupe d’opinion est alors représenté par un amas de nœuds reliés entres eux par des connexions
binaires, et la distribution de taille de ces amas peut être obtenue à l’aide de résultats standards en théorie
des graphes.
En supposant que la probabilité que deux agents soient appariés est p = c/N, ce qui garantit que le
nombre moyen d’agents avec lequel un agent donné est relié reste fini (est égal à c) quand N tend

5.2. Modèles d’opinion contre modèles de marché 153
vers l’infini, les résultats de Erdös et Renyi (1960) permettent de montrer que la distribution de taille des
coalitions est une loi de puissance d’exposant 3/2 lorsque c = 1, et que cette loi de puissance est tronquée
exponentiellement lorsque c < 1. Ainsi, lorsque le système se trouve au seuil de percolation c = 1, ce
modèle permet de rendre compte de la décroissance en loi de puissance observée pour la distribution
des rendements des actifs financiers. Cependant, l’accord n’est que qualitatif, puisque l’indice de queue
généralement observé est plutôt de l’ordre de 3 − 4, alors que ce modèle prédit un exposant égal à 3/2.
Une des critiques élevée contre ce modèle est qu’a priori, le système n’a aucune raison de se trouver
au seuil c = 1. C’est pour cela que Stauffer et Sornette (1999) ont proposé que le paramètre c puisse
varier, ce qui permet de rendre compte de l’évolution des opinions au cours du temps. Moyennant cela,
la distribution des rendements est une loi de puissance dont l’exposant peut être ajusté pour satisfaire les
observations expérimentales. De plus, cette version dynamique du modèle de Cont et Bouchaud (2000)
est compatible avec la lente décroissante de la corrélation de la volatilité des rendements.
Ce modèle est clairement un modèle d’opinion puisque les décisions des agents sont indépendantes du
prix de l’actif qu’ils s’échangent. En particulier, le fait que les agents choisissent de placer un ordre sur
le marché ou non ne dépend pas du prix du marché, ce qui est assez peu réaliste. Cela montre comment
rendre de façon simple le prix endogène dans ce type de modèle, permettant ainsi de d’obtenir un réel
modèle de marché. Il suffit pour cela de rendre les probabilités d’entrer ou non dans le marché et d’être
acheteur ou vendeur dépendantes du prix de l’actif. A notre connaissance cela n’a pas encore été fait
pour ce type de modèle.
Cette méthode d’endogénéisation du prix est généralement celle suivie dans tous les modèles de marché.
Ainsi, dans les modèles proposés par Kirman (1991), Brock (1993), Lux (1995) ou encore Farmer (1998)
parmi bien d’autres, il existe deux classes d’agents : certains sont supposés fondamentalistes et prennent
donc position en fonction de l’écart entre le prix du marché et un prix fondamental, tandis que d’autres
sont chartistes et suivent la tendance qu’ils essaient de détecter dans les cours. Le seuil de déclenchement
du placement d’ordre est donc fonction du prix du marché.
Dans le même esprit, mais à un niveau macroscopique cette fois, Bouchaud et Cont (1998) - dans une
version linéaire - puis Ide et Sornette (2002) - dans une généralisation non linéaire - ont repris cette
approche afin de décrire non plus l’excès de demande individuelle mais l’excès de demande agrégée des
agents, ce qui leur a permis - dans l’approche non linéaire - de rendre compte de la croissance superexponentielle
des bulles spéculatives ainsi que des oscillations log-périodiques précédant les krachs.
Il convient de noter que dans tous ces modèles que nous venons de citer, les fondamentalistes jouent le
rôle d’une force de rappel qui tend à ramener le prix de marché vers le prix fondamental, alors que les
chartistes tendent à l’en écarter, et ce d’autant plus vite que le prix de marché augmente rapidement.
Ceci donne alors naissance à une succession de phases calmes, où les fondamentalistes sont en majorité,
alternant avec des phases de forte volatilité, où les chartistes dominent le marché.
Bien d’autres exemples de modèles, tel celui de Brock et Hommes (1997) ou celui de Lux (1998), puis
Lux et Marchesi (1999), reposent sur ces mêmes idées, mais dans tous les cas, l’endogénéisation du prix
s’effectue de la même manière : les seuils de déclenchement des actions des agents sur le marché sont
rendus dépendants du prix, ce qui est à la fois la manière la plus simple et la plus naturelle de concevoir
la chose.

154 5. Approche comportementale des marchés financiers : l’apport des modèles d’agents
5.3 Modèles à agents adaptatifs contre modèles à agents non adaptatifs
Parmi les modèles que nous venons de présenter, certains sont dits à agents non adaptatifs et d’autres à
agents adaptatifs. Pour illustrer cette distinction, considérons par exemple le modèle de Farmer (1998).
Les agents y sont soit fondamentalistes soit chartistes et le nombre d’agents appartenant à chacune de
ces catégories, fixé au départ, n’évolue pas au cours du temps. Ainsi, les agents choisissent certes d’être
acheteur ou vendeur selon le prix du marché (le modèle de Farmer (1998) est un modèle de marché
et non un simple modèle d’opinion), mais ils ne peuvent pas adapter leur stratégie - fondamentaliste
ou chartiste- aux modifications de leur environnement. Le modèle de Farmer (1998) est donc qualifié de
modèle à agents non adaptatifs. Le modèle de Bouchaud et Cont (1998) et son extension proposée par Ide
et Sornette (2002), de même que le modèle de Lévy, Lévy et Solomon (1995), par exemple, appartiennent
eux aussi à cette catégorie.
Au contraire, le modèle de Lux (1998) ou Lux et Marchesi (1999), fait partie des modèles à agents adaptatifs.
En effet, à la différence du modèle de Farmer (1998), les différents agents communiquent entre
eux et comparent les résultats des différentes stratégies afin de se déterminer. Un agent peut donc passer
d’une stratégie chartiste à fondamentaliste (ou vice-versa) s’il a l’espoir que cela lui permette de réaliser
un gain plus important. Il est donc sensible à son environnement et tend notamment à imiter les agents
avec lesquels il rentre en contact si cela lui est profitable. Ces changements de stratégie permettent de
rendre compte des bouffées de volatilité caractéristiques des séries financières. En effet, lorsque les fondamentalistes
sont majoritaires, le marché est calme et les cours ne présentent que des fluctuations de
faibles amplitudes autour de leur prix fondamental. Au contraire, dès que les chartistes sont en nombre
suffisant, une tendance peut apparaître, puis s’amplifier, conduisant alors à une augmentation de la volatilité,
qui ne retombera que lorsque l’enthousiasme des chartistes aura disparu et que les fondamentalistes
reprendront le dessus. Ainsi, ce modèle permet d’interpréter en terme microstructurel pourquoi les séries
financières présentent des bouffées de volatilité, et justifie l’idée selon laquelle la dynamique des marchés
financiers présente des changements de régime (Susmel 1996).
Le modèle de Cont et Bouchaud (2000) dans la version généralisée de Stauffer et Sornette (1999) peut
lui aussi être considéré, à la limite, comme un modèle à agents adaptatifs. En effet, même si le processus
d’adaptation n’est pas clairement précisé et est spécifié de manière totalement ad hoc, les agents peuvent
changer de coalition au cours du temps, marquant ainsi leur adhésion à telle ou telle opinion et conduisant
à une évolution de l’opinion globale du système.
Dans les deux modèles précédents, les agents n’ont le choix qu’entre un petit nombre de stratégies,
l’hétérogénéité de ces modèles est donc faible. C’est pourquoi il est important de mentionner deux autres
types de modèles à agents adaptatifs, complètement différents de tous ceux évoqués jusqu’ici, à savoir
les jeux de minorité et le modèle d’agents “génétiques” développé au Santa-Fe Intitut (Palmer, Arthur,
Holland, LeBaron et Taylor 1994, Arthur, Holland, LeBaron, Palmer et Taylor 1997, LeBaron, Arthur et
Palmer 1999). Dans ce dernier modèle, les agents sont représentés par des algorithmes génétiques, petits
programmes informatiques imitant le processus de sélection naturelle des gènes en compétition pour leur
survie et leur développement. Ainsi, ces agents, doués de facultés d’adaptation très poussées, forment
des prédictions sur les prix futurs et placent des ordres sur le marché en fonction de leurs prévisions. Les
règles de sélection et d’évolution des agents sont ensuite déterminées par leur performance, de sorte qu’à
chaque génération, les agents les moins performants disparaissent au profit des plus performants. Il est
alors observé que dans certaines conditions, ces agents apprennent à coopérer et de leurs actions individuelles
émerge un équilibre rationnel dynamique conforme à ce qui est prévu par la théorie économique.
Le modèle de Farmer (1998) que nous avons évoqué plus haut constitue en fait une version simplifiée de
cette approche, qui exploite l’analogie entre le marché et un écosystème où se développe une compétition
entre diverses stratégies favorisant ainsi l’émergence de nouvelles méthodes tirant parti des inefficacités

5.4. Conséquences des phénomènes d’imitation et d’antagonisme... 155
des anciennes qu’elles finissent par remplacer.
Les modèles de minorité, introduits par Challet et Zhang (1997), procède d’une toute autre approche.
Considèrons N agents en compétition pour une ressource limitée et dont les choix se résument à deux
alternatives mutuellement exclusives, que l’on pourra noter 0 ou 1. Chaque agent effectue un choix, et les
gagnants sont ceux qui sont en minorité (on peut reconnaître ici le célèbre problème du bar d’El Farol introduit
par Arthur (1994)). Un tel modèle est pertinent dans un contexte financier, car un agent qui décide
de vendre, par exemple, ne va empocher un bénéfice que si la majorité des autres agents sont acheteurs,
de sorte que l’excès de demande soit positif et que le prix augmente (voir Challet, Chessa, Marsili et
Zhang (2001) pour une discussion de l’application des jeux de minorité aux marchés financiers).
Concrètement, le jeu est répété un grand nombre de fois, si bien que chaque agent dispose de la même
information : la succession des décisions minoritaires, soit une séquence de 0 et de 1. En supposant que
les agents ne disposent que d’une mémoire de taille limitée m, tous les agents connaissent donc les m
derniers choix gagnants. Une séquence de longueur m représente donc l’histoire des événements et il y
a évidemment 2m histoires possibles. Chaque agent établit des stratégies visant à prédire l’état futur du
système en se fondant sur l’histoire passée. Il existe donc en tout 22m stratégies. Tous les agents possèdent
un même nombre de stratégies, ces stratégies étant différentes d’un agent à l’autre. Cette approche permet
de rendre compte de l’extraordinaire diversité des acteurs économiques présents sur les marchés
réels. En effet, ceux-ci ont des objectifs différents, aussi bien en terme de gains espérés que d’horizon
d’investissements mais aussi d’aversion au risque et présentent donc des stratégies extrêmement variées.
Les résultats fournis par ce type de modèles sont satisfaisants dans la mesure où ils permettent de rendre
compte des distributions de rendement en loi de puissance et de la lente décroissance de la corrélation de
la volatilité lorsque le rapport m/N est voisin d’une certaine valeur critique. Cependant, comme relevé
récemment par Andersen et Sornette (2002b) et Giardina et Bouchaud (2002), le principe de minorité
n’est pas toujours le plus réaliste. En effet, un agent a intérêt à se trouver dans la minorité lorsqu’un
renversement de tendance est observé sur les prix, mais lorsque la tendance se poursuit, il faut être dans
la majorité pour profiter de celle-ci. Dans le cadre de ce modèle, la trajectoire de la richesse des agents
est notablement différente de celle observée dans les jeux de minorités standards, mais ce type de modèle
rend tout aussi bien compte des distributions de rendement à queue épaisse que de la corrélation à longue
portée de la volatilité.
5.4 Conséquences des phénomènes d’imitation et d’antagonisme sur la
forme de la trajectoire des prix d’un actif.
Tous les modèles microscopiques que nous avons décrits jusqu’à présent permettent de rendre compte
de la distribution des rendements ainsi que de la corrélation à longue portée de la volatilité, qui sont les
faits stylisés les plus marquants observés au sujet des rentabilités boursières. Pour autant, aucun de ces
modèles n’est capable d’expliquer la croissance super-exponentielle des prix que l’on observe lorsqu’une
bulle spéculative se développe.
Afin de pouvoir modéliser ce phénomène, il convient de comprendre ce que recouvre une croissance
super-exponentielle du prix d’un actif. Lorsque le rendement d’un actif est constant, son prix augmente
naturellement de façon exponentielle, puisque l’actualisation est un processus multiplicatif. Une croissance
super-exponentielle traduit donc le fait que le rendement de l’actif considéré augmente au fur et
à mesure que son prix lui-même augmente. Un mécanisme possible est le suivant : lorsqu’une tendance
haussière suffisamment forte apparaît, le nombre d’agents acheteurs augmente, attiré par la promesse de
gains importants, ce qui en retour intensifie la tendance du fait de la forte pression acheteuse, validant du

156 5. Approche comportementale des marchés financiers : l’apport des modèles d’agents
même coup la croyance en un gain important, et attire en conséquence toujours plus d’acheteurs.
Nous pouvons donc conclure qu’un effet d’aubaine ou un effet de foule semble tout à fait à même de justifier
la croissance super-exponentielle des prix observés en présence de bulles spéculatives. Ces effets de
foules ont été observés sur les marchés spéculatifs (Arthur 1987, Orléan 1992, Shiller 2000, notamment)
et traduisent le phénomène d’imitation entre agents économiques. Il est à noter que ce phénomène d’imitation
est parfaitement rationnel, comme le souligne Orléan (1989), et découle simplement des règles
d’apprentissage bayésien.
Si l’on s’en tient au comportement mimétique et auto-référentiel exposé ci-dessus, la bulle spéculative
n’a aucune raison de prendre fin. Il est donc raisonnable de penser qu’à un certain moment, les agents
réalisent que les prix sont devenus trop éloignés des fondamentaux de l’économie ou que les liquidités
vont venir à manquer et donc choisissent de vendre pour engranger leurs bénéfices tant que l’afflux
d’acheteurs demeure suffisamment important pour pouvoir absorber leurs ordres de vente sans que les
cours ne chutent. Ainsi, durant cette phase, certains agents choisissent de prendre le marché à contrecourant,
et présentent un comportement antagoniste par rapport à l’opinion majoritaire à ce moment
là. On peut noter que dans l’esprit, ceci est assez proche du modèle de Andersen et Sornette (2002b)
évoqué plus haut, et il apparaît clairement pourquoi le principe de minorité ne saurait parvenir à décrire
complètement le fonctionnement des marchés financiers.
Cette approche heuristique permet de comprendre que les comportements mimétiques et antagonistes
des agents peuvent suffire à expliquer la croissance super-exponentielle des bulles spéculatives, ce qui
nous a conduit à développer le modèle simple exposé dans Corcos, Eckmann, Malaspinas, Malevergne et
Sornette (2002). Dans ce modèle, nous considérons un marché où s’échange un seul actif et un ensemble
de N agents pouvant être soit acheteurs soit vendeurs. Chacun interroge m de ses voisins et peut choisir
de suivre l’opinion majoritaire qu’il a relevée ou bien d’en prendre le contre-pied. Par exemple, un agent
acheteur changera d’avis et deviendra vendeur si la proportion d’agents vendeurs parmi les m agents
qu’il a interrogés dépasse un certain seuil ρhb, ce qui correspond à un comportement mimétique, ou si la
proportion d’agents acheteurs est jugée trop importante et dépasse un autre seuil ρhh, ce qui correspond
à un comportement antagoniste, tandis qu’il conserve son opinion si aucun de ces seuils n’est dépassé.
Nous montrons alors que, comme attendu, le prix présente des bulles spéculatives dont la croissance est
d’abord exponentielle, puis accélère et tend à présenter une singularité en temps fini. En fait, à cause
du mécanisme de ré-injection lié à l’existence de comportements antagonistes, la bulle éclate avant que
la divergence ait lieu. Ainsi donc ce modèle met en évidence l’importance des phénomènes d’imitation
et d’antagonisme, qui à eux seuls suffisent à expliquer la croissance super-exponentielle des bulles
spéculatives suivies par des krachs. Par ailleurs, ce modèle rend aussi compte, de manière qualitative,
des distributions de rendements à queues épaisses. Cependant, il conduit à des rendements beaucoup
trop corrélés pour être réalistes.
Ce défaut est dû au caractère faiblement stochastique de notre modèle et peut être facilement corrigé.
En fait, d’après notre classification, ce modèle n’est pas un modèle de marché mais un simple modèle
d’opinion. Il convient donc d’en endogénéiser le prix, ce qui est réalisable par une généralisation simple
du modèle précédent. Pour cela, nous pouvons considérer que les N agents ne sont pas tous en position
sur le marché à un instant donné. Plus précisément, chaque agent estime la valeur du prix fondamental
de l’actif et décide d’entrer sur le marché si (et seulement si) l’écart entre le prix de marché de cet actif
et son fondamental dépasse un certain seuil. Etant entré sur le marché, l’agent interroge m agents euxmêmes
sur le marché et peut alors se comporter comme un fondamentaliste s’il ne note aucune majorité
forte permettant de soutenir une tendance ou au contraire, peut choisir de suivre cette tendance s’il pense
qu’elle est viable et donc qu’il peut en tirer profit.
Par rapport au modèle originel, cette généralisation apporte un couplage entre le volume et la volatilité,

5.5. Conclusion 157
ce qui induit une beaucoup plus grande variabilité dans la dynamique des prix. Nous espérons donc que
ceci permettra de guérir les carences observées sur le modèle d’origine.
5.5 Conclusion
Les modèles d’agents en interaction se sont développés pour palier aux insuffisances de l’approche standard
en terme d’agent économique représentatif, qui n’est rien d’autre qu’une approximation de champ
moyen mettant l’accent sur l’équilibre. Cette nouvelle façon d’aborder la modélisation des marchés financiers
semble prometteuse et à notre avis présente un double intérêt.
D’une part, elle offre un moyen de cerner les comportements minimaux des agents nécessaires à la
restitution des faits stylisés des séries financières. En cela, elle permet d’aborder l’aspect comportemental
des marchés financiers, aspect qui nous semble essentiel à la détermination de certaines composantes du
risque associé à l’activité financière. En effet, l’un des apports des modèles d’agents en interaction est de
pouvoir traiter cette dimension comportementale non plus du seul point de vue qualitatif mais également
de manière quantitative.
D’autre part, une fois un modèle crédible établi, il peut permettre de réaliser des expériences, visant par
exemple à étudier l’influence de l’évolution de l’environnement réglementaire, mais aussi de simuler
divers scénarii et donc apporter de nouveaux moyens de gestion des risques.

158 5. Approche comportementale des marchés financiers : l’apport des modèles d’agents

Chapitre 6
Comportements mimétiques et
antagonistes : bulles hyperboliques, krachs
et chaos
Les comportements mimétiques et antagonistes sont deux attitudes opposées typiques des investisseurs
sur les marchés financiers. Nous introduisons un modèle simple nous permettant d’étudier le rôle de
chacun de ces comportements sur un marché financier où les agents ne peuvent choisir qu’entre deux
alternatives : être acheteur ou vendeur. Chaque agent acheteur (resp. vendeur) interroge m “amis” et
change d’opinion pour devenir vendeur (resp. acheteur) si
– au moins m · ρhb (resp. m · ρbh) parmi les m agents consultés sont vendeurs (resp. acheteurs),
– ou si au moins m · ρhh > m · ρhb (resp. m · ρbb > m · ρbh) parmi les m agents consultés sont acheteurs
(resp. vendeurs).
Ces conditions correspondent respectivement à des comportements mimétique et antagoniste.
Dans la limite où le nombre N d’agents est infini, la dynamique de la fraction d’agents acheteurs est
déterministe et présente un comportement chaotique dans un domaine significatif de l’espace des paramètres
{ρhb, ρbh, ρhh, ρbb, m}. Une trajectoire chaotique typique est caractérisée par des phases intermittentes
de chaos, de comportement quasi-périodique et de développement super-exponentiel de bulles
ponctuées par des krachs. Une bulle commence par croître initialement à un rythme exponentiel puis
accélère jusqu’à conduire à une singularité en temps fini. Le mécanisme de réinjection dû à la présence
d’agents antagonistes introduit un effet de taille finie, prévenant cette singularité et conduisant au chaos.
Nous documentons les principaux faits stylisés de ce modèle dans les cas symétrique et asymétrique. Il
est à noter que ce modèle est l’un des rares modèles d’agents donnant naissance à une dynamique non
triviale dans la limite “thermodynamique” - c’est-à-dire lorsque le nombre N d’agents tend vers l’infini.
Nous discutons aussi du cas où le nombre d’agents est fini, ce qui introduit une source endogène de bruit
qui se superpose à la dynamique chaotique.
Reprint from : A. Corcos, J.-P. Eckmann, A. Malaspinas, Y. Malevergne et D. Sornette (2002), “Imitation
and contrarian behavior : hyperbolic bubbles, crashes and chaos”, Quantitative Finance 2, 264-281.
Erratum : page 164, l’équation Q = prob({x < m(1 − ρbh)} ∪ {x > mρbb}) doit être remplacée par
Q = prob({x < m(1 − ρbb)} ∪ {x > mρbh}).
159

160 6. Comportements mimétiques et antagonistes : bulles hyperboliques, krachs et chaos

RESEARCH PAPER Q UANTITATIVE F INANCE V OLUME 2 (2002) 264–281
quant.iop.org I NSTITUTE OF P HYSICS P UBLISHING
Imitation and contrarian behaviour:
hyperbolic bubbles, crashes and chaos
A Corcos 1 , J-P Eckmann 2,3 , A Malaspinas 2 , Y Malevergne 4,5
and D Sornette 4,6
1 CRIISEA, Université de Picardie, BP 2716, 80027 Amiens, France
2 Dépt. de Physique Théorique, Université deGenève, CH-1211 Genève 4,
Switzerland
3 Section de Mathématiques, Université deGenève, CH-1211 Genève 4,
Switzerland
4 Laboratoire de Physique de la Matière Condensée, CNRS UMR6622 and
Université de Nice-Sophia Antipolis, BP 71, Parc Valrose,
06108 Nice Cedex 2, France
5 Institut de Science Financière et d’Assurances—Université Lyon I, 43,
Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
6 Institute of Geophysics and Planetary Physics and Department of Earth and
Space Science, University of California, Los Angeles, CA 90095, USA
Received 25 September 2001, in final form 26 November 2001
Published 2 August 2002
Online at stacks.iop.org/Quant/2/264
Abstract
Imitative and contrarian behaviours are the two typical opposite attitudes of
investors in stock markets. We introduce a simple model to investigate their
interplay in a stock market where agents can take only two states, bullish or
bearish. Each bullish (bearish) agent polls m ‘friends’ and changes her
opinion to bearish (bullish) if (i) at least mρhb (mρbh) among the m agents
inspected are bearish (bullish) or (ii) at least mρhh >mρhb (mρbb >mρbh)
among the m agents inspected are bullish (bearish). The condition (i) ((ii))
corresponds to imitative (antagonistic) behaviour. In the limit where the
number N of agents is infinite, the dynamics of the fraction of bullish agents
is deterministic and exhibits chaotic behaviour in a significant domain of the
parameter space {ρhb,ρbh,ρhh,ρbb,m}. A typical chaotic trajectory is
characterized by intermittent phases of chaos, quasi-periodic behaviour and
super-exponentially growing bubbles followed by crashes. A typical bubble
starts initially by growing at an exponential rate and then crosses over to a
nonlinear power-law growth rate leading to a finite-time singularity. The
reinjection mechanism provided by the contrarian behaviour introduces a
finite-size effect, rounding off these singularities and leads to chaos. We
document the main stylized facts of this model in the symmetric and
asymmetric cases. This model is one of the rare agent-based models that give
rise to interesting non-periodic complex dynamics in the ‘thermodynamic’
limit (of an infinite number N of agents). We also discuss the case of a finite
number of agents, which introduces an endogenous source of noise
superimposed on the chaotic dynamics.
264 1469-7688/02/040264+18$30.00 © 2002 IOP Publishing Ltd PII: S1469-7688(02)29456-7
161

162 6. Comportements mimétiques et antagonistes : bulles hyperboliques, krachs et chaos
Q UANTITATIVE F INANCE Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos
‘Human behaviour is a main factor in how markets
act. Indeed, sometimes markets act quickly, violently
with little warning. [...] Ultimately, history tells us
that there will be a correction of some significant dimension.
I have no doubt that, human nature being
what it is, that it is going to happen again and again.’
Alan Greenspan, Chairman of the Federal Reserve
of the USA, before the Committee on Banking and
Financial Services, US House of Representatives, 24
July 1998.
1. Introduction
In recent economic and finance research, there is a growing
interest in incorporating ideas from social sciences to account
for the fact that markets reflect the thoughts, emotions and
actions of real people as opposed to the idealized economic
investor whose behaviour underlies the efficient market and
random walk hypothesis. This was captured by the now
famous pronouncement of Keynes (1936) that most investors’
decisions ‘can only be taken as a result of animal spirits—of
a spontaneous urge to action rather than inaction, and not the
outcome of a weighed average of benefits multiplied by the
quantitative probabilities’. A real investor may intend to be
rational and may try to optimize his actions, but that rationality
tends to be hampered by cognitive biases, emotional quirks
and social influences. ‘Behavioural finance’ is a growing
research field (Thaler 1993, De Bondt and Thaler 1995, Shefrin
2000, Shleifer 2000, Goldberg and von Nitzsch 2001), which
uses psychology, sociology and other behavioural theories
to attempt to explain the behaviour of investors and money
managers. The behaviour of financial markets is thought to
result from varying attitudes towards risk, the heterogeneity in
the framing of information, from cognitive errors, self-control
and lack thereof, from regret in financial decision-making and
from the influence of mass psychology. Assumptions about the
frailty of human rationality and the acceptance of such drives
as fear and greed are underlying the recipes developed over
decades by so-called technical analysts.
There is growing empirical evidence for the existence
of herd or ‘crowd’ behaviour in speculative markets (Arthur
1987, Bikhchandani et al 1992, Johansen et al 1999, 2000,
Orléan 1986, 1990, 1992, Shiller 1984, 2000, Topol 1991,
West 1988). Herd behaviour is often said to occur when
many people take the same action, because some mimic the
actions of others. Herding has been linked to many economic
activities, such as investment recommendations (Graham and
Dodd 1934, Scharfstein and Stein 1990), price behaviour of
IPOs (Initial Public Offering) (Welch 1992) fads and customs
(Bikhchandani et al 1992), earnings forecasts (Trueman 1994),
corporate conservatism (Zwiebel 1995) and delegated portfolio
management (Maug and Naik 1995).
Here, we introduce arguably the simplest model capturing
the interplay between mimetic and contrarian behaviour in
a population of N agents taking only two possible states,
‘bullish’ or ‘bearish’ (buying or selling). In the limit of
an infinite number N → ∞ of agents, the key variable,
which is the fraction p of bullish agents, follows a chaotic
deterministic dynamics on a subspace of positive measure in
the parameter space. Before explaining and analysing the
model in subsequent sections, we compare it in three respects
to standard theories of economic behaviour.
1. Since in the limit N →∞, the model operates on
a purely deterministic basis, it actually challenges the purely
external and unpredictable origin of market prices. Our model
exploits the continuous mimicry of financial markets to show
that the disordered and random aspect of the time series of
prices can be in part explained not only by the advent of
‘random’ news and events, but can also be generated by the
behaviour of the agents fixing the prices.
In the limit N → ∞, the dynamics of prices in our
model is deterministic and derives from the theory of chaotic
dynamical systems, which have the feature of exhibiting
endogenously perturbed motion. After the first papers on
the theory of chaotic systems, such as Lorenz (1963), May
(1976), (see, e.g. Collet and Eckmann (1980) for an early
exposition), a series of economic papers dealt with models
mostly of growth (Benhabib and Day 1981, Day 1982, 1983,
Stutzer 1980). Later, a vast and varied number of fields of
economics were analysed in the light of the theory of chaos
(Grandmont 1985, 1987, Grandmont and Malgrange 1986).
They extend from macro-economics—business cycles, models
of class struggles, political economy—to micro-economics—
models with overlapping generations, optimizing behaviour—
and touch subjects such as game theory and the theory
of finance. The applicability of these theories has been
thoroughly tested on the stock market prices (Brock 1988,
Brock et al 1987, 1991, Brock and Dechert 1988, LeBaron
1988, Hsieh 1989, Scheinkman and LeBaron 1989a, 1989b)
in studies which tried to detect signs of nonlinear effects and
to nail down the deterministic nature of these prices. While
the theoretical models (Van Der Ploeg 1986, De Grauwe and
Vansanten 1990, De Grauwe et al 1993) seem to agree on
the relevance of chaotic deterministic dynamics, the empirical
studies (Eckmann et al 1988, Hsieh and LeBaron 1988, Hsieh
1989, 1991, 1992, LeBaron 1988, Scheinkman and LeBaron
1989a, 1989b) are less clear-cut, mostly because of lack of
sufficiently long time series (Eckmann and Ruelle 1992) or
because the deterministic component of market behaviour is
necessarily overshadowed by the inevitable external effects.
An additional source of ‘noise’ is found to result from the
finiteness of the number N of agents. For finite N, the
deterministically chaotic dynamics of the price is replaced by a
stochastic dynamics shadowing the corresponding trajectories
obtained for N →∞.
The model presented here shows a mechanism of price
fixing—decisions to buy or sell dictated by comparison with
other agents—which is at the origin of an instability of
prices. From one period to the next, and in the absence of
information other than the anticipations of other agents, prices
can continuously exhibit erratic behaviour and never stabilize,
without diverging. Thus, the model questions the fundamental
hypothesis that equilibrium prices have to converge to the
intrinsic value of an asset.
2. We can also consider our model in the context of the
excess market volatility of financial markets. The volatility of
265

A Corcos et al Q UANTITATIVE F INANCE
prices generated by our chaotic model could give the beginning
of an explanation of the excess volatility observed on financial
markets (Grossman and Shiller 1981, Fama 1965, Flavin 1983,
Shiller 1981, West 1988) which traditional models, such as
ARCH, try to incorporate (Engle 1982, Bollerslev et al 1991,
Bollerslev 1987).
3. Finally, we can see speculative bubbles in our model
as a natural consequence of mimetism. We can compare
this to the two basic trends in explaining the problem of
bubbles. The first makes reference to rational anticipations
(Muth 1961) and rests on the hypothesis of efficient markets.
With fixed information, and knowing the dynamics of prices,
the recurrence relation for the price is seen to depend on the
fundamental value and a self-referential component, which
tends to cause a deviation from the fundamental value: this
is a speculative bubble (Blanchard and Watson 1982). This
theory of rational speculative bubbles fails to explain the birth
of such events, and even less their collapse, which it does
not predict either. Recent developments improve on these
traditional approaches by combining the rational agents in
the economy with irrational ‘noise’ traders (Johansen et al
1999, 2000, Sornette and Johansen 2001). These noise traders
are imitative investors who reside on an interaction network.
Neighbours of an agent on this network can be viewed as the
agent’s friends or contacts, and an agent will incorporate his
neighbours’ views regarding the stock into his own view. These
noise traders are responsible for triggering crashes. Sornette
and Andersen (2001) develop a similar model in which the
noise traders induce a nonlinear positive feedback in the stock
price dynamics with an interplay between nonlinearity and
multiplicative noise. The derived hyperbolic stochastic finitetime
singularity formula transforms a Gaussian white noise
into a rich time series possessing all the stylized facts of
empirical prices, as well as accelerated speculative bubbles
preceding crashes.
The second trend purports to explain speculative bubbles
by a limitation of rationality (Shiller 1984, 2000, West 1988,
Topol 1991). It allows us to incorporate notions which the
neo-classical analysis does not take into account: asymmetry
of information, inefficiency of prices, heterogeneity of
anticipations (Grossman 1977, Grossman and Stiglitz 1980,
Grossman 1981, Radner 1972, 1979). In our approach, which
follows the second trend, the agents act without knowing the
actual effect of their behaviour: this contrasts with the position
of a model-builder (Orléan 1986, 1989, 1990, 1992). This, in
turn, can lead to prices which disconnect from the fundamental
indicators of economics.
In the present paper we show that self-referred behaviour
in financial markets can generate chaos and speculative
bubbles. They will be seen to be caused by mimetic behaviour:
bubbles will form due to imitative behaviour and collapse when
certain agents believe in the advent of a turn of trend, while
they observe the behaviour of their peers.
Our work can also be seen as a dynamical generalization
of Galam and Moscovici (1991) and Galam (1997) who
have introduced the idea of a universal behaviour in group
decision making, independently of the nature of the decision, in
situations where two opposite choices are proposed. Using the
266
163
formalism of the Ising model with the condition of minimal
conflict specifying the interactions between members of the
group, Galam and Moscovici (1991) and Galam (1997) have
established a static phase diagram of possible behaviours.
Local pressure and anticipation have been taken into account
by a local random field and a mean field respectively, in the
context of the Ising formalism.
Section 2 defines the model. Section 3 provides
a qualitative understanding and analysis of its dynamical
properties. Section 4 extends it with a quantitative analysis
of the phases of speculative bubbles in the symmetric case.
Section 5 describes the statistical properties of the price returns
derived from its dynamics in the symmetric case. Section 6
discusses the asymmetric case. Section 7 explores some effects
introduced by the finiteness N

164 6. Comportements mimétiques et antagonistes : bulles hyperboliques, krachs et chaos
Q UANTITATIVE F INANCE Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos
value investors (Ide and Sornette 2001, Sornette and Ide 2001).
We build on this insight and construct a very simple model
of price dynamics, which puts emphasis on the fundamental
nonlinear behaviour of both classes of agents.
These well-known principles generate different kinds of
risks between which agents choose by arbitrage. The former
is a competing risk (Keynes 1936, Orléan 1989) which leads
agents to imitate the collective point of view since the market
price includes it. Thus, it is assumed that Keynes’ animal
spirits may exist. More simply, there is the risk of mistaken
expectation: agents believe in a price different from the market
price. Keynes uses his famous beauty contest as a parable
for stock markets. In order to predict the winner of a beauty
contest, objective beauty is not very important, but knowledge
or prediction of others’ prediction of beauty is. In Keynes’
view, the optimal strategy is not to pick those faces the player
thinks the prettiest, but those the other players are likely to
think the average opinion will be, or those the other players
will think the others will think the average opinion will be, or
even further along this iterative loop.
On the other hand, in the latter case, the emerging
price is not necessarily in harmony with economic reality
and fundamental value. Self-referred decisions and selfvalidation
phenomena can then indeed lead to speculative
bubbles or sunspots (i.e. external random events) (Azariadis
1981, Azariadis and Guesnerie 1982, Blanchard and Watson
1982, Jevons 1871, Kreps 1977). Thus, the latter risk is
the result of precaution. It addresses the fitting of market
price to fundamental value and, by extension, collapse of the
speculative bubble.
Both attitudes are likely to be important and are integrated
in decision rules. Agents realize an arbitrage between the two
kinds of risk we have described. That is why they have both a
mimetic behaviour and an antagonistic one: they either follow
the collective point of view or they have reversed expectations.
We are now going to put these assumptions into the
simplest possible mathematical form. We assume that, at
any given time t, the population is divided into two parts.
Agents are explicitly differentiated as being bullish or bearish
in proportions pt and qt = 1 − pt respectively. The first ones
expect an increase of the price, while the bearish ones expect
a decrease. The agents then form their opinion for time t +1
by sampling the expectations of m other agents at time t, and
modifying their own expectations accordingly. The number m
of agents polled by a given agent to form her opinion at time
t + 1 is the first important parameter in our model.
We then introduce threshold densities ρhb and ρhh. We
assume 0 ρhb ρhh 1. A bullish agent will change
opinion if at least one of the following propositions is true.
(1) At least m · ρhb among the m agents inspected are bearish.
(2) At least m · ρhh among the m agents inspected are bullish.
The first case corresponds to ‘following the crowd’, while
the second case corresponds to the ‘antagonistic behaviour’.
The quantity ρhb is thus the threshold for a bullish agent
(‘haussier’) to become bearish (‘baissier’) for mimetic reasons,
and similarly, ρhh is the threshold for a bullish agent to
become bearish because there are ‘too many’ bullish agents.
One reason for this behaviour is, as we said, that the deviation
of the market price from the fundamental value is felt to be
unsustainable. Another reason is that if many managers tell
you that they are bullish, it is probable that they have large
‘long’ positions in the market: they therefore tell you to buy,
hoping to be able to unfold in part their position in favourable
conditions with a good profit.
The deviation of the threshold ρhb above the symmetric
value 1/2 is a measure of the ‘stubbornness’ (or ‘buy-and-hold’
tendency) of the agent to keep her position. For ρhb = 1/2,
the agent strictly endorses without delay the opinion of the
majority and believes in any weak trend. This corresponds to a
reversible dynamics. A value ρhb > 1/2 expresses a tendency
towards conservatism: a large ρhb means that the agent will
rarely change opinion. She is risk-adverse and would like to
see an almost unanimity appearing before changing her mind.
Her future behaviour has thus a strong memory of her past
position. ρhb − 1/2 can be called the bullish ‘buy-and-hold’
index.
The deviation of the threshold ρhh below 1 quantifies the
strength of disbelief of the agent in the sustainability of a
speculative trend. For ρhh = 1, she always follows the crowd
and is never contrarian. For ρhh close to 1/2, she has little faith
in trend-following strategies and is closer to a fundamentalist,
expecting the price to revert rapidly to its fundamental value.
1 − ρhh can be called the bullish reversal index.
Putting the above rules into mathematical equations we
see that the probability P for an agent who is bullish at time t
to change his opinion at time t + 1 is:
P = prob({x mρhh}), (1)
where x is the number of bullish agents found in the sample of
m agents.
In an entirely similar way, we introduce thresholds ρbh,
and ρbb. The thresholds ρbh and ρbb have completely
symmetric roles when the agent is initially bearish. ρbh − 1/2
can be called the bearish ‘buy-and-hold’ index. 1 − ρbb can
be called the bearish reversal index. The probability Q for a
bearish agent at time t to become bullish at time t + 1 is:
Q = prob({x mρbb}).
We can combine these two rules into a dynamical law
governing the time evolution of the populations. Denoting pt
the proportion of bullish agents in the population at time t,we
can find the new proportion, pt+1, at time t + 1, by taking into
account those agents which have changed opinion according to
the deterministic law given above. To simplify notation, we let
pt+1 = p ′ and pt = p. Then, the above statements are easily
used to express p ′ in terms of p, by using the probability of
finding j bullish people among m (Corcos 1993):
p ′ = p − p
m
p
j
m−j (1 − p) j
+ (1 − p)
jm·ρhb
or j

A Corcos et al Q UANTITATIVE F INANCE
Figure 1. The family of functions Fρ,m(p) for ρhb = ρbh = 0.72
and ρhh = ρbb = 0.85. The curves are for
m = 13 + j · 26,j = 0,...,13. Note the convergence to the
function Gρ, (indicated by m =∞).
where ρ ={ρhb,ρbh,ρhh,ρbb}. Thus, the function Fρ,m(p)
completely characterizes the dynamics of the proportion of
bullish and bearish populations.
3. Qualitative analysis of the dynamical
properties
3.1. The limit m →∞
The law given by equation (2) is not easy to analyse, and we
give in figure1afewsample curves Fρ,m. We see that as m gets
larger, the curves seem to tend to a limiting curve. Using this
observation, our conceptual understanding of the dynamics can
be drastically simplified if we consider the problem for a large
number m of polled partners. Indeed, it is most convenient
to first study the unrealistic problem m =∞and to view the
large m case as a perturbation of this limiting case. The main
ingredient in the study of the case m =∞is the Law of Large
Numbers, which we use in a form given in Feller (1966):
Lemma. Let g be a continuous function on [0, 1]. Then, for
p ∈ [0, 1],
m
m
lim p
m→∞ j
j (1 − p) m−j g(j/m) = g(p). (3)
j=0
We apply this lemma to the (piecewise continuous)
function g = fh, where fh is the indicator function of the
set defining P :
1, if x ρhb or x

166 6. Comportements mimétiques et antagonistes : bulles hyperboliques, krachs et chaos
Q UANTITATIVE F INANCE Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos
p t
p t
p t
p t
1.0
0.5
0.0
1.0
0.5
0.0
1.0
0.5
0.0
1.0
0.5
0.0
r bb = 0.75
r bb = 0.76
r bb = 0.77
r bb = 0.85
Figure 3. The time series for the same parameter values as in
figure 2. Note that, for ρhh = ρbb = 0.75, one has convergence to a
bullish equilibrium, for 0.76 a bullish period 2, for 0.77 a bullish, but
chaotic behaviour. The most interesting case is ρhh = ρbb = 0.85,
where calm periods alternate in a seemingly random fashion with
speculative bubbles ending in crashes and anti-crashes.
(2) The next, more interesting case, is the appearance of a limit
cycle (of period 2): at successive times, the population
of bullish and bearish agents will oscillate between two
different values. This happens, e.g. for ρhh = ρbb = 0.76,
with the other parameters as before (see second panel of
figure 3).
(3) But for certain values of the parameters, e.g. ρhh = ρbb =
0.85, the sequence of values of pt is a chaotic sequence,
with positive Lyapunov exponent (compare with Eckmann
and Ruelle 1985). The mechanism for this is really a
combination of sufficiently strong buy-and-hold index
ρhb − 1/2 and of sufficiently weak reversal index 1 − ρhh.
This regime thus occurs when the opinion of a trader has
a strong memory of her past positions and changes it only
when a strong majority appears. This regime also requires
a weak belief of the agent in fundamental valuation, as
she will believe until very late that a strong bullish or
bearish speculative trend is sustainable. Fundamentally, it
is this self-referential behaviour of the anticipations alone
which is responsible for a deterministic, but seemingly
erratic evolution of the population of bullish and bearish
agents. No external noise is needed to make this happen,
and in general, we view external stimuli as acting on top of
the intrinsic mechanism which we exhibit here (Eckmann
1981). Note that the set of parameter values ρ for which
chaos is expected (say, near the values used at the bottom
of figure 3) has positive Lebesgue measure.
t
t
t
t
What should be a reasonable value for ρhh = ρbb to
describe empirical data? We believe that this is an illposed
question and, conditioned on the use of the present
very simplified model, ρhh = ρbb is probably not constant
in time but is a (possibly slowly varying) function of time.
As a consequence, the market dynamics may explore all the
possible different regimes described above: stability of price,
oscillations or cycles, chaotic behaviour (of course each of
them being decorated by noise capturing a finite N as discussed
below as well as other external sources of uncertainties not
captured by the model). The idea of a time-varying control
parameter ρhh = ρbb is similar to that advocated in Stauffer and
Sornette (1999) to cure the percolation model of stock market
prices (Cont and Bouchaud 2000), which needed to be tuned
very artificially to close to the critical percolation threshold
in order to exhibit interesting reasonable statistics. Here, we
shall not explore further this possibility but will instead go on
investigating the properties of the model with fixed parameters.
We next consider in more detail the time evolution of
pt for the parameter values of the last frame of figure 3,
which are typical for the abundant set of ‘chaotic’ parameter
values, and we will show how the time evolution exhibits
‘speculative bubbles’. This phenomenon is akin to the
notion of intermittency (of ‘type I’) as known to physicists,
see e.g. Manneville (1991) for an exposition. Indeed, we
can distinguish two distinct behaviours in the last frame of
figure 3, which occur repeatedly with more or less pronounced
separation. The first process is the ‘laminar phase’, which
is seen to occur when the population pt is near 0.5. Then,
the evolution of the population is slow, and the population
grows slowly away from 0.5, either monotonically or through
an oscillation of period 2, depending on ρ. This motion is
slower when the inspected sample size (m) is larger, reflecting
a more stable evolution for less independent agents. When
the distance from 0.5 is large, erratic behaviour sets in, which
persists until the population reaches again a value of about 0.5,
at which point the whole scenario repeats. The determinism of
the model is reflected by ‘equal causes lead to equal effects’,
while its chaotic nature is reflected by the erratic length of the
laminar periods, as well as of the bubbles of wild behaviour.
Having analysed qualitatively the evolution of the number
of bullish agents, we next describe how the price πt+1 of an
asset at time t + 1 is related to the proportion pt of bullish
agents. One can argue (Corcos 1993, Bouchaud and Cont
1998, Farmer 1998) that the price change πt+1 − πt from one
period to the next is a monotonic function of pt (and, perhaps,
of πt). This function is positive when pt > 1/2 and negative
when pt < 1/2. If the reaction to a change in pt is reflected
in the prices in the next period, then a bubble in pt will lead to
a speculative bubble in the prices in the next period. Thus, our
model predicts the occurrence of bubbles from the behaviour
of the agents alone. Furthermore, for quite general laws of the
form
πt+1 = H(πt,pt), (7)
a simple application of the chain rule of differentiation leads
to the observation that the variable πt has the same Lyapunov
exponent as pt. In fact, this will be the case if 0

A Corcos et al Q UANTITATIVE F INANCE
p t
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
t
Figure 4. Evolution of the system over 10 000 time steps for
N =∞, m = 60 polled agents and the parameters
ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85. Note the existence of
accelerations ending in crashes and anti-crashes (fast jumps).
and ∂pH >c>0, where λ is the Lyapunov exponent for pt,as
follows from δπt+1 = ∂πH · δπt + ∂pH · δpt. This condition is,
in particular, satisfied for a law of the form πt+1 = πt + G(pt),
where G is strictly monotonic. Thus, chaotic behaviour of
bullish agents leads to chaotic behaviour of prices.
In the following, we shall take the simplest form of a
log-difference of the price linearly proportional to the order
unbalance (Farmer 1998), leading to
ln πt+1 − ln πt ≡ rt+1 = γ(pt − 1
), (8)
2
showing that the return rt calculated over one period is
proportional to the imbalance pt − 1
. Thus, the properties
2
of the return time series can be derived directly from those of
pt as we document below. This linear function of (pt − 1
) can 2
be improved for instance by the hyperbolic tangent function
tanh(pt − 1
) documented in Plerou et al (2001).
2
To summarize this qualitative analysis of the case of an
infinite number N of agents, we observe a time evolution
which, while satisfying certain criteria of randomness (such
as possessing an absolutely continuous invariant measure
and exhibiting a positive Lyapunov exponent (compare with
Eckmann and Ruelle 1985)) at the same time exhibits some
regularities on short time scales, since it is deterministic.
Our model thus establishes that straightforward fundamental
conditions may suffice to generate chaotic stock market
behaviour, depending on the parameter values. If the market
adjusts present market price on the basis of expectations and
mimicry—self-referred behaviour—then chaotic evolution of
the population will also imply chaotic evolution of prices.
4. Quantitative analysis of the
speculative bubbles within the chaotic
regime in the symmetric case
For an infinite number N of agents and in the symmetric case
ρhb = ρbh ≡ ρ1 and ρhh = ρbb ≡ ρ2, let us rewrite the
270
p t –1/2
p t –1/2
0.25
167
0.20
0.15
0.10
0.05
0.00
0 100 200 300 400 500 600
10 0
t
10 –1
10 –2
10 –3
10 –4
0 100 200 300
t
400 500 600
Figure 5. The first bubble of figure 4 for N =∞agents with
m = 60 polled agents and parameters ρhb = ρbh = 0.72 and
ρhh = ρbb = 0.85.
dynamical evolution (2) of the system as
p ′ m
m
= p − p p
j j=0
m−j (1 − p) j
j
f
m
m
m
+ (1 − p) (1 − p)
j j=0
m−j p j
j
f ,
m
where
(9)
1,
f(x)=
0,
if x ρ1 or x

168 6. Comportements mimétiques et antagonistes : bulles hyperboliques, krachs et chaos
Q UANTITATIVE F INANCE Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos
Fr,m(p)–p
10 0
10 –1
10 –2
10 –3
10 –4
10 –5
10 –6
10 –7
m = 30
m = 60
m = 100
10 –3 10 –2
slope m(m = 60)=3
p–1/2
10 –1
Figure 6. The logarithm of Fm(p) − p versus the logarithm of
p − 1/2 for three different values of m = 30, 60 and 100, with
ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85. Note the transition from a
slope 1 to a large effective slope before the reinjection due to the
contrarian mechanism.
κ>0) followed by a super-exponential growth accelerating
so much as to give the impression of reaching a singularity in
finite-time.
To understand this phenomenon, we plot the logarithm of
Fm(p)−p versus the logarithm of p −1/2 in figure 6 for three
different values of m = 30, 60 and 100. Two regimes can be
observed.
(1) For small p − 1/2, the slope of log 10 (Fm(p) − p) versus
log 10 (p − 1/2) is 1, i.e.
10 0
p ′ − p ≡ Fm(p) − p α(m)(p − 1
). (13)
2
This expression (13) explains the exponential growth
observed at early times in figure 5.
(2) For larger p − 1/2, the slope of log 10 (Fm(p) − p) versus
log 10 (p − 1/2) increases above 1 and stabilizes to a
value µ(m) before decreasing again due to the reinjection
produced by the contrarian mechanism. The interval in
p − 1/2 in which the slope is approximately stabilized at
the value µ(m) enables us to write
Fm(p)−p β(m)(p − 1
2 )µ(m) , with µ>1. (14)
These two regimes can be summarized in the following
phenomenological expression for Fm(p):
Fm(p) = 1
+ (1 − 2gm(1/2)
2
− g ′ 1
1
m (1/2))(p − ) + β(m)(p − 2 2 )µ(m) , (15)
= 1
2
+ (p − 1
2
) + α(m)(p − 1
2 )
+ β(m)(p − 1
2 )µ(m) with µ>1, (16)
and
α(m) =−2gm(1/2) − g ′ m (1/2). (17)
Figure 7. Approximation of the function Fm(p) − 1
2
function f(p)= [1 + α](p + 1
2
by the
) + β(p + 1
2 )µ over different
p-intervals, for m = 60 interacting agents and parameters
ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85.
Table 1. Optimized parameters α, β and µ for several optimization
intervals with m = 60 interacting agents and ρhb = ρbh = 0.72 and
ρhh = ρbb = 0.85.
Optimization domain α β µ
0 p − 1
0.05 2 0.011 11.67 3.27
0 p − 1
0.10 2 0.013 43.66 3.77
0 p − 1
0.15 2
0 p −
0.014 60.32 3.91
1
0.20 2 0.004 30.64 3.54
This expression can be obtained as an approximation of the
exact expansion derived in the appendix.
In order to check the hypothesis (16), we numerically solve
the following problem
min
{α,β,µ} Fm(p) − 1
2
1
1
− [1 + α](p − ) − β(p − 2 2 )µ 2 , (18)
which amounts to constructing the best approximation of
the exact map Fm(p) in terms of an effective power-law
acceleration (see (20) below). The results obtained for m =
60 interacting agents and ρhb = ρbh = 0.72 and ρhh =
ρbb = 0.85 are given in table 1 and shown in figure 7.
The numerical values of α are in good agreement with the
theoretical prediction: α(m) = F ′ m (1/2) − 1 which yields
α(m) 0.011 in the present case (m = 60, ρhb = ρbh = 0.72
and ρhh = ρbb = 0.85). As a first approximation, we
can consider that the exponent µ is fixed over the interval
of interest, which is reasonable according to the very good
quality of the fits shown in figure 7. We can conclude from
this numerical investigation that µ(m) ∈ [3, 4]. A finer
analysis shows, however, that the exponent µ is in fact not
perfectly constant but shifts slowly from about 3 to 4 as p
increases. This should be expected as the function Fm(p)
contains many higher-order terms. We can also note that the
parameter pc = (β/α) −1/µ , which defines the typical scale
of the crossover remains constant and equal to pc 0.70 for
271

A Corcos et al Q UANTITATIVE F INANCE
all the fits (except for the largest interval p − 1/2 < 0.2,
for which pc = 0.8). In sum, the procedure (18) and its
results show that the effective power-law representation (16) is
a crossover phenomenon: it is not dominated by the ‘critical’
value ρhb = ρbh of the jump of the map obtained in the limit
of large m.
Introducing the notation ɛ = p − 1/2, the dynamics
associated with the effective map (16) can be rewritten
ɛ ′ − ɛ = α(m)ɛ + β(m)ɛ µ(m) , (19)
which, in the continuous time limit, yields
dɛ
dt = α(m)ɛ + β(m)ɛµ(m) . (20)
Thus, for small ɛ, we obtain an exponential growth rate
while for large enough ɛ
ɛt ∝ e α(m)t , (21)
1
−
ɛt ∝ (tc − t) µ(m)−1 . (22)
Of course, these behaviours become invalid for t too close to
tc for which the reinjection mechanism operates and ensures
that the probability remains less than one.
For example, for m = 60 with ρhb = ρbh = 0.72 and
ρhh = ρbb = 0.85, we can check on figure 6 that µ(m) = 3,
which yields for large ɛ:
pt − 1
2 ∝
1
√ . (23)
tc − t
The prediction (23) implies that plotting (pt − 1/2) −2 as a
function of t should be a straight line in this regime. This
non-parametric test is checked in figure 8 on five successive
bubbles. This provides a confirmation of the effective powerlaw
representation (16) of the map. The fact that it is the lowest
estimate µ ≈ 3 shown in table 1 which dominates in figure 8
results simply from the fact that it is the longest transient
corresponding to the regime where p is closest to the unstable
fixed point 1/2. This is visualized in figure 8 by the horizontal
dashed lines indicating the levels p−1/2 = 0.05, 0.01 and 0.2.
This demonstrates that most of the visited values are close to
the unstable fixed point, which determines the effective value
of the nonlinear exponent µ ≈ 3.
With the price dynamics (8), the prediction (22) implies
that the returns rt should increase in an accelerating superexponential
fashion at the end of a bubble, leading to a price
trajectory
πt = πc − C(tc − t) µ(m)−2
µ(m)−1 , (24)
where πc is the culminating price of the bubble reached at
t = tc when µ(m) > 2, such that the finite-time singularity
in rt gives rise only to an infinite slope of the price trajectory.
The behaviour (24) with an exponent 0 < µ(m)−2
< 1 has been
µ(m)−1
documented in many bubbles (Sornette et al 1996, Johansen
et al 1999, 2000, Johansen and Sornette 1999, 2000, Sornette
and Johansen 2001, Sornette and Andersen 2001, Sornette
2001). The case m = 60 with ρhb = ρbh = 0.72 and
272
169
1
Figure 8. (pt −1/2) 2 versus t to qualify the finite-time singularity
predicted by (23) for m = 60 with ρhb = ρbh = 0.72 and
ρhh = ρbb = 0.85. The points are obtained from the time series pt
and the straight continuous lines are the best linear fits. The
horizontal dashed lines indicate the levels p − 1/2 = 0.05, 0.01 and
0.2 to demonstrate that most of the visited values are close to the
unstable fixed point, which determines the effective value of the
nonlinear exponent µ ≈ 3.
ρhh = ρbb = 0.85 shown in figure 6 leads to µ(m)−2
= 1/2,
µ(m)−1
which is in reasonable agreement with previously reported
values.
Interpreted within the present model, the exponent µ(m)−2
µ(m)−1
of the price singularity gives an estimation of the ‘connectivity’
number m through the dependence of µ on m documented
in figure 6. Such a relationship has already been argued by
Johansen et al (2000) at a phenomenological level using a
mean-field equation in which the exponent is directly related
to the number of connections to a given agent.
5. Statistical properties of price returns
in the symmetric case
Using the price dynamics (8), the distribution of p − 1/2 is the
same as the distribution of returns, which is the first statistical
property analysed in econometric work (Campbell et al 1997,
Lo and MacKinlay 1999, Lux 1996, Pagan 1996, Plerou et al
1999, Laherrère and Sornette 1998). Note that the distribution
of p − 1/2 is nothing but the invariant measure of the chaotic
map p ′ (p) which can be shown to be continuous with respect to
the Lebesgue measure (Eckmann and Ruelle 1985). Figure 9
shows the cumulative distribution of rt ∝ pt −1/2. Notice the
two breaks at |p − 1/2| =0.28, which are due to the existence
of weakly unstable periodic orbits corresponding to a transient
oscillation between bullish and bearish states.
Figure 10 plots in double-logarithmic scales the survival
(i.e. complementary cumulative) distribution of rt ∝ pt − 1/2
for m = 30, 60 and 100. For m = 60, we can observe an
approximate power-law tail but the exponent is smaller than
1 in contradiction to the empirical evidence which suggests
a tail of the survival probability with exponents 3–5. In

170 6. Comportements mimétiques et antagonistes : bulles hyperboliques, krachs et chaos
Q UANTITATIVE F INANCE Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos
Probability
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Effect of periodic orbits
0.0
–0.4 –0.3 –0.2 –0.1 0.0 0.1 0.2 0.3 0.4
p–1/2
Figure 9. Cumulative distribution for m = 60 polled agents and the
parameters ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85.
the other cases, we cannot conclude on the existence of a
power-law regime, but it is obvious that the tail behaviour of
the distribution function depends on the number m of polled
agents.
Figure 11 shows the behaviour of the autocorrelation
function for m = 60 and 100, with the same values of the
other parameters ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85.
For m = 100, the presence of the weakly unstable orbits is felt
much stronger, which is reflected in (i) a very strong periodic
component of the correlation function and (ii) its slow decay.
Even for m = 60, the correlation function does not decay
fast enough compared to the typical duration of speculative
bubbles to be in quantitative agreement with empirical data.
This anomalously large correlation of the returns is obviously
related to the deterministic dynamics of the returns. We thus
expect that including stochastic noise due to a finite number N
of agents (see below) and adding external noise due to ‘news’
will whiten rt significantly.
Figure 12 compares the correlation function for the returns
time series rt ∝ pt − 1/2 and the volatility time series defined
as |rt|. The volatility is an important measure of risks and thus
plays an important role in portfolio managements and option
pricing and hedging. Note that taking the absolute value of the
return removes the one source of irregularity stemming from
the change of sign of rt ∝ pt − 1/2 to focus on the local
amplitudes. We observe in figure 12 a significantly longer
correlation time for the volatility. Moreover, the correlation
function of the volatility first decays exponentially and then as a
power law. This behaviour has previously been documented in
many econometric works (Ding et al 1993, Ding and Granger
1996, Müller et al 1997, Dacorogna et al 1998, Arneodo et al
1998, Ballocchi et al 1999, Muzy et al 2001).
6. Asymmetric cases
We have seen that the symmetric case ρhb = ρbh and ρhh = ρbb
is plagued by the weakly unstable periodic orbits which put a
Probability
10 0
10 –1
10 –5
m = 30
m = 60
m = 100
10 –4 10 –3
Survival distribution
10 –2
1/2–p
10 –1
Figure 10. Survival (i.e. complementary cumulative) distribution
for m = 30, 60 and 100 polled ‘friends’ per agent and parameters
ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85. The total number N of
agents is infinite.
strong and unrealistic imprint on the statistical properties of
the return time series. It is natural to argue that breaking the
symmetry will destroy the strength of these periodic orbits.
From a behavioural point of view, it is also quite clear that
the attitude of an investor is not symmetric. One can expect
a priori a stronger bullish buy-and-hold index ρhb − 1/2 than
bearish buy-and-hold index ρbh − 1/2: one is a priori more
prone to hold a position in a bullish market than in a bearish one.
Similarly, we expect a smaller bullish reversal index 1 − ρhh
than bearish reversal index 1 − ρbb: speculative bubbles are
rarely seen on downward trends as it is much more common
that increasing prices are favourably perceived and can be sustained
much longer without reference to the fundamental price.
Such an asymmetry has been clearly demonstrated
empirically in the difference between the rate of occurrence
and size of extreme drawdowns compared to drawups in stock
market time series (Johansen and Sornette 2001). Drawdowns
(drawups) are defined as the cumulative losses (gains) from
the last local maximum (minimum) to the next local minimum
(maximum). Drawdowns and drawups are very interesting
because they offer a more natural measure of real market risks
than the variance, the value-at-risk or other measures based
on fixed time scale distributions of returns. For the major
stock market indices, there are very large drawdowns which are
‘outliers’ while drawups do not exhibit such a drastic change
of regime. For major companies, drawups of amplitude larger
than 15% occur at a rate about twice as large as the rate of
drawdowns, but with lower absolute amplitude. An asymmetry
ρhh = ρbb could result from the mechanism of asymmetric
information, in which actors on one side of the market have
better information than those on the other (see Riley (2001) for
a survey of developments in the economics of information over
the last 25 years and The Bank of Sweden Prize in Economics
Sciences in Memory of Alfred Nobel 2001, document entitled
‘Markets with Asymmetric Information’, 10 October 2001).
10 0
273

A Corcos et al Q UANTITATIVE F INANCE
p t
Autocorrelation
1.0
m = 60
1.0
m = 100
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000
t t
1.0
1.0
0.5
0.5
0.0
–0.5
–1.0
0 20 40 60 80 100 120 140 160 180
200
p t
Autocorrelation
0.0
–0.5
–1.0
0 20 40 60 80 100 120 140 160 180 200
Time lag Time lag
Figure 11. The upper panels represent the time series pt for m = 60 (left) and m = 100 (right). The lower panels represent the
corresponding autocorrelation function of rt ∝ p − 1/2 for m = 60 (left) and m = 100 (right) with the same parameters ρhb = ρbh = 0.72
and ρhh = ρbb = 0.85.
1.2
1.0
0.8
0.6
0.4
0.2
0.0
Exponential
decay
Power-law
decay
Return
Volatility
0 50 100 150 200 250 300
Figure 12. Autocorrelation function of the returns and of the
volatility for m = 60 polled agents and the parameters
ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85.
Figure 13 compares the dynamics for the symmetric
system (upper panel (a)) and for the asymmetric system
(lower panel (b)). It is clear that, as expected, the number
of periodic orbits decreases significantly in the asymmetric
system. However, there are still an unrealistic number of
negative bubbles. It is not possible to increase the asymmetry
sufficiently strongly without exiting from the chaotic regime.
This unrealistic feature is thus an intrinsic property and
limitation of the present model. We shall indicate in the
conclusion possible extensions and remedies.
Figure 14 compares the cumulative distributions of p−1/2
for m = 60 for the symmetric and asymmetric cases. The
strong effect of the weakly unstable periodic orbits observed
in the periodic case has disappeared. In addition, the tail of
274
171
(a)
1.0
0.8
0.6
0.4
0.2
0.0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
(b)
t
1.0
0.8
0.6
0.4
0.2
0.0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
t
p t
p t
Figure 13. Time evolution of pt over 100 00 time steps for m = 60
polled agents in (a) a symmetric case ρhb = ρbh = 0.72 and
ρhh = ρbb = 0.85 and (b) an asymmetric case ρhb = 0.72,
ρbh = 0.74, ρhh = 0.85 and ρbb = 0.87.
the distribution decays faster in the asymmetric case, in better
(but still not very good) agreement with empirical data.
Figure 15 shows the correlation function of the returns for
a symmetric and an asymmetric case. In the asymmetric case,
there is no trace of oscillations but the decay is slightly slower.
7. Finite-size effects
Until now, our analysis has focused on the limit of an infinite
number N →∞of agents, in which each agent polls randomly
m agents among N. In this limit, we have shown that, for a large
domain in the parameter space, the dynamics of the returns is
chaotic with interesting and qualitatively realistic properties.

172 6. Comportements mimétiques et antagonistes : bulles hyperboliques, krachs et chaos
Q UANTITATIVE F INANCE Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos
Probability
1.0
Symmetric dynamics
0.9
0.8
0.7
0.6
0.5
Asymmetric dynamics
0.4
0.3
0.2
0.1
0.0
Effect of periodic
orbits
–0.5 –0.4 –0.3 –0.2 –0.1 0.0
p–1/2
0.1 0.2 0.3 0.4 0.5
Figure 14. Cumulative distribution function of p − 1/2 for m = 60
polled agents and parameters ρhb = ρbh = 0.72 and
ρhh = ρbb = 0.85 (dashed line) and ρhb = 0.72, ρbh = 0.74,
ρhh = 0.85 and ρbb = 0.87 (continuous line). Note that the
asymmetric dynamics has the effect of shifting the cumulative
distribution to the right.
7.1. Finite-size effects in other models
We now investigate finite-size effects resulting from a finite
number N of interacting agents trading on the stock market.
This issue of the role of the number of agents has recently been
investigated vigorously with surprising results.
Hellthaler (1995) studied the N-dependence of the
dynamical properties of price time series of the Levy et al
model (1995, 2000). Egenter et al (1999) did the same for the
Kim and Markowitz (1989) and the Lux and Marchesi (1999)
models. They found that, if this number N goes to infinity,
nearly periodic oscillations occur and the statistical properties
of the price time series become completely unrealistic. Stauffer
(1999) reviewed this work and others such as the Levy et al
(1995, 2000) model: realistically looking price fluctuations
are obtained for N ∝ 102 , but for N ∝ 106 the prices
vary smoothly in a nearly periodic and thus unrealistic way.
The model proposed by Farmer (1998) suffers from the same
problem: with a few hundred investors, most investors are
fundamentalists during calm times, but bursts of high volatility
coincide with large fractions of noise traders. When N
becomes much larger, the fraction of noise traders goes to zero
in contradiction to reality. On a somewhat different issue,
Huang and Solomon (2001) have studied finite-size effects
in dynamical systems of price evolution with multiplicative
noise. They find that the exponent of the Pareto law
obtained in stochastic multiplicative market models is crucially
affected by a finite N and may cause, in the absence of
an appropriate social policy, extreme wealth inequality and
market instability. Another model (apart from ours) where
the market may stay realistic even for N →∞seems to be
the Cont and Bouchaud percolation model (2001). However,
this only occurs for an unrealistic tuning of the percolation
concentration to its critical value. Thus, in most cases, the
limit N →∞leads to a behaviour of the simulated markets
Correlation
1.2
1.0
0.8
0.6
0.4
0.2
0.0
–0.2
Symmetric dynamics
Asymmetric dynamics
0 20 40 60 80 100 120 140 160 180 200
Time lag
Figure 15. Correlation function for m = 60 polled agents and
parameters ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85 (dashed line)
and ρhb = 0.72, ρbh = 0.74, ρhh = 0.85 and ρbb = 0.87 (continuous
line). Note that the correlation function of returns goes to zero
(within the noise level) at time lags larger than about 100. This
unrealistic feature of a long-range correlation in the returns makes it
unnecessary to show the even longer-range correlation of the
absolute values of the returns.
which becomes quite smooth or periodic and thus predictable,
in contrast to real markets. Our model, which remains
(deterministically) chaotic, is thus a significant improvement
upon this behaviour. We trace this improvement on the highly
nonlinear behaviour resulting from the interplay between
the imitative and contrarian behaviour. It has thus been
argued (Stauffer 1999) that, if these previous models are good
descriptions of markets, then real markets with their strong
random fluctuations are dominated by a rather limited number
of large players: this amounts to the assumption that the
hundred most important investors or investment companies
have much more influence than the millions of less wealthy
private investors.
There is another class of models, the minority games
(Challet and Zhang 1997), in which the dynamics remains
complex even in the limit N →∞. It has been established
that the fluctuations of the sum of the aggregate demand
have an approximate scaling with similar sized fluctuations
(volatility/standard deviation) for any N and m for the scale
scaled variable 2 m /N, where m is the memory length (Challet
et al 2000). In a generalization, the so-called grand canonical
version of the minority game (Jefferies et al 2001), where the
agents have a confidence threshold that prevents them from
playing if their strategies have not been successful over the
last T turns, the dynamics can depend more sensitively on N:
as N becomes small, the dynamics can become quite different.
For large N, the complexity remains.
The difference between the limit N →∞considered up
to now in this paper and the case of finite N is that pt is no
longer the fraction of bullish agents. For finite N, pt must
be interpreted as the probability for an agent to be bullish.
Of course, in the limit of large N, the law of large numbers
275

A Corcos et al Q UANTITATIVE F INANCE
(a)
1.0
p t
0.5
0.0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
(b)
t
1.0
p t
p t
0.5
0.0
(c)
0.4
0.2
0
–0.2
–0.4
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
t
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
t
Figure 16. Time evolution of pt over 10 000 time steps for m = 60
polled agents with (a) N =∞, (b) N = m +1= 61 agents and
parameters ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85. Panel
(c) represents the noise due to the finite size of the system and is
obtained by subtracting the time series in panel (a) from the time
series in panel (b).
ensures that the fraction of bullish agents becomes equal to the
probability for an agent to be bullish. There are several ways
to implement a finite-size effect. We here discuss only the two
simplest ones.
7.2. Finite external sampling of an infinite system
Consider a system with an infinite number of agents for which
the fraction pt of bullish agents is governed by the deterministic
dynamics (2). At each time step t, let us sample a finite number
N of them to determine the fraction of bullish agents. We get
a number n, which is in general close but not exactly equal to
Npt due to statistical fluctuations. The probability of finding n
bullish agents among N agents is indeed given by the binomial
law
N
Pr(n) = p
n
n (1 − p) N−n . (25)
This shows that the observed proportion ˜p = n/N of bullish
agents is asymptotically normal with mean p and standard
deviation 1/ √ p(1 − p)N :Pr( ˜p) ∝ N (p, 1/ √ p(1 − p)N).
Iterating the sampling among N agents at each time step gives
a noisy dynamics ˜pt shadowing the true deterministic one.
Figure 16 compares the dynamics of the deterministic pt
corresponding to N →∞(panel (a)) with ˜pt for a number
N = m +1 = 61 of sampled agents among the infinite
ensemble of them (panel (b)). Panel (c) is the ‘noise’ time
series defined as ˜pt − pt, i.e. by subtracting the time series
of panel (a) from the time series of panel (b). The noise time
series of panel (c) thus represents the statistical fluctuations
due to the finite sampling of agents’ opinions. Figure 16(b)
shows the characteristic volatility clusters which is one of the
most important stylized properties of empirical time series.
276
Correlation
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
–0.1
173
0 5 10 15 20 25 30 35 40 45 50
Time lag
Figure 17. Correlation function for m = 60 polled agents with
N =∞(thin curve), N = 600 (dashed curve) and N = 61
(continuous curve) agents and parameters ρhb = ρbh = 0.72 and
ρhh = ρbb = 0.85.
For large N, we can write
˜pt = pt +
1
√ pt(1 − pt)N Wt
(26)
where {Wt} are independent and identically distributed
Gaussian variables with zero mean and unit variance.
Therefore, the correlation function corrN(τ) at lag τ = 0is
obtained from that for N →∞by multiplication by a constant
factor:
Nvar(p)
corrN(τ) =
E[1/{p(1 − p)}]+Nvar(p)
× corr∞(τ) and τ = 0, (27)
corr∞(τ) for large N, (28)
where E[x] denotes the expectation of x with respect to the
continuous invariant measure of the dynamical system (2).
Note that E[1/{p(1 − p)}] always exists for m

174 6. Comportements mimétiques et antagonistes : bulles hyperboliques, krachs et chaos
Q UANTITATIVE F INANCE Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos
1.0
0.9
0.8
0.7
0.6
p¢ 0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
p
Figure 18. Return map of the fraction of bullish agents for m = 60
polled agents among N = 61 agents (points) and the deterministic
trajectory (continuous curve) corresponding to N =∞agents. The
parameters are ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85.
agents in the market (this is realistic) and they are in contact
with only m other agents that they poll at each time period. Not
knowing the true value of N but assuming it to be large, it is
rational for them to develop the best predictor of the dynamics
by assuming the ideal case of an infinite number of agents with
m polled agents and thus use the deterministic dynamics (2) as
their best predictor.
At each time period t, each agent thus chooses randomly
m agents that she polls. She then counts the number of bullish
and bearish agents among her polled sample of m agents.
This number divided by m gives her an estimation ˆpt of
the probability pt being bullish at time t. Introducing this
estimation in the deterministic equation (2), the agent obtains
a forecast ˆp ′ of the true probability p ′ being bullish at the next
time step.
Results of the simulations of this model are shown in figure
21. We observe a significantly stronger ‘noise’ compared
to the previous section, which is expected since the noise is
itself injected in the dynamical equation at each time step. As
a consequence, the correlation function of the returns and of
the volatility decay faster than their deterministic counterpart.
The correlation of the volatility still decays about ten times
slower than the correlation of the returns, but this clustering of
volatility is not sufficiently strong compared to empirical facts.
As in the definition of the model presented in section 2,
we use here again a ‘mean-field’ approximation, valid only if
each agent selects randomly her m agents to watch, with a new
selection completely independent of the previous one made at
every new time step. This unrealistic but simplifying feature
is the one we have followed in our simulations. If each agent
has some neighbouring agents which she will watch again
and again, then correlations will build up which are ignored
completely here; some correlations of this type are presumably
more realistic. This is left for future works.
Other more realistic models of a finite number of agents
can be introduced. For instance, at time t, consider an agent
1.0
0.9
0.8
0.7
0.6
p¢
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5
p
0.6 0.7 0.8 0.9 1.0
Figure 19. Return map of the fraction of bullish agents for m = 60
polled agents among N = 600 agents (points) and the deterministic
trajectory (continuous curve) corresponding to N =∞agents. The
parameters are ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85.
among the N. She chooses m other agents randomly and
polls them. Each of them is either bullish or bearish as a
result of decisions taken during the previous time period. She
then counts the number of bullish agents among the m, and
then determines her new attitude using the rules (1). If she is
polled at time t + 1 by another agent, her attitude will be the
one determined from t to t + 1. In this way, we never refer
to the deterministic dynamics pt but only to its underlying
rules. As a consequence, this deterministic dynamics does
not exert an attraction that minimizes the effect of statistical
fluctuations due to finite sizes. This approach is similar to
going from a Fokker–Planck equation (equation (2)) to a
Langevin equation with finite-size effects. This class of models
will be investigated elsewhere.
8. Conclusions
The traditional concept of stock market dynamics envisions a
stream of stochastic ‘news’ that may move prices in random
directions. This paper, in contrast, demonstrates that certain
types of deterministic behaviour—mimicry and contradictory
behaviour alone—can already lead to chaotic prices.
If the market prices are assumed to follow the pt
behaviour, our description refers to the well-known evolution
of the speculative bubbles. Such apparent regularities often
occur in the stock market and form the basis of the so-called
‘technical analysis’ whereby traders attempt to predict future
price movements by extrapolating certain patterns from recent
historical prices. Our model provides an explanation of birth,
life and death of the speculative bubbles in this context.
While the traditional theory of rational anticipations
exhibits and emphasizes self-reinforcing mechanisms, without
either predicting their inception nor their collapse, the strength
of our model is to justify the occurrence of speculative
bubbles. It allows for their collapse by taking into account
277

A Corcos et al Q UANTITATIVE F INANCE
p t
p t
p t
0.5
0.0
–0.5
0 1 2 3 4 5 6 7 8 9 10
2.5
2.0
1.5
1.0
0.5
0.0
t
0 1 2 3 4 5
t
6 7 8 9 10
10 1
10 0
10 –1
0 1 2 3 4 5
t
6 7 8 9 10
Figure 20. Upper panel: return trajectory ˜rt = γ ˜pt − 1/2 for
m = 100, N = 100, ρhb = ρbh = 0.72 and ρhh = ρbb = 0.85 and
γ = 0.01. Middle panel: price trajectory obtained by
πt = πt−1 exp[˜rt]. Lower panel: same as the middle panel with πt
shown on a logarithmic scale. Note the ‘flat trough–sharp peak’
structure of the log-price trajectory (Roehner and Sornette 1998).
the combination of mimetic and antagonistic behaviour in the
formation of expectations about prices.
The specific feature of the model is to combine these two
Keynesian aspects of speculation and enterprise and to derive
from them behavioural rules based on collective opinion: the
agents can adopt an imitative and gregarious behaviour or,
on the contrary, anticipate a reversal of tendency, thereby
detaching themselves from the current trend. It is this duality,
the continuous coexistence of these two elements, which is at
the origin of the properties of our model: chaotic behaviour
and the generation of bubbles.
It is a common wisdom that deterministic chaos leads to
fundamental limits of predictability because the tiny inevitable
fluctuations in those chaotic systems quickly snowball in
unpredictable ways. This has been investigated in relation
to, for instance, long-term weather patterns. However, in
the context of our models, we have shown that the chaotic
dynamics of the returns alone cannot be the limiting factor
for predictability, as it contains too many residual correlations.
Endogenous fluctuations due to finite-size effects and external
news (noise) seem to be needed as important factors leading to
the observed randomness of stock market prices. The relation
between these extrinsic factors and the intrinsic ones studied
in this paper will be explored elsewhere.
Remarks and acknowledgements
This paper is an outgrowth and extension of unpublished work
(1994) by three of us (AC, JPE, AM) which was in turn based
on the PhD of Anne Corcos in 1993. We are grateful to
J V Andersen for useful discussions. This work was partially
supported by the Fonds National Suisse (JPE and AM) and
278
Figure 21. Evolution of the system over 10 000 time steps for
m = 60 polled agents with (upper panel) N =∞, (second panel)
N = m +1= 61 and parameters ρhb = ρbh = 0.72 and
ρhh = ρbb = 0.85. The lower panel represents the ‘noise’
introduced by the finite size of the system and is obtained by
subtracting the upper panel from the second panel.
175
by the James S Mc Donnell Foundation 21st century scientist
award/studying complex system (DS).
Appendix
We expand Fm(p) around the fixed point p = 1/2, so that,
using the symmetry of Fm(p)
Fm(p) = 1
2 +F ′ m
1
(1/2)(p− 2
(3)
1
)+F m (1/2)(p− 2 )3 +···. (29)
First of all, it is obvious to show by recursion that
F ′ m (1/2) = 1 − 2gm(1/2) − g ′ m (1/2)
F
(30)
(2k+1)
m (1/2) =−2(2k +1)g (2k)
m (1/2)
− g (2k+1)
m (1/2) if k>0. (31)
The problem thus amounts to calculating the derivatives of gm.
Some simple algebraic manipulations allow us to obtain
g ′ m−1
m − 1
m (p) = m
p
j
j=0
m1−j (1 − p) j
j j +1
× f − f
m m
m−1
m − 1
=−m
p
j
j=0
(32)
m1−j (1 − p) j 1fm(j), (33)
where 1fm(·) is the first-order discrete derivative of f( ·
m ),
which yields
1
=−
2
m
2m−1 m−1
m − 1
1fm(j). (34)
j
g ′ m
j=0

176 6. Comportements mimétiques et antagonistes : bulles hyperboliques, krachs et chaos
Q UANTITATIVE F INANCE Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos
By recursion, it is easy to prove that
g (k)
m
1
2
= (−1)k m!
2m−k m−k
m − k
kfm(j) (35)
k! j
j=0
and kfm(·) is the k th order discrete derivative of f( ·
m ):
Finally,
kfm(j) =
k
i=0
k
(−1)
i
i f
j + i
m
. (36)
F (2k+1)
m!
m (1/2) =
2m−2k−1 (2k)!
m−2k−1
1
m − 2k − 1
×
2k+1fm(j).
2k +1
j
j=0
m−2k
m − 2k
− (2k +1)
2kfm(j) . (37)
j
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281

Deuxième partie
Etude des propriétés de dépendances entre
actifs financiers
179

Chapitre 7
Etude de la dépendance à l’aide des
copules
Nous abordons maintenant l’étude de la dépendance entres actifs financiers. Nous avons déjà évoqué aux
chapitres 1 et 4 le problème de la dépendance temporelle pour un actif donné, et c’est pourquoi nous
nous bornerons ici à présenter l’aspect que l’on peut qualifier - en référence au langage de la physique -
de “spatial”, et qui consiste en l’étude des interactions entre différents actifs à un instant donné.
Jusqu’à une période récente, le coefficient de corrélation était l’outil de prédilection utilisé pour quantifier
l’intensité de la dépendance entre deux actifs. Ceci est bien sûr à rattacher au fait que l’on pensait
que non seulement la distribution marginale des rendements était gaussienne, mais aussi, par une extension
somme toute très naturelle, leur distribution multivariée. L’hypothèse gaussienne s’effondrant
peu à peu (voir Richardson et Smith (1993) pour un test spécifique du caractère gaussien multivarié
des distributions de rendements), la pertinence du coefficient corrélation comme mesure de dépendance
était elle-même remise en cause, si bien qu’il semble désormais admis qu’une telle mesure est totalement
insuffisante pour donner une idée juste des propriétés de dépendance entre deux actifs (Blum, Dias
et Embrechts 2002, Embrechts, McNeil et Straumann 2002) et qu’il est nécessaire d’avoir recours à la
distribution jointe des probabilités de leurs rendements 1 pour obtenir une description aussi complète
que possible de leur structure de dépendance 2 . En fait, cela est d’autant plus crucial que l’on souhaite
s’intéresser aux événements rares (et donc extrêmes).
Il convient toutefois de prendre conscience que la distribution bivariée des rendements contient deux
types d’information : d’une part de l’information sur la distribution individuelle (monovariée) des rendements
de chacun des actifs et d’autre part de l’information concernant la seule dépendance entre ces deux
actifs, indépendamment de leur comportement individuel. Il va donc être souhaitable de pouvoir séparer
ces deux sources d’information, afin de ne conserver que celle qui nous intéresse ici, à savoir l’information
sur le comportement collectif (ou comportement joint) des deux actifs. Ceci est rendu possible grâce
à l’emploi d’objets mathématiques dénommés copules, dont le rôle est précisément de décrire de manière
complète et généralement unique la structure de dépendance présentée par les deux actifs financiers, à
l’exclusion de tout autre information sur leur comportement marginal.
Cette approche de la dépendance spatiale en terme de copule s’est abondamment développée ces dernières
1 Nous l’avons déjà fait remarquer dans la partie précédente, les rendements des actifs financiers semblent être mieux adaptés
que les prix eux-mêmes du fait du manque de stationnarité flagrant dont font preuve ces derniers. Ainsi, dans toute la suite de
cette partie la distribution bivariée des rendements sera un des objets fondamentaux de notre étude.
2 Pour des raisons de simplicité et de clarté de l’exposé, nous limitons notre présentation au cas où seuls deux actifs sont
présents. Il va de soit que cela n’est en rien restrictif et que le cas général pourrait être traité de la même manière.
181

182 7. Etude de la dépendance à l’aide des copules
années et semble prometteuse aussi bien d’un point de vue pratique que théorique. En effet, sur le plan
pratique, la dépendance entre les actifs est l’un des piliers de la gestion des risques, puisque c’est de cette
étude de la dépendance que va découler la stratégie de gestion à mettre en œuvre selon que l’on souhaite
diversifier les risques en les agrégeant sous forme de portefeuilles, ou bien que l’on décide de couvrir
ces risques à l’aide de produits dérivés, ou enfin que l’on choisisse de les titriser, c’est-à-dire les mettre
sur le marché, ce qui revient à les céder à une tierce partie. Or, l’apport des copules est de permettre une
meilleure compréhension et quantification de l’effet de l’interaction entre actifs par l’étude de l’influence
de diverses structures de dépendance entre les multiples sources de risques. Les exemples d’applications
sont nombreux, aussi bien en finance 3 que dans d’autres domaines tels que l’assurance 4 .
Sur le plan théorique, l’étude de la dépendance entre actifs est fondamentale car elle permet de sonder
la structure sous-jacente des mécanismes de marché. En effet, il est très raisonnable de penser que la
dépendance entre actifs résulte, du moins en partie 5 , de l’interaction entre les agents agissant sur ces
marchés. De part les positions qu’ils prennent, ceux-ci sont responsables de l’évolution individuelle
(en fait, des fluctuations) des prix des actifs, mais aussi de part les choix qu’ils opèrent - vendant ou
achetant tel titre plutôt que tel autre - ils créent de la dépendance entre ces actifs. Donc, l’étude de la
dépendance entre actifs doit permettre de compléter la compréhension des mécanismes en jeu sur les
marchés financiers et ainsi de mieux saisir les interactions entre les agents. Elle doit aussi amener à
cerner les paramètres macroscopiques auxquels les agents sont sensibles.
Avant d’entrer dans le vif du sujet et de faire tout d’abord quelques rappels sur la notion de copule et
quelques unes de leurs propriétés fondamentales, puis de présenter certaines familles de copules dont
nous aurons à nous servir par la suite, et enfin d’en venir aux résultats empiriques que nous avons pu
obtenir sur la dépendance entre actifs financiers, nous souhaitons préciser une hypothèse importante sur
laquelle nous nous appuierons dans la suite. En effet, nous avons affirmé qu’il était naturel de découpler
l’étude de la dépendance spatiale de celle de la dépendance temporelle. Or, pour demeurer dans un cadre
complètement général, il convient de garder à l’esprit que la dépendance entre deux actifs peut tout
à fait évoluer au cours du temps, elle n’a aucune raison de rester constante (Patton 2001, Rockinger et
Jondeau 2001), donc il conviendrait aussi d’étudier sa dynamique. Cela dit, dans un cadre aussi vaste, une
étude aussi bien empirique que théorique devient extrêmement délicate. C’est pourquoi nous avons choisi
de faire une hypothèse simplificatrice en considérerant que la dépendance temporelle est entièrement
capturée par l’évolution marginale de chaque actif et donc que leurs propriétés de dépendance spatiale
demeurent les mêmes à tout instant.
7.1 Les copules
Nous allons exposer brièvement dans ce paragraphe les propriétés principales des copules. Pour une
présentation complète du sujet le lecteur est invité à se référer aux ouvrages de Joe (1997) et Nelsen
(1998) notamment et aux articles de revues de Frees et Valdez (1998) ou Bouyé, Durrleman, Nikeghbali,
Riboulet et Roncalli (2000) pour une présentation plus orientée vers les applications financières et
3
Voir notamment Embrechts, Hoeing et Juri (2001) pour les conséquences sur le calcul de VaR et la gestion de portefeuille ou
encore Cherubini et Luciano (2000) et Coutant, Durrleman, Rapuch et Roncalli (2001) pour une application au calcul d’options
et Frey et McNeil (2001), Frey, McNeil et Nyfeler (2001) pour ce qui est du risque de crédit.
4
Voir notamment Frees et Valdez (1998), Valdez (2001), ou Cossette, Gaillardetz, Marceau et Rioux (2002) pour certaines
applications
5
Bien évidemment, la dépendance observée entre différents actifs n’a pas pour seule et unique source l’action des agents
économiques sur les marchés financiers. Les facteurs macroéconomiques jouent aussi un rôle important en ce qui concerne
cette dépendance, qui est par exemple d’autant plus forte que les actifs considérés appartiennent au même secteur économique
et sont donc de ce fait sensibles aux mêmes variables, aux mêmes évolutions de l’environnement économique.

7.2. Quelques familles de copules 183
actuarielles.
Commençons par rappeler les propriétés mathématiques que vérifient les copules :
DÉFINITION 1 (COPULE)
Une fonction C : [0, 1] × [0, 1] −→ [0, 1] est une copule si elle satisfait les propriétés suivantes :
– ∀u ∈ [0, 1], C(1, u) = C(u, 1) = u ,
– ∀ui ∈ [0, 1], C(u1, u2) = 0 si au moins un des ui est nul,
– C est croissante dans le sens où le C-volume de chaque rectangle dont les sommets se situent dans
[0, 1] 2 est positif.
Cela signifie simplement qu’une copule n’est rien d’autre qu’une distribution multivariée dont les distributions
marginales sont uniformes. Leur principal intérêt provient en fait du théorème de représentation
de Sklar (1959), qui stipule que toute distribution multivariée peut être exprimée comme une fonction de
ses marginales :
THÉORÈME 1 (SKLAR (1959))
Etant donné deux variables aléatoires X et Y dont la fonction de distribution est notée F et dont les
distributions marginales sont FX et FY , il existe une copule C : [0, 1] 2 −→ [0, 1] telle que :
F (x, y) = C(FX(x), FY (y)) . (7.1)
Dans le cas où les distributions marginales des variables X et Y sont continues, cette copule est unique
et C(·, ·) est alors la copule du couple de variables aléatoires (X, Y ). Bien évidemment, ce théorème
s’étend au cas d’un nombre quelconque de variables aléatoires. Une de ces conséquences immédiates est
que la fonction F (FX −1 (x), FY −1 (y)) est une copule, et plus précisément la copule de (X, Y ). Donc,
partant de n’importe quelle distribution jointe, il est aisé d’en dériver une copule. C’est une des méthodes
les plus employées pour construire des copules.
Enfin, cela permet aisément de comprendre que la copule de deux variables aléatoires est invariante par
changement de variable strictement croissant, ce qui est démontré dans Lindskog (2000) notamment.
Ce résultat est particulièrement important car il justifie que la copule est une mesure intrinsèque de
la dépendance entre variables aléatoires. En effet, par changement de variable strictement croissant,
l’ancienne et la nouvelle variable sont comonotones. Or, il est naturel de requérir d’une mesure de la
dépendance entre deux variables aléatoires qu’elle soit indifférente à la substitution par une variable
comonotone : si X et X ′ sont deux variables comonotones, il est normal que la structure de dépendance
entre (X, Y ) d’une part et (X ′ , Y ) d’autre part soit la même. C’est exactement ce que traduit ce théorème
et dont rend compte la copule. Bien entendu, des quantités telles que le coefficient de corrélation, qui
sont fonction à la fois de la copule et des marginales, ne sont pas invariantes par une telle transformation
et ne constituent donc pas des mesures de la seule dépendance.
En résumé, les copules sont des objets permettant de décrire de manière complète et généralement unique
les propriétés de dépendance entre deux variables aléatoires. En cela, elles autorisent l’étude séparée des
distributions marginales de chaque actif financier et de leur dépendance, ce qui va être le point auquel
nous allons nous intéresser maintenant. Mais avant cela, nous allons présenter quelques copules d’usage
courant dont nous aurons à traiter dans la suite.
7.2 Quelques familles de copules
Comme nous l’avons indiqué après avoir rappelé le théorème de Sklar (1959), il est possible de dériver
une copule de toute distribution multivariée. Leur nombre est donc considérable. Cependant quelques co-

184 7. Etude de la dépendance à l’aide des copules
pules ou familles de copules occupent une place plus importante que d’autres et nous allons en présenter
quelques unes.
7.2.1 Les copules gaussiennes et copules de Student
Comme leurs noms l’indiquent ces deux classes de copules sont respectivement issues de la distribution
gaussienne et des distributions de Student. Et, de même que la distribution gaussienne est un cas limite de
distribution de Student, la copule gaussienne est aussi un cas limite de copule de Student. C’est pourquoi
nous traitons ces copules, a priori différentes, dans le même paragraphe. Une autre raison vient de ce
que l’une comme l’autre sont des copules elliptiques, c’est-à-dire des copules dérivées des distributions
elliptiques.
L’intérêt de cette classe de copules vient simplement du fait qu’elle ouvre la voie à la généralisation des
distributions gaussiennes et elliptiques à des distributions dites “méta-gaussiennes” - dont l’introduction
est originellement due à Krzysztofowicz et Kelly (1996) et dont l’intérêt pratique a été démontré par Karlen
(1998) dans le domaine du traitement d’expériences de physique des particules et par Sornette, Simonetti
et Andersen (2000) pour ce qui est de la finance - et “méta-elliptiques” (Fang, Fang et Kotz 2002).
Ces “méta-distributions” possèdent la même structure de dépendance que les distributions gaussiennes
ou elliptiques mais en différent par leurs marginales qui peuvent être quelconques. Or, on sait que les
distributions elliptiques peuvent aisément être obtenues par des modèles à volatilité stochastiques, ce qui
fait des copules elliptiques en général et des copules gaussiennes et de Student en particulier, des copules
tout à fait pertinentes pour la modélisation de la dépendance entre actifs financiers.
Utilisant le théorème de Sklar (1959), on obtient simplement leurs expressions, même s’il n’est pas
possible d’en donner une forme fermée :
DÉFINITION 2 (COPULE GAUSSIENNE)
Soit Φ la distribution gaussienne standard et Φρ,N la distribution Gaussienne en dimension N, dont la
matrice de corrélation est ρ. La copule gaussienne de matrice de corrélation ρ est alors :
et sa densité est donnée par
avec yk(u) = Φ −1 (uk).
Cρ,N(u1, · · · , uN) = Φρ,N(Φ −1 (u1), · · · , Φ −1 (uN)) (7.2)
cρ,N(u1, · · · , uN) =
1
√ exp −
det ρ 1
2 yt (u) (ρ−1
− Id)y (u)
DÉFINITION 3 (COPULE DE STUDENT)
Soit Tν la distribution de Student avec ν degrés de liberté et Tν,ρ,N la distribution de Student avec ν
degrés de liberté en dimension N, dont la matrice de corrélation est ρ. La copule de Student avec ν
degrés de liberté et de matrice de corrélation ρ est alors :
et sa densité est donnée par
avec yk(u) = T −1
ν (uk).
(7.3)
Cρ,ν,N (u1, · · · , uN) = Tν,ρ,N (T −1
ν (u1), · · · , Tnu −1 (uN)) (7.4)
cρ,ν,N (u1, · · · , uN) =
1 Γ
√
det ρ
ν+N ν
2 Γ
ν+1 Γ 2
N−1 2
N
N
k=1
1 + y2 k
ν
1 + yt ρy
ν
ν+1
2
ν+N
2
, (7.5)

7.2. Quelques familles de copules 185
Par construction même, les copules de Student et la copule gaussienne sont très proches dans leur région
centrale, et l’on observe que le domaine où ces deux copules sont quasiment identiques s’étend de plus
en plus au fur et à mesure que le nombre de degrés de liberté de la copule de Student augmente. En
conséquence, il est parfois délicat de les distinguer, même avec des tests assez fins. Or, nous verrons plus
loin en détail que la confusion entre ces deux copules à des conséquences importantes pour l’étude des
risques extrêmes.
Enfin, signalons que ces copules présentent en outre l’intérêt d’être aisément synthétisables, ce qui est
très utile en pratique pour la simulation numérique ou l’étude de scénarii. En effet, il est facile de générer
des variables aléatoires gaussiennes ou de Student, ce qui, après un changement de variable (croissant),
permet d’obtenir les distributions marginales recherchées tout en conservant la copule inchangée.
7.2.2 Les copules archimédiennes
Les copules d’Archimède revêtent un intérêt tout particulier dans la mesure où un très grand nombre de
copules appartiennent à cette classe, qui de plus jouit d’un certain nombre de propriétés intéressantes. En
outre, comme souligné par Frees et Valdez (1998), nombre de modèles - essentiellement issus du domaine
de l’assurance - visant à rendre compte de la dépendance entre diverses sources de risques conduisent
à des copules archimédiennes. Une exception notable cependant est celle des modèles à facteurs, qui
jouent un rôle fondamental dans la description phénoménologique de l’interaction entre actifs financiers,
et dont les copules archimédiennes ne suffisent à rendre compte.
Avant d’aller plus loin, commençons par définir ce qu’est une copule archimédienne :
DÉFINITION 4 (COPULE ARCHIMÉDIENNE)
Soit ϕ une fonction continue, strictement décroissante de [0, 1] dans [0, ∞] et telle que ϕ(1) = 0. Soit
ϕ [−1] le pseudo-inverse de ϕ :
ϕ [−1]
ϕ−1 (t), si 0 ≤ t ≤ ϕ(0) ,
(t) =
(7.6)
0, si t ≥ ϕ(0) ,
alors la fonction
est une copule dite archimédienne de générateur ϕ.
C(u, v) = ϕ [−1] (ϕ(u) + ϕ(v)) (7.7)
On remarque donc que cette classe de copules est particulièrement simple dans la mesure où toute la
complexité de la structure de dépendance, décrite de manière générale par une fonction de deux (ou N)
variables, est totalement contenue dans l’expression du générateur ϕ, qui n’est qu’une fonction d’une
seule variable. On passe ainsi d’un problème multidimensionnel - et donc généralement délicat - à un
problème unidimensionnel, et ainsi beaucoup plus simple.
Parmi les nombreux membres de cette famille, citons par exemple la copule de Clayton :
Cθ(u, v) = max u θ + v θ
−1/θ
− 1 , 0 , (7.8)
mais aussi la copule de Gumbel, qui joue un rôle particulier pour la description de la dépendance dans la
théorie des valeurs extrêmes :
Cθ(u, v) = exp − (− ln u) θ + (− ln v) θ 1/θ
, (7.9)

186 7. Etude de la dépendance à l’aide des copules
ou encore la copule de Frank :
Cθ(u, v) = − 1
θ ln
1 + (e−θu − 1)(e−θv − 1)
e−θ
. (7.10)
− 1
Notons que les copules gaussiennes ou de Student ne font pas partie de cette famille. En fait, on peut
montrer que toutes les copules archimédiennes sont d’une part associatives 6 , ce que clairement les copules
gaussiennes et de Student ne satisfont pas, et d’autre part leurs valeurs sur la première bissectrice
vérifient C(u, u) < u pour tout u ∈]0, 1[. Réciproquement, on peut démontrer que (Nelsen 1998) toute
copule possédant ces deux propriétés est archimédienne, ce qui donne une idée de la généralité de cette
famille.
Enfin, précisons que Juri et Wüthrich (2002) ont établi, pour cette classe de copules, un théorème limite
équivalent au théorème de Gnedenko - Pikand - Balkema - de Haan (Embrechts et al. 1997, par exemple),
montrant la convergence vers la copule de Clayton (7.8) de la copule de la distribution des excédents de
plusieurs variables aléatoires au-dessus d’un seuil, quand celui-ci tend vers l’infini. Ainsi, la copule de
Clayton joue un rôle similaire (en dimension N) à celui des distributions de Pareto généralisées (un
dimension un). Ceci est particulièrement intéressant lorsque l’on s’intéresse à la statistique multivariée
des extrêmes.
7.3 Tests empiriques
Les tests empiriques visent à déterminer la nature de la copule permettant de décrire la structure de
dépendance entre deux actifs et peuvent être soit paramétriques soit non paramétriques. Cette dernière
méthode est bien évidemment beaucoup plus générale puisqu’elle ne requière pas le choix d’un modèle
a priori, et n’est donc pas sujette à l’erreur de spécification. Cependant, si le modèle est correctement
spécifié, l’estimation paramétrique est beaucoup plus précise. De plus, le petit nombre de paramètres intervenant
dans la description de la copule sélectionnée apparaît alors comme l’ensemble des paramètres
pertinents pour résumer les propriétés de dépendance entre actifs. Que l’on pense, par exemple, à la
représentation gaussienne (ou plus généralement à toute représentation en terme de distribution elliptique)
dont la dépendance est complètement capturée par le coefficient de corrélation linéaire. Ainsi, ces
paramètres vont pouvoir jouer un rôle particulier et il est tentant de les interpréter comme des variables
macroscopiques (ou phénoménologiques) synthétisant l’ensemble des interactions microscopiques entre
les agents économiques dont résulte la dépendance observée. Or, l’identification des “ bonnes variables”
est la première étape de la construction de modèles dont on peut espérer qu’ils seront mieux à même de
rendre compte des phénomènes observés. L’intérêt de l’estimation paramétrique est donc grand et c’est
pour cela que nous nous sommes tout particulièrement focalisés sur cette approche.
7.3.1 Tests paramétriques
Le paradigme gaussien a longtemps été en vigueur en finance. Certes, l’on a vu que les distributions
marginales ne sauraient être modélisées par des distributions gaussiennes dont les queues sont trop fines
pour rendre compte des grandes déviations observées sur les distributions des rendements, mais a priori,
la structure de dépendance entre deux actifs étant fort peu connue, rien ne permet de rejeter le fait que la
copule gaussienne soit à même de décrire correctement cette structure de dépendance. De plus, suivant
une idée initialement développée par Karlen (1998) et Sornette, Simonetti et Andersen (2000), il apparaît
6 Une copule est associative si C(u, C(v, w)) = C(C(u, v), w).

7.3. Tests empiriques 187
Y
5
4
3
2
1
0
−1
−2
−3
−4
Copule Gaussienne
−5
−5 −4 −3 −2 −1 0
X
1 2 3 4 5
Y
5
4
3
2
1
0
−1
−2
−3
−4
Copule de Student
−5
−5 −4 −3 −2 −1 0
X
1 2 3 4 5
FIG. 7.1 – Représentation des réalisations de deux variables aléatoires dont les marginales sont gaussiennes
et la copule est soit gaussienne (figure de gauche) soit de Student avec trois degrés de liberté
(figure de droite), le coefficient de corrélation étant le même : ρ = 0.8.
que la copule gaussienne est obtenue de manière naturelle à partir du principe de maximum d’entropie 7 .
En outre, cette copule est sans doute la copule la plus simple de la classe des copules elliptiques, puisqu’elle
est entièrement spécifiée par son seul coefficient de corrélation et ne dépend donc que d’un seul
paramètre, contrairement à la copule de Student, par exemple, qui est aussi fonction d’un certain nombre
de degrés de liberté. Ainsi, la copule gaussienne semble être un point de départ logique pour l’étude de la
dépendance entre actifs et nous avons donc choisi de tester sa capacité à rendre compte des données (voir
Malevergne et Sornette (2001b), présenté au chapitre suivant). Les tests ont été conduits sur différents
types d’actifs : des actions, des taux de change et des matières premières (métaux). Pour ces deux derniers,
la copule gaussienne ne semble pas une très bonne approximation. En fait, pour les métaux, elle est
quasi systématiquement rejetée, tandis que pour les monnaies la situation est plus complexe. Il semble
en effet que pour les monnaies pour lesquelles n’existe pas de mécanisme de régulation de la parité l’hypothèse
de copule gaussienne soit raisonnable alors que dans le cas contraire (cf. les monnaies des pays
membres du SME) cette hypothèse est fortement rejetée, ce qui, somme toute, est parfaitement cohérent.
Pour ce qui est des actions, la copule gaussienne semble fournir une bonne approximation, y compris
pour des actions appartenant à un même secteur d’activité.
D’autres tests paramétriques ont été menés par Klugman et Parsa (1999) sur des données touchant aux
domaines de l’assurance, où il semble que la famille des copules de Frank soient à même de rendre
compte de la dépendance pour ce genre de données. Patton (2001) s’est quant à lui intéressé plus en
détail à la dépendance entre taux de change, et conclut que la copule gaussienne n’est pas la mieux
adaptée. Au contraire, la copule de Clayton fournit une meilleure description des données.
En fait, une des limites des tests que nous avons conduits sur les copules gaussiennes est de ne pas
pouvoir distinguer une copule de Student d’une copule gaussienne dès lors que le nombre de degrés
de liberté de celle-là devient trop grand. Typiquement, lorsque le nombre de degrés de liberté dépasse
quinze ou vingt, les tests que nous avons utilisés ne sont plus capables de distinguer entre ces deux types
de copules. Ceci n’est pas très grave tant que l’on ne s’occupe que des événements “normaux” et pas des
événements extrêmes. En effet, comme nous l’avons déjà signalé, les copules de Student et les copules
gaussiennes sont très semblables dans leur cœur, et ce n’est que lorsque l’on s’en éloigne que l’on devient
7 Pour d’autres exemples de détermination de distributions à l’aide du principe de maximum d’entropie, voir notamment
Rockinger et Jondeau (2002)

188 7. Etude de la dépendance à l’aide des copules
sensible à la différence. Concrètement, la distinction essentielle entre ces deux familles provient du fait
que les copules gaussiennes produisent des événements extrêmes indépendants tandis que les copules de
Student produisent des événements extrêmes concomitants avec une probabilité non nulle, celle-ci étant
d’autant plus importante que le nombre de degrés de liberté de la copule est faible et que la corrélation
est importante. Ceci est clairement illustré par la figure 7.1. où l’on a représenté les réalisations de deux
variables aléatoires dont les marginales sont gaussiennes et la copule est soit gaussienne soit de Student
avec trois degrés de liberté, le coefficient de corrélation étant le même : ρ = 0.8.
Pour quantifier cette propension des extrêmes à se produire simultanément, il est commode d’introduire
le coefficient de dépendance de queue λ défini comme la probabilité que l’une des variables, X par
exemple, soit extrême conditionnée au fait que l’autre variable, Y , soit elle-même extrême. Ceci nous
conduit à la définition mathématique suivante :
λ = lim
u→1 Pr{X > FX −1 (u) | Y > FY −1 (u)} (7.11)
= lim
u→1
1 − 2u + C(u, u)
, (7.12)
1 − u
qui démontre que le coefficient de dépendance de queue n’est fonction que de la copule et non des
marginales.
Dans le cas de la copule gaussienne, il est démontré que la dépendance de queue est nulle pour toute
valeur du coefficient de corrélation (différent de un), tandis que pour les copules de Student, elle dépend
à la fois de la corrélation et du nombre de degrés de liberté. Dans le premier cas, on dit que les extrêmes se
produisent asymptotiquement indépendamment alors que dans le second cas ils sont asymptotiquement
dépendants. Cette différence de comportement est illustrée sur la figure 7.1, où dans le cas de la copule
de Student on remarque que les points se répartissent à l’intérieur de fuseaux de plus en plus étroits au fur
et à mesure que l’on considère les réalisations de plus en plus extrêmes. On observe que ce phénomène
se produit non seulement dans les quadrans inférieur-gauche et supérieur-droit, mais aussi dans les deux
autres quadrans, ce qui vient du fait que le coefficient de dépendance de queue n’est pas nul pour les
coefficients de corrélation négatifs (voir figure 1 dans Malevergne et Sornette (2001b) page 214).
L’utilisation du coefficient de dépendance de queue nous a permis d’estimer le risque potentiel qu’il y
a à considérer une modélisation en terme de copule gaussienne, et il s’avère que pour des corrélations
élevées entre actifs il est dangereux d’utiliser cette représentation tant que l’on n’a pas d’idée plus précise
sur la dépendance de queue, quantité à laquelle ne nous donne pas accès les tests paramétriques. En effet,
choisir une forme paramétrique de la copule revient à fixer a priori l’existence ou non de dépendance de
queue et ne permet donc pas de trancher cette question. Donc, pour aller plus avant, une approche non
paramétrique pourrait s’avérer utile.
7.3.2 Tests non paramétriques
Les tests non paramétriques ont, par rapport aux tests paramétriques, l’avantage d’être beaucoup plus
généraux puisqu’ils ne conduisent pas à imposer a priori de modèle. Donc, on peut espérer qu’ils apporteront
une réponse au problème auquel nous sommes désormais confrontés. Diverses méthodes d’estimation
non paramétriques ont vu le jour. Citons notamment les approches basées sur l’estimation à
l’aide de familles de polynômes, tels les polynômes de Bernstein (Li, Mikusinski et Taylor 1998), et que
Durrleman, Nikeghbali et Roncalli (2000) passent en revue. Citons encore les méthodes s’appuyant sur
des estimateurs à noyau, telle celle développée par Scaillet (2000b).
Toutes ces méthodes ont l’avantage de produire des copules estimées qui sont lisses et dérivables en
tout point, ce qui présente un grand nombre d’avantages lorsque l’on souhaite les réemployer à des

7.4. Conclusion 189
fins plus appliquées. Cela permet notamment de réaliser des études de sensibilité, ou de simuler des
variables aléatoires ayant la même copule que celle qui a été estimée (Embrechts et al. 2002). Cependant,
cet avantage est en même temps le principal défaut de ces méthodes. En effet, on peut montrer très
simplement (Durrleman et al. 2000) que toute copule dérivable dans le voisinage du point (1, 1) (ou du
point (0, 0)) n’admet pas de dépendance de queue. En effet, il est nécessaire que la copule ne soit pas
dérivable au voisinage de (1, 1) pour que λ soit non nul 8 .
Ainsi, les méthodes d’estimation non paramétriques de la copule que nous venons de mentionner souffrent
toutes du même défaut : négliger les événements extrêmes concomitants. Nous avons cherché
d’autres méthodes d’estimation des copules qui ne pâtissent pas de ce problème, mais nos recherches
se sont révélées infructueuses. Aussi avons-nous choisi d’attaquer le problème de l’estimation de la
dépendance de queue de manière directe. En effet, cela nous est apparu être le seul moyen de décider
si la copule gaussienne, qui est une bonne approximation dans le cœur, demeurait satisfaisante dans les
extrêmes ou bien s’il valait mieux considérer les copules de Student, dont nous avons rappelé qu’elles
étaient très proches de la copule gaussienne dans la région centrale mais qu’elles conduisaient à des
réalisations extrêmes pouvant se produirent simultanément.
7.4 Conclusion
Les copules fournissent le moyen le plus simple, le plus complet et le plus naturel de décrire et de mesurer
la dépendance entre plusieurs variables aléatoires. Parmi la très grande variété de copules, quelques
familles présentent un intérêt particulier pour la modélisation des interactions entre actifs financiers,
comme par exemple la famille des copules elliptiques, dont les copules gaussiennes et les copules de
Student sont les exemples les plus célèbres, mais aussi la famille des copules archimédiennes. Le rôle
prépondérant de ces deux familles vient du fait qu’elles peuvent être reliées simplement à des modèles
phénoménologiques standards en sciences financières (ou actuarielles).
Ceci nous a incité à tenter d’estimer la copule décrivant l’interaction entre actifs financiers. Il nous est
apparu que la copule gaussienne pouvait être un bon candidat, ce que nos tests ont confirmé, du moins
pour ce qui est des actions. Cependant, une forte incertitude demeure sur le fait de savoir si une copule
de Student avec un nombre de degrés de liberté élevés ne serait pas mieux adaptée. Tout l’enjeu de cette
question est de savoir si les extrêmes ont tendance à se produire de manière simultanée pour les divers
actifs ou indépendamment, point crucial pour la gestion des risques et la stratégie à mettre en œuvre pour
les diversifier. Ceci n’ayant pas pu être résolu par une approche globale, nous allons devoir trouver une
méthode permettant l’estimation directe de la dépendance de queue, ce qui sera l’objet du chapitre 9.
8 Notons au passage que cette condition nécessaire n’est cependant pas suffisante comme le prouve l’exemple de la copule
gaussienne qui ne présente pas de dépendance de queue, mais n’en est pas pour autant dérivable en (1, 1).

190 7. Etude de la dépendance à l’aide des copules

Chapitre 8
Tests de copule gaussienne
Nous utilisons la propriété d’invariance des copules par changement de variable strictement croissant
pour tester l’hypothèse nulle selon laquelle la dépendance entre actifs financiers peut-être modélisée à
l’aide d’une copule gaussienne. Nous trouvons que certaines devises ainsi que la plupart des principales
actions cotées sur le NYSE sont compatibles avec cette hypothèse, tandis qu’elle est fortement rejetée
quand il s’agit de décrire la dépendance entre matières premières (métaux). Cependant, en dépit de
l’apparente qualification de l’hypothèse de copule gaussienne pour la plupart des actions et devises, une
copule non gaussienne, telle qu’une copule de Student, ne peut être rejetée par nos tests si elle a un
nombre de degrés de liberté suffisamment grand . En conséquence, il peut être dangereux d’accepter
aveuglément l’hypothèse de copule gaussienne, tout particulièrement quand le coefficient de corrélation
entre les paires d’actifs est important de sorte que l’existence d’une dépendance de queue, totalement
ignorée par la copule gaussienne, peut conduire à négliger des événements extrêmes se produisant de
manière concomitante.
191

192 8. Tests de copule gaussienne

Testing the Gaussian Copula Hypothesis
for Financial Assets Dependences ∗
Y. Malevergne 1,2 and D. Sornette 1,3
1 Laboratoire de Physique de la Matière Condensée CNRS UMR 6622
Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France
2 Institut de Science Financière et d’Assurances - Université Lyon I
43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex
3 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science
University of California, Los Angeles, California 90095, USA
Corresponding author: D. Sornette
Institute of Geophysics and Planetary Physics
University of California, Los Angeles, California 90095, USA
email: sornette@ess.ucla.edu tel: (310) 825 28 63 Fax: (310) 206 30 51
Submitted to Quantitative Finance
Abstract
Using one of the key property of copulas that they remain invariant under an arbitrary monotonous
change of variable, we investigate the null hypothesis that the dependence between financial assets
can be modeled by the Gaussian copula. We find that most pairs of currencies and pairs of major
stocks are compatible with the Gaussian copula hypothesis, while this hypothesis can be rejected for
the dependence between pairs of commodities (metals). Notwithstanding the apparent qualification
of the Gaussian copula hypothesis for most of the currencies and the stocks, a non-Gaussian copula,
such as the Student’s copula, cannot be rejected if it has sufficiently many “degrees of freedom”. As a
consequence, it may be very dangerous to embrace blindly the Gaussian copula hypothesis, especially
when the correlation coefficient between the pair of asset is too high as the tail dependence neglected
by the Gaussian copula can be as large as 0.6, i.e., three out five extreme events which occur in unison
are missed.
JEL Classification: C12, C15, F31, G19
Keywords: Copulas, Dependence Modelisation, Risk Management, Tail Dependence.
∗ We acknowledge helpful discussions and exchanges with J. Andersen, P. Embrechts, J.P. Laurent, F. Lindskog, V. Pisarenko
and R. Valkanov. This work was partially supported by the James S. Mc Donnell Foundation 21st century scientist
award/studying complex system.
1
193

194 8. Tests de copule gaussienne
1 Introduction
The determination of the dependence between assets underlies many financial activities, such as risk
assessment and portfolio management, as well as option pricing and hedging. Following (Markovitz
1959), the covariance and correlation matrices have, for a long time, been considered as the main tools
for quantifying the dependence between assets. But the dimension of risk captured by the correlation
matrices is only satisfying for elliptic distributions and for moderate risk amplitudes (Sornette et al.
2000a). In all other cases, this measure of risk is severely incomplete and can lead to a very strong
underestimation of the real incurred risks (Embrechts et al. 1999).
Although the unidimensional (marginal) distributions of asset returns are reasonably constrained
by empirical data and are more or less satisfactorily described by a power law with tail index ranging
between 2 and 4 (De Vries 1994, Lux 1996, Pagan 1996, Guillaume et al. 1997, Gopikrishnan et al. 1998)
or by stretched exponentials (Laherrère and Sornette 1998, Gouriéroux and Jasiak 1999, Sornette et
al. 2000a, Sornette et al. 2000b), no equivalent results have been obtained for multivariate distributions
of asset returns. Indeed, a brute force determination of multivariate distributions is unreliable due to the
limited data set (the curse of dimensionality), while the sole knowledge of marginals (one-point statistics)
of each asset is not sufficient to obtain information on the multivariate distribution of these assets which
involves all the n-points statistics.
Some progress may be expected from the concept of copulas, recently proposed to be useful for financial
applications (Embrechts et al. 2001, Frees and Valdez 1998, Haas 1999, Klugman and Parsa 1999).
This concept has the desirable property of decoupling the study of the marginal distribution of each asset
from the study of their collective behavior or dependence. Indeed, the dependence between assets is
entirely embedded in the copula, so that a copula allows for a simple description of the dependence structure
between assets independently of the marginals. For instance, assets can have power law marginals
and a Gaussian copula or alternatively Gaussian marginals and a non-Gaussian copula, and any possible
combination thereof. Therefore, the determination of the multivariate distribution of assets can be
performed in two steps : (i) an independent determination of the marginal distributions using standard
techniques for distributions of a single variable ; (ii) a study of the nature of the copula characterizing
completely the dependence between the assets. This exact separation between the marginal distributions
and the dependence is potentially very useful for risk management or option pricing and sensitivity analysis
since it allows for testing several scenarios with different kind of dependences between assets while
the marginals can be set to their well-calibrated empirical estimates. Such an approach has been used by
(Embrechts et al. 2001) to provide various bounds for the Value-at-Risk of a portfolio made of depend
risks, and by (Rosenberg 1999) or (Cherubini and Luciano 2000) to price and to analyse the pricing
sensitivity of binary digital options or options on the minimum of a basket of assets.
A fundamental limitation of the copula approach is that there is in principle an infinite number of possible
copulas (Genest and MacKay 1986, Genest 1987, Genest and Rivest 1993, Joe 1993, Nelsen 1998)
and, up to now, no general empirical study has determined the classes of copulas that are acceptable for
financial problems. In general, the choice of a given copula is guided both by the empirical evidences
and the technical constraints, i.e., the number of parameters necessary to describe the copula, the possibility
to obtain efficient estimators of these parameters and also the possiblity offered by the chosen
parameterization to allow for tractable analytical calculation. It is indeed sometimes more advantageous
to prefer a simplest copula to one that fit better the data, provided that we can clearly quantify the effects
of this substitution.
In this vein, the first goal of the present article is to show that, in most cases, the Gaussian copula
2

can provide an approximation of the unknown true copula that is sufficiently good so that it cannot be
rejected on a statistical basis. Our second goal is to draw the consequences of the parameterization
involved in the Gaussian copula in term of potential over/underestimation of the risks, in particular for
large and extreme events.
The paper is organized as follows.
In section 2, we first recall some important general definitions and theorems about copulas that will
be useful in the sequel. We then introduce the concept of tail dependence that will allow us to quantify
the probability that two extreme events might occur simultaneously. We define and describe the two
copulas that will be at the core of our study : the Gaussian copula and the Student’s copula and compare
their properties particularly in the tails.
In section 3, we present our statistical testing procedure which is applied to pairs of financial time
series. First of all, we determine a test statistics which leads us to compare the empirical distribution of
the data with a χ 2 -distribution using a bootstrap method. We also test the sensitivity of our procedure
by applying it to synthetic multivariate Student’s time series. This allows us to determine the minimum
statistical test value needed to be able to distinguish between a Gaussian and a Student’s copula, as a
function of the number of degrees of freedom and of the correlation strength.
Section 4 presents the empirical results obtained for the following assets which are combined pairwise
in the test statistics:
• 6 currencies,
• 6 metals traded on the London Metal Exchange,
• 22 stocks choosen among the largest companies quoted on the New York Stocks Exchange.
We show that the Gaussian copula hypothesis is very reasonnable for most stocks and currencies, while
it is hardly compatible with the description of multivariate behavior for metals.
Section 5 summarizes our results and concludes.
2 Generalities about copulas
2.1 Definitions and important results about copulas
This section does not pretend to provide a rigorous mathematical exposition of the concept of copula. We
only recall a few basic definitions and theorems that will be useful in the following (for more information
about the concept of copula, see for instance (Lindskog 1999, Nelsen 1998)).
We first give the definition of a copula of n random variables.
DEFINITION 1 (COPULA)
A function C : [0, 1] n −→ [0, 1] is a n-copula if it enjoys the following properties :
• ∀u ∈ [0, 1], C(1, · · · , 1, u, 1 · · · , 1) = u ,
• ∀ui ∈ [0, 1], C(u1, · · · , un) = 0 if at least one of the ui equals zero ,
3
195

196 8. Tests de copule gaussienne
• C is grounded and n-increasing, i.e., the C-volume of every boxes whose vertices lie in [0, 1] n is
positive.
It is clear from this definition that a copula is nothing but a multivariate distribution with support
in [0,1] n and with uniform marginals. The fact that such copulas can be very useful for representing
multivariate distributions with arbitrary marginals is seen from the following result.
THEOREM 1 (SKLAR’S THEOREM)
Given an n-dimensional distribution function F with continuous marginal (cumulative) distributions
F1, · · · , Fn, there exists a unique n-copula C : [0, 1] n −→ [0, 1] such that :
F (x1, · · · , xn) = C(F1(x1), · · · , Fn(xn)) . (1)
This theorem provides both a parameterization of multivariate distributions and a construction scheme
for copulas. Indeed, given a multivariate distribution F with marginals F1, · · · , Fn, the function
C(u1, · · · , un) = F F −1
1 (u1), · · · , F −1
n (un)
is automatically a n-copula. This copula is the copula of the multivariate distribution F . We will use this
method in the sequel to derive the expressions of standard copulas such as the Gaussian copula or the
Student’s copula.
A very powerful property of copulas is their invariance under arbitrary strictly increasing mapping
of the random variables :
THEOREM 2 (INVARIANCE THEOREM)
Consider n continuous random variables X1, · · · , Xn with copula C. Then, if g1(X1), · · · , gn(Xn) are
strictly increasing on the ranges of X1, · · · , Xn, the random variables Y1 = g1(X1), · · · , Yn = gn(Xn)
have exactly the same copula C.
It is this result that shows us that the full dependence between the n random variables is completely
captured by the copula, independently of the shape of the marginal distributions. This result is at the
basis of our statistical study presented in section 3.
2.2 Dependence between random variables
The dependence between two time series is usually described by their correlation coefficient. This measure
is fully satisfactory only for elliptic distributions (Embrechts et al. 1999), which are functions of a
quadratic form of the random variables, when one is interested in moderately size events. However, an
important issue for risk management concerns the determination of the dependence of the distributions in
the tails. Practically, the question is whether it is more probable that large or extreme events occur simultaneously
or on the contrary more or less independently. This is refered to as the presence or abscence
of “tail dependence”.
The tail dependence is also an interesting concept in studying the contagion of crises between markets
or countries. These questions have recently been addressed by (Ang and Cheng 2001, Longin and Solnik
2001, Starica 1999) among several others. Large negative moves in a country or market are often found
to imply large negative moves in others.
Technically, we need to determine the probability that a random variable X is large, knowing that
the random variable Y is large.
4
(2)

DEFINITION 2 (TAIL DEPENDENCE 1)
Let X and Y be random variables with continuous marginals FX and FY . The (upper) tail dependence
coefficient of X and Y is, if it exists,
lim
u→1
Pr{X > F −1
X
197
−1
(u)|Y > F (u)} = λ ∈ [0, 1] . (3)
In words, given that Y is very large (which occurs with probability 1 − u), the probability that X is very
large at the same probability level u defines asymptotically the tail dependence coefficient λ.
It turns out that this tail dependence is a pure copula property which is independent of the marginals. Let
C be the copula of the variables X and Y , then
THEOREM 3
if the bivariate copula C is such that
lim
u→1
¯C(u, u)
1 − u
Y
= λ (4)
exists (where ¯ C(u, u) = 1 − 2u − C(u, u)), then C has an upper tail dependence coefficient λ.
If λ > 0, the copula presents tail dependence and large events tend to occur simultanously, with the
probabilty λ. On the contrary, when λ = 0, the copula has no tail dependence in this sense and large
events appear to occur essentially independently. There is however a subtlety in this definition of tail
dependence. To make it clear, first consider the case where for large X and Y the distribution function
F (x, y) factorizes such that
lim
x,y→∞
F (x, y)
= 1 . (5)
FX(x)FY (y)
This means that, for X and Y sufficiently large, these two variables can be considered as independent. It
is then easy to show that
lim
u→1
Pr{X > F −1
X
−1
(u)|Y > F (u)} = lim
Y
u→1
so that independent variables really have no tail dependence, as one can expect.
−1
1 − FX(FX (u)) (6)
= lim
u→1 1 − u = 0, (7)
Unfortunatly, the converse does not holds : a value λ = 0 does not automatically imply true independence,
namely that F (x, y) satisfies equation (5). Indeed, the tail independence criterion λ = 0 may
still be associated with an absence of factorization of the multivariate distribution for large X and Y .
In a weaker sense, there may still be a dependence in the tail even when λ = 0. Such behavior is for
instance exhibited by the Gaussian copula, which has zero tail dependence according to the definition 2
but nevertheless does not have a factorizable multivariate distribution, since the non-diagonal term of the
quadratic form in the exponential function does not become negligible in general as X and Y go to infinity.
To summarize, the tail independence, according to definition 2, is not equivalent to the independence
in the tail as defined in equation (5).
After this brief review of the main concepts underlying copulas, we now present two special families
of copulas : the Gaussian copula and the Student’s copula.
5

198 8. Tests de copule gaussienne
2.3 The Gaussian copula
The Gaussian copula is the copula derived from the multivariate Gaussian distribution. Let Φ denote
the standard Normal (cumulative) distribution and Φρ,n the n-dimensional Gaussian distribution with
correlation matrix ρ. Then, the Gaussian n-copula with correlation matrix ρ is
whose density
reads
−1
Cρ(u1, · · · , un) = Φρ,n Φ (u1), · · · , Φ −1 (un) , (8)
cρ(u1, · · · , un) =
cρ(u1, · · · , un) = ∂Cρ(u1, · · · , un)
∂u1 · · · ∂un
1
√ exp −
det ρ 1
2 yt (u) (ρ−1
− Id)y (u)
with yk(u) = Φ −1 (uk). Note that theorem 1 and equation (2) ensure that Cρ(u1, · · · , un) in equation (8)
is a copula.
As we said before, the Gaussian copula does not have a tail dependence :
lim
u→1
¯Cρ(u, u)
1 − u
(9)
(10)
= 0, ∀ρ ∈ (−1, 1). (11)
This results is derived for example in (Embrechts et al. 2001). But this does not mean that the Gaussian
copula goes to the independent (or product) copula Π(u1, u2) = u1 · u2 when (u1, u2) goes to one.
Indeed, consider a distribution F (x, y) with Gaussian copula :
Its density is
where fX and fY are the densities of X and Y . Thus,
F (x, y) = Cρ(FX(x), FY (y)). (12)
f(x, y) = cρ(FX(x), FY (y)) · fX(x) · fY (y), (13)
f(x, y)
lim
= lim
(x,y)→∞ fX(x) · fY (y) (x,y)→∞ cρ(FX(x), FY (y)), (14)
which should equal 1 if the variables X and Y were independent in the tail. Reasoning in the quantile
space, we set x = F −1
−1
X (u) and y = FY (u), which yield
f(x, y)
lim
= lim
(x,y)→∞ fX(x) · fY (y) u→1 cρ(u, u). (15)
Using equation (10), it is now obvious to show that cρ(u, u) goes to one when u goes to one, if and
only if ρ = 0 which is equivalent to Cρ=0(u1, u2) = Π(u1, u2) for every (u1, u2). When ρ > 0, cρ(u, u)
goes to infinity, while for ρ negative, cρ(u, u) goes to zero as u → 1. Thus, the dependence structure
described by the Gaussian copula is very different from the dependence structure of the independent
copula, except for ρ = 0.
The Gaussian copula is completly determined by the knowledge of the correlation matrix ρ. The
parameters involved in the description of the Gaussian copula are very simple to estimate, as we shall
see in the following.
6

In our tests presented below, we focus on pairs of assets, i.e., on Gaussian copulas involving only
two random variables. Testing the Gaussian copula hypothesis for two random variables gives useful
information for a larger number of dependent variables constituting a large basket or portfolio. Indeed,
let us assume that each pair (a, b), (b, c) and (c, a) have a gaussian copula. Then, the triplet (a, b, c) has
also a Gaussian copula. This result generalizes to an arbitrary number of random variables.
2.4 The Student’s copula
The Student’s copula is derived from the Student’s multivariate distribution. Given a multivariate Student’s
distribution Tρ,ν with ν degrees of freedom and a correlation matrix ρ
1 Γ
Tρ,ν(x) = √
det ρ
ν+n
2
Γ
ν
2 (πν) N/2
the corresponding Student’s copula reads :
Cρ,ν(u1, · · · , un) = Tρ,ν
x1
−∞
xN
· · ·
−∞
dx
1 + xt ρx
ν
ν+n
2
199
, (16)
t −1
ν (u1), · · · , t −1
ν (un) , (17)
where tν is the univariate Student’s distribution with ν degrees of freedom. The density of the Student’s
copula is thus
where yk = t −1
ν (uk).
cρ,ν(u1, · · · , un) =
1 Γ
√
det ρ
ν+n ν
2 Γ
ν+1 Γ 2
n−1 2
n
n
k=1
1 + y2 k
ν
1 + yt ρy
ν
ν+1
2
ν+n
2
, (18)
Since the Student’s distribution tends to the normal distribution when ν goes to infinity, the Student’s
copula tends to the Gaussian copula as ν → +∞. In contrast to the Gaussian copula, the Student’s
copula for ν finite presents a tail dependence given by :
λν(ρ) = lim
u→1
¯Cρ,ν(u, u)
1 − u
= 2¯tν+1
√ √
ν + 1 1 − ρ
√
1 + ρ
, (19)
where ¯tν+1 is the complementary cumulative univariate Student’s distribution with ν + 1 degrees of
freedom (see (Embrechts et al. 2001) for the proof). Figure 1 shows the upper tail dependence coefficient
as a function of the correlation coefficient ρ for different values of the number ν of degrees of freedom.
As expected from the fact that the Student’s copula becomes identical to the Gaussian copula for ν →
+∞ for all ρ = 1, λν(ρ) exhibits a regular decay to zero as ν increases. Moreover, for ν sufficiently large,
the tail dependence is significantly different from 0 only when the correlation coefficient is sufficiently
close to 1. This suggests that, for moderate values of the correlation coefficient, a Student’s copula with
a large number of degrees of freedom may be difficult to distinguish from the Gaussian copula from a
statistical point of view. This statement will be made quantitative in the following.
Figure 2 presents the same information in a different way by showing the maximum value of the
correlation coefficient ρ as a function of ν, below which the tail dependence λν(ρ) of a Student’s copula
is smaller than a given small value, here taken equal to 1%, 2.5%, 5% and 10%. The choice λν(ρ) = 5%
for instance corresponds to 1 event in 20 for which the pair of variables are asymptotically coupled. At
7

200 8. Tests de copule gaussienne
the 95% probability level, values of λν(ρ) ≤ 5% are undistinguishable from 0, which means that the
Student’s copula can be approximated by a Gaussian copula.
The description of a Student’s copula relies on two parameters : the correlation matrix ρ, as in the
Gaussian case, and in addition the number of degrees of freedom ν. The estimation of the parameter ν
is rather difficult and this has an important impact on the estimated value of the correlation matrix. As a
consequence, the Student’s copula is more difficult to calibrate and use than the Gaussian copula.
3 Testing the Gaussian copula hypothesis
In view of the central role that the Gaussian paradigm has played and still plays in particular in finance, it
is natural to start with the simplest choice of dependence between different random variables, namely the
Gaussian copula. It is also a natural first step as the Gaussian copula imposes itself in an approach which
consists in (1) performing a nonlinear transformation on the random variables into Normal random variables
(for the marginals) which is always possible and (2) invoking a maximum entropy principle (which
amounts to add the least additional information in the Shannon sense) to construct the multivariable distribution
of these Gaussianized random variables (Sornette et al. 2000a, Sornette et al. 2000b, Andersen
and Sornette 2001).
In the sequel, we will denote by H0 the null hypothesis according to which the dependence between
two (or more) random variables X and Y can be described by the Gaussian copula.
3.1 Test Statistics
We now derive the test statistics which will allow us to reject or not our null hypothesis H0 and state the
following proposition:
PROPOSITION 1
Assuming that the N-dimensionnal random vector x = (x1, · · · , xN) with distribution function F and
marginals Fi, satisfies the null hypothesis H0, then, the variable
where the matrix ρ is
z 2 =
N
j,i=1
follows a χ 2 -distribution with N degrees of freedom.
Φ −1 (Fi(xi)) (ρ −1 )ij Φ −1 (Fj(xj)), (20)
ρij = Cov[Φ −1 (Fi(xi)), Φ −1 (Fj(xj))], (21)
To proove the proposition above, first consider an N-dimensionnal random vector x = (x1, · · · , xN).
Let us denote by F its distribution function and by Fi the marginal distribution of each xi. Let us now
assume that the distribution function F satisfies H0, so that F has a Gaussian copula with correlation
matrix ρ while the Fi’s can be any distribution function. According to theorem 1, the distribution F can
be represented as :
F (x1, · · · , xN) = Φρ,N(Φ −1 (F1(x1)), · · · , Φ −1 (FN(xN))) . (22)
Let us now transform the xi’s into Normal random variables yi’s :
yi = Φ −1 (Fi(xi)) . (23)
8

Since the mapping Φ −1 (Fi(·)) is obviously increasing, theorem 2 allows us to conclude that the copula
of the variables yi’s is identical to the copula of the variables xi’s. Therefore, the variables yi’s have
Normal marginal distributions and a Gaussian copula with correlation matrix ρ. Thus, by definition, the
multivariate distribution of the yi’s is the multivariate Gaussian distribution with correlation matrix ρ :
201
G(y) = Φρ,N(Φ −1 (F1(x1)), · · · , Φ −1 (FN(xN))) (24)
= Φρ,N(y1, · · · , yN), (25)
and y is a Gaussian random vector. From equations (24-25), we obviously have
Consider now the random variable
ρij = Cov[Φ −1 (Fi(xi)), Φ −1 (Fj(xj))]. (26)
z 2 = y t ρ −1 y =
N
yi (ρ −1 )ij yj , (27)
i,j=1
where · t denotes the transpose operator. This variable has already been considered in (Sornette et al.
2000a) in preliminary statistical tests of the transformation (23). It is well-known that the variable z 2
follows a χ 2 -distribution with N degrees of freedom. Indeed, since y is a Gaussian random vector with
covariance matrix 1 ρ, it follows that the components of the vector
˜y = ρ −1/2 y, (28)
are independent Normal random variables. Here, ρ −1/2 denotes the square root of the matrix ρ −1 , which
can be obtain by the Cholevsky decomposition, for instance. Thus, the sum ˜y t ˜y = z 2 is the sum of the
squares of N independent Normal random variables, which follows a χ 2 -distribution with N degrees of
freedom.
3.2 Testing procedure
The testing procedure used in the sequel is now described. We consider two financial series (N = 2) of
size T : {x1(1), · · · , x1(t), · · · , x1(T )} and {x2(1), · · · , x2(t), · · · , x2(T )}. We assume that the vectors
x(t) = (x1(t), x2(t)), t ∈ {1, · · · , T } are independent and identicaly distributed with distribution F ,
which implies that the variables x1(t) (respectively x2(t)), t ∈ {1, · · · , T }, are also independent and
identically distributed, with distributions F1 (respectively F2).
The cumulative distribution ˆ Fi of each variable xi, which is estimated empirically, is given by
ˆFi(xi) = 1
T
T
1 {xi

202 8. Tests de copule gaussienne
The sample covariance matrix ˆρ is estimated by the expression :
which allows us to calculate the variable
ˆz 2 (k) =
ˆρ = 1
T
2
i,j=1
T
ˆy(i) · ˆy(i) t
i=1
(31)
ˆyi(k) ( ˆ
ρ −1 )ij ˆyj(k) , (32)
as defined in (27) for k ∈ {1, · · · , T }, which should be asymptotically distributed according to a χ 2 -
distribution if the Gaussian copula hypothesis is correct.
The usual way for comparing an empirical with a theoretical distribution is to measure the distance
between these two distributions and to perform the Kolmogorov test or the Anderson-Darling (Anderson
and Darling 1952) test (for a better accuracy in the tails of the distribution). The Kolmogorov distance
is the maximum local distance along the quantile which most often occur in the bulk of the distribution,
while the Anderson-Darling distance puts the emphasis on the tails of the two distributions by a suitable
normalization. We propose to complement these two distances by two additional measures which are
defined as averages of the Kolmogorov distance and of the Anderson-Darling distance respectively:
Kolmogorov : d1 = max |Fz2(z z
2 ) − Fχ2(z 2 )|
(33)
average Kolmogorov : d2 = |Fz2(z 2 ) − Fz2(z 2 )| dFχ2(z 2 ) (34)
Anderson − Darling : d3 = max
z
average Anderson − Darling : d4 =
|F z 2(z 2 ) − F χ 2(z 2 )|
F χ 2(z 2 )[1 − F χ 2(z 2 )]
(35)
|Fz2(z 2 ) − Fχ2(z2 )|
Fχ2(z2 )[1 − Fχ2(z2 dFχ2(z )]
2 ) (36)
The Kolmogorov distance d1 and its average d2 are more sensitive to the deviations occurring in the bulk
of the distributions. In contrast, the Anderson-Darling distance d3 and its average d4 are more accurate
in the tails of the distributions. We present our statistical tests for these four distances in order to be as
complete as possible with respect to the different sensitivity of the tests.
The distances d2 and d4 are not of common use in statistics, so let us justify our choice. One usually
uses distances similar to d2 and d4 but which differ by the square instead of the modulus of F z 2(z 2 ) −
F χ 2(z 2 ) and lead respectively to the ω-test and the Ω-test, whose statitics are theoretically known. The
main advantage of the distances d2 and d4 with respect to the more usual distances ω and Ω is that they
are simply equal to the average of d1 and d3. This averaging is very interesting and provides important
information. Indeed, the distances d1 and d3 are mainly controlled by the point that maximizes the
argument within the max(·) function. They are thus sensitive to the presence of an outlier. By averaging,
d2 and d4 become less sensitive to outliers, since the weight of such points is only of order 1/T (where
T is the size of the sample) while it equals one for d1 and d3. Of course, the distances ω and Ω also
perform a smoothing since they are averaged quantities too. But they are the average of the square of
d1 and d3 which leads to an undesired overweighting of the largest events. In fact, this weight function
is chosen as a convenient analytical form that allows one to derive explicitely the theoretical asymptotic
statistics for the ω and Ω-tests. In contrast, using the modulus of F z 2(z 2 ) − F χ 2(z 2 ) instead of its
square in the expression of d2 and d4, no theoretical test statistics can be derived analytically. In other
10

words, the presence of the square instead of the modulus of F z 2(z 2 ) − F χ 2(z 2 ) in the definition of
the distances ω and Ω is motivated by mathematical convenience rather than by statistical pertinence.
In sum, the sole advantage of the standard distances ω and Ω with respect to the distances d2 and d4
introduced here is the theoretical knowledge of their distributions. However, this advantage disappears
in our present case in which the covariance matrix is not known a priori and needs to be estimated from
the empirical data: indeed, the exact knowledge of all the parameters is necessary in the derivation of
the theoretical statistics of the ω and Ω-tests (as well as the Kolmogorov test). Therefore, we cannot
directly use the results of these standard statistical tests. As a remedy, we propose a bootstrap method
(Efron and Tibshirani 1986), whose accuracy is proved by (Chen and Lo 1997) to be at least as good
as that given by asymptotic methods used to derive the theoretical distributions. For the present work,
we have determined that the generation of 10,000 synthetic time series was sufficient to obtain a good
approximation of the distribution of distances described above. Since a bootstrap method is needed to
determine the tests statistics in every case, it is convenient to choose functional forms different from the
usual ones in the ω and Ω-tests as they provide an improvement with respect to statistical reliability, as
obtained with the d2 and d4 distances introduced here.
To summarize, our test procedure is as follows.
1. Given the original time series x(t), t ∈ {1, · · · , T }, we generate the Gaussian variables ˆy(t),
t ∈ {1, · · · , T }.
2. We then estimate the covariance matrix ˆρ of the Gaussian variables ˆy, which allows us to compute
the variables ˆz 2 and then measure the distance of its estimated distribution to the χ 2 -distribution.
3. Given this covariance matrix ˆρ, we generate numerically a time series of T Gaussian random
vectors with the same covariance matrix ˆρ.
4. For the time series of Gaussian vectors synthetically generated with covariance matrix ˆρ, we estimate
its sample covariance matrix ˜ρ.
5. To each of the T vectors of the synthetic Gaussian time series, we associate the corresponding
realization of the random variable z 2 , called ˜z 2 (t).
6. We can then construct the empirical distribution for the variable ˜z 2 and measure the distance
between this distribution and the χ 2 -distribution.
7. Repeating 10,000 times the steps 3 to 6, we obtain an accurate estimate of the cumulative distribution
of distances between the distribution of the synthetic Gaussian variables and the theoretical
χ 2 -distribution.
8. Then, the distance obtained at step 2 for the true variables can be transformed into a significance
level by reading the value of this synthetically determined distribution of distances between the
distribution of the synthetic Gaussian variables and the theoretical χ 2 -distribution as a function
of the distance: this provides the probability to observe a distance smaller than the chosen or
empirically determined distance.
3.3 Sensitivity of the method
Before presenting the statistical tests, it is important to investigate the sensitivity of our testing procedure.
More precisely, can we distinguish for instance between a Gaussian copula and a Student’s copula with
11
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204 8. Tests de copule gaussienne
a large number of degrees of freedom, for a given value of the correlation coefficient? Formaly, denoting
by Hν the hypothesis according to which the true copula of the data is the Student’s copula with ν degrees
of freedom, we want to determine the minimum significance level allowing us to distinguish between H0
and Hν.
3.3.1 Importance of the distinction between Gaussian and Student’s copulas
This question has important practical implications because, as discussed in section 2.4, the Student’s
copula presents a significant tail dependence while the Gaussian copula has no asymptotic tail dependence.
Therefore, if our tests are unable to distinguish between a Student’s and a Gaussian copula,
we may be led to choose the later for the sake of simplicity and parsimony and, as a consequence, we
may underestimate severely the dependence between extreme events if the correct description turns out
to be the Student’s copula. This may have catastrophic consequences in risk assessment and portfolio
management.
Figure 1 provides a quantification of the dangers incurred by mistaking a Student’s copula for a
Gaussian one. Consider the case of a Student’s copula with ν = 20 degrees of freedom with a correlation
coefficient ρ lower than 0.3 ∼ 0.4 ; its tail dependence λν(ρ) turns out to be less than 0.7%, i.e., the
probability that one variable becomes extreme knowing that the other one is extreme is less than 0.7%.
In this case. the Gaussian copula with zero probability of simultaneous extreme events is not a bad
approximation of the Student’s copula. In contrast, let us take a correlation ρ larger than 0.7 − 0.8 for
which the tail dependence becomes larger than 10%, corresponding to a non-negligible probability of
simultaneous extreme events. The effect of tail dependence becomes of course much stronger as the
number ν of degrees of freedom decreases.
These examples stress the importance of knowing whether our testing procedure allows us to distinguish
between a Student’s copula with ν = 20 (or less) degrees of freedom and a given correlation
coefficient ρ = 0.5, for instance, and a Gaussian copula with an appropriate correlation coefficient ρ ′ .
3.3.2 Statistical test on the distinction between Gaussian and Student’s copulas
To address this question, we have generated 1,000 pairs of time series of size T = 1250, each pair of
random variables following a Student’s bivariate distribution with ν degrees of freedom and a correlation
coefficient ρ between the two simultaneous variables of the same pair, while the variables along the time
axis are all independent. We have then applied the previous testing procedure to each of the pairs of time
series.
Specifically, for each pair of time series, we construct the marginals distributions and transform the
Student’s variables xi(k) into their Gaussian counterparts yi(k) via the transformation (23). For each
pair (y1(k), y2(k)), k ∈ {1, · · · , T }, we estimate its correlation matrix, then construct the time series
with T realizations of the random variable z 2 (k) defined in (27). The set of T variables z 2 then allows
us to construct the distribution of z 2 (with N = 2) and to compare it with the χ 2 -distribution with two
degrees of freedom. We then measure the distances d1, d2, d3 and d4 defined by (33-36) between the
distribution of z 2 and the χ 2 -distribution. Using the 1,000 pairs of such time series with the same ν
and ρ, we then construct the distribution Di(di), i ∈ {1, 2, 3, 4} of each of these distances di. Using
the previously determined distribution of distances expected for the synthetic Gaussian variables, we can
translate each distance d obtained for the Student’s vectors into a corresponding Gaussian probability
p: p is the probability that pairs of Gaussian random variables with the correlation coefficient ρ have
12

a distance equal to or larger than the distance obtained for the Student’s vector time series. A small
p corresponds to a clear distinction between Student’s and Gaussian vectors, as it is improbable that
Gaussian vectors exhibit a distance larger than found for the Student’s vectors. The “distribution of
probabilities” D(p) ≡ D(p(d)) then assesses how often this “improbable” event occurs among the set of
1,000 Student’s vectors, i.e., attempts to quantify the rarety of such large deviations. In other words, the
“distribution of probabilities” D(p) gives the number of Student’s vectors that exhibit the value p for the
probability that Gaussian vectors can have a similar or larger distance. Then, fixing a confidence level
D ∗ , this procedure allows us to reject or not the null hypothesis that the empirical vector of returns is
described by a Gaussian copula: this will occur when the observed p gives a “distribution of probabilities”
D(p) larger than D ∗ .
The “distributions of probabilities” D(p) for each of the four distances di, i ∈ {1, 2, 3, 4} are shown
in figure 3 for ν = 4 degrees of freedom and in figure 4 for ν = 20 degrees of freedom, for 5 different
values of the correlation coefficient ρ = 0.1, 0.3, 0.5, 0.7 and 0.9. The very steep increase observed for
almost all cases in figure 3 reflects the fact that most of the 1,000 Student’s vectors with ν = 4 degrees of
freedom have a small p, i.e., their copula is easily distinguishable from the Gaussian copula. The same
cannot be stated for Student’s vectors with ν = 20 degrees of freedom. Note also that the distances d1,
d2 and d4 give essentially the same result while the Anderson-Darling distance d3 is more sensitive to ρ,
especially for small ν.
Fixing for instance the confidence level at D ∗ = 95%, we can read from each of these curves in
figures 3 and 4 the minimum p 95%-value necessary to distinguish a Student’s copula with a given ν from
a Gaussian copula. This p 95% is the abscissa corresponding to the ordinate D(p 95%) = 0.95. These
values p 95% are reported in table 1, for different values of the number ν of degrees of freedom ranging
from ν = 3 to ν = 50 and correlation coefficients ρ = 0.1 to 0.9. The values of p 95%(ν, ρ) reported
in table 1 are the maximum values that the probability p should take in order to be able to reject the
hypothesis that a Student’s copula with ν degrees and correlation ρ can be mistaken with a Gaussian
copula at the 95% confidence level.
The results of the table 1 are depicted in figures 5-6 and represent the conventional “power/size”
statistics. The statistical “power” is usually defined as the rejection of null hypothesis when false. When
the null hypothesis H0 and the alternative hypothesis Hν are identical, the power should be equal to
= 0.05, corresponding to the 95% confidence level. In our framework, this amounts to plot the abscissa
as the inverse ν −1 of the number ν of degrees of freedom, which provides a natural “distance” between
the Gaussian copula hypothesis H0 and the Student’s copula hypothesis Hν. In the ordinate, the “power”
is represented by the minimum significance level (1−p 95%) necessary to distinguish between H0 and Hν.
The typical shape of these curves is a sigmoid, starting from a very small value for ν −1 → 0, increasing
as ν −1 increases and going to 1 as ν −1 becomes large enough. This typical shape simply expresses the
fact that it is easy to separate a Gaussian copula from a Student’s copula with a small number of degrees
of freedom, while it is difficult and even impossible for too large a number of degrees of freedom.
The figure 5 shows us that the distances d1, d2 and d3 are not sensitive to the value of the correlation
coefficient ρ, while the discriminating power of d3 increases with ρ. On figure 6, we note that d2 and d4
have the same discriminating power for all ρ’s (which makes them somewhat redundant) and that they
are the most efficient to differentiate Hν from H0 for small ρ. When ρ is about 0.5, d2, d3 and d4 (and
maybe d1) are equivalent with respect to the differential power, while for large ρ, d3 becomes the most
discriminating one with high significance.
This study of the test sensitivity involves a non-parametric approach and the question may arise why
it should be prefered to a direct parametric test involving for instance the calibration of the Student
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206 8. Tests de copule gaussienne
copula. First, a parametric test of copulas would face the “curse of dimensionality”, i.e., the estimation
of functions of several variables. With the limited data set available, this does not seem a reasonable
approach. Second, we have taken the Student copula as an example of an alternative to the Gaussian
copula. However, our tests are independent of this choice and aim mainly at testing the rejection of
the Gaussian copula hypothesis. They are thus of a more general nature than would be a parametric
test which would be forced to choose one family of copulas with the problem of excluding others. The
parametric test would then be exposed to the criticism that the rejection of a given choice might not be
of a general nature.
In the sequel, we will choose the level of 95% as the level of rejection, which leads us to neglect
one extreme event out of twenty. This is not unreasonable in view of the other significant sources of
errors resulting in particular from the empirical determination of the marginals and from the presence of
outliers for instance.
4 Empirical results
We investigate the following assets :
• foreign exchange rates,
• metals traded on the London Metal Exchange,
• stocks traded on the New York Stocks Exchange.
4.1 Currencies
The sample we have considered is made of the daily returns for the spot foreign exchanges for 6 currencies
2 : the Swiss Franc (CHF), the German Mark (DEM), the Japanese Yen (JPY), the Malaysian Ringgit
(MYR), the Thai Baht (THA) and the Bristish Pound (UKP). All the exchange rates are expressed against
the US dollar. The time interval runs over ten years, from January 25, 1989 to December 31, 1998, so
that each sample contains 2500 data points.
We apply our test procedure to the entire sample and to two sub-samples of 1250 data points so that
the first one covers the time interval from January 25, 1989 to January 11, 1994 and the second one from
January 12, 1994 to December 31, 1998. The results are presented in tables 2 to 4 and depicted in figures
7 to 9.
Tables 2-4 give, for the total time interval and for each of the two sub-intervals, the probability p(d) to
obtain from the Gaussian hypothesis a deviation between the distribution of the z 2 and the χ 2 -distribution
with two degrees of freedom larger than the observed one for each of the 15 pairs of currencies according
to the distances d1-d4 defined by (33)-(36).
The figures 7-9 organize the information shown in the tables 2-4 by representing, for each distance
d1 to d4, the number of currency pairs that give a test-value p within a bin interval of width 0.05. A
clustering close to the origin signals a significant rejection of the Gaussian copula hypothesis.
At the 95% significance level, table 2 and figure 7 show that only 40% (according to d1 and d3) but
60% (according to d2 and d4) of the tested pairs of currencies are compatible with the Gaussian copula
2 The data come from the historical database of the Federal Reserve Board.
14

hypothesis over the entire time interval. During the first half-period from January 25, 1989 to Januray
11, 1994 (table 3 and figure 8), 47% (according to d3) and up to about 75 % (according to d2 and d4)
of the tested currency pairs are compatible with the assumption of Gaussian copula, while during the
second sub-period from January 12, 1994 to December 31, 1998 (table 4 and figure 9), between 66%
(according to d1) and about 75% (according to d2, d3 and d4) of the currency pairs remain compatible
with the Gaussian copula hypothesis. These results raise several comments both on a statistical and an
economic point of view.
We first note that the most significant rejection of the Gaussian copula hypothesis is obtained for
the distance d3, which is indeed the most sensitive to the events in the tail of the distributions. The test
statistics given by this distance can indeed be very sensitive to the presence of a single large event in the
sample, so much so that the Gaussian copula hypothesis can be rejected only because of the presence
of this single event (outlier). The difference between the results given by d3 and d4 (the averaged d3)
are very significant in this respect. Consider for instance the case of the German Mark and the Swiss
Franc. During the time interval from January 12, 1994 to December 31, 1998, we check on table 4 that
the non-rejection probability p(d) is very significant according to d1, d2 and d4 (p(d) ≥ 31%) while
it is very low according to d3: p(d) = 0.05%, and should lead to the rejection of the Gaussian copula
hypothesis. This suggests the presence of an outlier in the sample.
To check this hypothesis, we show in the upper panel of figure 10 the function
f3(t) = |Fz2(z2 (t) − Fχ2(χ2 (t))|
Fχ2(χ2 )[1 − Fχ2(χ2 )]
207
, (37)
used in the definition of the Anderson-Darling distance d3 = maxz f3(z) (see definition (35)), expressed
in terms of time t rather than z 2 . The function have been computed over the two time sub-intervals
separately.
Apart from three extreme peaks occurring on June 20, 1989, August 19, 1991 and September 16,
1992 during the first time sub-interval and one extreme peak on September 10, 1997 during the second
time sub-interval, the statistical fluctuations measured by f3(t) remain small and of the same order.
Excluding the contribution of these outlier events to d3, the new statistical significance derived according
to d3 becomes similar to that obtained with d1, d2 and d4 on each sub-interval. From the upper pannel
of figure 10, it is clear that the Anderson-Darling distance d3 is equal to the height of the largest peak
corresponding to the event on August 19, 1991 for the the first period and to the event on September 10,
1997 for the second period. These events are depicted by a circled dot in the two lower panels of figure
10, which represent the return of the German Mark versus the return of the Swiss Franc over the two
considered time periods.
The event on August 19, 1991 is associated with the coup against Gorbachev in Moscow: the German
mark (respectively the Swiss franc) lost 3.37% (respectively 0.74%) in daily annualized value against the
US dollar. The 3.37% drop of the German Mark is the largest daily move of this currency against the
US dollar over the whole first period. On September 10, 1997, the German Mark appreciated by 0.60%
against the US dollar while the Swiss Franc lost 0.79% which represents a moderate move for each
currency, but a large joint move. This event is related to the contradictory announcements of the Swiss
National Bank about the monetary policy, which put an end to a rally of the Swiss Franc along with the
German mark against the US dollar.
Thus, neglecting the large moves associated with major historical events or events associated with
unexpected incoming information, which cannot be taken into account by a statistical study, we obtain,
for d3, significance levels compatible with those obtained with the other distances. We can thus conclude
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208 8. Tests de copule gaussienne
that, according to the four distances, during the time interval from January 12, 1994 to December 31,
1998 the Gaussian copula hypothesis cannot be rejected for the couple German Mark / Swiss Franc.
However, the non-rejection of the Gaussian copula hypothesis does not always have minor consequences
and may even lead to serious problem in stress scenarios. As shown in section 3.3, the nonrejection
of the Gaussian copula hypothesis does not exclude, at the 95% significance level, that the
dependence of the currency pairs may be accounted for by a Student’s copula with adequate values of ν
and ρ. Still considering the pair German Mark / Swiss Franc, we see in table 1 that, according to d1, d2
and d4, a Student’s copula with about five degrees of freedom allows to reach the test values given in table
4. But, with the correlation coefficient ρ = 0.92 for the German Mark/Swiss Franc couple, the Gaussian
copula assumption could lead to neglect a tail dependence coefficient λ5(0.92) = 63% according to the
Student’s copula prediction. Such a large value of λ5(0.92) means that when an extreme event occurs
for the German Mark it also occurs for the Swiss Franc with a probabilty equals to 0.63. Therefore, a
stress scenario based on a Gaussian copula assumption would fail to account for such coupled extreme
events, which may represent as many as two third of all the extreme events, if it would turn out that the
true copula would be the Student’s copula with five degrees of freedom. In fact, with such a value of the
correlation coefficient, the tail dependence remains high even if the number of degrees of fredom reach
twenty or more (see figure 1).
The case of the Swiss Franc and the Malaysian Ringgit offers a striking difference. For instance,
in the second half-period, the test statistics p(d) are greater than 70% and even reach 91% while the
correlation coefficient is only ρ = 0.16, so that a Student’s copula with 7-10 degrees of freedom can be
mistaken with the Gaussian copula (see table 1). Even in the most pessimistic situation ν = 7, the choice
of the Gaussian copula amounts to neglecting a tail dependence coefficient λ5(0.16) = 4% predicted by
the Student’s copula. In this case, stress scenarios based on the Gaussian copula would predict uncoupled
extreme events, which would be shown wrong only once out of twenty five times.
These two examples show that, more than the number of degrees of freedom of the Student’s copula
necessary to describe the data, the key parameter is the correlation coefficient.
¿From an economic point of view, the impact of regulatory mechanisms between currencies or monetary
crisis can be well identified by the rejection or absence of rejection of our null hypothesis. Indeed,
consider the couple German Mark / British Pound. During the first half period, their correlation coefficient
is very high (ρ = 0.82) and the Gaussian copula hypothesis is strongly rejected according to the
four distances. On the contrary, during the second half period, the correlation coefficient significantly
decreases (ρ = 0.56) and none of the four distances allows us to reject our null hypothesis. Such a
non-stationarity can be easily explained. Indeed, on January 1, 1990, the British Pound entered the European
Monetary System (EMS), so that the exchange rate between the German Mark and the Bristish
Pound was not allowed to fluctuate beyond a margin of 2.25%. However, due to a strong speculative
attack, the British Pound was devaluated on September 1992 and had to leave the EMS. Thus, between
January 1990 and September 1992, the exchange rate of the German Mark and the British Pound was
confined within a narrow spread, incompatible with the Gaussian copula description. After 1992, the
British Pound exchange rate floated with respect to German Mark, the dependence between the two currencies
decreased, as shown by their correlation coefficient. In this regime, we can no more reject the
Gaussian copula hypothesis.
The impact of major crisis on the copula can be also clearly identified. Such a case is exhibited by the
couple Malaysian Ringgit/Thai Baht. Indeed, during the period from Januray 1989 to January 1994, these
two currencies have only undergone moderate and weakly correlated (ρ = 0.29) fluctuations, so that our
null hypothesis cannot be rejected at the 95% significance level. On the contrary, during the period from
16

January 1994 to October 1998, the Gaussian copula hypothesis is strongly rejected. This rejection is
obviously due to the persistent and dependent (ρ = 0.44) shocks incured by the Asian financial and
monetary markets during the seven months of the Asian Crisis from July 1997 to January 1998 (Baig and
Goldfajn 1998, Kaminsky and Schlmukler 1999).
These two cases show that the Gaussian copula hypothesis can be considered reasonable for currencies
in absence of regulatory mechanisms and of strong and persistent crises. They also allows us to
understand why the results of the test over the entire sample are so much weaker than the results obtained
for the two sub-intervals: the time series are strongly non-stationnary.
4.2 Commodities: metals
We consider a set of 6 metals traded on the London Metal Exchange: aluminium, copper, lead, nickel,
tin and zinc. Each sample contains 2270 data points and covers the time interval from January 4, 1989
to December 30, 1997. The results are synthetized in table 5 and in figure 11.
Table 5 gives, for each of the 15 pairs of commodities, the probability p(d) to obtain from the Gaussian
hypothesis a deviation between the distribution of the z 2 and the χ 2 -distribution with two degrees
of freedom larger than the observed one for the commodity pair according to the distances d1-d4 defined
by (33)-(36).
The figure 11 organizes the information shown in table 5 by representing, for each distance, the
number of commodity pairs that give a test-value p within a bin interval of width 0.05. A clustering
close to the origin signals a significant rejection of the Gaussian copula hypothesis.
According to the three distances d1, d2 and d4, at least two third and up to 93% of the set of 15
pairs of commodities are inconsistent with the Gaussian copula hypothesis. Surprisingly, according to
the distance d3, at the 95% significance level, two third of the set of 15 pairs of commodities remain
compatible with the Gaussian copula hypothesis. This is the reverse to the previous situation found for
currencies. These test values lead to globally reject the Gaussian copula hypothesis.
Moreover, the largest value obtained for the distance d3 is p = 65% for the pair copper-tin, which is
significantly smaller than the 80% or 90% reached for some currencies over a similar time interval. Thus,
even in the few cases where the Gaussian copula assumption is not rejected, the test values obtained are
not really sufficient to distinguish between the Gaussian copula and a Student’s copula with ν = 5 ∼ 6
degrees of freedom. In such a case, with correlation coefficients ranging between 0.31 and 0.46, the
tail dependence neglected by keeping the Gaussian copula is no less than 10% and can reach 15%. One
extreme event out of seven or ten might occur simultaneously on both marginals, which would be missed
by the Gaussian copula.
To summarize, the Gaussian copula does not seem a reasonnable assumption for metals, and it has
not appeared necessary to test these data over smaller time interval.
4.3 Stocks
We now study the daily returns distibutions for 22 stocks among the largest compagnies quoted on the
New York Stock Exchange 3 : Appl. Materials (AMAT), AT&T (T), Citigroup (C), Coca Cola (KO),
EMC, Exxon-Mobil (XOM), Ford (F), General Electric (GE), General Motors (GM), Hewlett Packard
3 The data come from the Center for Research in Security Prices (CRSP) database.
17
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210 8. Tests de copule gaussienne
(HPW), IBM, Intel (INTC), MCI WorldCom (WCOM), Medtronic (MDT), Merck (MRK), Microsoft
(MSFT), Pfizer (PFE), Procter&Gamble (PG), SBC Communication (SBC), Sun Microsystem (SUNW),
Texas Instruments (TXN), Wal Mart (WMT).
Each sample contains 2500 data points and covers the time interval from February 8, 1991 to December
29, 2000 and have been divided into two sub-samples of 1250 data points, so that the first one
covers the time interval from February 8, 1991 to January 18, 1996 and the second one from January
19, 1996 to December 20, 2000. The results of fifteen randomly chosen pairs of assets are presented in
tables 6 to 8 while the results obtain for the entire set are represented in figures 12 to 14.
At the 95% significance level, figure 12 shows that 75% of the pairs of stocks are compatible with the
Gaussian copula hypothesis. Figure 13 shows that over the time interval from February 1991 to January
1996, this percentage becomes larger than 99% for d1, d2 and d4 while it equals 94% according to d3. It
is striking to note that, during this period, according to d1, d2 and d4, more than a quarter of the stocks
obtain a test-value p larger than 90%, so that we can assert that they are completely inconsistent with the
Student’s copula hypothesis for Student’s copulas with less than 10 degrees of freedom. Among this set
of stocks, not a single one has a correlation coefficient larger than 0.4, so that a scenario based on the
Gaussian copula hypothesis leads to neglecting a tail dependence of less than 5% as would be predicted
by the Student’s copula with 10 degrees of freedom. In addition, about 80% of the pairs of stocks lead
to a test-value p larger than 50% according to the distances d1, d2 and d4, so that as much as 80% of
the pairs of stocks are incompatible with a Student’s copula with a number of degrees of freedom less
than or equal to 5. Thus, for correlation coefficients smaller than 0.3, the Gaussian copula hypothesis
leads to neglecting a tail dependence less than 10%. For correlation coefficients smaller than 0.1 which
corresponds to 13% of the total number of pairs, the Gaussian copula hypothesis leads to neglecting a
tail dependence less than 5%.
Figure 14 shows that, over the time interval from January 1996 to December 2000, 92% of the pairs
of stocks are compatible with the Gaussian copula hypothesis according to d1, d2 and d4 and more than
79% according to d3. About a quarter of the pair of stocks have a test-value p larger than 50% according
to the four measures and thus are inconsistent with a Student’s copula with less than five degrees of
freedom.
For completeness, we present in table 9 the results of the tests performed for five stocks belonging
to the computer area : Hewlett Packard, IBM, Intel, Microsoft and Sun Microsystem. We observe that,
during the first half period, all the pairs of stocks qualify the Gaussian copula Hypothesis at the 95%
significance level. The results are rather different for the second half period since about 40% of the pairs
of stocks reject the Gaussian copula hypothesis according to d1, d2 and d3. This is probably due to the
existence of a few shocks, notably associated with the crash of the “new economy” in March-April 2000.
On the whole, it appears however that there is no systematic rejection of the Gaussian copula hypothesis
for stocks within the same industrial area, notwithstanding the fact that one can expect stronger
correlations between such stocks than for currencies for instance.
5 Conclusion
We have studied the null hypothesis that the dependence between pairs of financial assets can be modeled
by the Gaussian copula.
Our test procedure is based on the following simple idea. Assuming that the copula of two assets
18

X and Y is Gaussian, then the multivariate distribution of (X, Y ) can be mapped into a Gaussian multivariate
distribution, by a transformation of each marginal into a normal distribution, which leaves the
copula of X and Y unchanged. Testing the Gaussian copula hypothesis is therefore equivalent to the
more standard problem of testing a two-dimensional multivariate Gaussian distribution. We have used
a bootstrap method to determine and calibrate the test statistics. Four different measures of distances
between distributions, more or less sensitive to the departure in the bulk or in the tail of distributions,
have been proposed to quantify the probability of rejection of our null hypothesis.
Our tests have been performed over three types of assets: currencies, commodities (metals) and
stocks. In most cases, for currencies and stocks, the Gaussian copula hypothesis can not be rejected at
the 95% confidence level. For currencies, according to three of the four distances at least,
• 40% of the pairs of currencies, over a 10 years time interval (due to non-stationnary data),
• 67% of the pairs of currencies, over the first 5 years time interval,
• 73% of the pairs of currencies, over the second 5 years time interval,
are compatible with the Gaussian copula hypothesis. For stocks, we have shown that
• 75% of the pairs of stocks, over a 10 years time interval,
• 93% of the pairs of stocks, over the first 5 years time interval,
• 92% of the pairs of stocks, over the second 5 years time interval,
are compatible with the Gaussian copula hypothesis. In constrast, the Gaussian copula hypothesis cannot
be considered as reasonable for metals : between 66% and 93% of the pairs of metals reject the null
hypothesis at the 95% confidence level.
Notwithstanding the apparent qualification of the Gaussian copula hypothesis for most of the currencies
and the stocks we have analyzed, we must bear in mind the fact that a non-Gaussian copula cannot
be rejected. In particular, we have shown that a Student’s copula can always be mistaken for a Gaussian
copula if its number of degrees of freedom is sufficiently large. Then, depending on the correlation coefficient,
the Student’s copula can predict a non-negligible tail dependence which is completely missed
by the Gaussian copula assumption. In other words, the Gaussian copula predicts no tail dependences
and therefore does not account for extreme events that may occur simultaneously but nevertheless too
rarely to modify the test statistics. To quantify the probability for neglecting such events, we have investigated
the situations when one is unable to distinguish between the Gaussian and Student’s copulas for
a given number of degrees of freedom. Our study leads to the conclusion that it may be very dangerous
to embrace blindly the Gaussian copula hypothesis when the correlation coefficient between the pair of
asset is too high as the tail dependence neglected by the Gaussian copula can be as large as 0.6. In this
respect, the case of the Swiss Franc and the German Mark is striking. The test values p obtained are very
significant (about 33%), so that we cannot mistake the Gaussian copula for a Student’s copula with less
than 5-7 degrees of freedom. However, their correlation coefficient is so high (ρ = 0.9) that a Student’s
copula with, say ν = 30 degrees of freedom, still has a large tail dependence.
This remark shows that it is highly desirable to develop tests that are specific to the detection of a
possible tail dependence between two time series. This task is very difficult but we hope to report useful
progress in the near future. Another approach is to test for other non-Gaussian copulas, such as the
Student’s copula.
19
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212 8. Tests de copule gaussienne
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21
213

214 8. Tests de copule gaussienne
λ ν (ρ)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
ν=3
ν=5
ν=10
ν=20
ν=50
ν=100
0
−1 −0.8 −0.6 −0.4 −0.2 0
ρ
0.2 0.4 0.6 0.8 1
Figure 1: Upper tail dependence coefficient λν(ρ) for the Student’s copula with ν degrees of freedom as
a function of the correlation coefficient ρ, for different values of ν.
22

ρ
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
λ ν (ρ) = 1%
λ ν (ρ) = 2.5%
λ ν (ρ) = 5%
λ ν (ρ) = 10%
−1
0 10 20 30 40 50
ν
60 70 80 90 100
Figure 2: Maximum value of the correlation coefficient ρ as a function of ν, below which the tail
dependence λν(ρ) of a Student’ copula is smaller than a given small value, here taken equal to
λν(ρ) = 1%, 2.5%, 5% and 10%. The choice λν(ρ) = 5% for instance corresponds to 1 event in 20
for which the pair of variables are asymptotically coupled. At the 1 − λν(ρ) probability level, values of
λ ≤ λν(ρ) are undistinguishable from 0, which means that the Student’s copula can be approximated by
a Gaussian copula.
23
215

216 8. Tests de copule gaussienne
1
0.8
0.6
0.4
ρ=0.1
ρ=0.3
0.2
ρ=0.5
ρ=0.7
ρ=0.9
0
0 0.2 0.4 0.6 0.8 1
1
0.8
0.6
0.4
ρ=0.1
ρ=0.3
0.2
ρ=0.5
ρ=0.7
ρ=0.9
0
0 0.2 0.4 0.6 0.8 1
d 1
d 3
1
0.8
0.6
0.4
ρ=0.1
ρ=0.3
0.2
ρ=0.5
ρ=0.7
ρ=0.9
0
0 0.2 0.4 0.6 0.8 1
1
0.8
0.6
0.4
ρ=0.1
ρ=0.3
0.2
ρ=0.5
ρ=0.7
ρ=0.9
0
0 0.2 0.4 0.6 0.8 1
Figure 3: Cumulative “distribution of probabilities” D(p) ≡ D(p(d)) obtained as the fraction of Student’s
pairs with ν = 4 degrees of freedom that exhibit the value p for the probability that Gaussian
vectors can have a similar or larger distance. See the text for a detailled description of how D(p) is defined
and constructed. Each panel corresponds to one of the four distances di, i ∈ {1, 2, 3, 4}, defined in
the text by equations (33-36). In each panel, we construct the cumulative “distribution of probabilities”
D(p) for 5 different values of the correlation coefficient ρ = 0.1, 0.3, 0.5, 0.7 and 0.9 of the Student’s
copula.
24
d 2
d 4

1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
1
0.8
0.6
0.4
0.2
ρ=0.1
ρ=0.3
ρ=0.5
ρ=0.7
ρ=0.9
ρ=0.1
ρ=0.3
ρ=0.5
ρ=0.7
ρ=0.9
d 1
d 3
0
0 0.2 0.4 0.6 0.8 1
1
0.8
0.6
0.4
0.2
ρ=0.1
ρ=0.3
ρ=0.5
ρ=0.7
ρ=0.9
d 2
0
0 0.2 0.4 0.6 0.8 1
1
0.8
0.6
0.4
0.2
ρ=0.1
ρ=0.3
ρ=0.5
ρ=0.7
ρ=0.9
d 4
0
0 0.2 0.4 0.6 0.8 1
Figure 4: Same as figure 3 for Student’s distributions with ν = 20 degrees of freedom.
25
217

218 8. Tests de copule gaussienne
ν = 3
ν = 5
ν = 8
ν = 20
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.07 0.08 0.07 0.04 0.07
d2 0.03 0.03 0.07 0.04 0.06
d3 0.22 0.17 0.08 0.03 0.01
d4 0.03 0.03 0.08 0.03 0.04
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.46 0.47 0.46 0.52 0.52
d2 0.36 0.34 0.39 0.44 0.43
d3 0.52 0.54 0.47 0.30 0.14
d4 0.37 0.36 0.43 0.45 0.45
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.85 0.86 0.87 0.88 0.89
d2 0.85 0.84 0.86 0.87 0.88
d3 0.91 0.91 0.91 0.81 0.70
d4 0.86 0.85 0.90 0.89 0.90
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.97 0.99 0.97 0.99 0.99
d2 0.99 0.99 0.97 0.99 0.99
d3 0.99 0.99 0.98 0.99 0.97
d4 0.99 0.99 0.98 0.99 0.99
ν = 4
ν = 7
ν = 10
ν = 50
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.28 0.26 0.32 0.30 0.29
d2 0.18 0.17 0.21 0.21 0.24
d3 0.36 0.33 0.26 0.15 0.03
d4 0.18 0.17 0.23 0.21 0.21
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.78 0.81 0.81 0.81 0.86
d2 0.71 0.78 0.76 0.77 0.82
d3 0.80 0.81 0.82 0.73 0.52
d4 0.75 0.81 0.79 0.80 0.83
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.92 0.93 0.96 0.95 0.94
d2 0.93 0.92 0.95 0.96 0.94
d3 0.96 0.96 0.96 0.95 0.88
d4 0.94 0.94 0.96 0.97 0.95
ρ 0.1 0.3 0.5 0.7 0.9
d1 0.99 0.99 0.99 0.99 0.99
d2 0.99 0.99 0.99 0.99 0.99
d3 0.99 0.99 0.99 0.99 0.99
d4 0.99 0.99 0.99 0.99 0.99
Table 1: The values p 95%(ν, ρ) shown in this table give the maximum values that the probability p should
take in order to be able to reject the hypothesis that a Student’s copula with ν degrees and correlation
ρ is undistinguishable from a Gaussian copula at the 95% confidence level. p 95% is the abscissa corresponding
to the ordinate D(p 95%) = 0.95 shown in figures 3 and 4. p is the probability that pairs of
Gaussian random variables with the correlation coefficient ρ have a distance (between the distribution of
z 2 and the theoretical χ 2 distribution) equal to or larger than the corresponding distance obtained for the
Student’s vector time series. A small p corresponds to a clear distinction between Student’s and Gaussian
vectors, as it is improbable that Gaussian vectors exhibit a distance larger than found for the Student’s
vectors. Different values of the number ν of degrees of freedom ranging from ν = 3 to ν = 50 and of
the correlation coefficient ρ = 0.1 to 0.9 are shown. Let us take for instance the example with ν = 4 and
ρ = 0.3. The table indicates that p should be less than about 0.3 (resp. 0.2) according to the distances d1
and d3 (resp. d2 and d4) for being able to distinguish this Student’s copula from the Gaussian copula at
the 95% confidence level. This means that less than 20−30% of Gaussian vectors should have a distance
for their z 2 larger than the one found for the Student’s. See text for further explanations.
26

1−p 95%
1−p 95%
1
0.8
0.6
0.4
0.2
ρ=0.1
ρ=0.3
ρ=0.5
ρ=0.7
ρ=0.9
d 1
0
0 0.1 0.2
1/ν
0.3
1
0.8
0.6
0.4
0.2
ρ=0.1
ρ=0.3
ρ=0.5
ρ=0.7
ρ=0.9
d 3
0
0 0.1 0.2
1/ν
0.3 0.4
1−p 95%
1−p 95%
1
0.8
0.6
0.4
0.2
ρ=0.1
ρ=0.3
ρ=0.5
ρ=0.7
ρ=0.9
d 2
0
0 0.1 0.2
1/ν
0.3
1
0.8
0.6
0.4
0.2
ρ=0.1
ρ=0.3
ρ=0.5
ρ=0.7
ρ=0.9
d 4
0
0 0.1 0.2
1/ν
0.3
Figure 5: Graph of the minimun significance level (1 − p 95%) necessary to distinguish the Gaussian
copula hypothesis H0 from the hypothesis of a student copula with ν degrees of freedom, as a function
of 1/ν, for a given distance di and various correlation coefficients ρ = 0.1, 0.3, 0.5, 0.7 and 0.9.
27
219

220 8. Tests de copule gaussienne
1−p 95%
1−p 95%
1
0.8
0.6
0.4
0.2
d
1
d
2
d
3
d
4
ρ =0.1 − 0.3
0
0 0.1 0.2
1/ν
0.3 0.4
1
0.8
0.6
0.4
0.2
d
1
d
2
d
3
d
4
ρ =0.7
0
0 0.1 0.2
1/ν
0.3 0.4
1−p 95%
1−p 95%
1
0.8
0.6
0.4
0.2
d
1
d
2
d
3
d
4
ρ =0.5
0
0 0.1 0.2
1/ν
0.3 0.4
1
0.8
0.6
0.4
0.2
d
1
d
2
d
3
d
4
ρ =0.9
0
0 0.1 0.2
1/ν
0.3 0.4
Figure 6: Same as figure 5 but comparing different distances for the same correlation coefficient ρ.
28

ˆρ d1 d2 d3 d4
CHF DEM 0.92 1.01e-02 6.70e-03 0.00e+00 7.20e-03
CHF JPY 0.53 3.44e-01 2.71e-01 2.32e-02 2.83e-01
CHF MYR 0.23 7.27e-01 8.71e-01 5.77e-01 9.26e-01
CHF THA 0.21 3.08e-02 9.47e-02 3.31e-02 9.52e-02
CHF UKP 0.69 2.80e-03 1.80e-03 6.00e-04 1.30e-03
DEM JPY 0.54 2.26e-02 1.33e-01 1.00e-01 1.51e-01
DEM MYR 0.26 4.25e-01 6.77e-01 6.22e-01 7.35e-01
DEM THA 0.24 6.53e-02 1.35e-01 3.26e-02 1.32e-01
DEM UKP 0.72 1.70e-03 4.00e-04 0.00e+00 4.00e-04
JPY MYR 0.31 2.45e-02 6.34e-02 2.26e-01 6.86e-02
JPY THA 0.34 0.00e+00 0.00e+00 3.24e-02 0.00e+00
JPY UKP 0.41 2.85e-02 3.72e-02 5.22e-02 3.09e-02
MYR THA 0.40 0.00e+00 0.00e+00 2.22e-02 0.00e+00
MYR UKP 0.21 6.94e-01 7.94e-01 6.23e-01 8.31e-01
THA UKP 0.15 5.22e-01 6.23e-01 3.21e-02 7.05e-01
Table 2: Each row gives the statistics of our test for each of the 15 pairs of currencies over a 10 years
time interval from January 25, 1989 to December 31, 1998. The column ˆρ gives the empirical correlation
coefficient for each pair determined as in section 3.1 and defined in (31). The columns d1, d2, d3 and d4
gives the probability to obtain, from the Gaussian hypothesis, a deviation between the distribution of the
z 2 and the χ 2 -distribution with two degrees of freedom larger than the observed one for the currency pair
according to the distances d1-d4 defined by (33)-(36).
29
221

222 8. Tests de copule gaussienne
ˆρ d1 d2 d3 d4
CHF DEM 0.92 1.73e-02 1.33e-02 0.00e+00 1.31e-02
CHF JPY 0.55 1.34e-01 1.49e-01 3.83e-01 1.41e-01
CHF MYR 0.32 8.47e-01 7.00e-01 3.56e-01 7.40e-01
CHF THA 0.17 4.40e-01 7.10e-01 3.53e-02 7.11e-01
CHF UKP 0.79 3.10e-03 1.00e-03 0.00e+00 5.00e-04
DEM JPY 0.56 2.46e-02 9.43e-02 1.63e-01 9.26e-02
DEM MYR 0.35 9.32e-01 7.95e-01 3.51e-01 7.95e-01
DEM THA 0.21 4.36e-01 8.77e-01 3.47e-02 8.74e-01
DEM UKP 0.82 0.00e+00 0.00e+00 0.00e+00 0.00e+00
JPY MYR 0.34 4.90e-01 5.49e-01 3.66e-01 5.94e-01
JPY THA 0.27 3.89e-01 3.06e-01 3.37e-02 3.59e-01
JPY UKP 0.53 9.00e-04 1.66e-02 6.72e-02 1.67e-02
MYR THA 0.29 1.08e-01 8.71e-02 3.42e-02 9.30e-02
MYR UKP 0.33 1.12e-01 2.86e-01 3.54e-01 3.45e-01
THA UKP 0.21 4.34e-01 8.62e-01 3.13e-02 8.67e-01
Table 3: Same as table 2 for currencies over a 5 years time interval from January 25, 1989 to Januay 11,
1994.
30

ˆρ d1 d2 d3 d4
CHF DEM 0.92 3.15e-01 3.11e-01 5.00e-04 3.41e-01
CHF JPY 0.52 5.84e-01 6.44e-01 1.98e-02 6.74e-01
CHF MYR 0.16 7.11e-01 9.15e-01 8.83e-01 9.22e-01
CHF THA 0.25 1.10e-02 3.87e-02 1.05e-01 3.34e-02
CHF UKP 0.53 9.75e-02 1.03e-01 2.33e-01 9.29e-02
DEM JPY 0.53 3.63e-01 5.40e-01 1.77e-02 6.54e-01
DEM MYR 0.18 3.55e-01 5.00e-01 5.84e-01 5.67e-01
DEM THA 0.28 1.28e-02 2.18e-02 1.08e-01 1.51e-02
DEM UKP 0.56 1.15e-01 1.10e-01 3.02e-01 1.06e-01
JPY MYR 0.29 7.63e-02 2.14e-01 6.67e-02 2.23e-01
JPY THA 0.38 0.00e+00 2.00e-04 3.09e-02 2.00e-04
JPY UKP 0.28 4.62e-01 2.30e-01 1.23e-01 2.07e-01
MYR THA 0.44 5.00e-04 1.20e-03 5.34e-02 1.20e-03
MYR UKP 0.11 5.94e-01 7.44e-01 6.95e-01 7.82e-01
THA UKP 0.12 1.26e-02 7.66e-02 1.19e-01 6.51e-02
Table 4: Same as table 2 for currencies over a 5 years time interval from January 12, 1994 to December
31, 1998.
31
223

224 8. Tests de copule gaussienne
9
8
7
6
5
4
3
2
1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 7: For each distance d1-d4 defined in equations (33)-(36), this figure shows the number of currency
pairs that give a given p (shown on the abscissa) within a bin interval of width 0.05 for different currencies
over a 10 years time interval from January 25, 1989 to December 31, 1998. p is the probability that
pairs of Gaussian random variables with the same correlation coefficient ρ have a distance (between the
distribution of z 2 and the theoretical chi 2 distribution) equal to or larger than the corresponding distance
obtained for each currency pair. A clustering close to the origin signals a significant rejection of the
Gaussian copula hypothesis.
32
d
1
d
2
d
1
d
4

8
7
6
5
4
3
2
1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 8: Same as figure 7 for currencies over a 5 years time interval from January 25, 1989 to January
11, 1994.
33
d
1
d
2
d
1
d
4
225

226 8. Tests de copule gaussienne
5
4
3
2
1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 9: Same as figure 7 for currencies over a 5 years time interval from January 12, 1994 to December
1998.
34
d
1
d
2
d
1
d
4

ˆρ d1 d2 d3 d4
aluminium copper 0.46 6.46e-02 4.48e-02 1.45e-02 4.00e-02
aluminium lead 0.35 1.14e-01 5.01e-02 1.70e-01 4.59e-02
aluminium nickel 0.36 3.30e-03 5.10e-03 3.41e-02 6.20e-03
aluminium tin 0.34 1.34e-01 1.38e-01 1.25e-02 1.59e-01
aluminium zinc 0.36 2.30e-03 2.20e-03 6.21e-02 2.30e-03
copper lead 0.35 4.71e-02 1.74e-02 1.79e-01 1.34e-02
copper nickel 0.38 4.91e-02 4.60e-02 1.48e-01 3.80e-02
copper tin 0.32 1.94e-01 1.35e-01 6.53e-01 1.47e-01
copper zinc 0.40 3.24e-02 2.05e-02 1.75e-01 1.94e-02
lead nickel 0.32 6.71e-02 3.78e-02 2.74e-01 3.62e-02
lead tin 0.33 7.86e-02 4.04e-02 4.91e-02 3 .31e-02
lead zinc 0.42 2.00e-04 1.00e-04 4.59e-02 3.00e-04
nickel tin 0.35 9.10e-03 9.20e-03 8.70e-02 7.60e-03
nickel zinc 0.33 8.00e-04 3.40e-03 8.91e-02 3.50e-03
tin zinc 0.31 5.30e-03 2.02e-02 1.03e-01 1.75e-02
Table 5: Same as table 2 for metals over a 9 years time interval from January 4, 1989 to December 30,
1997.
35
227

228 8. Tests de copule gaussienne
f 3
German Mark
4
3
2
1
0
Jan 89 Jan 94 Dec 98
4
2
0
−2
20/06/1989
19/08/1991
−4
−4 −2 0
Swiss Franc
2 4
16/09/1992
German Mark
3
2
1
0
−1
−2
10/09/97
−3
−6 −4 −2 0 2 4
Swiss Franc
Figure 10: The upper panel represents the graph of the function f3(t) defined in (37) used in the definition
of the distance d3 for the couple Swiss Franc/German Mark as a function of time t, over the time intervals
from January 25, 1989 to January 11, 1994 and from January 12, 1994 to December 31, 1998. The two
lower panels represent the scatter plot of the return of the German Mark versus the return of the Swiss
Franc during the two previous time periods. The circled dot, in each figure, shows the pair of returns
responsible for the largest deviation of f3 during the considered time interval.
36

14
12
10
8
6
4
2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 11: Same as figure 7 for metals over a 9 years time interval from January 4, 1989 to December
30, 1997.
37
d
1
d
2
d
3
d
4
229

230 8. Tests de copule gaussienne
ˆρ d1 d2 d3 d4
amat pfe 0.15 7.41e-02 1.12e-01 8.40e-03 1.14e-01
c sunw 0.28 2.56e-01 4.87e-01 1.09e-01 5.39e-01
f ge 0.33 2.52e-01 2.74e-01 1.15e-01 2.90e-01
gm ibm 0.21 1.49e-01 3.85e-01 1.62e-01 4.18e-01
hwp sbc 0.12 4.23e-01 1.69e-01 2.52e-01 1.72e-01
intc mrk 0.17 2.48e-01 1.09e-01 6.46e-01 1.04e-01
ko sunw 0.14 1.41e-01 1.01e-01 2.12e-01 9.35e-02
mdt t 0.16 1.21e-01 2.81e-01 8.41e-02 2.98e-01
mrk xom 0.19 1.54e-01 1.50e-01 1.12e-01 1.45e-01
msft sunw 0.44 3.40e-02 1.85e-02 2.60e-03 1.74e-02
pfe wmt 0.27 4.24e-02 4.12e-02 1.54e-01 3.74e-02
t wcom 0.27 5.67e-02 8.02e-02 5.44e-02 9.07e-02
txn wcom 0.28 4.79e-01 3.77e-01 1.52e-01 3.75e-01
wmt xom 0.20 3.20e-03 0.00e+00 6.02e-02 0.00e+00
Table 6: Same as table 2 for stocks over a 10 years time interval from February 8, 1991 to December 29,
2000.
38

ˆρ d1 d2 d3 d4
amat pfe 0.10 5.83e-01 5.81e-01 1.18e-01 6.38e-01
c sunw 0.23 4.66e-01 5.94e-01 4.34e-01 6.16e-01
f ge 0.31 8.73e-01 7.87e-01 1.54e-01 8.48e-01
gm ibm 0.21 6.00e-01 6.53e-01 1.03e-01 5.27e-01
hwp sbc 0.11 8.73e-01 8.06e-01 2.84e-01 8.59e-01
intc mrk 0.13 8.59e-01 8.21e-01 5.48e-02 8.65e-01
ko sunw 0.20 3.53e-01 5.98e-01 4.51e-01 6.79e-01
mdt t 0.14 9.09e-01 8.98e-01 1.68e-01 9.15e-01
mrk xom 0.12 5.36e-01 6.21e-01 1.20e-01 6.18e-01
msft sunw 0.40 2.68e-01 1.38e-01 1.60e-01 1.39e-01
pfe wmt 0.23 2.94e-01 4.66e-01 1.41e-01 5.23e-01
t wcom 0.19 7.92e-01 9.36e-01 4.95e-02 9.49e-01
txn wcom 0.23 9.10e-01 9.83e-01 1.00e-01 9.93e-01
wmt xom 0.22 7.16e-01 6.71e-01 7.35e-02 6.89e-01
Table 7: Same as table 2 for stocks over a 5 years time interval from February 8, 1991 to January 18,
1996.
39
231

232 8. Tests de copule gaussienne
ˆρ d1 d2 d3 d4
amat pfe 0.19 2.96e-01 3.39e-01 3.10e-02 3.95e-01
c sunw 0.31 7.12e-01 6.58e-01 9.47e-01 7.08e-01
f ge 0.34 3.80e-01 2.36e-01 3.22e-01 2.18e-01
gm ibm 0.21 3.05e-02 1.79e-01 2.37e-01 2.19e-01
hwp sbc 0.11 3.47e-01 6.13e-01 7.17e-01 6.40e-01
intc mrk 0.20 1.31e-01 2.06e-01 5.57e-01 2.05e-01
ko sunw 0.10 6.89e-01 3.44e-01 8.59e-01 3.52e-01
mdt t 0.19 4.28e-01 6.11e-01 5.01e-01 5.79e-01
mrk xom 0.23 3.57e-01 6.64e-01 1.13e-01 7.38e-01
msft sunw 0.46 5.79e-02 7.60e-02 8.00e-04 8.07e-02
pfe wmt 0.30 2.31e-01 2.12e-01 5.59e-01 1.98e-01
t wcom 0.33 1.20e-01 1.37e-01 1.73e-01 1.40e-01
txn wcom 0.31 5.63e-01 4.06e-01 4.64e-01 4.17e-01
wmt xom 0.19 1.61e-01 5.38e-02 3.78e-02 4.94e-02
Table 8: Same as table 2 for stocks over a 5 years time interval from January 19, 1996 to December 29,
2000.
40

70
60
50
40
30
20
10
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 12: Same as figure 7 for stocks over a 10 years time interval from February 8, 1991 to December
29, 2000.
41
d
1
d
2
d
3
d
4
233

234 8. Tests de copule gaussienne
60
50
40
30
20
10
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 13: Same as figure 7 for stocks over a 5 years time interval from February 8, 1991 to January 18,
1996.
42
d
1
d
2
d
3
d
4

50
45
40
35
30
25
20
15
10
5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 14: Same as figure 7 for stocks over a 5 years time interval from January 19, 1996 to December
30, 2000.
43
d
1
d
2
d
3
d
4
235

236 8. Tests de copule gaussienne
Time interval from
Frebruary 8, 1991
to January 18, 1996
Time interval from
January 19, 1996 to
December 29, 2000
ˆρ d1 d2 d3 d4
hwp ibm 0.34 3.36e-01 2.26e-01 3.33e-01 2.35e-01
hwp intc 0.46 3.01e-01 4.73e-01 5.12e-01 5.21e-01
hwp msft 0.41 7.63e-01 4.72e-01 3.23e-01 4.53e-01
hwp sunw 0.40 2.96e-01 2.98e-01 7.66e-01 3.54e-01
ibm intc 0.30 4.81e-01 3.54e-01 4.18e-02 3.34e-01
ibm msft 0.24 3.93e-01 6.61e-01 5.88e-01 7.07e-01
ibm sunw 0.29 9.65e-01 9.71e-01 3.46e-01 9.86e-01
intc msft 0.47 2.59e-01 1.45e-01 4.50e-02 1.53e-01
intc sunw 0.40 4.81e-01 3.86e-01 4.47e-02 3.95e-01
msft sunw 0.40 2.68e-01 1.38e-01 1.66e-01 1.39e-01
ˆρ d1 d2 d3 d4
hwp ibm 0.46 2.02e-02 3.21e-02 9.60e-03 3.96e-02
hwp intc 0.44 2.88e-02 4.89e-02 6.00e-04 5.80e-02
hwp msft 0.37 5.23e-02 9.88e-02 3.36e-01 1.18e-01
hwp sunw 0.45 5.66e-01 5.65e-01 1.08e-01 6.23e-01
ibm intc 0.43 5.34e-02 3.31e-02 1.68e-02 2.44e-02
ibm msft 0.39 1.00e-02 9.50e-03 2.28e-02 8.80e-03
ibm sunw 0.46 2.35e-01 1.56e-01 3.38e-01 1.49e-01
intc msft 0.57 3.18e-01 1.61e-01 1.15e-01 1.71e-01
intc sunw 0.50 6.68e-02 3.55e-02 1.00e-04 4.37e-02
msft sunw 0.46 5.79e-02 7.60e-02 8.00e-04 8.07e-02
Table 9: Same as table 2 for stocks belonging to the informatic sector, over two time intervals of 5 years.
44

Chapitre 9
Mesure de la dépendance extrême entre
deux actifs financiers
Les deux chapitres précédents nous ont permis de comprendre l’importance de l’étude de la dépendance
extrême entre deux actifs financiers. En effet, nous avons montré qu’il était très difficile, par une estimation
directe de la copule, de pouvoir obtenir une mesure fiable de ces propriétés de dépendances
extrêmes. Or, celles-ci ont un impact très important sur la gestion des risques, et plus particulièrement
des grands risques, puisque selon que les extrêmes auront tendance à se produire ensemble ou de manière
indépendante, la constitution de portefeuilles agrégeant ces risques permettra, ou non, de les diversifier.
Nous avons mentionné, au chapitre 7, le coefficient de dépendance de queue comme moyen de quantifier
la propension des extrêmes à se produire de manière concomitante. Cependant, bien d’autres mesures de
dépendances entre les événements de grandes amplitudes peuvent être imaginées comme par exemple des
coefficients de corrélation conditionnés, et il convient donc de se demander pourquoi choisir telle mesure
plutôt que telle autre. Ce sera l’objet de la première partie de ce chapitre, où nous comparons ces diverses
méthodes de mesures des dépendances extrêmes et montrons qu’elles conduisent à des résultats parfois
surprenants et contradictoires. Ceci nous permettra de justifier notre choix concernant le coefficient de
dépendance de queue, dont nous passons également en revue les différents moyens d’estimation, ce qui
est en fait assez délicat et ne peut être réalisé de manière (relativement) précise que dans quelques cas
particuliers.
Une des méthodes d’estimation à laquelle nous nous sommes particulièrement intéressés s’appuie sur
les propriétés de la classe des modèles à facteurs, dont on sait qu’elle joue un rôle prépondérant dans la
modélisation des rendements des actifs financiers. Cette méthode est présentée en détail dans la seconde
partie de ce chapitre. Elle nous a permis d’estimer les coefficients de dépendance de queue d’un ensemble
d’actions sous l’hypothèse que l’indice de marché peut être considéré comme le facteur commun
principal expliquant la dépendance entre ces divers actifs.
Enfin, dans la dernière partie de ce chapitre, nous utilisons ces résultats pour répondre à la question que
nous nous posions à la fin du chapitre 7 sur la pertinence de l’approximation par la copule gaussienne, et
plus généralement elliptique, de la structure de dépendance entre actifs.
237

238 9. Mesure de la dépendance extrême entre deux actifs financiers
9.1 Les différentes mesures de dépendances extrêmes
Nous allons étudier l’information relative contenue dans diverses mesures de dépendance entre deux
variables aléatoires X et Y pour des événements grands et extrêmes et différents modèles de séries
financières. Les mesures considérées comprennent à la fois des quantités conditionnées et non conditionnées
telles que le coefficient de corrélation au-delà d’un seuil donné ou le coefficient de dépendance
de queue. Nous présentons des formules analytiques explicites ainsi que des résultats numériques et des
estimations empiriques de ces mesures de dépendance, ce qui nous permet de prouver quantitativement
que les dépendances conditionnées peuvent être très différentes des dépendances non conditionnées. Cet
effet de conditionnement fournit un mécanisme simple et général pour expliquer les changements de
corrélations basés sur des changements de volatilité ou de tendance. Ainsi, les outils basés sur des quantités
conditionnées doivent être utilisés avec précaution puisque le conditionnement à lui seul induit des
changements dans la structure de dépendance, qui n’ont rien à voir avec de réels changements de la
dépendance non conditionnée. Pour cela, le coefficient de dépendance de queue nous semble devoir être
préféré aux corrélations conditionnées. De plus, ces mesures de dépendances présentent des comportements
différents et parfois opposés, suggérant que les propriétés de dépendances extrêmes possèdent un
caractère multidimensionnel qui peut être quantifié de diverses manières conduisant à des conclusions
différentes.
Nous appliquons nos résultats théoriques au problème controversé de la contagion entre les marchés
sud-américains durant les périodes de turbulence associées aux crises financières mexicaine de 1994
et argentine débutée en 2001 1 . Notre analyse des différentes mesures de dépendance entre les marchés
argentin, brésilien, chilien et mexicain montre que les effets de conditionnement n’expliquent pas totalement
le comportement des marchés sud-américains ce qui confirme l’existence d’une possible contagion.
Nous trouvons que la crise mexicaine de 1994 s’est étendue aux marchés argentin et brésilien par
un mécanisme de contagion alors que de simples co-mouvements suffisent à expliquer les fluctuations
concomitantes du marché chilien. Au sujet de la récente crise argentine débutée en 2001, nous ne trouvons
pas de preuve de contagion vers les autres marchés sud-américains (excepté peut-être le Brésil)
mais identifions des co-mouvements significatifs.
1 Cette étude empirique semble déconnectée des autres études menées jusqu’ici, et l’on peut se demander pourquoi nous
n’avons pas continué à étudier le même jeu d’actifs que précédemment. Ceci est simplement dû à l’exigence du referee de la
revue à laquelle le papier a été soumis.

9.1. Les différentes mesures de dépendances extrêmes 239
Investigating Extreme Dependences: Conditioning Effect Versus
Contagion in Latin-American Crises ∗
Y. Malevergne 1,2 and D. Sornette 1,3
1 Laboratoire de Physique de la Matière Condensée CNRS UMR 6622
Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France
2 Institut de Science Financière et d’Assurances - Université Lyon I
43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex
3 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science
University of California, Los Angeles, California 90095, USA
email: Yannick.Malevergne@unice.fr and sornette@unice.fr
fax: (33) 4 92 07 67 54
Abstract
We investigate the relative information content of several measures of dependence between two random
variables X and Y for large or extreme events in various models of financial time series. The considered
measures involve both conditional and unconditional quantities such as the conditional correlation
coefficient over a given threshold or the coefficient of tail dependence. We offer explicit analytical formulas
as well as numerical and empirical estimations for these measures of dependence, which allow us
to provide a quantitative proof that conditional dependences may be very different from the unconditional
ones. This conditioning effect provides a straightforward and general mechanism for explaining changes
of correlations based on changes of volatility or of trends. Thus, tools based upon conditional quantities
should be used with caution since conditioning alone induces a change in the dependence structure
which has nothing to do with a genuine change of unconditional dependence. In this respect, for its
stability, the coefficient of tail dependence should be prefered to the conditional correlations. Moreover,
the various measures of dependence exhibit different and sometimes opposite behaviors, suggesting that
extreme dependence properties possess a multidimensional character that can be quantified in various
ways leading to different conclusions. We apply our theoretical results to the controversial contagion
problem across Latin American markets during the turmoil period associated with the Mexican crisis in
1994 and with the Argentina crisis that started in 2001. Our analysis of several measures of dependence
between the Argentina, Brazilian, Chilean and Mexican markets shows that the conditioning effect does
not fully explain the behavior of the Latin American stock indexes, confirming the existence of a possible
contagion. We find that the 1994 Mexican crisis has spread over to Argentina and Brazil through
contagion mechanisms and to Chile only through co-movements. Concerning the recent Argentina crisis
starting in 2001, we find no evidence of contagion to the other Latin American countries (except perhaps
in the direction of Brazil) but identify significant co-movements.
∗ We acknowledge helpful discussions and exchanges with J.P. Laurent, F. Lindskog and V. Pisarenko. This work was partially
supported by the James S. Mc Donnell Foundation 21st century scientist award/studying complex system.
1

240 9. Mesure de la dépendance extrême entre deux actifs financiers
Introduction
The October 19, 1987, stock-market crash stunned Wall Street professionals, hacked about $1 trillion off
the value of all U.S. stocks, and elicited predictions of another Great Depression. On “Black Monday,” the
Dow Jones industrial average plummeted 508 points, or 22.6 percent, to 1, 738.74. Contrary to common
belief, the US was not the first to decline sharply. Non-Japanese Asian markets began a severe decline on
October 19, 1987, their time, and this decline was echoed first on a number of European markets, then in
North American, and finally in Japan. However, most of the same markets had experienced significant but
less severe declines in the latter part of the previous week. With the exception of the US and Canada, other
markets continued downward through the end of October, and some of these declines were as large as the
great crash on October 19.
On December 19, 1994, the Mexican government, facing a solvency crisis, chose to devaluate the peso and
abandoned its exchange rate parity. This devaluation plunged the country into a major financial crisis which
quickly propagated to the rest of the Latin American countries.
From July 1997 to December 1997, several East Asian markets crashed, starting with the Thai market on
July 2, 1997 and ending with the Hong Kong market on October 17, 1997. After this regional event, the
turmoil spread over to the American and European markets.
The “slow” crash and in particular the turbulent behavior of the stock markets worldwide starting midaugust
1998 are widely associated with and even attributed to the plunge of the Russian financial markets,
the devaluation of its currency and the default of the government on its debts obligations.
The Nasdaq Composite index dropped precipiteously with a low of 3227 on April 17, 2000, corresponding
to a cumulative loss of 37% counted from its all-time high of 5133 reached on March 10, 2000. The drop
was mostly driven by the so-called “New Economy” stocks which have risen nearly fourfold over 1998 and
1999 compared to a gain of only 50% for the S&P500 index. And without technology, the benchmark would
be flat.
All these events epitomize the observation often reported by market professionals that, “during major market
events, correlations change dramatically” (Bookstaber 1997). The possible existence of changes of correlation,
or more precisely of changes of dependence, between assets and between markets in different market
phases has obvious implications in risk assessment, portfolio management and in the way policy and regulation
should be performed. Concerning portfolio management, (Ang and Bekaert 2001, Ang and Chen 2001)
for instance, have stressed that these questions related to state-varying-dependence are important for practical
applications since in such a case the optimal portfolio will also become state-dependent, and neglecting
this point can lead to very inefficient asset allocations. In this spirit, the recent Argentine crisis in 2001 has
triggered fears of a contagion to other Latin American market. Also, the Enron financial scandal at the end
of 2001 seems to have opened a flux of similar bankrupcies in other “new economy” companies.
From an academic perspective, all these manifestations of propagating crisis have given birth to an intense
activity concerning the notion of contagion (see (Claessens et al. 2001) for a review) which is defined,
according to the most commonly accepted definition, as an increase in the correlation (or linkage) across
markets during turmoil periods. In fact, as we shall see, there are two distinct classes of mechanisms for
understanding “changes of correlations”, not necessarily mutually exclusive.
• It is possible that there are genuine changes with time of the unconditional (with respect to amplitudes)
correlations and thus of the underlying structure of the dynamical processes, as observed by identifying
shifts in ARMA-ARCH/GARCH processes (Silvapulle and Granger 2001), in regime-switching
models (Ang and Bekaert 2001, Ang and Chen 2001) or in contagion models
2

9.1. Les différentes mesures de dépendances extrêmes 241
(Quintos 2001, Quintos et al. 2001). (Longin and Solnik 1995, Tsui and Yu 1999) and many others
have shown that the hypothesis of a constant conditional correlation for stock returns or international
equity returns must be rejected. In fact, there is strong evidence that the correlations are not only
time dependent but also state dependent. Indeed, as shown by (King and Wadhwani 1990, Ramchand
and Susmel 1998), the correlations increase in periods of large volatility. Moreover, (Longin
and Solnik 2001) have proved that the correlations across international equity markets are also trend
dependent.
• In contrast, a second class of explanation is that correlations between two variables conditioned on
signed exceedance (one-sided) or on absolute value (volatility) exceedance of one or both variables
may deviate significantly from the unconditional correlation (Boyer et al. 1997, Loretan 2000, Loretan
and English 2000). In other words, with a fixed unconditional correlation ρ, the measured correlation
conditioned of a given bullish trend, bearish trend, high or low market volatility, may in general differ
from ρ and be a function of the specific market phase. According to this explanation, changes of
correlation may be only a fallacious appearance that stems from a change of volatility or a change of
trend of the market and not from a real change of unconditional correlation or dependence.
The existence of the second class of explanation is appealing by its parsimony, as it posits that observed
“changes of correlation” may simply result from the way the measure of dependence is performed. This
approach has been followed by several authors but is often open to misinterpretation,
as stressed by (Forbes and Rigobon 2002). In addition, it may also be misleading since it does not provide
a signature or procedure to identify the existence of a genuine contagion phenomenon, if any. Therefore,
in order to clarify the situation and eventually develop more adequate tools for probing the dependences
between assets and between markets, it is highly desirable to characterize the different possible ways with
which higher or lower conditional dependence can occur in models with constant unconditional dependence.
In order to make progress, it is necessary to first distinguish between the different measures of dependence
between two variables for large or extreme events that have been introduced in the literature, because the
conclusions that one can draw about the variability of dependence are sensitive to the choice of its measure.
These measures include
1. the correlation conditioned on signed exceedance of one or both variables (Boyer et al. 1997, Loretan
2000, Loretan and English 2000, Cizeau et al. 2001), that we call respectively ρ + v and ρu, where u
and v denote the thresholds above which the exceedances are calculated,
2. the correlation conditioned on absolute value exceedance (or large volatility), above the threshold
v, of one or both variables (Boyer et al. 1997, Loretan 2000, Loretan and English 2000, Cizeau et
al. 2001), that we call ρ s v (for a condition of exceedance on one variable),
3. the tail-dependence parameter λ, which has a simple analytical expression when using copulas
(Embrechts et al. 2001, Lindskog 1999) such as the Gumbel copula (Longin and Solnik 2001),
and whose estimation provides useful information about the occurrence of extreme co-movements
(Malevergne and Sornette 2001, Poon et al. 2001, Juri and Wüthrich 2002),
4. the spectral measure associated with the tail index (assumed to be the same of all assets) of extreme
value multivariate distributions (Davis et al. 1999, Starica 1999, Hauksson et al. 2001),
5. tail indices of extremal correlations defined as the upper or lower correlation of exceedances of ordered
log-values (Quintos 2001),
6. confidence weighted forecast correlations (Bhansali and Wise 2001) or algorithmic complexity measures
(Mansilla 2001).
3

242 9. Mesure de la dépendance extrême entre deux actifs financiers
Our contribution to the literature is both methological and empirical. On the methodological front, firstof-all,
we review the existing tools available for probing the dependence between large or extreme events
for several models of interest for financial time series; second, we provide explicit analytical expressions
for these measures of dependence between two variables; third, this allows us to quantify the misleading
intrepretations of certain conditional coefficients commomly used for exploring the evolution of the dependence
associated with a change in the market conditions (an increase of the volatility, for instance). On the
empirical front, we apply our theoretical results to the controversial problem of the occurrence or not of a
contagion phenomenon across Latin American markets during the turmoil period associated with the Mexican
crisis in 1994 or with the recent Argentina crisis. In this purpose, we use the novel insight derived from
our analysis on several measures of dependence and apply them to the question of a possible evolution of
the dependence between the Argentina, Brazilian, Chilean and Mexican markets with respect to the market
conditions.
The dependence measures we study are the conditional correlation coefficients ρ + v , ρ s v, ρu, the conditional
Spearman’s rho ρs(v) and the tail dependence coefficients λ and ¯ λ, whose properties are investigated for
several models among which are the bivariate Gaussian distribution, the bivariate Student’s distribution,
and the one factor model for various distributions of the factor. Initially, we hoped to show the existence
of logical links between some of these measures, such as a vanishing tail-dependence parameter λ implies
vanishing asymptotic conditional correlation coefficients. In fact, we will show that this turns out to be
wrong and one can construct simple examples for which all possible combinations occur. Therefore, each
of these measures probe a different quality of the dependence between two variables for large or extreme
events. In addition, even if the conditional correlation coefficients are asymptotically zero, they decay in
general extremely slowly, as inverse powers of the value of the threshold, and may thus remain significant
for most pratical applications. These results will allow us to assert that, somewhat similarly to risks whose
adequate characterization requires an extension beyond the restricted one-dimensional measure in terms of
the variance (volatility) to include all higher order cumulants or more generally the knowledge of the full
distribution (Sornette et al. 2000a, Sornette et al. 2000b, Andersen and Sornette 2001), our results suggest
that tail-dependence has also a multidimensional character.
The paper is organized as follows.
Section 1 describes three conditional correlation coefficients, namely the correlation ρ + v conditioned on
signed exceedance of one variable, or on both variables (ρu) and the correlation ρ s v conditioned on absolute
value exceedance (or large volatility) of one variable. (Boyer et al. 1997) have already provided the general
expression of ρ + v and ρ s v for the Gaussian bivariate model, which are used to derive their v dependence for
large v, and to show that, for a given distribution, the conditional correlation coefficient changes even if the
unconditional correlation is let unchanged and the nature of this change depends on the conditioning set.
We then provide the general expression of ρ + v and ρ s v for the Student’s bivariate model with ν degrees of
freedom and for the factor model X = αY + ɛ, for which we give a general expression of the conditional
correlation coefficient whatever the distributions of Y and ɛ may be. This leads us to conclude by comparision
with the Gaussian model that, for a fixed conditioning set, the behavior of the conditional correlation
change dramatically from a distribution to another one. Conditioning now on both variables, we are able to
provide the asymptotic dependence of ρu only for the bivariate Gaussian model and show that it essentially
behaves like ρ + v . We then apply these results to show that we cannot entirely explain the behavior of the conditional
correlation coefficient of the Latin American stock indexes by the conditioning effect, suggesting
the existence of a possible contagion.
In section 2, to account for several deficiencies of the correlation coefficient, we propose an alternative
measure of dependence, the conditional Spearman’s rho, which is related to the probability of concordance
and discordance of several events drawn from the same probability distribution. This measure provides
an important improvement with respect to the correlation coefficient since it only takes into account the
4

9.1. Les différentes mesures de dépendances extrêmes 243
dependence structure of the variable and is not sensitive to the marginal behavior of each variable. We
perform numerical computations to derive the behavior of the conditional Spearman’s rho, denoted by ρs(v).
This allows us to prove that there is no direct relation between the Spearman’s rho conditioned on large
values and the correlation coefficient conditioned on the same values. Therefore, each of these coefficients
quantifies a different kind of extreme dependence. Then, calibrating our models on the Latin American
market data, we confirm that the conditional effect cannot fully explain the observed dependence and that
contagion can therefore be invoked. This results are much clearer for the conditional Spearman’s rho than
for the condition (linear) correlation coefficient, due to the much larger impact of large statistical fluctuations
in the later.
Section 3 discusses the tail-dependence parameters λ and ¯ λ. We first recall their definitions and their values
for Gaussian and Student’s bivariate distributions of X and Y , already known in the literature. For the
Gaussian factor model, it is trivial to show that λ = 0. A non-trivial result is obtained for the Student’s factor
model: λ is found non-zero and a function only of α and of the scale factor of ɛ. More generally, a theorem
established in (Malevergne and Sornette 2002) allows one to calculate the coefficient of tail dependence
for any distribution of the factor and shows that λ vanishes for any rapidly varying factor. We then apply
the (Poon et al. 2001)’s procedure to estimate non-parametrically the tail dependence coefficients. We find
them significant and thus conclude that with or without contagion mechanism, extreme co-movements must
naturally occur on the various Latin American markets as soon as one of them enters a crisis.
Section 4 provides a synthesis and comparison between these different results. A first important message is
that there is no unique measure of extreme dependence. Each of the coefficients of extreme dependence that
we have studied provides a specific quantification that is sensitive to a certain combination of the marginals
and of the copula of the two random variables. Similarly to risks whose adequate characterization requires
an extension beyond the restricted one-dimensional measure in terms of the variance (volatility) to include
the knowledge of the full distribution, tail-dependence has also a multidimensional character. A second
important message is that the increase of some of the conditional coefficients of extreme dependence as
one goes more in the tails does not necessarily signals a genuine increase of the unconditional correlation
or dependence between the two variables. Our calculations firmly confirm that this increase is a general
and unvoidable result of the statistical properties of many multivariate models of dependence. From the
standpoint of the contagion across Latin American markets, our theoretical and empirical results suggest an
asymmetric contagion phenomenon from Chile and Mexico onto Argentina and Brazil: large moves of the
Chilean and Mexican markets tend to propagate to Argentina and Brazil through contagion mechanisms, i.e.,
with a change in the dependence structure, while the converse does not hold. As a consequence, our study
seems to prove that the 1994 Mexican crisis have spread over to Argentina and Brazil through contagion
mechanisms and to Chile only through co-movements. Concerning the recent Argentina crisis starting in
2001, we find no evidence of contagion to the other Latin American countries (except perhaps in the direction
of Brazil) but identify only co-movements.
1 Conditional correlation coefficient
In this section, we discuss the properties of the correlation coefficient conditioned on one variable. We study
the difference between conditioning on the signed values or on absolute values of the variable (conditioning
on the absolute value of the variable of interest is only meaningful when its distribution is symmetric).
This allows us to conclude that conditioning on signed values generally provides more information than
conditioning on absolute values, and that, as already underlined by (Boyer et al. 1997, for instance), the
conditional correlation coefficient suffers from a bias which forbids its use as a measure of a change in the
correlation between two assets when the volatility increases, as seen in many papers about contagion. We
5

244 9. Mesure de la dépendance extrême entre deux actifs financiers
then present an empirical illustration of the evolution of the correlation between several stock indexes of
Latin American markets.
1.1 Definition
We study the correlation coefficient ρA of two real random variables X and Y conditioned on Y ∈ A, where
A is a subset of R such that Pr{Y ∈ A} > 0.
By definition, the conditional correlation coefficient ρA is given by
ρA =
Cov(X, Y | Y ∈ A)
Var(X | Y ∈ A) · Var(Y | Y ∈ A) . (1)
Applying this general expression of the conditional correlation coefficient, we will give closed formula for
several standard distributions and models. This will allow us to investigate the influence of the conditionning
set and the underlying model on the behavior of ρA.
1.2 Influence of the conditioning set
Let the variables X and Y have a multivariate Gaussian distribution with (unconditional) correlation coefficient
ρ. The following result have been proved by (Boyer et al. 1997) :
ρ
. (2)
ρA =
ρ2 + (1 − ρ2 Var(Y )
) Var(Y | Y ∈A)
We can note that ρ and ρA have the same sign, that ρA = 0 if and only if ρ = 0 and that ρA does not depend
directly on Var(X). Note also that ρA can be either greater or smaller than ρ since Var(Y | Y ∈ A) can be
either greater or smaller than Var(Y ). We will illustrate this property in the two following examples, where
we consider a conditioning on large positive (or negative) returns and a conditioning on large volatility.
The difference comes from the fact that in the first case, we account for the trend while we neglect this
information in the second case.
Example 1: conditioning on large (positive) returns. Let us first consider the conditioning set A =
[v, +∞), with v ∈ R+. Thus ρA is the correlation coefficient conditioned on the returns Y larger than a
given positive threshold v. It will be denoted by ρ + v in the sequel. Assuming for simplicity, but without loss
of generality, that Var(Y ) = 1, we can easily show (see appendix A.1.1 for an exact calculation) that for
large v
ρ + v ∼v→∞
ρ 1
· , (3)
1 − ρ2 v
which slowly goes to zero as v goes to infinity. Obviously, by symmetry, the conditional correlation coefficient
ρ − v , conditioned on Y smaller than v, obeys the same formula.
Example 2: conditioning on large volatilities. Let now the conditioning set be A = (−∞, −v] ∪
[v, +∞), with v ∈ R+. Thus ρA is the correlation coefficient conditioned on |Y | larger than v, i.e., it
is conditioned on large volatility of Y . Still assuming Var(Y ) = 1, we denote this correlation coefficient by
ρ s v and, as shown in appendix A.1.2, we can conclude that, for large v,
ρ s v ∼v→∞
ρ
ρ2 + 1−ρ2
2+v2 , (4)
6

9.1. Les différentes mesures de dépendances extrêmes 245
which goes to one as v goes to infinity as 1 − ρ s v ∼v→∞ 1−ρ2
ρ 2
These two simple examples clearly show that, in the case of two Gaussian random variables, the two conditional
correlation coefficients ρ + v and ρ s v exhibit opposite behavior since the conditional correlation coefficient
ρ + v is a decreasing function of v which goes to zero as v → +∞ while the conditional correlation
coefficient ρ s v is an increasing function of v and goes to one as v → ∞. These opposite behaviors are very
general and do not depend on the particular choice of the joint distribution of X and Y , namely the Gaussian
distribution studied until now, as it will be seen in the sequel.
This result underlines the importance of the choice of the conditioning set with the following two caveats.
First, as already stressed by many authors, the conditional correlation ρ + v ou ρ s v change with the value of the
threshod v even if the unconditional correlation ρ remains unchanged. Thus, the observation of a change
in the conditional correlation does not provide a reliable signature of a change in the true (unconditional)
correlation. Second, the conditional correlations can exhibit really opposite behaviors depending on the
conditioning sets. Specifically, we have seen that accounting for a signed trend or only for its amplitude
may yield a decrease or an increase of the conditional correlation with respect to the unconditional one, so
that these changes cannot be interpreted as a strengthening or a weakening of the correlations.
v −2 .
1.3 Influence of the underlying distribution for a given conditioning set
For a fixed conditioning set defining a specific conditional correlation coefficient like ρ + v or ρ s v, the behavior
of these coefficients can be dramatically different from a pair of random variables to another one, depending
on their underlying joint distribution. As an example, let the variables X and Y have a multivariate Student’s
distribution with ν degrees of freedom and an (unconditional) correlation coefficient ρ. According to the
proposition stated in appendix B.1, we have the exact formula
ρA =
ρ
ρ2 + E[E(X2 | Y )−ρ2Y 2 . (5)
| Y ∈A]
Var(Y | Y ∈A)
Appendix B.1 gives the explicit formulas of E[E(X 2 | Y ) − ρ 2 Y 2 | Y ∈ A] and Var(Y | Y ∈ A).
Expression (5) is the analog for a Student bivariate distribution to (2) derived above for the Gaussian bivariate
distribution. Again, ρ and ρA share the following properties: they have the same sign, ρA equals zero if and
only if ρ equals zero and ρA can be either greater or smaller than ρ. We now apply this general formula (5)
to the calculus of ρ + v and ρ s v and we find (see appendices B.3 and B.4) that, conditioning on large returns,
ρ + v −→
while when conditionning on large volatility,
ρ s v −→
ρ 2 + (ν − 1)
ρ 2 + 1
(ν−1)
ρ
ν−2
ν (1 − ρ2 )
ρ
ν−2
ν (1 − ρ 2 )
, (6)
. (7)
In both cases, ρ + v and ρ s v goes, at infinity, to non vanishing constant (exepted for ρ = 0). Moreover, for ν
larger than νc 2.839, this constant is smaller than the unconditional correlation coefficient ρ, for all value
of ρ, in the case of ρ + v , while for ρ s v it is always larger than ρ, whatever ν (larger than two) may be.
These results show that, conditioned on large returns, ρ + v is a decreasing function of the threshold v (at
least when ν ≥ 2.839), while, conditioned on large volatilities, ρ s v is an increasing function of v. Thus, for
7

246 9. Mesure de la dépendance extrême entre deux actifs financiers
Student’s random variables, ρ + v and ρ s v exhibit behaviors exactly opposite to those observed for Gaussian
random variables, namely decreasing ρ + v and increasing ρ s v, as shown in the previous paragraph.
To give another example, let us now assume that X and Y are two random variables following the equation
X = αY + ɛ , (8)
where α is a non random real coefficient and ɛ an idiosyncratic noise independent of Y , whose distribution
admits a centered moment of second order σ 2 ɛ . Let us also denote by σ 2 y the second centered moment of the
variable Y . This kind of relation between X and Y is the so-called one factor model whose applications
in finance can be traced back to (Ross 1976). This one factor model with independence between Y and
ɛ is of course naive for concrete applications, as it neglects the potential influence of other factors in the
determination of X. However, it has been argued to be a useful model in the context of the contagion,
and several studies have been based upon it (see (Baig and Goldfajn 1998) or (Forbes and Rigobon 2002),
for instance). Moreover for our purpose, it provides a simple illustrative model with rich and somewhat
surprising results.
Appendix C shows that the conditional correlation coefficient of X and Y is
ρ
ρA =
ρ2 + (1 − ρ2 )
Var(y)
, (9)
where
α · σy
ρ =
α2 · σ2 y + σ2 ɛ
Var(y | y∈A)
denotes the unconditional correlation coefficient of X and Y . Note that the term σ 2 ɛ in the expression (10)
of ρ is the only place where the influence of the idiosynchratic noise is felt.
Expression (9) is the same as (2) for the bivariate Gaussian situation studied in 1.2. This is not surprising
since, in the case where Y and ɛ have univariate Gaussian distributions, the joint distribution of X and
Y is a bivariate Gaussian distribution. The new fact is that this expression (9) remains true whatever the
distribution of Y and ɛ, provided that their second moments exist.
We now present the asymptotic expression of ρA for Y with a Gaussian or a Student’s distribution. Note
that the expression of ρA is simple enough to allow for exact calculations for a larger class of distributions,
but for our purpose, these two simple case will be sufficient.
Assuming that Y has a Gaussian distribution, while the distribution of ɛ can be everything (provided that
E[ɛ 2 ] < ∞) allows us to show that the same results as those given by equations (3) and (4) still hold, so that
ρ + v goes to zero, while ρ s v goes to one.
On the contrary, assuming that Y has a Student’s distribution yields both for ρ + v and ρ s v:
ρ +,s
v
∼
1
1 + K
v 2
(10)
, (11)
where K is a positive constant. ρ +,s
v thus goes to 1 as v goes to infinity with 1 − ρ +,s
v ∝ 1/v2 , which shows
that they can have similar behavior.
1.4 Conditional correlation coefficient on both variables as an alternative?
Since our intensive exploration of the behavior of the correlation coefficient conditionned on only one variable
clearly indicates that it exhibits any kind of behavior, it is natural to look for the effect of a more
8

9.1. Les différentes mesures de dépendances extrêmes 247
constraining conditioning. To this aim, we consider two random variables X and Y and define their conditional
correlation coefficient ρA,B, conditioned upon X ∈ A and Y ∈ B, where A and B are two subsets of
R such that Pr{X ∈ A, Y ∈ B} > 0, by
ρA,B =
Cov(X, Y | X ∈ A, Y ∈ B)
Var(X | X ∈ A, Y ∈ B) · Var(Y | X ∈ A, Y ∈ B) . (12)
In this case, it is much more difficult to obtain general results for any specified class of distributions compared
to the previous case of conditioning on a single variable. We have only been able to give the asymptotic
behavior for a Gaussian distribution in the situation detailed below, using the expressions in (Johnson and
Kotz 1972, p 113) or proposition A.1 of (Ang and Chen 2001).
Let us assume that the pair of random variables (X,Y) has a Normal distribution with unit unconditional variance
and unconditional correlation coefficient ρ. The subsets A and B are both choosen equal to [u, +∞),
with u ∈ R+, so that we focus on the correlation coefficient conditional on the returns of both X and Y
larger than the threshold u. Denoting by ρu the correlation coefficient conditional on this particular choice
for the subsets A and B, we are able to show (see eppendix A.2) that, for large u:
ρu ∼u→∞ ρ
1 + ρ
1 − ρ
1
· , (13)
u2 which goes to zero. This decay is faster than in the case governed by (3) resulting from the conditioning on
a single variable leading to ρ + v ∼v→+∞ 1/v, but, unfortunately, we do not observe a qualitative change.
Thus, the correlation coefficient conditioned on both variables does not yield new significant information
and does not provide any special improvement with respect to the correlation coefficient conditioned on a
single variable.
1.5 Empirical evidence
We consider four national stock markets in Latin America, namely Argentina (MERVAL index), Brazil
(IBOV index), Chile (IPSA index) and Mexico (MEXBOL index). We are particularly interested in the
contagion effects which may have occurred across these markets. We will study this question for the market
indexes expressed in US Dollar to emphasize the effect of the devaluations of local currencies and so to
account for monetary crises. Doing so, we follow the same methodology as in most contagion papers (see
(Forbes and Rigobon 2002), for instance). Our sample contains the daily (log) returns of each stock in
local currency and US dollar during the time interval from January 15, 1992 to June 15, 2002 and thus
encompasses both the Mexican crisis as well as the current Argentina crisis.
Before applying the theoretical results derived above, we first need to check whether we are allowed to do
so. Namely, we have to test whether the index returns distributions are not too fat tailed. Indeed, it its well
known that the correlation coefficient exists if and only if the tail of the distribution decays faster than a
power law with tail index α = 2, and its estimator given by the Pearson’s coefficient is well behaved if at
least the fourth moment of the distribution is finite.
Figure 1 represents the complementary distribution of the positive and negative tails of the index returns in
US dollar. We observe that the positive tail clearly decays faster than a power law with tail index µ = 2.
In fact, Hill’s estimator provides a value ranging between 3 and 4 for the four indexes. The case of the
negative tail is slightly different, particularly for the Brazilian index. Indeed, for the Argentina, the Chilean
and the Mexican indexes, the negative tail behaves almost like the positive one, but for the Brazilian index,
the negative tail exponent is hardly larger than two, as confirmed by Hill’s estimator. This means that, in
9

248 9. Mesure de la dépendance extrême entre deux actifs financiers
the Brazilian case, the estimates of the correlation coefficient will be particularly noisy and thus of weak
statistical value.
We have checked that the fat tailness of the indexes expressed in US dollar comes from the impact of the
exchange rates. Thus, an alternative should be to consider the indexes in local currency, following (Longin
and Solnik 1995) and (Longin and Solnik 2001)’s methodology, but it would lead to focus on the linkages
between markets only and to neglect the impact of the devaluations, which is precisely the main concern of
many studies about the contagion in Latin America.
Figures 2, 4 and 6 give the conditional correlation coefficient ρ +,−
v
(plain thick line) for the pairs Ar-
gentina / Brazil, Brazil / Chile and Chile / Mexico while figures 3, 5 and 7 show the conditional correlation
coefficient ρ s v for the same pairs. For each figure, the dashed thick line gives the theoretical curve obtained
under the bivariate Gaussian assumption whose analytical expressions can be found in appendices A.1.1
and A.1.2. The unconditional correlation coefficient of the Gaussian model is set to the empirically estimated
unconditional correlation coefficent. The two dashed thin lines represent the interval within which
we cannot reject, at the 95% confidence level, the hypothesis according to which the estimated conditional
correlation coefficient is equal to the theoretical one. This confidence interval has been estimated using the
Fisher’s statistics. Similarly, the thick dotted curve graphs the theoretical curve obtained under the bivariate
Student’s assumption with ν = 3 degrees of freedom (whose expressions are given in appendices B.3
and B.4) and the two thin dotted lines are its 95% confidence level. Here, the Fisher’s statistics cannot be
applied, since it requires at least that the fourth moment of the distribution exists. In fact, (Meerschaert and
Scheffler 2001) have shown that, in such a case, the distribution of the sample correlation converges to a
stable law with index 3/2, which justifies why the confidence interval for the Student’s model with three
degres of freedom is much larger than the confidence interval for the Gaussian model. In the present study,
we have used a bootstrap method to derive this confidence interval since the scale factor intervening in the
stable law is difficult to calculate.
In figures 2, 4 and 6, we observe that the change in the conditional correlation coefficients ρ +,−
v
are not sig-
nificantly different, at the 95% confidence level, from those obtained with a bivariate Student’s model with
three degrees of freedom. In contrast, the Gaussian model is often rejected. In fact, similar results hold (but
are not depicted here) for the three others pairs Argentina / Chile, Argentina / Mexico and Brazil / Mexico.
Thus, these observations should lead us to conclude that, in these cases, no change in the correlations, and
therefore no contagion mechanism, needs to be invoked to explain the data, since they are compatible with
a Student’s model with constant correlation.
Let us now discuss the results obtained for the correlation coefficient conditioned on the volatility. Figures
3 and 7 show that the estimated correlation coefficients conditioned on volatility remain consistent with the
Student’s model with three degres of freedom, while they still reject the Gaussian model. In contrast, figure
5 shows that the increase of the correlation cannot be explained by any of the Gaussian or Student models,
when conditioning on the Mexican index volatilty. Indeed, when the Mexican index volatility becomes larger
than 2.5 times its standard deviation, none of our models can account for the increase of the correlation. The
same discrepancy is observed for the pairs Argentina / Chile, Argentina / Mexico and Brazil / Mexico which
have not been represented here. In each case, the Chilean or the Mexican market have an impact on the
Argentina or the Brazilian market which cannot be accounted for by neither the Gaussian or Student models
with constant correlation.
To conclude this empirical part, there is no significant increase in the real correlation between Argentina
and Brazil in the one hand and between Chile and Mexico in the other hand, when the volatility or the
returns exhibit large moves. But, in period of high volatility, the Chilean and Mexican market may have an
genuine impact on the Argentina and Brazilian markets. Thus, a priori, this should lead us to conclude on
the existence of a contagion across these markets. However, this conclusion is based on the investigation of
10

9.1. Les différentes mesures de dépendances extrêmes 249
only two theoretical models and it would be a little bit too hasty to conclude on the existence of contagion
on the sole basis of these results. This is all the more so since our theoretical models are all symmetric in
their positive and negative tails, a crucial property needed for the derivation of the expressions of ρ s v, while
the empirical sample distributions are certainly not symmetric, as shown in figure 1.
1.6 Summary
The previous sections have shown that the conditional correlation coefficients can exhibit any behavior,
depending on their conditioning set and the underlying distributions of returns. More precisely, we have
shown that the correlation coefficients, conditioned on large returns or volatility above a threshold v, can be
either increasing or decreasing functions of the threshold, can go to any value between zero and one when
the threshold goes to infinity and can produce contradictory results in the sense that accounting for a trend or
not can lead to conclude on an absence of linear correlation or on a perfect linear correlation. Moreover, due
to the large statistical fluctuations of the empirical estimates, one should be very careful when concluding
on an increase or decrease of the genuine correlations.
Thus, from the general standpoint of the study of extreme dependences, but more particularly for the specific
problem of the contagion across countries, the use of conditional correlation does not seem very informative
and is sometimes misleading since it leads to spurious changes in the observed correlations: even when
the unconditional correlation remains constant, conditional correlations yield artificial changes as we have
forcefully shown. Since one of the most commonly accepted and used definition of contagion is the detection
of an increase of the conditional correlations during a period of turmoil, namely when the volatility increases,
our results cast serious shadows on previous results. In this respect, the conclusions of (Calvo and Reinhart
1996) about the occurrence of contagion across Latin American markets during the 1994 Mexican crisis but
more generally also the results of (King and Wadhwani 1990), or (Lee and Kim 1993) on the effect of the
October 1987 crash on the linkage of national markets, must be considered with some caution. It is quite
desirable to find a more reliable tool for studying extreme dependences.
2 Conditional concordance measures
The (conditional) correlation coefficients, which have just been investigated, suffer from several theoretical
as well as empirical deficiencies. From the theoretical point of view, it is only a measure of linear dependence.
Thus, as stressed by (Embrechts et al. 1999), it is fully satisfying only for the description of the
dependence of variables with elliptical distributions. Moreover, the correlation coefficient aggregates the
information contained both in the marginal and in the collective behavior. The correlation coefficient is
not invariant under an increasing change of variable, a transformation which is known to let unchanged the
dependence structure. From the empirical standpoint, we have seen that, for some considered data, the correlation
coefficient may not always exist, and even when it exits, it cannot always be accuratly estimated, due
to sometimes “wild” statistical fluctuations. Thus, it is desirable to find another measure of the dependence
between two assets or more generally between two random variables, which, contrarily to the linear correlation
coefficient, is always well-defined and only depends on the copula properties. This ensures that this
measure is not affected by a change in the marginal distributions (provided that the mapping is increasing).
It turns out that this desirable property is shared by all measures of concordance. Among these measures
are the well-known Kendall’s tau, Spearman’s rho or Gini’s beta (see (Nelsen 1998) for details).
However, these concordance measures are not well-adapted, as such, to the study of extreme dependence,
because they are functions of the whole distribution, including the moderate and small returns. A simple idea
11

250 9. Mesure de la dépendance extrême entre deux actifs financiers
to investigate the extreme concordance properties of two random variables is to calculate these quantities
conditioned on values larger than a given threshold and let this threshold go to infinity.
In the sequel, we will only focus on the Spearman’s rho which can be easily estimated empirically. It offers a
natural generalization of the (linear) correlation coefficient. Indeed, the correlation coefficient quantifies the
degree of linear dependence between two random variables, while the Spearman’s rho quantifies the degree
of functional dependence, whatever the functional dependence between the two random variables may be.
This represents a very interesting improvement. Perfect correlations (respectively anti-correlation) give a
value 1 (respectively −1) both for the standard correlation coefficient and for the Spearman’s rho. Otherwise,
there is no general relation allowing us to deduce the Spearman’s rho from the correlation coefficient and
vice-versa.
2.1 Definition
The Spearman’s rho, denoted ρs in the sequel, measures the difference between the probability of concordance
and the probability of discordance for the two pairs of random variables (X1, Y1) and (X2, Y3),
where the pairs (X1, Y1), (X2, Y2) and (X3, Y3) are three independent realizations drawn from the same
distribution:
ρs = 3 (Pr[(X1 − X2)(Y1 − Y3) > 0] − Pr[(X1 − X2)(Y1 − Y3) < 0]) . (14)
The Spearman’s rho can also be expressed with the copula C of the two variables X and Y (see (Nelsen
1998), for instance):
ρs = 12
1 1
0
0
C(u, v) du dv − 3, (15)
which allows us to easily calculate ρs when the copula C is known in closed form.
Denoting U = FX(X) and V = FY (V ), it is easy to show that ρs is nothing but the (linear) correlation
coefficient of the uniform random variables U and V :
Cov(U, V )
ρs = . (16)
Var(U)Var(V )
This justifies its name as a correlation coefficient of the rank, and shows that it can easily be estimated.
An attractive feature of the Spearman’s rho is to be independent of the margins, as we can see in equation
(15). Thus, contrarily to the linear correlation coefficient, which aggregates the marginal properties
of the variables with their collective behavior, the rank correlation coefficient takes into account only the
dependence structure of the variables.
Using expression (16), we propose a natural definition of the conditional rank correlation, conditioned on V
larger than a given threshold ˜v:
ρs(˜v) =
whose expression in term of the copula C(·, ·) is given in appendix D.
2.2 Example
Cov(U, V | V ≥ ˜v)
Var(U | V ≥ ˜v)Var(V | V ≥ ˜v) , (17)
Contrarily to the conditional correlation coefficient, we have not been able to obtain analytical expressions
for the conditional Spearman’s rho, at least for the distributions that we have considered up to now. Obviously,
for many families of copulas known in closed form, equation (17) allows for an explicit calculation
12

9.1. Les différentes mesures de dépendances extrêmes 251
of ρs(v). However, most copulas of interest in finance have no simple closed form, so that it is necessary to
resort to numerical computations.
As an example, let us consider the bivariate Gaussian distribution (or copula) with unconditional correlation
coefficent ρ. It is well-known that its unconditional Spearman’s rho is given by
ρs = 6 ρ
· arcsin . (18)
π 2
The left panel of figure 8 shows the conditional Spearman’s rho ρs(v) defined by (17) obtained from a
numerical integration. We observe the same bias as for the conditional correlation coefficient, namely the
conditional rank correlation changes with v eventhough the unconditional correlation is fixed to a constant
value. Nonetheless, this conditional Spearman’s rho seems more sensitive than the conditional correlation
coefficient since we can observe in the left panel of figure 8 that, as v goes to one, the conditional Spearman’s
rho ρs(v) does not go to zero for all values of ρ (at the precision of our bootstrap estimates), as previously
observed with the conditional correlation coefficient (see equation (3)).
The right panel of figure 8 depicts the conditional Spearman’s rho of the Student’s copula with three degres
of freedom. The results are the same concerning the bias, but this time ρs(v) goes to zero for all value
of ρ when v goes to one. Thus, here again, many different behaviours can be observed depending on the
underlying copula of the random variables. Moreover, these two examples show that the quantification of
extreme dependence is a function of the tools used to quantify this dependence. Here, the conditional Spearman’s
ρ goes to a non-vanishing constant for the Gaussian model, while the conditional (linear) correlation
coefficient goes to zero, contrarily to the Student’s distribution for which the situation is exactly the opposite.
2.3 Empirical evidence
Figures 9, 10 and 11 give the conditionnal Spearman’s rho respectively for the Argentina / Brazilian stock
markets, the Brazilian / Chilean stock markets and the Chilean / Mexican stock markets. As previously, the
plain thick line refers to the estimated correlation, while the dashed lines refer to the Gaussian copula and
its 95% confidence levels and and dotted lines to the Student’s copula with three degrees of freedom and its
95% confidence levels.
We first observe that contrarily to the cases of the conditional (linear) correlation coefficient exhibited in
figures 2, 4 and 6, the empirical conditional Spearman’s ρ does not always comply with the Student’s model
(neither with the Gaussian one), and thus confirm the discrepancies observed in figures 3, 5 and 7. In all
cases, for thresholds v larger than the quantile 0.5 corresponding to the positive returns, the Student’s model
with three degrees of freedom is always sufficient to explain the data. In contrast, for the negative returns
and thus thresholds v lower then the quantile 0.5, only the interaction between the Chilean and the Mexican
markets is well described by the Student copula and does not needed any additional ingredient such as the
contagion mechanism. For all other pairs, none of our models explain the data satisfyingly. Therefore, for
these cases and from the perspective of our models, the contagion hypothesis seems to be needed.
There are however several caveats. First, even though we have considered the most natural financial models,
there may be other models, that we have ignored, with constant dependence structure which can account for
the observed evolutions of the conditional Spearman’s ρ. If this is true, then the contagion hypothesis would
not be needed. Second, the discrepancy between the empirical conditional Spearman’s ρ and the prediction
of the the Student’s model does not occur in the tails the distribution, i.e for large and extreme movements,
but in the bulk. Thus, during periods of turmoil, the Student’s model with three degrees fo freedom seems to
remain a good model of co-movements. Third, the contagion effect is never necessary for upwards moves.
Indeed, we observe the same asymmetry or trend dependence as found by (Longin and Solnik 2001) for five
13

252 9. Mesure de la dépendance extrême entre deux actifs financiers
major equity markets. This was apparent in figures 2, 4 and 6 for ρ +,−
v , and is strongly confirmed on the
conditional Spearman’s ρ.
Interestingly, there is also an asymmetry or directivity in the mutual influence between markets. For instance,
the Chilean and Mexican markets have an influence on the Argentina and Brazilian markets, but the later do
not have any impact on the Chile and Mexican markets. Chile and Mexico have no contagion effect on each
other while Argentina and Brazil have.
These empirical results on the conditional Spearman’s ρ are different from and often opposite to the conclu-
sion derived from the conditional correlation coefficients ρ +,−
v . This puts in light the difficulty in obtaining
reliable, unambiguous and sensitive estimations of conditional correlation measures. In particular, the Pearson’s
coefficient usually employed to estimate the correlation coefficient between two variables is known
to be not very efficient when the variables are fat-tailed and when the estimation is performed on a small
sample. Indeed, with small samples, the Pearson’s coefficient is very sensitive to the largest value, which
can lead to an important bias in the estimation. Moreover, even with large sample sizes, (Meerschaert and
Scheffler 2001) have shown that the nature of convergence as the sample size T tends to infinity of the Pearson’s
coefficient of two times series with tail index µ towards the theoretical correlation is sensitive to the
existence and strength of the theoretical correlation. If there is no theoretical correlation between the two
times series, the sample correlation tends to zero with Gaussian fluctuations. If the theoretical correlation is
non-zero, the difference between the sample correlation and the theoretical correlation times T 1−2/µ converges
in distribution to a stable law with index µ/2. These large statistical fluctuations are responsible for
the lack of accuracy of the estimated conditional correlation coefficient encountered in the previous section.
Thus, we think that the conditional Spearman’s ρ provides a good alternative both from a theoretical and an
empirical viewpoint.
3 Tail dependence
For the sake of completeness, and since it is directly related to the multivariate extreme values theory, we
study the so-called coefficient of tail dependence λ. To our knowledge, its interest for financial applications
has been first underlined by (Embrechts et al. 2001).
The coefficient of tail dependence characterizes an important property of the extreme dependence between
X and Y , using the (original or unconditional) copula of X and Y . In constrast, the conditional spearman’s
rho is defined in terms of a conditional copula, and can be seen as the “unconditional Spearman’s rho” of the
copula of X and Y conditioned on Y larger than the threshold v. This copula of X and Y conditioned on
Y larger than the threshold v is not the true copula of X and Y because it is modified by the conditioning.
In this sense, the tail dependence parameter λ is a more natural property directly related to the copula of X
and Y .
To begin with, we recall the definition of the coefficient λ as well as of ¯ λ (see below) which allows one to
quantify the amount of dependence in the tail. Then, we present several results concerning the coefficient λ
of tail dependence for various distributions and models, and finally, we discuss the problems encountered in
the estimation of these quantities.
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9.1. Les différentes mesures de dépendances extrêmes 253
3.1 Definition
The concept of tail dependence is appealing by its simplicity. By definition, the (upper) tail dependence
coefficient is:
−1
λ = lim
(u)|Y > F (u)] , (19)
Pr[X > F
u→1 −1
X
and quantifies the probability to observe a large X, assuming that Y is large itself. For a survey of the
properties of the tail dependence coefficient, the reader is refered to (Coles et al. 1999, Embrechts et al.
2001, Lindskog 1999), for instance. In words, given that Y is very large (which occurs with probability
1 − u), the probability that X is very large at the same probability level u defines asymptotically the tail
dependence coefficient λ. As an example, considering that X and Y represent the volatility of two different
national markets, the coefficient of tail dependence λ gives the probabilty that both markets exhibit together
very high volatility.
One of the appeal of this definition of tail dependence is that it is a pure copula property, i.e., it is independent
of the margins of X and Y . Indeed, let C be the copula of the variables X and Y , then if the bivariate copula
C is such that
Y
1 − 2u + C(u, u)
log C(u, u)
lim
= lim 2 − = λ (20)
u→1 1 − u
u→1 log u
exists, then C has an upper tail dependence coefficient λ (see (Coles et al. 1999, Embrechts et al. 2001,
Lindskog 1999)).
If λ > 0, the copula presents tail dependence and large events tend to occur simultaneously, with the
probability λ. On the contrary, when λ = 0, the copula has no tail dependence and the variables X and Y
are said asymptotically independent. There is however a subtlety in this definition (19) of tail dependence.
To make it clear, first consider the case where for large X and Y the cumulative distribution function H(x, y)
factorizes such that
F (x, y)
lim
= 1 , (21)
x,y→∞ FX(x)FY (y)
where FX(x) and FY (y) are the margins of X and Y respectively. This means that, for X and Y sufficiently
large, these two variables can be considered as independent. It is then easy to show that
lim
u→1 Pr{X > FX −1 (u)|Y > FY −1 (u)} = lim 1 − FX(FX
u→1 −1 (u)) (22)
= lim
u→1 1 − u = 0, (23)
so that independent variables really have no tail dependence λ = 0, as one can expect.
However, the result λ = 0 does not imply that the multivariate distribution can be automatically factorized
asymptotically, as shown by the Gaussian example. Indeed, the Gaussian multivariate distribution does not
have a factorizable multivariate distribution, even asymptotically for extreme values, since the non-diagonal
term of the quadratic form in the exponential function does not become negligible in general as X and Y go
to infinity. Therefore, in a weaker sense, there may still be a dependence in the tail even when λ = 0.
To make this statement more precise, following (Coles et al. 1999), let us introduce the coefficient
¯λ = lim
u→1
2 log Pr{X > FX −1 (u)}
log Pr{X > FX −1 (u), Y > FY −1 (u)}
− 1 (24)
2 log(1 − u)
= lim
− 1 . (25)
u→1 log[1 − 2u + C(u, u)]
It can be shown that the coefficient ¯ λ = 1 if and only if the coefficient of tail dependence λ > 0, while ¯ λ
takes values in [−1, 1) when λ = 0, allowing us to quantify the strength of the dependence in the tail in such
15

254 9. Mesure de la dépendance extrême entre deux actifs financiers
a case. In fact, it has been established that, when ¯ λ > 0, the variables X and Y are simultaneously large more
frequently than independent variables, while simultaneous large deviations of X and Y occur less frequenlty
than under independence when ¯ λ < 0 (the interested reader is refered to (Ledford and Tawn 1996, Ledford
and Tawn 1998)).
To summarize, independence (factorization of the bivariate distribution) implies no tail dependence λ = 0.
But λ = 0 is not sufficient to imply factorization and thus true independence. It also requires, as a necessary
condition, that ¯ λ = 0.
We will first recall the expression of the tail dependence coefficient for usual distributions, and then calculate
it in the case of a one-factor model for different distributions of the factor.
3.2 Tail dependence for Gaussian distributions and Student’s distributions
Assuming that (X, Y ) are normally distributed with correlation coefficient ρ, (Embrechts et al. 2001) shows
that for all ρ ∈ [−1, 1), λ = 0, while (Heffernan 2000) gives ¯ λ = ρ, which expresses, as one can expect,
that extremes appear more likely together for positively correlated variables.
In constrast, if (X, Y ) have a Student’s distribution, (Embrechts et al. 2001) shows that the tail dependence
coefficient is
λ = 2 · ¯
√ν 1 − ρ
Tν+1 + 1 ,
1 + ρ
(26)
which is greater than zero for all ρ > −1, and thus ¯ λ = 1. This last example proves that extremes appear
more likely together whatever the correlation coefficient may be, showing that, in fact, there is no general
relationship between the asymptotic dependence and the linear correlation coefficient.
The Gaussian and Student’s distributions are elliptical, for which the following general result is known:
(Hult and Lindskog 2001) have shown that ellipticaly distributed random variables presents tail dependence
if and only if they are regularly varing, i.e., behaves asymptotically like power laws with some exponent
ν > 0. In such a case, for every regularly varying pair of random variables elliptically distributed, the
coefficent of tail dependence λ is given by expression (26). This result is very natural since the correlation
coefficient is an invariant quantity within the class of elliptical distributions and since the coefficient of tail
dependence is only determined by the asymptotic behavior of the distribution, so that it does not matter that
the distribution is a Student’s distribution with ν degrees of freedom or any other elliptical distribution as
long as they have the same asymptotic behavior in the tail.
3.3 Tail dependence generated by a factor model
Consider the one-factor model
X1 = α1 · Y + ɛ1, (27)
X2 = α2 · Y + ɛ2, (28)
where the ɛi’s are random variables independent of Y and the αi’s non-random positive coefficients.
(Malevergne and Sornette 2002) have shown that the tail dependence λ of X1 and X2 can be simply expressed
as the minimum of the tail dependence coefficients λ1 and λ2 between the two random variables X1
and Y and X2 and Y respectively:
λ = min{λ1, λ2}. (29)
16

9.1. Les différentes mesures de dépendances extrêmes 255
To understand this result, note that the tail dependence between X1 and X2 is created only through the
common factor Y . It is thus natural that the tail dependence between X1 and X2 is bounded from above
by the weakest tail dependence between the Xi’s and Y while deriving the equality requires more work
(Malevergne and Sornette 2002). Thus, it it only necessary to focus our study on the tail dependence
between any Xi and Y . So, in order to simplify the notations, we neglect the subscripts 1 or 2, since
they are irrelevant for the dependence of X1 (or X2) and Y .
A general result concerning the tail dependence generated by factor models for every kind of factor and
noise distributions can be found in (Malevergne and Sornette 2002). It has been proved that the coefficient
of (upper) tail dependence between X and Y is given by
∞
where, provided that they exist,
λ =
max{1, l
α}
l = lim
u→1
dx f(x) , (30)
FX −1 (u)
FY −1 , (31)
(u)
t · PY (t · x)
f(x) = lim
. (32)
t→∞ ¯FY (t)
As a direct consequence, one can show that any rapidly varying factor, which encompasses the Gaussian, the
exponential or the gamma distributed factors for instance, leads to a vanishing coefficient of tail dependence,
whatever the distribution of the idiosyncratic noise may be. This resut is obvious when both the factor and
the idiosyncratic are Gaussianly distributed, since then X and Y follow a bivariate Gaussian distibution,
whose tail dependence has been said to be zero.
On the contrary, regularly vaying factors, like the Student’s distributed factors, lead to a tail dependence,
provided that the distribution of the idiosycratic noise does not become fatter-tailed than the factor distribution.
One can thus conclude that, in order to generate tail dependence, the factor must have a sufficiently
‘wild’ distribution. To present an explicit example, let us assume now that the factor Y and the idiosyncratic
noise ɛ have centered Student’s distributions with the same number ν of degrees of freedom and scale factors
respectively equal to 1 and σ. The choice of the scale factor equal to 1 for Y is not restrictive but only
provides a convenient normalization for σ. Appendix E shows that the tail dependence coefficient is
1
λ =
1 +
σ ν . (33)
α
As is reasonable intuitively, the larger the typical scale σ of the fluctuation of ɛ and the weaker is the coupling
coefficient α, the smaller is the tail dependence.
Let us recall that the unconditional correlation coefficient ρ can be writen as ρ = (1+ σ2
α 2 ) −1/2 , which allows
us to rewrite the coefficient of upper tail dependence as
λ =
ρν ρν + (1 − ρ2 . (34)
) ν/2
Surprinsingly, λ does not go to zero for all ρ’s as ν goes to infinity, as one would expect intuitively. Indeed,
a natural reasoning would be that, as ν goes to infinity, the Student’s distribution goes to the Gaussian
distribution. Therefore, one could a priori expect to find again the result given in the previous section for
the Gaussian factor model. We note that λ → 0 when ν → ∞ for all ρ’s smaller than 1/ √ 2. But, and
here lies the surprise, λ → 1 for all ρ larger than 1/ √ 2 when ν → ∞. This counter-intuitive result is due
17

256 9. Mesure de la dépendance extrême entre deux actifs financiers
to a non-uniform convergence which makes the order to two limits non-commutative: taking first the limit
u → 1 and then ν → ∞ is different from taking first the limit ν → ∞ and then u → 1. In a sense, by taking
first the limit u → 1, we always ensure somehow the power law regime even if ν is later taken to infinity.
This is different from first “sitting” on the Gaussian limit ν → ∞. It then is a posteriori reasonable that the
absence of uniform convergence is made strongly apparent in its consequences when measuring a quantity
probing the extreme tails of the distributions.
As an illustration, figure 12 represents the coefficient of tail dependence for the Student’s copula and Student’s
factor model as a function of ρ for various value of ν. It is interesting to note that λ equals zero for all
negative ρ in the case of the factor model, while λ remains non-zero for negative values of the correlation
coefficient for bivariate Student’s variables.
If Y and ɛ have different numbers νY and νɛ of degrees of freedom, two cases occur. For νY < νɛ, ɛ is
negligible asymptotically and λ = 1. For νY > νɛ, X becomes asymptotically identical to ɛ. Then, X and
Y have the same tail-dependence as ɛ and Y , which is 0 by construction.
3.4 Estimation of the coefficient of tail dependence
It would seem that the coefficient of tail dependence could provide a useful measure of the extreme dependence
between two random variables which could then be useful for the analysis of contagion between
markets. Indeed, either the whole data set does not exhibit tail dependence, and a contagion mechanism
seems necessary to explain the occurrence of concomitant large movements during turmoil periods, or it
exhibits tail dependence so that the usual dependence structure is such that it is able to produce by itself
concomitant extremes.
Unfortunately, the empirical estimation of the coefficient of tail dependence is a strenuous task. Indeed, a
direct estimation of the conditional probability Pr{X > FX −1 (u) | Y > FY −1 (u)}, which should tend
to λ when u → 1 is impossible to put in practice due to the combination of the curse of dimensionality
and the drastic decrease of the number of realisations as u become close to one. A better approach consists
in using kernel estimators, which generally provide smooth and accurate estimators (Kulpa 1999, Li et
al. 1998, Scaillet 2000). However, these smooth estimators lead to differentiable estimated copulas which
have automatically vanishing tail dependence. Indeed, in order to obtain a non-vanishing coefficient of tail
dependence, it is necessary for the corresponding copula to be non-differentiable at the point (1, 1) (or at
(0, 0)). An alternative is then the fully parametric approach. One can choose to model dependence via a
specific copula, and thus to determine the associated tail dependence (Longin and Solnik 2001, Malevergne
and Sornette 2001, Patton 2001). The problem with such a method is that the choice of the parameterization
of the copula amounts to choose a priori whether or not the data presents tail dependence.
In fact, there exist three ways to estimate the tail dependence coefficient. The two first ones are specific to a
class of copulas or of models, while the last one is very general, but obvioulsy less accurate. The first method
is only reliable when it is known that the underlying copula is Archimedian (see (Joe 1997) or (Nelsen 1998)
for the definition). In such a case, a limit theorem established by (Juri and Wüthrich 2002) allows to estimate
the tail dependence. The problem is that it is not obvious that the Archimedian copulas provide a good
representation of the dependence structure for financial assets. For instance, the Achimedian copulas are
generally inconsistent with a representation of assets by factor models. In such case, a second method
provided by (Malevergne and Sornette 2002) offers good results allowing to estimate the tail dependence in
a semi-parametric way, which solely relies on the estimation of marginal distributions, a significantly easier
task.
When none of these situations occur, or when the factors are too difficult to extract, a third and fully non-
18

9.1. Les différentes mesures de dépendances extrêmes 257
parametric method exists, which is based upon the mathematical results by (Ledford and Tawn 1996, Ledford
and Tawn 1998) and (Coles et al. 1999) and has recently been applied by (Poon et al. 2001). The method
consists in tranforming the original random variables X and Y into Fréchet random variables denoted by S
and T respectively. Then, considering the variable Z = min{S, T }, its survival distribution is:
Pr{Z > z} = d · z 1/η
as z → ∞, (35)
and ¯ λ = 2 · η − 1, with λ = 0 if ¯ λ < 1 or ¯ λ = 1 and λ = d. The parameters η and d can be estimated
by maximum likelihood, and deriving their asymptotic statistics allows one to test whether the hypothesis
¯λ = 1 can be rejected or not, and consequently, whether the data present tail dependence or not.
We have implemented this procedure and the estimated values of the coefficient of tail dependence are given
in table 1 both for the positive and the negative tails. Our tests show that we cannot reject the hypothesis of
tail dependence between the four considered Latin American markets (Argentina, Brazil, Chile and Mexico).
Notice that the positive tail dependence is almost always slightly smaller than the negative one, which could
be linked with the trend asymmetry of (Longin and Solnik 2001), but it turns out that these differences are
not statistically significant. These results indicate that, according to this analysis of the extreme dependence
coefficient, the propensity of extreme co-movements is almost the same for each pair of stock markets:
even if the transmission mechanisms of a crisis are different from a country to another one, the propagation
occur with the same probability overall. Thus, the subsequent risks are the same. In table 2, we also
give the coefficients of tail dependence estimated under the Student’s copula (or in fact any copula derived
from an elliptical distribution) with three dregrees of freedom, given by expression (26). One can observe
a remarkable agreement between these values and the non-parametric estimates given in table 1. This
is consistent with the results given by the conditional Spearman’s ρ, for which we have shown that the
Student’s copula seems to reasonably account for the extreme dependence.
4 Summary and Discussion
Table 3 summarizes the asymptotic dependences for large v and u of the signed conditional correlation
coefficient ρ + v , the unsigned conditional correlation coefficient ρ s v and the correlation coefficient ρu conditioned
on both variables for the bivariate Gaussian, the Student’s model, the Gaussian factor model and
the Student’s factor model. Our results provide a quantitative proof that conditioning on exceedance leads
to conditional correlation coefficients that may be very different from the unconditional correlation. This
provides a straightforward mechanism for fluctuations or changes of correlations, based on fluctuations of
volatility or changes of trends. In other words, the many reported variations of correlation structure might
be in large part attributed to changes in volatility (and statisical uncertainty).
We also suggest that the distinct dependences as a function of exceedance v and u of the conditional correlation
coefficients may offer novel tools for characterizing the statistical multivariate distributions of extreme
events. Since their direct characterization is in general restricted by the curse of dimensionality and the scarsity
of data, the conditional correlation coefficients provide reduced robust statistics which can be estimated
with reasonable accuracy and reliability. In this respect, our empirical results encourage us to assert that
a Student’s copula, or more generally and elliptical copula, with a tail index of about three seems able to
account for the main extreme dependence properties investigated here.
Table 4 gives the asymptotic values of ρ + v , ρ s v and ρu for v → +∞ and u → ∞ in order to compare them
with the tail-dependence λ.
These two tables only scratch the surface of the rich sets of measures of tail and extreme dependences. We
have shown that complete independence implies the absence of tail dependence: λ = 0. But λ = 0 does not
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258 9. Mesure de la dépendance extrême entre deux actifs financiers
implies independence, at least in the intermediate range, since it is only an asymptotic property. Conversely,
a non-zero tail dependence λ implies the absence of asymptotic independence. Nonetheless, it does not
imply necessarily that the conditional correlation coefficients ρ + v=∞ and ρ s v=∞ are non-zero, as one would
have expected naively.
Note that the examples of Table 4 are such that λ = 0 seems to go hand-in-hand with ρ + v→∞ = 0. However,
the logical implication (λ = 0) ⇒ (ρ + v→∞ = 0) does not hold in general. A counter example is offered by
the Student’s factor model in the case where νY > νɛ (the tail of the distribution of the idiosynchratic noise
is fatter than that of the distribution of the factor). In this case, X and Y have the same tail-dependence
as ɛ and Y , which is 0 by construction. But, ρ + v=∞ and ρ s v=∞ are both 1 because a large Y almost always
gives a large X and the simultaneous occurrence of a large Y and a large ɛ can be neglected. The reason for
this absence of tail dependence (in the sense of λ) coming together with asymptotically strong conditional
correlation coefficients stems from two facts:
• first, the conditional correlation coefficients put much less weight on the extreme tails that the taildependence
parameter λ. In other words, ρ + v=∞ and ρ s v=∞ are sensitive to the marginals, i.e., there are
determined by the full bivariate distribution, while, as we said, λ is a pure copula property independent
of the marginals. Since ρ + v=∞ and ρ s v=∞ are measures of tail dependence weighted by the specific
shapes of the marginals, it is natural that they may behave differently.
• Secondly, the tail dependence λ probes the extreme dependence property of the original copula of the
random variables X and Y . On the contrary, when conditioning on Y , one changes the copula of X
and Y , so that the extreme dependence properties investigated by the conditional correlations are not
exactly those of the original copula. This last remark explains clearly why we observe what (Boyer et
al. 1997) call a “bias” in the conditional correlations. Indeed, changing the dependence between two
random variables obviously leads to changing their correlations.
The consequences are of importance. In such a situation, one measure (λ) would conclude on asymptotic
tail-independence while the other measures ρ + v=∞ and ρ s v=∞ would conclude the opposite. Thus, before
concluding on a change in the dependence structure with respect to a given parameter - the volatility or
the trend, for instance - one should check that this change does not result from the tool used to probe the
dependence. In this respect, our study allows us to shed new lights on recent controversial results about the
occurrence or abscence of contagion during the Latin American crises. As in every previous works, we find
no contagion evidence between Chile and Mexico, but contrarily to (Forbes and Rigobon 2002), we think it
is difficult to ignore the possibility of contagion towards Argentina and Brazil, and in this respect we agree
with (Calvo and Reinhart 1996).
In fact, we think that most of the discrepancies between these different studies stem from the fact that
the conditional correlation coefficient does not provide an accurate tool for probing the potential changes
of dependence. Indeed, even when the bias have been accounted for, the fat-tailness of the distributions
of returns are such that the Pearson’s coefficient is subjected to very strong statistical fluctuations which
forbid an accurate estimation of the correlation. Moreover, when studying the dependence properties, it is
interesting to free oneself from the marginal behavior of each random variable. This is why the conditional
Spearman’s rho seems a good tool: it only depends on the copula and is statistically well-behaved.
The conditional Spearman’s rho has allowed us to identify a change in the dependence structure during
downward trends in Latin American markets, similar to that found by (Longin and Solnik 2001) in their
study of the contagion across five major equity markets. It has also enabled us to put in light the asymmetry
in the contagion effects: Mexico and Chile can be potential source sof contagion toward Argentina and
Brazil, while reverse does not seem to hold. This phenomenom has been observed during the 1994 Mexican
20

9.1. Les différentes mesures de dépendances extrêmes 259
crisis and appears to remain true in the recent Argentina crisis, for which only Brazil seems to exhibit the
signature of a possible contagion.
We suggest that a possible origin for the discovered asymmetry may lie in the difference in the more marketoriented
countries versus more state intervention oriented economies, giving rise to either currency floating
regimes adapted to an important manufacturing sector which tend to deliver more competitive real exchange
rates (Chile and Mexico) or fixed rate pegs (Argentina until the 2001 crisis and Brazil until the early 1999
crisis) (Frieden 1992, Frieden et al. 2000a, Frieden et al. 2000b). The asymmetry of the contagion is compatible
with the view that fixed echange rates tighten more strickly an economy and its stock market to external
shocks (case of Argentina and Brazil) while a more flexible exchange rate seems to provide a cushion allowing
a decoupling between the stock market and external influences.
Finally, the absence of contagion does not imply necessarily the absence of “contamination.” Indeed, the
study of the coefficient of tail dependence has proven that with or without contagion mechanisms - i.e.,
increase in the linkage between markets during crisis - the probability of extreme co-movements during the
crisis - i.e., the contamination - is almost the same for all pairs of markets. Thus, whatever the propagation
mechanism may be - historically closed relationship or irrational fear and herd behavior - the observed
effects are the same: the propagation of the crisis. From the practical perspective of risk management or
regulatory policy, this last point is may be more important than the real knowledge of the occurrence or not
of contagion.
21

260 9. Mesure de la dépendance extrême entre deux actifs financiers
Figure legends
Figure 1: The upper (respectively lower) panel graphs the complementary distribution of the positive (respectively
the minus negative) returns in US dollar. The straight line represents the slope of a power law
with tail exponent α = 2.
Figure 2: In the upper panel, the thick plain curve depicts the correlation coefficient between the Argentina
stock index daily returns and the Mexican stock index daily returns conditional on the Mexican stock index
daily returns larger than (smaller than) a given positive (negative) value v (after normalization by the stan-
dard deviation). The thick dashed curve represents the theoretical conditional correlation coefficient ρ +,−
v
calculated for a bivariate Gaussian model, while the two thin dashed curves represent the area within which
we cannot consider, at the 95% confidence level, that the estimated correlation coefficient is significantly
different from its Gaussian theoretical value. The dotted curves provide the same information under the
assumption of a bivariate Student’s model with ν = 3 degrees of freedom.
The lower panel gives the same kind of information for the correlation coefficient conditioned on the Argentina
stock index daily returns larger than (smaller than) a given positive (negative) value v.
Figure 3: In the upper panel, the thick plain curve gives the correlation coefficient between the Argentina
stock index daily returns and the Mexican stock index daily returns conditioned on the Mexican stock index
daily volatility larger than a give value v (after normalization by the standard deviation). The thick dashed
curve represents the theoretical conditional correlation coefficient ρ +,−
v calculated for a bivariate Gaussian
model, while the two thin dashed curves represent the area within which we cannot consider, at the 95%
confidence level, that the estimated correlation coefficient is significantly different from its Gaussian theoretical
value. The dotted curves provide the same information under the assumption of a bivariate Student’s
model with ν = 3 degrees of freedom.
The lower panel gives the same kind of information for the correlation coefficient conditioned on the Argentina
stock index daily volatility larger than a given value v.
Figure 4: In the upper panel, the thick plain curve gives the correlation coefficient between the Brazilian
stock index daily returns and the Chilean stock index daily returns conditioned on the Chilean stock index
daily returns larger than (smaller than) a given positive (negative) value v (after normalization by the stan-
dard deviation). The thick dashed curve represents the theoretical conditional correlation coefficient ρ +,−
v
calculated for a bivariate Gaussian model, while the two thin dashed curves represent the area within which
we cannot consider, at the 95% confidence level, that the estimated correlation coefficient is significantly
different from its Gaussian theoretical value. The dotted curves provide the same information under the
assumption of a bivariate Student’s model with ν = 3 degrees of freedom.
The lower panel gives the same kind of information for the correlation coefficient conditioned on the Brazilian
stock index daily returns larger than (smaller than) a given positive (negative) value v.
Figure 5: In the upper panel, the thick plain curve shows the correlation coefficient between the Brazilian
stock index daily returns and the Chilean stock index daily returns conditional on the Chilean stock index
daily volatility larger than a given value v (after normalization by the standard deviation). The thick dashed
curve represents the theoretical conditional correlation coefficient ρ +,−
v calculated for a bivariate Gaussian
model, while the two thin dashed curves represent the area within which we cannot consider, at the 95%
confidence level, that the estimated correlation coefficient is significantly different from its Gaussian theoretical
value. The dotted curves provide the same information under the assumption of a bivariate Student’s
model with ν = 3 degrees of freedom.
The lower panel gives the same kind of information for the correlation coefficient conditioned on the Brazil-
22

9.1. Les différentes mesures de dépendances extrêmes 261
ian stock index daily volatility larger than a given value v.
Figure 6: In the upper panel, the thick plain curve shows the correlation coefficient between the Chilean
stock index daily returns and the Mexican stock index daily returns conditioned on the Mexican stock index
daily returns larger than (smaller than) a given positive (negative) value v (after normalization by the standard
deviation). The thick dashed curve represents the theoretical conditional correlation coefficient ρ +,−
v
calculated for a bivariate Gaussian model, while the two thin dashed curves represent the area within which
we cannot consider, at the 95% confidence level, that the estimated correlation coefficient is significantly
different from its Gaussian theoretical value. The dotted curves provide the same information under the
assumption of a bivariate Student’s model with ν = 3 degrees of freedom.
The lower panel gives the same kind of information for the correlation coefficient conditioned on the Chilean
stock index daily returns larger than (smaller than) a given positive (negative) value v.
Figure 7: In the upper panel, the thick plain curve depicts the correlation coefficient between the Chilean
stock index daily returns and the Mexican stock index daily returns conditioned on the Mexican stock index
daily volatility larger than a give value v (after normalization by the standard deviation). The thick dashed
curve represents the theoretical conditional correlation coefficient ρ +,−
v calculated for a bivariate Gaussian
model, while the two thin dashed curves represent the area within which we cannot consider, at the 95%
confidence level, that the estimated correlation coefficient is significantly different from its Gaussian theoretical
value. The dotted curves provide the same information under the assumption of a bivariate Student’s
model with ν = 3 degrees of freedom.
The lower panel gives the same kind of information for the correlation coefficient conditioned on the Chilean
stock index daily volatility larger than a given value v.
Figure 8: Conditional Spearman’s rho for a bivariate Gaussian copula (left panel) and a Student’s copula
with three degrees of freedom (right panel), with an unconditional linear correlation coefficient ρ =
0.1, 0.3, 0.5, 0.7, 0.9.
Figure 9: In the upper panel, the thick curve shows the Spearman’s rho between the Argentina stock index
daily returns and the Brazilian stock index daily returns. Above the quantile v = 0.5, the Spearman’s rho is
conditioned on the Brazilian index daily returns whose quantiles are larger than v, while below the quantile
v = 0.5 it is conditioned on the Brazilian index daily returns whose quantiles are smaller than v. As in the
above figures for the correlation coefficients, the dashed lines refer to the prediction of the Gaussian copula
and its 95% confidence levels and the dotted lines to the Student’s copula with three degrees of freedom
and its 95% confidence levels. The lower panel gives the same kind of information for the Spearman’s rho
conditioned on the realizations of the Argentina index daily returns.
Figure 10: In the upper panel, the thick curve shows the Spearman’s rho between the Brazilian stock index
daily returns and the chilean stock index daily returns. Above the quantile v = 0.5, the Spearman’s rho is
conditioned on the Chilean index daily returns whose quantiles are larger than v, while below the quantile
v = 0.5 it is conditioned on the Mexican index daily returns whose quantiles are smaller than v. The dashed
lines refer to the prediction of the Gaussian copula and its 95% confidence levels and the dotted lines to the
Student’s copula with three degrees of freedom and its 95% confidence levels. The lower panel gives the
same kind of information for the Spearman’s rho conditioned on the realizations of the Brazilian index daily
returns.
Figure 11: In the upper panel, the thick curve depicts the Spearman’s rho between the Chilean stock index
daily returns and the Mexican stock index daily returns. Above the quantile v = 0.5, the Spearman’s rho is
conditioned on the Mexican index daily returns whose quantiles are larger than v, while below the quantile
v = 0.5 it is conditioned on the Mexican index daily returns whose quantiles are smaller than v. The dashed
lines refer to the prediction of the Gaussian copula and its 95% confidence levels and the dotted lines to the
23

262 9. Mesure de la dépendance extrême entre deux actifs financiers
Student’s copula with three degrees of freedom and its 95% confidence levels. The lower panel gives the
same kind of information for the Spearman’s rho conditioned on the realizations of the Chilean index daily
returns.
Figure 12: Coefficient of upper tail dependence as a function of the correlation coefficient ρ for various
values of the number of degres of freedomn ν for the student’s Copula (left panel) and the Student’s factor
model (right panel).
Figure 13: The graph of the function 1
(1+ ɛu
) x0 ν
− 1
(thick solid line), the string which gives an upper bound
of the function within 1−x0
ɛ , 0 (dashed line) and the tangent in 0 + which gives an upper bound of the
function within 0, x0
ɛ (dash dotted line).
24

9.1. Les différentes mesures de dépendances extrêmes 263
A Conditional correlation coefficient for Gaussian variables
Let us consider a pair of Normal random variables (X, Y ) ∼ N (0, Σ) where Σ is their covariance matrix
with unconditional correlation coefficient ρ. Without loss of generality, and for simplicity, we shall assume
Σ with unit unconditional variances.
A.1 Conditioning on one variable
A.1.1 Conditioning on Y larger than v
Given a conditioning set A = [v, +∞), v ∈ R+, ρA = ρ + v is the correlation coefficient conditioned on Y
larger than v:
ρ + v =
ρ 2 +
ρ
1−ρ 2
Var(Y | Y >v)
√ v
πe 2 = v +
v√2
2 erfc
1
v
. (A.1)
We start with the calculation of the first and the second moment of Y conditioned on Y larger than v:
E(Y | Y > v) =
√
2
2 1
− + O
v3 v5
, (A.2)
E(Y 2 | Y > v) = 1 +
√ 2v
√ v
πe 2
v√2
2 erfc
which allows us to obtain the variance of Y conditioned on Y larger than v:
Var(Y | Y > v) = 1 +
which, for large v, yields:
√ 2v
√ v
πe 2
v√2
2 erfc
A.1.2 Conditioning on |Y | larger than v
⎛
− ⎝
ρ + v ∼v→∞
= v 2 + 2 − 2
1
+ O
v2 v4
, (A.3)
√ 2
√ v
πe 2
v√2
2 erfc
⎞
⎠
2
= 1
1
+ O
v2 v4
, (A.4)
ρ 1
· . (A.5)
1 − ρ2 v
Given a conditioning set A = (−∞, −v] ∪ [v, +∞), v ∈ R+, ρA = ρ s v is the correlation coefficient
conditioned on |Y | larger than v:
ρ s v =
ρ 2 +
ρ
1−ρ 2
Var(Y | |Y |>v)
. (A.6)
The first and second moment of Y conditioned on |Y | larger than v can be easily calculated:
E(Y | |Y | > v) = 0, (A.7)
E(Y 2 √
2v
| |Y | > v) = 1 +
= v 2 + 2 − 2
1
+ O
v2 v4
. (A.8)
√ v
πe 2
v√2
2 erfc
25

264 9. Mesure de la dépendance extrême entre deux actifs financiers
Expression (A.8) is the same as (A.4) as it should. This gives the following conditional variance:
Var(Y | |Y | > v) = 1 +
√
2v
1
v2
, (A.9)
and finally yields, for large v,
A.1.3 Intuitive meaning
ρ s v ∼v→∞
ρ
ρ 2 + 1−ρ2
2+v 2
√ v
πe 2 = v
v√2
2 erfc
2 + 2 + O
∼v→∞ 1 − 1
2
1 − ρ 2
ρ 2
1
. (A.10)
v2 Let us provide an intuitive explanation (see also (Longin and Solnik 2001)). As seen from (A.1), ρ + v is
controlled by the dependence Var(Y | Y > v) ∝ 1/v 2 derived in Appendix A.1.1. In contrast, as seen from
(A.6), ρ s v is controlled by Var(Y | |Y | > v) ∝ v 2 given in Appendix A.1.2. The difference between ρ + v and
ρ s v can thus be traced back to that between Var(Y | Y > v) ∝ 1/v 2 and Var(Y | |Y | > v) ∝ v 2 for large v.
This results from the following effect. For Y > v, one can picture the possible realizations of Y as those of
a random particle on the line, which is strongly attracted to the origin by a spring (the Gaussian distribution
that prevents Y from performing significant fluctuations beyond a few standard deviations) while being
forced to be on the right to a wall at Y = v. It is clear that the fluctuations of the position of this particle
are very small as it is strongly glued to the unpenetrable wall by the restoring spring, hence the result
Var(Y | Y > v) ∝ 1/v 2 . In constrast, for the condition |Y | > v, by the same argument, the fluctuations
of the particle are hindered to be very close to |Y | = v, i.e., very close to Y = +v or Y = −v. Thus, the
fluctuations of Y typically flip from −v to +v and vice-versa. It is thus not surprising to find Var(Y | |Y | >
v) ∝ v 2 .
This argument makes intuitive the results Var(Y | Y > v) ∝ 1/v2 and Var(Y | |Y | > v) ∝ v2 for large
v and thus the results for ρ + v and for ρs v if we use (A.1) and (A.6). We now attempt to justify ρ + v ∼v→∞ 1
v
and 1 − ρs v ∼v→∞ 1/v2 directly by the following intuitive argument. Using the picture of particles, X
and Y can be visualized as the positions of two particles which fluctuate randomly. Their joint bivariate
Gaussian distribution with non-zero unconditional correlation amounts to the existence of a spring that ties
them together. Their Gaussian marginals also exert a spring-like force attaching them to the origin. When
Y > v, the X-particle is teared off between two extremes, between 0 and v. When the unconditional
correlation ρ is less than 1, the spring attracting to the origin is stronger than the spring attracting to the
wall at v. The particle X thus undergoes tiny fluctuations around the origin that are relatively less and less
attracted by the Y -particle, hence the result ρ + v ∼v→∞ 1
v → 0. In constrast, for |Y | > v, notwithstanding
the still strong attraction of the X-particle to the origin, it can follow the sign of the Y -particle without
paying too much cost in matching its amplitude |v|. Relatively tiny fluctuation of the X-particle but of the
same sign as Y ≈ ±v will result in a strong ρs v, thus justifying that ρs v → 1 for v → +∞.
A.2 Conditioning on both X and Y larger than u
By definition, the conditional correlation coefficient ρu, conditioned on both X and Y larger than u, is
ρu =
=
Cov[X, Y | X > u, Y > u]
Var[X | X > u, Y > u] Var[Y | X > u, Y > u] , (A.11)
m11 − m10 · m01
√
m20 − m10 2√ , (A.12)
m02 − m01
2
26

9.1. Les différentes mesures de dépendances extrêmes 265
where mij denotes E[X i · Y j | X > u, Y > u].
Using the proposition A.1 of (Ang and Chen 2001) or the expressions in (Johnson and Kotz 1972, p 113),
we can assert that
m10 L(u, u; ρ) =
1 − ρ
(1 + ρ) φ(u) 1 − Φ
1 + ρ u
m20 L(u, u; ρ) =
, (A.13)
(1 + ρ 2
1 − ρ
) u φ(u) 1 − Φ
1 + ρ u
+ ρ 1 − ρ2
2
√ φ
2π 1 + ρ u
m11 L(u, u; ρ) =
+ L(u, u; ρ),(A.14)
1 − ρ
2ρ u φ(u) 1 − Φ
1 + ρ u
1 − ρ2 2
+ √ φ
2π 1 + ρ u
+ ρ L(u, u; ρ) , (A.15)
where L(·, ·; ·) denotes the bivariate Gaussian survival (or complementary cumulative) distribution:
1
L(h, k; ρ) =
2π 1 − ρ2 ∞ ∞
dx dy exp −
h k
1 x
2
2 − 2ρxy + y2 1 − ρ2
, (A.16)
φ(·) is the Gaussian density:
and Φ(·) is the cumulative Gaussian distribution:
A.2.1 Asymptotic behavior of L(u, u; ρ)
φ(x) = 1 x2
− √ e 2 , (A.17)
2π
Φ(x) =
x
−∞
We focus on the asymptotic behavior of
1
L(u, u; ρ) =
2π 1 − ρ2 ∞ ∞
dx
u u
du φ(u). (A.18)
dy exp − 1
2
x 2 − 2ρxy + y 2
1 − ρ 2
for large u. Performing the change of variables x ′ = x − u and y ′ = y − u, we can write
u2
e− 1+ρ
L(u, u; ρ) =
2π 1 − ρ2 ∞
dx
0
′
∞
0
dy ′
exp −u x′ + y ′
1 + ρ
exp − 1
2
x ′2 − 2ρx ′ y ′ + y ′2
1 − ρ 2
, (A.19)
. (A.20)
Using the fact that
exp − 1 x
2
′2 − 2ρx ′ y ′ + y ′2
1 − ρ2
= 1− x′2 − 2ρx ′ y ′ + y ′2
2(1 − ρ2 +
)
(x′2 − 2ρx ′ y ′ + y ′2 ) 2
8(1 − ρ2 ) 2 − (x′2 − 2ρx ′ y ′ + y ′2 ) 3
48(1 − ρ2 ) 3 +· · · ,
(A.21)
and applying theorem 3.1.1 in (Jensen 1995, p 58) (Laplace’s method), equations (A.20) and (A.21) yield
and
(1 + ρ)2
L(u, u; ρ) =
2π 1 − ρ
u2
e− 1+ρ
·
2
1/L(u, u; ρ) = 2π u2 1 − ρ 2
(1 + ρ) 2
u 2
(2 − ρ)(1 + ρ)
1 − ·
1 − ρ
1
u2 + (2ρ2 − 6ρ + 7)(1 + ρ) 2
(1 − ρ) 2
−3 (12 − 13ρ + 8ρ2 − 2ρ 3 )(1 + ρ) 3
(1 − ρ) 3
· e u2
(2 − ρ)(1 + ρ)
1+ρ 1 + ·
1 − ρ
1
+ (16 − 13ρ + 10ρ2 − 3ρ 3 )(1 + ρ) 3
(1 − ρ) 3
27
· 1
u4 · 1
1
+ O
u6 u8
, (A.22)
u2 − 3 − 2ρ + ρ2 )(1 + ρ) 2
(1 − ρ) 2
· 1
u4 · 1
1
+ O
u6 u8
. (A.23)

266 9. Mesure de la dépendance extrême entre deux actifs financiers
A.2.2 Asymptotic behavior of the first moment m10
The first moment m10 = E[X | X > u, Y > u] is given by (A.13). For large u,
1 − ρ
1 − Φ
1 + ρ u
= 1
2 erfc
1 − ρ
2(1 + ρ) u
=
1 + ρ
1 − ρ
1 + ρ
−15
1 − ρ
1−ρ
−
e 2(1+ρ) u2
√
2π u
3
(A.24)
2 1 + ρ 1 1 + ρ
1 − · + 3 ·
1 − ρ u2 1 − ρ
1
u4 · 1
1
+ O
u6 u8 , (A.25)
so that multiplying by (1 + ρ) φ(u), we obtain
u2
−
(1 + ρ)2 e 1+ρ
2
1 + ρ 1 1 + ρ
m10 L(u, u; ρ) = 1 − · + 3 ·
1 − ρ2 2π u 1 − ρ u2 1 − ρ
1
3 1 + ρ
− 15 ·
u4 1 − ρ
1
1
+ O
u6 u8 .
(A.26)
Using the result given by equation (A.22), we can conclude that
m10 = u + (1 + ρ) · 1
u − (1 + ρ)2 (2 − ρ)
·
(1 − ρ)
1
u3 + (10 − 8ρ + 3ρ2 )(1 + ρ) 3
(1 − ρ) 2
In the sequel, we will also need the behavior of m10 2 :
m10 2 = u 2 + 2 (1 + ρ) − (1 + ρ)2 (3 − ρ)
(1 − ρ)
A.2.3 Asymptotic behavior of the second moment m20
· 1
u2 + 2(8 − 5ρ + 2ρ2 )(1 + ρ) 3
(1 − ρ) 2
· 1
1
+ O
u5 u7
. (A.27)
· 1
1
+ O
u4 u6
. (A.28)
The second moment m20 = E[X2 | X > u, Y > u] is given by expression (A.14). The first term in the
right hand side of (A.14) yields
(1 + ρ 2
1 − ρ
) u φ(u) 1 − Φ
1 + ρ u
= (1 + ρ 2 u2
−
1 + ρ e 1+ρ
2
1 + ρ 1 1 + ρ
)
1 − · + 3 ·
1 − ρ 2π 1 − ρ u2 1 − ρ
1
u4 3 1 + ρ
−15 ·
1 − ρ
1
1
+ O
u6 u8 (A.29) ,
while the second term gives
ρ 1 − ρ 2
√ 2π
2
φ
1 + ρ u
= ρ 1 − ρ
2 e− u2
1+ρ
2π
. (A.30)
Putting these two expressions together and factorizing the term (1 + ρ)/(1 + ρ 2 ) allows us to obtain
m20 L(u, u; ρ) =
(1 + ρ)2
1 − ρ 2
u2
−
e 1+ρ
2π
1 + ρ2 1
1 − ·
1 − ρ
−15 (1 + ρ2 )(1 + ρ) 2
(1 − ρ) 3
28
u2 + 3(1 + ρ2 )(1 + ρ)
(1 − ρ) 2
· 1
+ O
u6 1
u 8
· 1
u 4
+ L(u, u; ρ) , (A.31)

9.1. Les différentes mesures de dépendances extrêmes 267
which finally yields
m20 = u 2 (1 + ρ)2 1
+ 2 (1 + ρ) − 2 ·
1 − ρ u2 + 2(5 + 4ρ + ρ3 )(1 + ρ) 2
(1 − ρ) 2
1 1
+ O
u4 u6
. (A.32)
A.2.4 Asymptotic behavior of the cross moment m11
The cross moment m11 = E[X · Y | X > u, Y > u] is given by expression (A.15). The first and second
terms in the right hand side of (A.15) respectively give
2ρ u φ(u)[1 − Φ(u)] = 2ρ
1 − ρ 2
√ 2π
1 + ρ
1 − ρ
which, after factorization by (1 + ρ)/ρ, yields
and finally
m11 L(u, u; ρ) =
(1 + ρ)2
1 − ρ 2
u2
−
e 1+ρ
2π
2 1 + ρ 1 1 + ρ
1 − · + 3 ·
1 − ρ u2 1 − ρ
1
u4 3 1 + ρ
−15 ·
1 − ρ
1
1
+ O
u6 u8 , (A.33)
2
φ
1 + ρ u
= 1 − ρ
−30
m11 = u 2 + 2 (1 + ρ) − (1 + ρ)2 (3 − ρ)
(1 − ρ)
u2
−
e 1+ρ
2π
ρ
1 − 2
1 − ρ
ρ(1 + ρ)2 1
·
(1 − ρ) 3 u
A.2.5 Asymptotic behavior of the correlation coefficient
6 + O
2 e− u2
1+ρ
2π
· 1
u2 + (16 − 9ρ + 3ρ2 )(1 + ρ) 3
(1 − ρ) 2
, (A.34)
1 ρ(1 + ρ) 1
· + 6 ·
u2 (1 − ρ) 2 u4
1
u8
+ ρ L(u, u; ρ), (A.35)
· 1
1
+ O
u4 u6
. (A.36)
The conditional correlation coefficient conditioned on both X and Y larger than u is defined by (A.12).
Using the symmetry between X and Y , we have m10 = m01 and m20 = m02, which allows us to rewrite
(A.12) as follows
ρu = m11 − m10 2
. (A.37)
m20 − m10
2
Putting together the previous results, we have
which proves that
m20 − m10 2 =
m11 − m10 2 = ρ
ρu = ρ
(1 + ρ)2
u 2
(1 + ρ)3
1 − ρ
1 + ρ
1 − ρ
− 2 (4 − ρ + 3ρ2 + 3ρ3 )(1 + ρ) 2
·
1 − ρ
1
1
+ O
u4 u6
, (A.38)
1 1
· + O
u4 u6
, (A.39)
1 1
· + O
u2 u4
29
and ρ ∈ [−1, 1). (A.40)

268 9. Mesure de la dépendance extrême entre deux actifs financiers
B Conditional correlation coefficient for Student’s variables
B.1 Proposition
Let us consider a pair of Student’s random variables (X, Y ) with ν degrees of freedom and unconditional
correlation coefficient ρ. Let A be a subset of R such that Pr{Y ∈ A} > 0. The correlation coefficient of
(X, Y ), conditioned on Y ∈ A defined by
can be expressed as
with
and
⎡
ρA =
Cov(X, Y | Y ∈ A)
Var(X | Y ∈ A) Var(Y | Y ∈ A) . (B.41)
ρA =
⎢ν
− 1
Var(Y | Y ∈ A) = ν ⎣
ν − 2 ·
ρ
ρ 2 + E[E(x2 | Y )−ρ 2 Y 2 | Y ∈A]
Var(Y | Y ∈A)
ν Pr ν−2 Y ∈ A | ν − 2
Pr{Y ∈ A | ν}
E[E(X 2 | Y ) − ρ 2 Y 2 | Y ∈ A] = (1 − ρ 2 )
B.2 Proof of the proposition
ν
ν − 2 ·
ν Pr ν−2
, (B.42)
⎤
2 ⎥ dy y · ty(y)
y∈A
− 1⎦
−
, (B.43)
Pr{Y ∈ A | ν}
Y ∈ A | ν − 2
Pr{Y ∈ A | ν}
. (B.44)
Let the variables X and Y have a multivariate Student’s distribution with ν degrees of freedom and a
correlation coefficient ρ :
PXY (x, y) =
=
Γ
ν+2
2
νπ Γ
ν+1
2 1 − ρ2
ν + 1
ν + y2 1/2
1 + x2 − 2ρxy + y 2
ν (1 − ρ 2 )
1
1 − ρ2 tν(y)
ν + 1
· tν+1
ν + y2 where tν(·) denotes the univariate Student’s density with ν degrees of freedom
tν(x) =
Γ
ν+1
2
Γ
ν
2 (νπ) 1/2 ·
1
1 + x2
Cν
ν+1 =
2
ν 1 + x2
ν
Let us evaluate Cov(X, Y | Y ∈ A):
− ν+2
2
, (B.45)
1/2
x − ρy
, (B.46)
1 − ρ2 ν+1
2
. (B.47)
Cov(X, Y | Y ∈ A) = E(X · Y | Y ∈ A) − E(X | Y ∈ A) · E(Y | Y ∈ A), (B.48)
= E(E(X | Y ) · Y | Y ∈ A) − E(E(X | Y ) | Y ∈ A) · E(Y | Y ∈ A).(B.49)
As it can be seen in equation (B.46), E(X | Y ) = ρY , which gives
Cov(X, Y | Y ∈ A) = ρ · E(Y 2 | Y ∈ A) − ρ · E(Y | Y ∈ A) 2 , (B.50)
= ρ · Var(Y | Y ∈ A). (B.51)
30

9.1. Les différentes mesures de dépendances extrêmes 269
Thus, we have
ρA = ρ
Var(Y | Y ∈ A)
.
Var(X | Y ∈ A)
(B.52)
Using the same method as for the calculation of Cov(X, Y | Y ∈ A), we find
which yields
as asserted in (B.42).
Var(X | Y ∈ A) = E[E(X 2 | Y ) | Y ∈ A)] − E[E(X | Y ) | Y ∈ A)] 2 , (B.53)
= E[E(X 2 | Y ) | Y ∈ A)] − ρ 2 · E[Y | Y ∈ A] 2 , (B.54)
= E[E(X 2 | Y ) − ρ 2 Y 2 | Y ∈ A)] − ρ 2 · Var[Y | Y ∈ A], (B.55)
ρA =
ρ
ρ 2 + E[E(x2 | Y )−ρ 2 Y 2 | Y ∈A]
Var(Y | Y ∈A)
(B.56)
, (B.57)
To go one step further, we have to evaluate the three terms E(Y | Y ∈ A), E(Y 2 | Y ∈ A), and
E[E(X 2 | Y ) | Y ∈ A].
The first one is trivial to calculate :
y∈A dy y · ty(y)
E(Y | Y ∈ A) =
. (B.58)
Pr{Y ∈ A | ν}
The second one gives
so that
y∈A dy y2 · ty(y)
E(Y 2 | Y ∈ A) =
, (B.59)
Pr{Y ∈ A | ν}
⎡
⎢ν
− 1
= ν ⎣
ν − 2 ·
⎤
ν Pr ν−2 Y ∈ A | ν − 2
⎥
− 1⎦
, (B.60)
Pr{Y ∈ A | ν}
⎡
⎢ν
− 1
Var(Y | Y ∈ A) = ν ⎣
ν − 2 ·
ν Pr ν−2 Y ∈ A | ν − 2
Pr{Y ∈ A | ν}
⎤
2 ⎥ y∈A dy y · ty(y)
− 1⎦
−
. (B.61)
Pr{Y ∈ A | ν}
To calculate the third term, we first need to evaluate E(X2 | Y ). Using equation (B.46) and the results given
in (Abramovitz and Stegun 1972), we find
which yields
E(X 2 | Y ) =
= ν + y2
dx
ν + 1
ν + y2 1/2 x2 ν + 1
· tν+1
1 − ρ2 ν + y2
1/2
x − ρy
, (B.62)
1 − ρ2 ν − 1 (1 − ρ2 ) − ρ 2 y 2 , (B.63)
E[E(X 2 | Y ) − ρ 2 Y 2 | Y ∈ A] = ν
ν − 1 (1 − ρ2 1 − ρ2
) +
ν − 1 E[Y 2 | Y ∈ A] , (B.64)
31

270 9. Mesure de la dépendance extrême entre deux actifs financiers
and applying the result given in eqation (B.60), we finally obtain
E[E(X 2 | Y ) − ρ 2 Y 2 | Y ∈ A] = (1 − ρ 2 )
which concludes the proof.
B.3 Conditioning on Y larger than v
The conditioning set is A = [v, +∞), thus
Pr{Y ∈ A | ν} = ¯ Tν(v) = ν ν−1
2
y∈A
Cν
ν
ν − 2 ·
Pr
+ O
vν
ν
Pr Y ∈ A | ν − p =
ν − p ¯
ν − p
Tν−p v =
ν
ν ν−p
2
ν
dy y · ty(y) =
ν − 2 tν−2
ν − 2
v =
ν
ν ν
ν
ν−2 Y ∈ A | ν − 2
Pr{Y ∈ A | ν}
, (B.65)
v −(ν+2)
, (B.66)
(ν − p) 1
2
Cν−p
+ O v
vν−p −(ν−p+2)
, (B.67)
2
√ ν − 2
Cν−2
+ O
vν−1 v −(ν−3)
(B.68) ,
where tν(·) and ¯ Tν(·) denote respectively the density and the Student’s survival distribution with ν degrees
of freedom and Cν is defined in (B.47).
Using equation (B.42), one can thus give the exact expression of ρ + v . Since it is very cumbersomme, we will
not write it explicitely. We will only give the asymptotic expression of ρ + v . In this respect, we can show that
Thus, for large v,
Var(Y | Y ∈ A) =
E[E(X 2 | Y ) − ρ 2 Y 2 | Y ∈ A] =
B.4 Conditioning on |Y | larger than v
ν
(ν − 2)(ν − 1) 2 v2 + O(1) (B.69)
ν 1 − ρ
ν − 2
2
ν − 1 v2 + O(1) . (B.70)
ρ + ρ
v −→
ρ2
ν−2
+ (ν − 1) ν (1 − ρ2 . (B.71)
)
The conditioning set is now A = (−∞, −v]∪[v, +∞), with v ∈ R+. Thus, the right hand sides of equations
(B.66) and (B.67) have to be multiplied by two while
dy y · ty(y) = 0, (B.72)
y∈A
for symmetry reasons. So the equation (B.70) still holds while
Thus, for large v,
Var(Y | Y ∈ A) =
ρ s v −→
ν
(ν − 2) v2 + O(1) . (B.73)
ρ
ρ2 + 1
ν−2
(ν−1) ν (1 − ρ2 )
32
. (B.74)

9.1. Les différentes mesures de dépendances extrêmes 271
B.5 Conditioning on Y > v versus on |Y | > v
The results (B.71) and (B.74) are valid for ν > 2, as one can expect since the second moment has to
exist for the correlation coefficient to be defined. We remark that here, contrarily to the Gaussian case, the
conditioning set is not really important. Indeed with both conditioning set, ρ + v and ρ s v goes to a constant
different from zero and one, when v goes to infinity. This striking difference can be explained by the large
fluctuations allowed by the Student’s distribution, and can be related to the fact that the coefficient of tail
dependence for this distribution does not vanish even though the variables are anti-correlated (see section
3.2 below).
Contrarily to the Gaussian distribution which binds the fluctuations of the variables near the origin, the Student’s
distribution allows for ‘wild’ fluctuations. These properties are thus responsible for the result that,
contrarily to the Gaussian case for which the conditional correlation coefficient goes to zero when conditioned
on large signed values and goes to one when conditioned on large unsigned values, the conditional
correlation coefficient for Student’s variables have a similar behavior in both cases. Intuitively, the large
fluctuations of X for large v dominate and control the asymptotic dependence.
33

272 9. Mesure de la dépendance extrême entre deux actifs financiers
C Proof of equation (9)
We assume that X and Y are related by the equation
X = αY + ɛ , (C.75)
where α is a non random real coefficient and ɛ an idiosyncratic noise independent of Y , whose distribution
is assumed to admit a moment of second order σ 2 ɛ . Let us also denote by σ 2 y the second moment of the
variable Y .
We have
Cov(X, Y | Y ∈ A) = Cov(αY + ɛ, Y | Y ∈ A), (C.76)
since Y and ɛ are independent. We have also
= αVar(Y | Y ∈ A) + Cov(ɛ, Y | Y ∈ A), (C.77)
= αVar(Y | Y ∈ A), (C.78)
Var(X | Y ∈ A) = = α 2 Var(Y | Y ∈ A) + 2 Cov(ɛ, Y | Y ∈ A) + Var(ɛ | Y ∈ A), (C.79)
= α 2 Var(Y | Y ∈ A) + σ 2 ɛ , (C.80)
where, again, we have used the independence of Y and ɛ. This allows us to write
Since
we finally obtain
which conclude the proof.
ρA =
=
αVar(Y | Y ∈ A)
, (C.81)
Var(Y | Y ∈ A)(α2Var(Y | Y ∈ A) + σ2 ɛ )
sgn(α)
1 + σ2 ɛ
α2 1 · Var(Y | Y ∈A)
ρA =
ρ =
sgn(α)
1 + σ2 ɛ
α 2 · 1
Var(Y )
ρ
ρ2 + (1 − ρ2 )
34
. (C.82)
Var(y)
Var(y | y∈A)
, (C.83)
, (C.84)

9.1. Les différentes mesures de dépendances extrêmes 273
D Conditional Spearman’s rho
The conditional Spearman’s rho has been defined by
We have
ρs(˜v) =
Cov(U, V | V ≥ ˜v)
Var(U | V ≥ ˜v)Var(V | V ≥ ˜v) , (D.85)
1 1
˜v 0 · dC(u, v)
1 1 1
E[· | V ≥ ˜v] =
=
dC(u, v) 1 − ˜v ˜v 0
1
˜v
1
0
· dC(u, v) , (D.86)
thus, performing a simple integration by parts, we obtain
E[U | V ≥ ˜v] = 1 + 1
1
du C(u, ˜v) −
1 − ˜v 0
1
,
2
(D.87)
E[V | V ≥ ˜v]
E[U
=
1 + ˜v
,
2
(D.88)
2 | V ≥ ˜v] = 1 + 2
1
du u C(u, ˜v) −
1 − ˜v
1
,
3
(D.89)
which yields
so that
0
E[V 2 | V ≥ ˜v] = ˜v2 + ˜v + 1
, (D.90)
3
1 1
1
1 + ˜v 1
E[U · V | V ≥ ˜v] = + dv du C(u, v) + ˜v du C(u, ˜v) −
2 1 − ˜v
1
, (D.91)
2
Cov(U, V | V ≥ ˜v) =
ρs(˜v) =
Var(U | V ≥ ˜v) =
Var(V | V ≥ ˜v) =
1 − 4˜v + 24 (1 − ˜v) 1
˜v
0
1 1
1
dv du C(u, v) −
1 − ˜v ˜v 0
1
1
du C(u, ˜v) −
2 0
1
, (D.92)
4
1
1 − 4˜v 2
2˜v − 1
+ du u C(u, ˜v) +
12 (1 − ˜v) 2 1 − ˜v 0
(1 − ˜v) 2
1
du C(u, ˜v)
0
1
−
(1 − ˜v) 2
1
2
du C(u, ˜v) , (D.93)
(1 − ˜v)2
12
0
, (D.94)
12 1
1−˜v ˜v dv 1
0 du C(u, v) − 6 1
0
0 du u C(u, ˜v) + 12 (2˜v − 1) 1
0
0
du C(u, ˜v) − 3
du C(u, ˜v) − 12
E Tail dependence generated by the Student’s factor model
We consider two random variables X and Y , related by the relation
1
2
0 du C(u, ˜v)
(D.95)
X = αY + ɛ, (E.96)
35

274 9. Mesure de la dépendance extrême entre deux actifs financiers
where ɛ is a random variable independent of Y and α a non random positive coefficient. Assume that Y and
ɛ have a Student’s distribution with density:
Cν
PY (y) =
1 + y2
ν
Pɛ(ɛ) =
Cν
σ 1 + ɛ2
ν σ2 ν+1
2
ν+1
2
, (E.97)
. (E.98)
We first give a general expression for the probability for X to be larger than F −1
X (u) knowing that Y is
(u) :
larger than F −1
Y
LEMMA 1
The probability that X is larger than F −1
X
with
Pr[X > F −1
−1
X (u)|Y > FY (u)] = ¯ Fɛ(η) + α
∞
1 − u F −1
Y (u)
−1
(u) knowing that Y is larger than F (u) is given by :
η = F −1
X
Y
dy ¯ FY (y) · Pɛ[αF −1
Y (u) + η − αy] , (E.99)
−1
(u) − αF (u). (E.100)
The proof of this lemma relies on a simple integration by part and a change of variable, which are detailed
in appendix E.1.
Introducing the notation
we can show that
η = α 1 +
Y
˜Yu = F −1
Y (u) , (E.101)
σ
ν1/ν − 1 ˜Yu + O(
α
˜ Y −1
u ), (E.102)
which allows us to conclude that η goes to infinity as u goes to 1 (see appendix E.2 for the derivation of this
result). Thus, ¯ Fɛ(η) goes to zero as u goes to 1 and
∞ α
λ = lim dy
u→1 1 − u
¯ FY (y) · Pɛ(α ˜ Yu + η − αy) . (E.103)
Now, using the following result :
LEMMA 2
Assuming ν > 0 and x0 > 1,
1
lim
ɛ→0 ɛ
∞
1
˜Yu
dx 1
x ν
Cν
1 + x−x0
ɛ
2 ν+1
2
whose proof is given in appendix E.3, it is straigthforward to show that
The final steps of this calculation are given in appendix E.4.
= 1
xν , (E.104)
0
1
λ =
1 +
σ ν . (E.105)
α
36

9.1. Les différentes mesures de dépendances extrêmes 275
E.1 Proof of Lemma 1
By definition,
Pr[X > F −1
X
−1
(u), Y > F (u)] =
Let us perform an integration by part :
Defining η = F −1
X
Pr[X > F −1
−1
X (u), Y > F
we obtain the result given in (E.99)
Y
=
∞
F −1
X (u)
∞
F −1
Y (u)
dx
∞
F −1
Y (u)
dy PY (y) · Pɛ(x − αy) (E.106)
dy PY (y) · ¯ Fɛ[F −1
X (u) − αy]. (E.107)
Y (u)] = − ¯ FY (y) · ¯ Fɛ(F −1
X (u) − αy) ∞
+ α
∞
F −1
Y (u)
F −1
Y
(u) +
dy ¯ FY (y) · Pɛ(F −1
X (u) − αy) (E.108)
= (1 − u) ¯ Fɛ(F −1
−1
(u) − αF (u)) +
+ α
∞
F −1
Y (u)
−1
(u) − αF (u) (see equation(E.100)), and dividing each term by
E.2 Derivation of equation (E.102)
Y
X
Y
dy ¯ FY (y) · Pɛ(F −1
X (u) − αy) (E.109)
Pr[Y > F −1
Y (u)] = 1 − u, (E.110)
The factor Y and the idiosyncratic noise ɛ have Student’s distributions with ν degrees of freedom given by
(E.97) and (E.98) respectively. It follows that the survival distributions of Y and ɛ are :
and
¯FY (y) =
ν ν−1
2 Cν
y ν + O(y −(ν+2) ), (E.111)
¯Fɛ(ɛ) = σν ν ν−1
2 Cν
ɛ ν + O(ɛ −(ν+2) ), (E.112)
(E.113)
¯FX(x) = (αν + σ ν ) ν ν−1
2 Cν
x ν + O(x −(ν+2) ). (E.114)
Using the notation (E.101), equation (E.100) can be rewritten as
whose solution for large ˜ Yu (or equivalently as u goes to 1) is
σ
ν1/ν η = α 1 + − 1
α
¯FX(η + α ˜ Yu) = ¯ FY ( ˜ Yu) = 1 − u, (E.115)
˜Yu + O( ˜ Y −1
u ). (E.116)
To obain this equation, we have used the asymptotic expressions of ¯ FX and ¯ FY given in (E.114) and (E.111).
37

276 9. Mesure de la dépendance extrême entre deux actifs financiers
E.3 Proof of lemma 2
We want to prove that, assuming ν > 0 and x0 > 1,
The change of variable
gives
1
ɛ
∞
1
dx 1
x ν
1
lim
ɛ→0 ɛ
1 + 1
ν
∞
1
Cν
x−x0
ɛ
Consider the second integral. We have
which allows us to write
so that
∞
x 0
ɛ
du
1
dx 1
x ν
2 ν+1
2
1
1 + 1
ν
u =
=
(1 + u2 ) ν+1
2
(1 + ɛu
x0 )ν
Cν
(1 + u2 ) ν+1
2
The next step of the proof is to show that
Let us calculate
x0 ɛ
du
1−x 0
ɛ
1
(1 + ɛu
x0 )ν
x 0
ɛ
1−x 0
ɛ
Cν
du
(1 + u2
ν+1
ν ) 2
1
(1 + ɛu
x0 )ν
− 1
=
x − x0
ɛ
Cν
x−x0
ɛ
∞
1−x 0
ɛ
= 1
xν ∞
1−x0 0 ɛ
= 1
x ν 0
2 ν+1
2
= 1
xν . (E.117)
0
, (E.118)
x 0
ɛ
1−x0 ɛ
+ 1
xν ∞
x0 0 ɛ
1
du
(ɛu + x0) ν
du
du
du
1
(1 + ɛu
x0 )ν
1
(1 + ɛu
x0 )ν
1
(1 + ɛu
x0 )ν
Cν
(1 + u2
ν+1
ν ) 2
Cν
(1 + u2
ν+1
ν ) 2
Cν
(1 + u2
ν+1
ν ) 2
Cν
(1 + u2
ν+1
ν ) 2
+
(E.119)
(E.120)
. (E.121)
u ≥ x0
, (E.122)
ɛ
≤
Cν
ν ν+1
2 ɛ ν+1
x ν+1
0
≤ ν ν+1
2 ɛ ν+1
x ν+1
= ν ν+1
(1 + u2
ν+1
ν ) 2
x0 ɛ
0
2 ɛ ν
x ν 0
, (E.123)
∞
1
∞
x 0
ɛ
du
Cν
(1 + ɛu
x0 )ν
dv Cν
(1 + v) ν
(E.124)
(E.125)
= O(ɛ ν ). (E.126)
1−x 0
ɛ
38
du
−→ 1 as ɛ −→ 0. (E.127)
1
(1 + ɛu
x0 )ν
Cν
(1 + u2
ν+1
ν ) 2
−

9.1. Les différentes mesures de dépendances extrêmes 277
∞
Cν
− du
−∞ (1 + u2
ν+1
(E.128)
ν ) 2
x
0
ɛ 1
= du
1−x0 (1 +
ɛ
ɛu
Cν
− 1
)ν
x0 (1 + u2
ν+1 −
ν ) 2
1−x0 ɛ Cν
− du
−∞ (1 + u2
∞
Cν
ν+1 − du
x
ν ) 2
0
ɛ (1 + u2
ν+1
ν ) 2 (E.129)
x
0
ɛ 1
≤ du
1−x0 (1 +
ɛ
ɛu
Cν
− 1
)ν
x0 (1 + u2
ν+1
ν ) 2 +
1−x0 ɛ Cν
+ du
−∞ (1 + u2
ν+1
ν ) 2 +
∞
Cν
du
x0 ɛ (1 + u2
ν+1
ν ) 2 . (E.130)
The second and third integrals obviously behave like O(ɛν ) when ɛ goes to zero since we have assumed
x0
→ −∞ and ɛ → ∞ when ɛ → 0. For the first integral, we have
x0 ɛ
1
du
(1 + ɛu
− 1
)ν
x0
Cν
≤
x0 ɛ
1
du
(1
+ ɛu
− 1
)ν
x0
Cν
. (E.131)
x0 > 1 what ensures that 1−x0
ɛ
1−x 0
ɛ
(1 + u2
ν+1
ν ) 2
1−x 0
ɛ
The function
1
(1 + ɛu
− 1
)ν
x0
(1 + u2
ν+1
ν ) 2
(E.132)
vanishes at u = 0, is convex for u ∈ [ 1−x0
ɛ , 0] and concave for u ∈ [0, x0
ɛ ] (see also figure 13), so that there
are two constants A, B > 0 such that
1
(1
+ ɛu
− 1
)ν x0 ≤ −xν
1
(1
+
0 − 1
1 − x0
ɛ · u = −A · ɛ · u, ∀u ∈ , 0
x0 − 1 ɛ
(E.133)
ɛu
− 1
)ν x0
≤
νɛ
u = B · ɛ · u, ∀u ∈ 0,
x0
x0
.
ɛ
(E.134)
We can thus conclude that
x
0
ɛ 1
du
(1 + ɛu
− 1
)ν
x0
1−x 0
ɛ
Cν
(1 + u2
ν+1
ν ) 2
0
≤ −A · ɛ
1−x0 ɛ
x0 ɛ
+ B · ɛ du
0
du
u · Cν
(1 + u2
ν+1
ν ) 2
u · Cν
(1 + u2
ν+1
ν ) 2
(E.135)
= O(ɛ α ), (E.136)
with α = min{ν, 1}. Indeed, the two integrals can be perfomed exactly, which shows that they behave as
O(1) if ν > 1 and as O(ɛν−1 ) otherwise. Thus, we finally obtain
x0 ɛ 1
du
(1 + ɛu
x0 )ν
Cν
− 1
= O(ɛα ). (E.137)
1−x 0
ɛ
(1 + u2
ν+1
ν ) 2
39

278 9. Mesure de la dépendance extrême entre deux actifs financiers
Putting together equations (E.126) and (E.137) we obtain
∞
1
dx
ɛ
1
1
xν Cν
which concludes the proof.
1 + 1
ν
E.4 Derivation of equation (E.105)
From equation (E.111), we can deduce
¯FY (y) =
Using equations (E.98) and (E.102), we obtain
where
x−x0
ɛ
2 ν+1
2
Pɛ(α ˜ Yu + η − αy) = Pɛ(γ ˜ Yu − αy) ·
− 1
xν
= O(ɛ
0
min{ν,1} ) , (E.138)
ν−1
ν 2 Cν
yν −2
1 + O(y ) . (E.139)
γ = α 1 +
Putting together these results yields for the leading order
∞
dy ¯ FY (y) · Pɛ(α ˜ Yu + η − αy) =
˜Yu
= ν ν−1
1 + O( ˜ Y −2
u ) , (E.140)
σ
ν1/ν . (E.141)
α
∞
dy
˜Yu
2 Cν
ν
α ˜ Yu
ν−1
ν 2 Cν
yν ·
∞
1
σ
dx 1
·
xν where the change of variable x = y
has been performed in the last equation.
˜Yu
Cν
1 + (γ ˜ Yu−αy) 2
ν σ 2
1 + 1
ν
Cν α ˜ Yu
σ
x− γ
α
σ
α ˜ Yu
ν+1
2
2 ν+1
2
, (E.142)
We now apply lemma 2 with x0 = γ
σ
α > 1 and ɛ =
α ˜ which goes to zero as u goes to 1. This gives
Yu
which shows that
thus
which finally yields
∞
dy ¯ FY (y) · Pɛ(α ˜ Yu + η − αy) ∼u→1
˜Yu
Pr[X > F −1
−1
X (u), Y > FY (u)] ∼u→1 F −1
Y ( ˜ Yu)
Pr[X > F −1
X
−1
(u)|Y > F (u)] ∼u→1
λ =
Y
1
1 + σ
α
40
ν ν−1
2 Cν
α ˜ Y ν u
ν α
γ
ν α
= (1 − u)
γ
, (E.143)
, (E.144)
ν α
, (E.145)
γ
ν α
, (E.146)
γ
ν . (E.147)

9.1. Les différentes mesures de dépendances extrêmes 279
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Nelsen, R.B., 1998, An Introduction to Copulas. Lectures Notes in statistic 139 (Springer Verlag, New
York).
Patton, J.A., 2001, Estimation of copula models for time series of possibly different lengths, U of California,
Econ. Disc. Paper No. 2001-17.
Poon, S.H., M. Rockinger and J. Tawn, 2001, New extreme-value dependence measures and finance applications,
working paper.
Quintos, C.E., 2001, Estimating tail dependence and testing for contagion using tail indices, working paper.
Quintos, C.E., Z.H. Fan and P.C.B. Phillips, 2001, Structural change tests in tail behaviour and the Asian
crisis, Review of Economic Studies 68, 633-663.
Ramchand, L. and R. Susmel, 1998, Volatility and cross correlation across major stock markets, Journal of
Empirical Finance 5, 397-416.
Ross, S., 1976, The arbitrage theory of capital asset pricing, Journal of Economic Theory 17, 254-286.
Scaillet, O. 2000, Nonparametric estimation of copulas for time series, Working paper.
Sharpe, W., 1964, Capital assets prices: a theory of market equilibrium under conditions of risk, Journal of
Finance, 19, 425-442.
Silvapulle, P. and C.W.J. Granger, 2001, Large returns, conditional correlation and portfolio diversification:
a value-at-risk approach, Quantitative Finance 1, 542-551.
Sornette, D. P. Simonetti and J. V. Andersen, 2000, φ q -field theory for Portfolio optimization: “fat tails” and
non-linear correlations, Physics Report 335, 19-92.
Sornette, D., J.V. Andersen and P. Simonetti, 2000, Portfolio Theory for “Fat Tails”, International Journal
of Theoretical and Applied Finance 3, 523-535.
Starica, C., 1999, Multivariate extremes for models with constant conditional correlations, Journal of Empirical
Finance 6, 515-553.
Tsui, A.K. and Q. Yu, 1999, Constant conditional correlation in a bivariate GARCH model: evidence from
the stock markets of China, Mathematics and Computers in Simulation 48, 503-509.
43

282 9. Mesure de la dépendance extrême entre deux actifs financiers
Negative Tail
Argentina Brazil Chile Mexico
Argentina - 0.28 (0.04) 0.25 (0.04) 0.25 (0.05)
Brazil - 0.19 (0.03) 0.25 (0.05)
Chile - 0.24 (0.07)
Mexico -
Positive Tail
Argentina Brazil Chile Mexico
Argentina - 0.21 (0.06) 0.20 (0.04) 0.22 (0.04)
Brazil - 0.28 (0.04) 0.19 (0.04)
Chile - 0.19 (0.03)
Mexico -
Table 1: Coefficients of tail dependence between the four Latin American markets. The figure within
parenthesis gives the standard deviation of the estimated value derived under the assumption of asymptotic
normality of the estimators. Only the coefficients above the diagonal are indicated since they are symmetric.
44

9.1. Les différentes mesures de dépendances extrêmes 283
Student Hypothesis nu=3
Argentina Brazil Chile Mexico
Argentina - 0.24 0.25 0.27
Brazil - 0.24 0.27
Chile - 0.28
Mexico -
Table 2: Coefficients of tail dependence derived under the assumption of a Student’s copula with three
degrees of freedom
45

284 9. Mesure de la dépendance extrême entre deux actifs financiers
Bivariate Gaussian
Bivariate Student’s
√ ρ 1 ·
1−ρ2 v
ρ
ρ
2 +(ν−1)ν−2
(1−ρ ν
2 )
ρ + v ρ s v ρu
(3) 1 − 1
2
(6)
1−ρ 2
ρ 2
ρ
ρ
2 + 1
(1−ρ (ν−1)ν−2
ν
2 )
1
v2 (4) ρ 1+ρ 1
1−ρ · u2 (13)
(7) -
Gaussian Factor Model same as (3) same as (4) same as (13)
Student’s Factor Model 1 − K
v 2 (11) 1 − K
v 2 (11) -
Table 3: Large v and u dependence of the conditional correlations ρ + v (signed condition), ρ s v (unsigned
condition) and ρu (on both variables) for the different models studied in the present paper, described in the
first column. The numbers in parentheses give the equation numbers from which the formulas are derived.
The factor model is defined by (8), i.e., X = αY + ɛ. ρ is the unconditional correlation coefficient.
46

9.1. Les différentes mesures de dépendances extrêmes 285
ρ + v=∞ ρ s v=∞ ρu=∞ λ ¯ λ
Bivariate Gaussian 0 1 0 0 ρ
Bivariate Student’s see Table 3 see Table 3 - 2 · ¯ Tν+1
√ν + 1
1−ρ
1+ρ
Gaussian Factor Model 0 1 0 0 ρ
Student’s Factor Model 1 1 -
ρ ν
ρ ν +(1−ρ 2 ) ν/2
Table 4: Asymptotic values of ρ + v , ρs v and ρu for v → +∞ and u → ∞ and comparison with the taildependence
λ and ¯ λ for the four models indicated in the first column. The factor model is defined by (8),
i.e., X = αY + ɛ. ρ is the unconditional correlation coefficient. For the Student’s factor model, Y and ɛ
have centered Student’s distributions with the same number ν of degrees of freedom and their scale factors
are respectively equal to 1 and σ, so that ρ = (1 + σ2
α2 ) −1/2 . For the Bivariate Student’s distribution, we
refer to Table 1 for the constant values of ρ + v=∞ and ρs v=∞.
47
1
1

286 9. Mesure de la dépendance extrême entre deux actifs financiers
10 0
10 −1
10 −2
10 −3
10 −4
10 0
10 −1
10 −2
10 −3
10 −4
10 −4
10 −4
Argentina
Brazil
Chile
Mexico
Argentina
Brazil
Chile
Mexico
10 −3
10 −3
Positive Tail
10 −2
Negative Tail
10 −2
Figure 1:
48
μ=2
10 −1
μ=2
10 −1
10 0
10 0

9.1. Les différentes mesures de dépendances extrêmes 287
+,−
ρ ,y=Brazil
v
+,−
ρ ,y=Argentina
v
1
0.5
0
−0.5
Argentina−Brazil
−1
−5 −4 −3 −2 −1
v
0 1 2 3
1
0.5
0
−0.5
−1
−4 −3 −2 −1 0
v
1 2 3 4
Figure 2:
49

288 9. Mesure de la dépendance extrême entre deux actifs financiers
s
ρ ,y=Brazil
v
s
ρ ,y=Argentina
v
1.2
1
0.8
0.6
0.4
0.2
0
Argentina−Brazil
−0.2
0 0.5 1 1.5 2 2.5
v
3 3.5 4 4.5 5
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
0 0.5 1 1.5 2
v
2.5 3 3.5 4
Figure 3:
50

9.1. Les différentes mesures de dépendances extrêmes 289
+,−
ρ ,y=Chile
v
+,−
ρ ,y=Brazil
v
1
0.5
0
−0.5
Brazil−Chile
−1
−4 −3 −2 −1 0
v
1 2 3 4
1
0.5
0
−0.5
−1
−4 −3 −2 −1 0 1 2 3
v
Figure 4:
51

290 9. Mesure de la dépendance extrême entre deux actifs financiers
s
ρ ,y=Chile
v
s
ρ ,y=Brazil
v
1.2
1
0.8
0.6
0.4
0.2
0
Brazil−Chile
−0.2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
v
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
0 1 2 3
v
4 5 6
Figure 5:
52

9.1. Les différentes mesures de dépendances extrêmes 291
+,−
ρ ,y=Mexico
v
+,−
ρ ,y=Chile
v
1
0.5
0
−0.5
Chile−Mexico
−1
−4 −3 −2 −1 0
v
1 2 3 4
1
0.5
0
−0.5
−1
−4 −3 −2 −1 0
v
1 2 3 4
Figure 6:
53

292 9. Mesure de la dépendance extrême entre deux actifs financiers
s
ρ ,y=Mexico
v
s
ρ ,y=Chile
v
1.2
1
0.8
0.6
0.4
0.2
0
Chile−Mexico
−0.2
0 1 2 3
v
4 5 6
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
v
Figure 7:
54

9.1. Les différentes mesures de dépendances extrêmes 293
0.9
0.9
ρ=0.1
ρ=0.3
ρ=0.5
ρ=0.7
ρ=0.9
0.8
ρ=0.1
ρ=0.3
ρ=0.5
ρ=0.7
ρ=0.9
0.8
0.7
0.7
0.6
0.6
0.5
0.5
ρ s (v)
ρ s (v)
0.4
0.4
55
0.3
0.3
0.2
0.2
0.1
0.1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
v
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
v
Figure 8:

294 9. Mesure de la dépendance extrême entre deux actifs financiers
+,−
ρ ,y=Brazil
v
+,−
ρ ,y=Argentina
v
0.4
0.2
0
−0.2
−0.4
0.4
0.2
0
−0.2
Argentina−Brazil
0 0.1 0.2 0.3 0.4 0.5
v
0.6 0.7 0.8 0.9 1
−0.4
0 0.1 0.2 0.3 0.4 0.5
v
0.6 0.7 0.8 0.9 1
Figure 9:
56

9.1. Les différentes mesures de dépendances extrêmes 295
+,−
ρ ,y=Chile
v
+,−
ρ ,y=Brazil
v
0.6
0.4
0.2
0
−0.2
0.2
0
−0.2
−0.4
−0.6
Brazil−Chile
0 0.1 0.2 0.3 0.4 0.5
v
0.6 0.7 0.8 0.9 1
−0.8
0 0.1 0.2 0.3 0.4 0.5
v
0.6 0.7 0.8 0.9 1
Figure 10:
57

296 9. Mesure de la dépendance extrême entre deux actifs financiers
+,−
ρ ,y=Mexico
v
+,−
ρ ,y=Chile
v
0.5
0.4
0.3
0.2
0.1
0
−0.1
Chile−Mexico
−0.2
0 0.1 0.2 0.3 0.4 0.5
v
0.6 0.7 0.8 0.9 1
0.6
0.4
0.2
0
−0.2
0 0.1 0.2 0.3 0.4 0.5
v
0.6 0.7 0.8 0.9 1
Figure 11:
58

9.1. Les différentes mesures de dépendances extrêmes 297
Student’s Factor Model
Student’s Copula
1
1
ν=3
ν=5
ν=10
ν=20
ν=50
ν=100
0.9
0.9
0.8
ν=3
ν=5
ν=10
ν=20
ν=50
ν=100
0.8
0.7
0.7
0.6
0.6
0.5
λ
0.5
λ
0.4
0.4
59
0.3
0.3
0.2
0.2
0.1
0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
ρ
ρ
Figure 12:

298 9. Mesure de la dépendance extrême entre deux actifs financiers
0
(1−x )/ε
0
−
ν
x −1
0 ⋅ ε ⋅ u
x −1
0
0
u
Figure 13:
60
ν ⋅ ε
⋅ u
x
0
x 0 /ε

9.2. Estimation du coefficient de dépendance de queue 299
9.2 Estimation du coefficient de dépendance de queue
Utilisant le cadre des modèles à facteurs, nous étudions les co-mouvements extrêmes entre deux actifs
financiers ou entre un actif et le marché. A cette fin, nous établissons l’expression générale du coefficient
de dépendance de queue entre le marché et un actif (c’est-à-dire, la probabilité qu’un actif subisse une
perte extrême, sachant que le marché a lui-même subi une perte extrême) et entre deux actifs comme
une fonction des paramètres du modèle à facteur et des paramètres de queue des distributions du facteur
et du bruit idiosynchratique. Notre formule est valable pour des distributions marginales quelconques
et ne requièrt aucune paramètrisation de la distribution jointe du marché et des actifs. La détermination
de ce paramètre extrême, qui n’est pas accessible par inférence statistique directe, est rendue possible
par la mesure de paramètres dont l’estimation implique une quantité significative de données. Nos tests
empiriques démontrent un bon accord entre le coefficient de dépendance de queue calibré et les grandes
pertes réalisées entre 1962 et 2000. Néanmoins, un biais systématique est détecté, suggérant l’existence
d’un “ outlier” lors du krach d’octobre 1987 et pouvant inciter à penser que le modèle à un facteur
(CAPM) que nous avons considéré ne suffit pas totalement à rendre compte des propriétés extrêmes dans
certaines phases critiques de marché.

300 9. Mesure de la dépendance extrême entre deux actifs financiers
How to account for extreme co-movements between individual
stocks and the market ∗
Y. Malevergne 1,2 and D. Sornette 1,3
1 Laboratoire de Physique de la Matière Condensée CNRS UMR 6622
Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France
2 Institut de Science Financière et d’Assurances - Université Lyon I
43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
3 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science
University of California, Los Angeles, California 90095, USA
email: Yannick.Malevergne@unice.fr and sornette@unice.fr
fax: (33) 4 92 07 67 54
August 8, 2002
Abstract
Using the framework of factor models, we study the extreme co-movements between two
stocks and between a stock and the market. In this goal, we establish the general expression of
the coefficient of tail dependence between the market and a stock (that is, the probability that
the stock incurs a large loss, assuming that the market has also undergone a large loss) and
between two stocks as a function of the parameters of the underlying factor model and of the
tail parameters of the distributions of the factor and of the idiosyncratic noise of each stock.
Our formula holds for arbitrary marginal distributions and in addition does not require any
parameterization of the multivariate distributions of the market and stocks. The determination
of the extreme parameter, which is not accessible by a direct statistical inference, is made
possible by the measurement of parameters whose estimation involves a significant part of the
data with sufficient statistics. Our empirical tests find a good agreement between the calibration
of the tail dependence coefficient and the realized large losses over the period from 1962 to 2000.
Nevertheless, a bias is detected which suggests the presence of an outlier in the form of the crash
of October 1987.
∗ We acknowledge helpful discussions and exchanges with C.W.G. Granger, J.P. Laurent, V. Pisarenko, R. Valkanov
and D. Zajdenweber. This work was partially supported by the James S. Mc Donnell Foundation 21st century scientist
award/studying complex system.
1

9.2. Estimation du coefficient de dépendance de queue 301
Introduction
The concept of extreme or “tail dependence” probes the reaction of a variable to the realization
of another variable when this realization is of extreme amplitude and very low probability. The
dependence, and especially the extreme dependence, between two assets or between an asset and
any other exogeneous economic variable is an issue of major importance both for practioners and
for academics. The determination of extreme dependences is crucial for financial and for insurance
institutions involved in risk management. It is also fundamental for the establishment of a rational
investment policy striving for the best diversification of the various sources of risk. In all these
situations, the objective is to prevent or at least minimize the simultaneous occurrence of large
losses across the different positions held in the portfolio.
From an academic perspective, taking into account the extreme dependence properties provide
useful yardsticks and important constraints for the construction of models, which should not underestimate
or overestimate risks. From the point of view of univariate statistics, extreme values
theory provides the mathematical framework for the classification and quantification of very large
risks. This has been made possible by the existence of a “universal” behavior summarized by
the Gnedenko-Pickands-Balkema-de Haan theorem which gives a natural limit law for peak-overthreshold
values in the form of the Generalized Pareto Distribution (see Embrechts, Kluppelberg,
and Mikosh (1997, pp 152-168)). Moreover, most of these univariate extreme values results are robust
with respect to the time dependences observed in financial time series (see de Haan, Resnick,
Rootzen and de Vries (1989) or Starica (1999) for instance). In contrast, no such result is yet available
in the multivariate case. In such absence of theoretical guidelines, the alternative is therefore
to impose some dependence structure in a rather ad hoc and arbitrary way. This was the stance
taken for instance in Longin and Solnik (2001) in their study of the phenomenon of contagion across
international equity markets.
This approach, where the dependence structure is not determined from empirical facts or from
an economic model, is not fully satisfying. As a remedy, we propose a new approach, which does
not directly rely on multivariate extreme values theory, but rather derives the extreme dependence
structure from the characteristics of a financial model of assets. Specifically, we use the general
class of factor models, which is probably one of the most versatile and relevant one, and whose
introduction in finance can be traced back at least to Ross (1976). The factor models are now widely
used in many branches of finance, including stock return models, interest rate models (Vasicek
(1977), Brennan and Schwarz (1978), Cox, Ingersoll, and Ross (1985)), credit risks models (Carey
(1998), Gorby (2000), Lucas, Klaassen, Spreij and Straetmans (2001)), and so on, and are found
at the core of many theories and equilibrium models.
Here, we shall first focus on the characterization of the extreme dependence between stock returns
and the market return and then on the extreme dependence between two stocks which share a
common explaining factor. The role of the market return as a factor explaining the evolution of
individual stock returns is supported both by theoretical models such as the Capital Asset Pricing
Model (CAPM) (Sharpe (1964), Lintner (1965), Mossin (1966)) or the Arbitrage Pricing Theory
(APT) (Ross (1976)) and by empirical studies (Fama and Beth (1973), Kandel and Staumbaugh
(1987) among many others). It has even been shown in Roll (1988) that in certain dramatic
circumstances, such as the October 1987 stock-market crash, the (global) market was the sole
relevant factor needed to explain the stock market movements and the propagation of the crash
across countries. Thus, the choice of factor models is a very natural starting point for studying
extreme dependences from a general point of view. The main gain is that, without imposing any
a priori ad hoc dependences other than the definition of the factor model, we shall be able to
2

302 9. Mesure de la dépendance extrême entre deux actifs financiers
derive the general properties of extreme dependence between an asset and one of its factor and to
empirically determine these properties by a simple estimation of the factor model parameters.
Our results are directly relevant to a portfolio manager using any of the factor models such as the
CAPM or the APT to estimate the impact on her extreme risks upon the addition or removal of an
asset in her portfolio. In this framework, our results stated for single assets can easily be extended
to an entire portfolio, and some examples will be given. This problem is acute in particular in
funds of funds. From a more global perspective, our analysis of the tail dependence of two assets is
the correct setting for analyzing the strategic asset allocation facing a portofolio manager striving
to diversify between a portfolio of stocks and a portfolio of bonds or between portfolios constituted
of domestic and of international assets.
Our main addition to the literature is to provide a completely general analytical formula for the
extreme dependence between any two assets, which holds for any distribution of returns of these
two assets and of their common factor and which thus embodies their intrinsic dependence. Our
second innovation is to provide a novel and robust method for estimating empirically the extreme
dependence which we test on twenty majors stocks of the NYSE. Comparing with historical comovements
in the last forty years, we check that our prediction is validated out-of-sample and thus
provide an ex-ante method to quantify futur stressful periods, so that our results can be directly
used to construct a portfolio aiming at minimizing the impact of extreme events. We are also able
to detect an anomalous co-monoticity associated with the October 1987 crash.
The plan of our presentation is as follows. The first section defines the concepts needed for the
characterization and quantification of extreme dependences. In particular, we recall the definition
of the coefficient of tail dependence, which captures in a single number the properties of extreme
dependence between two random variables: the tail dependence is defined as the probability for
a given random variable to be large assuming that another random variable is large, at the same
probability level. We shall also need some basic notions on dependences between random variables
using the mathematical concept of copulas. In order to provide some perspective on the following
results, this section also contains the expression of some classical exemples of tail dependence
coefficients for specific multivariate distributions.
The second section states our main result in the form of a general theorem allowing the calculation
of the coefficient of tail dependence for any factor model with arbitrary distribution functions of
the factors and of the idiosyncratic noise. We find that the factor must have sufficiently “wild”
fluctuations (to be made precise below) in order for the tail dependence not to vanish. For normal
distributions of the factor, the tail dependence is identically zero, while for regularly varying distributions
(power laws), the tail dependence is in general non-zero. We also show that the most
interesting coefficients of tail dependence are those between each individual stock and their common
factor, since the tail dependence between any pair of assets is shown to be nothing but the
minimum of the tail dependence between each asset and their common factor.
The third section is devoted to the empirical estimation of the coefficients of tail dependence between
individual stock returns and the market return. The tests are performed for daily stock returns.
The estimated coefficients of tail dependence are found in good agreement with the fraction of
historically realized extreme events that occur simultaneously with any of the ten largest losses of
the market factor (these ten largest losses were not used to calibrate the tail dependence coefficient).
We also find some evidence for comonotonicity in the crash of Oct. 1987, suggesting that this event
is an “outlier,” providing additional support to a previous analysis of large and extreme drawdowns.
We summarize our results and conclude in the fourth section.
3

9.2. Estimation du coefficient de dépendance de queue 303
1 Intrinsic measure of casual and of extreme dependences
This section provides a brief informal summary of the mathematical concepts used in this paper to
characterize the normal and extreme dependences between asset returns.
1.1 How to characterize uniquely the full dependence between two random
variables?
The answer to this question is provided by the mathematical notion of “copulas,” initially introduced
by Sklar (1959) 1 , which allows one to study the dependence of random variables independently
of the behavior of their marginal distributions. Our presentation focuses on two variables
but is easily extended to the case of N random variables, whatever N may be. Sklar’s Theorem
states that, given the joint distribution function F (·, ·) of two random variables X and Y with
marginal distribution FX(·) and FY (·) respectively, there exists a function C(·, ·) with range in
[0, 1] × [0, 1] such that
F (x, y) = C(FX(x), FY (y)) , (1)
for all (x, y). This function C is the copula of the two random variables X and Y , and is unique if
the random variables have continous marginal distributions. Moreover, the following result shows
that copulas are intrinsic measures of dependence. If g1(X), g2(Y ) are strictly increasing on the
ranges of X, Y , the random variables ˜ X = g1(X), ˜ Y = g2(Y ) have exactly the same copula C
(see Lindskog (2000)). The copula is thus invariant under strictly increasing transformation of the
variables. This provides a powerful way of studying scale-invariant measures of associations. It is
also a natural starting point for construction of multivariate distributions.
1.2 Tail dependence between two random variables
A standard measure of dependence between two random variables is provided by the correlation
coefficient. However, it suffers from at least three deficiencies. First, as stressed by Embrechts,
McNeil, and Straumann (1999), the correlation coefficient is an adequate measure of dependence
only for elliptical distributions and for events of moderate sizes. Second, the correlation coefficient
measures only the degree of linear dependence and does not account of any other nonlinear functional
dependence between the random variables. Third, it agregates both the marginal behavior
of each random variable and their dependence. For instance, a simple change in the marginals
implies in general a change in the correlation coefficient, while the copula and, therefore the dependence,
remains unchanged. Mathematically speaking, the correlation coefficient is said to lack
the property of invariance under increasing changes of variables.
Since the copula is the unique and intrinsic measure of dependence, it is desirable to define measures
of dependences which depend only on the copula. Such measures have in fact been known for a long
time. Examples are provided by the concordance measures, among which the most famous are the
Kendall’s tau and the Spearman’s rho (see Nelsen (1998) for a detailed exposition). In particular,
the Spearman’s rho quantifies the degres of functional dependence between two random variables: it
equals one (minus one) when and only when the first variable is an increasing (decreasing) function
of the second variable. However, as for the correlation coefficient, these concordance measures do
1 The reader is refered to Joe (1997), Frees and Valdez (1998) or Nelsen (1998) for a detailed survey of the notion
of copulas and a mathematically rigorous description of their properties.
4

304 9. Mesure de la dépendance extrême entre deux actifs financiers
not provide a useful measure of the dependence for extreme events, since they are constructed over
the whole distributions.
Another natural idea, widely used in the contagion literature, is to work with the conditional correlation
coefficient, conditioned only on the largest events. But, as stressed by Boyer, Gibson,
and Lauretan (1997), such conditional correlation coefficient suffers from a bias: even for a constant
unconditional correlation coefficient, the conditional correlation coefficient changes with the
conditioning set. Therefore, changes in the conditional correlation do not provide a characteristic
signature of a change in the true correlations. The conditional concordance measures suffer from
the same problem.
In view of these deficiencies, it is natural to come back to a fundamental definition of dependence
through the use of probabilities. We thus study the conditional probability that the first variable is
large conditioned on the second variable being large too: ¯ F (x|y) = Pr{X > x|Y > y}, when x and
y goes to infinity. Since the convergence of ¯ F (x|y) may depend on the manner with which x and y
go to infinity (the convergence is not uniform), we need to specify the path taken by the variables
to reach the infinity. Recalling that it would be preferable to have a measure which is independent
of the marginal distributions of X and Y , it is natural to reason in the quantile space. This leads
to choose x = FX −1 (u) and y = FY −1 (u) and replace the conditions x, y → ∞ by u → 1. Doing so,
we define the so-called coefficient of upper tail dependence (see Coles, Heffernan, and Tawn (1999),
Lindskog (2000), or Embrechts, McNeil, and Straumann (2001)):
λ+ = lim
u→1 − Pr{X > FX −1 (u) | Y > FY −1 (u)} . (2)
As required, this measure of dependence is independent of the marginals, since it can be expressed
in term of the copula of X and Y as
λ+ = lim
u→1− 1 − 2u + C(u, u)
1 − u
. (3)
This representation shows that λ+ is symmetric in X and Y , as it should for a reasonable measure
of dependence.
In a similar way, we define the coefficient of lower tail dependence as the probability that X incurs
a large loss assuming that Y incurs a large loss at the same probability level
λ− = lim
u→0 + Pr{X < FX −1 (u) | Y < FY −1 (u)} = lim
u→0 +
C(u, u)
u
. (4)
This last expression has a simple interpretation in term of Value-at-Risk. Indeed, the quantiles
F −1
−1
X (u) and FY (u) are nothing but the Values-at-Risk of assets (or portfolios) X and Y at the
confidence level 1 − u. Thus, the coefficient λ− simply provides the probability that X exceeds the
VaR at level 1 − u, assuming that Y has exceeded the VaR at the same level confidence level 1 − u,
when this level goes to one. As a consequence, the probability that both X and Y exceed their VaR
at the level 1 − u is asymptotically given by λ− · (1 − u) as u → 0. As an example, consider a daily
VaR calculated at the 99% confidence level. Then, the probability that both X and Y undergo a
loss larger than their VaR at the 99% level is approximately given by λ−/100. Thus, when λ− is
about 0.1, the typical recurrence time between such concomitant large losses is about four years,
while for λ− ≈ 0.5 it is less than ten months.
The values of the coefficients of tail dependence are known explicitely for a large number of different
copulas. For instance, the Gaussian copula, which is the copula derived from de Gaussian multivariate
distribution, has a zero coefficient of tail dependence. In contrast, the Gumbel’s copula used
5

9.2. Estimation du coefficient de dépendance de queue 305
by Longin and Solnik (2001) in the study of the contagion between international equity markets,
which is defined by
Cθ(u, v) = exp − (− ln u) θ 1
+ (− ln v)
θ θ
, θ ∈ [0, 1], (5)
has an upper tail coefficient λ+ = 2 − 2 θ . For all θ’s smaller than one, λ+ is positive and the
Gumbel’s copula is said to present tail dependence, while for θ = 1, the Gumbel copula is said to
be asymptotically independent. One should however use this terminology with a grain of salt as
“tail independence” (quantified by λ+ = 0 or λ− = 0) does not imply necessarily that large events
occur independently (see Coles, Heffernan, and Tawn (1999) for a precise discussion of this point).
2 Tail dependence of factor models
2.1 Tail dependence between an asset and one of its explaining factors
Now we state the first part of our main theoretical result. Let us consider two random variables
X and Y of cumulative distribution functions FX(X) and FY (Y ), where X represents the return
of a single stock and Y is the market return for instance. Let us also introduce an idiosyncratic
noise ε, which is assumed independent of the market return Y . The factor model is defined by
the following relationship between the individual stock return X, the market return Y and the
idiosyncratic noise ε:
X = β · Y + ε . (6)
β is the usual coefficient introduced by the Capital Asset Pricing Model Sharpe (1964). Let us stress
that ε may embody other factors Y ′ , Y ′′ , . . ., as long as they remain independent of Y . Under such
conditions and a few other technical assumptions detailed in the theorem established in appendix
A.1, the coefficient of (upper) tail dependence between X and Y defined in (2) is obtained as
∞
λ+ =
max1, l β dx f(x) , (7)
where l denotes the limit, when u → 1, of the ratio of the quantiles of X and Y ,
l = lim
u→1
FX −1 (u)
FY −1 (u)
and f(x) is the limit, when t → +∞, of t · PY (tx)/ ¯ FY (t):
f(x) = lim
t→+∞ t PY (tx)
¯FY (t)
, (8)
. (9)
PY is the distribution density of Y and ¯ FY = 1 − FY is the complementary cumulative distribution
function of Y . A similar expression obviously holds, mutatis mutandis, for the coefficient of lower
tail dependence.
The measure of tail dependence given by equation (7) depends on two limits defined in (8) and
(9) and thus seems likely difficult to estimate. As it turns out, we will show that this is not the
case in the empirical section below. Indeed, the first limit (8) is nothing but a ratio of quantiles,
while the second limit (9) can be easily calculated for almost all distributions of the factor. For
6

306 9. Mesure de la dépendance extrême entre deux actifs financiers
instance, let us consider the Pareto distribution ¯ FY (y) = 1/(y/y0) µ defined for y ≥ y0, whose
density is equal to PY (y) = (µ/y0)/(y/y0) 1+µ ; the limit (9) gives f(x) = µ/x 1+µ . In contrast,
for the Poisson law ¯ FY (y) = e −ry defined for y ≥ 0 with density PY (y) = re −ry , the limit (9)
gives f(x) = limt→∞ r t e −rt(x−1) = 0 for x > 1. Thus an estimation of the tail of the factor
distribution is sufficient to infer the limit function f(x). Moreover, equation (7) has a rather simple
interpretation since it shows that a non-vanishing coefficient of tail dependence results from the
combination of two phenomena. First, the limit function f(x), which only depends on the behavior
of the factor distribution, must be non-zero. Second, the constant l must remain finite to ensure
that the integral in (7) does not vanish. Thus, the value of the coefficient of tail dependence is
controlled by f(x) solely function of the factor and a second variable l quantifying the competition
of the tails of the distribution of the factor Y and of the idiosyncratic noise ε.
The fundamental result (7) should be of vivid interest to financial economists because it provides a
general, rigorous and simple method for estimating one of the key variable embodying the occurrence
of and the risks associated with extremes in joint distributions. From a theoretical view point, it
also anchors the derivation and quantification of a key variable on extremes in the general class of
financial factor models, thus extending their use and relevance also to this rather novel domain of
extreme dependence, extreme risks and extreme losses.
Up to now, we have assumed that the factor Y and the idiosyncratic noise ε were independent. In
fact, it is important to stress for the sake of generality that the result (7) holds even when they are
dependent, provided that this dependence is not too strong, as explained and made specific at the
end of Appendix A.1.
We now derive two direct consequences of this result (7) (see corollary 1 and 2 in appendix B),
concerning rapidly varying and regularly varying factors 2 , which clearly illustrate the role the
factor itself and the impact of the trade off between the factor and the idiosyncratic noise.
2.2 Absence of tail dependence for rapidly varying factors
Let us assume that the factor Y and the idiosyncratic noise ε are normally distributed (the second
assumption is made for simplicity and will be relaxed below). As a consequence, the joint distribution
of (X, Y ) is the bivariate Gaussian distribution. Refering to the results stated in section 1.1.2,
we conclude that the copula of (X, Y ) is the Gaussian copula whose coefficient of tail dependence
is zero. In fact, it is easy to show that λ = 0 for any non-degenerated distribution of ε.
More generally, let us assume that the distribution of the factor Y is rapidly varying, which describes
the Gaussian, exponential and any distribution decaying faster than any power-law. Then, the
coefficient of tail dependence is identically zero. This result holds for any arbitrary distribution of
the idiosyncratic noise (see corollary 1 in appendix B). It also holds for mixtures of normals or other
distributions fatter than Gaussians, some of which are thought to be reasonable approximations to
empirical stock return distributions.
These statements are somewhat counter-intuitive since one could expect a priori that the coefficient
of tail dependence does not vanish as soon as the tail of the distribution of factor returns is fatter
than the tail the distribution noise returns. Said differently, when the standard deviation of the
idiosynchratic noise ε is small (but not zero), then the idiosynchratic noise component is small and
X and Y are practically identical, and it seems strange that their tail dependence can be equal
2 see Bigham, Goldie, and Teugel (1987) or Embrechts, Kluppelberg, and Mikosh (1997) for a survey of the
properties of rapidly and regularly varying functions
7

9.2. Estimation du coefficient de dépendance de queue 307
to zero. This non-intuitive result stems from the fact that the tail dependence is quantifying not
just a dependence but a specific dependence for extreme co-movements. Thus, in order to get
a non-vanishing tail-dependence, the fluctuations of the factor must be ‘wild’ enough, which is
not realized with rapidly varying distributions, irrespective of the relative values of the standard
deviations of the factor and the idosyncratic noise.
2.3 Coefficient of tail dependence for regularly varying factors
2.3.1 Example of the factor model with Student distribution
In order to account for the power-law tail behavior observed for the distributions of assets returns
it is logical to consider that the factor and the indiosyncratic noise also have power-law tailed
distributions. As an illustration, we will assume that Y and ε are distributed according to a
Student’s distribution with the same number of degrees of freedom ν (and thus same tail exponent
ν). Let us denote by σ the scale factor of the distribution of ε while the scale factor of the
distribution of Y is chosen equal to one 3 . Applying the theorem previously established, we find
that f(x) = ν/x ν+1 and l = β
1 +
σ
β
1
λ± =
1 +
ν 1/ν
, so that the coefficient of tail dependence is
σ
β
ν , and β > 0. (10)
As expected, the tail dependence increases as β increases and as σ decreases. Since the idiosyncratic
volatility of the asset increases when the scale factor σ increases, this results simply means that
the tail dependence decreases when the idiosyncratic volatility of a stock increases relative to the
market volatility. The dependence with respect to ν is less intuitive. In particular, let ν go to
infinity. Then, λ → 0 if σ > β and λ → 1 for σ < β. This is surprising as one could argue that, as
ν → ∞, the Student distribution tends to the Gaussian law. As a consequence, one would expect
the same coefficient of dependence λ± = 0 as for rapidly varying functions. The reason for the
non-certain convergence of λ± to zero as ν → ∞ is rooted in a subtle non-commutativity (and
non-uniform convergence) of the two limits ν → ∞ and u → 1. Indeed, when taking first the limit
u → 1, the result λ → 1 for β > σ indicates that a sufficiently strong factor coefficient β always
ensures the validity of the power law regime, whatever the value of ν. Correlatively, in this regime
β > σ, λ± is an increasing function of ν.
The result (10) is of interest for financial economics purpose because it provides a simple parametric
illustration and interpretation of how the risk of large co-movements is affected by the three key
parameters entering in the definition of the factor model. It allows one to weight how the ingredients
of the factor model impact on the large risks captured by λ± and thus links the financial basis
underlying the factor model to the extreme multivariate risks.
2.3.2 General result
We now provide the general result valid for any regularly varying distribution. Let the factor
Y follows a regularly varying distribution with tail index α: in other words, the complementary
cumulative distribution of Y is such that ¯ FY (y) = L(y) · y −α , where L(y) is a slowly varying
3 Such a choice is always possible via a rescaling of the coefficient β.
8

308 9. Mesure de la dépendance extrême entre deux actifs financiers
function, i.e:
Corollary 2 in appendix B.2 shows that
L(ty)
lim = 1, ∀y > 0. (11)
t→∞ L(t)
λ =
1
max
1, l
β
α , (12)
where l denotes the limit, when u → 1, of the ratio FX −1 (u)/FY −1 (u). In the case of particular
interest when the distribution of ε is also regularly varying with tail index α and if, in addition, we
have ¯ FY (y) ∼ Cy ·y −α and ¯ Fε(ε) ∼ Cε ·ε −α , for large y and ε, then the coefficient of tail dependence
is a simple function of the ratio Cε/Cy of the scale factors:
λ =
1
1 + β −α · Cε
Cy
. (13)
When the tail indexes αY and αε of the distribution of the factor and the residue are different,
then λ = 0 for αY < αε and λ = 1 for αY > αε.
The results (12) and (13) are very important both for a financial and and economic perseptive
because they express in the most general and straightforward way the risk of extreme co-movements
quantified by the tail dependence parameter λ within the important class of factor models. That λ
increases with the β factor is intuitively clear. Less obvious is the dependence of λ on the structure
of the marginal distributions of the factor and of the idiosynchratic noise, which is found to be
captured uniquely in terms of the ratio of their scale factors Cε and Cy. The scale factors Cε and
Cy together with the factor β thus replace the variance and covariance in their role as the sole
quantifiers of the extreme risks occurring in co-movements.
Until now, we have only considered a single asset X. Let us now consider a portfolio of assets Xi,
each of the assets following exactly the one factor model (6)
Xi = βi · Y + εi, (14)
with independent noises εi, whose scale factors are Cεi . The portfolio X = wiXi, with weights
wi, also follows the factor model with a parameter β = wiβi and noise ε, whose scale factor is
Cε = |wi| α · 4
Cεi . Thus, equation (13) shows that the tail dependence between the portfolio and
the factor is
|wi|
λ = 1 +
α · Cεi
( wiβi) α −1 . (15)
· CY
When unlimited short sells are allowed, one can follow a “market neutral” strategy yielding β = 0
and thus λ = 0. But in the more realistic case where only limited short sells are authorized, one
cannot reach β = 0, and the best portfolio, which is the less “correlated” with the large market
moves, has to minimze the tail dependence (15).
This simple example clearly shows that it is very different to minimize the extreme co-movements,
according to (15), and to minimize the (linear) correlation ρ between the portfolio and the market
factor given by
ρ =
1 +
w 2 i · V ar(εi)
( wiβi) 2 V ar(Y )
−1/2
. (16)
4 In the more realistic case where the εi’s are not independent but still embody one or several common factors
Y ′ , Y ′′ , · · ·, the resulting scale factor Cε can be calculated with the method described in Bouchaud, Sornette, Walter
and Aguilar (1998)
9

9.2. Estimation du coefficient de dépendance de queue 309
Thus, since the minimum of ρ may be very different from the minimum of λ, minimizing ρ almost
surely leads to accept a level of extreme risks which is not optimal.
2.4 Tail dependence between two assets related by a factor model
We now present the second part of our theoretical result. Let X1 and X2 be two random variables
(two assets) of cumulative distributions functions F1, F2 with a common factor Y . Let ε1 and ε2
be the idiosyncratic noises associated with these two assets X1 and X2. We allow the idiosyncratic
noises to be dependent random variables, as occurs for instance if they embody the effect of other
factors Y ′ , Y ′′ , ... which are independent of Y . Our essential assumption is that the distribution of
the factor Y must have a tail not thinner than the tail of the distributions of the other factors Y ′ ,
Y ′′ , ... This hypothesis is crucial in order to detect the existence of tail-dependence. This means
that, for purposes of characterizing tail dependencies in factor models, our model can always be
re-stated as a single factor model where the single factor is the factor with the thickest tail. This
makes our results quite general. Then, the model can be written as
X1 = β1 · Y + ε1 , (17)
X2 = β2 · Y + ε2 . (18)
We prove in appendix A.2, that the coefficient of (upper) tail dependence λ+ = limu→1 Pr{X1 >
F1 −1 (u) | X2 > F2 −1 (u)}, between the assets X1 and X2, is given by the expression
λ+ =
∞
maxl 1
β1 , l 2
β2 dx f(x) , (19)
which is very similar to that found for the tail dependence between an asset and one of its explaining
factor (see equation (7)). As previously, l1,2 denotes the limit, when u → 1, of the ratio
F1,2 −1 (u)/FY −1 (u), and f(x) is the limit, when t → +∞, of t · PY (tx)/ ¯ FY (t).
The result (19) can be cast in a different illuminating way. Let λ(X1, Y ) (resp. λ(X2, Y )) denote
the coefficient of tail dependence between the asset X1 (resp. X2) and their common factor Y . Let
λ(X1, X2) denote the tail dependence between the two assets. Equation (19) allows us to assert
that
λ(X1, X2) = min{λ(X1, Y ), λ(X2, Y )}. (20)
The tail dependence between the two assets X1 and X2 is nothing but the smallest tail dependence
between each asset and the common factor. Therefore, a decrease of the tail dependence between
the assets and the market will also lead automatically to a decrease of the tail dependence between
the two assets. This result also shows that it is sufficient to study the tail dependence between the
assets and their common factor to obtain the tail dependence between any pair of assets.
The result (20) is also useful in the context of portfolio analysis. Not only does it provide a tool
for assessing the probability of large losses of a portfolio composed of assets driven by a common
factor, it also allows us to define novel strategies of portfolio optimization based on the selection
and weighting of stocks chosen so as to balance to risks associated with extreme co-movements.
Such an approach has been tested in (Malevergne and Sornette 2002) with encouraging results.
3 Empirical study
We now apply our theoretical results to the daily returns of a set of stocks traded on the New York
Stock Exchange. In order to estimate the parameters of the factor model (6), the Standard and
10

310 9. Mesure de la dépendance extrême entre deux actifs financiers
Poor’s 500 index is chosen to represent the common “market factor.” It has been prefered over the
Dow Jones Industrial Averages Index for instance, because, although less diversified, it represents
about 80% of the total market capitalization.
We describe the set of selected stocks in the next sub-section. Next, we estimate the parameter
β in (6) and check the independence of the market returns and the residues. Then, applying the
commonly used hypothesis according to which the tail of the distribution of assets return is a power
law or at leastregularly varying (see Longin (1996), Lux (1996), Pagan (1996), or Gopikrishnan,
Meyer, Amaral, and Stanley (1998)), we estimate the tail index and the scale factor of these
distributions, which allows us to calculate the coefficients of tail dependence between each asset
return and the market return. Finally, we perform an analysis of the historical data to check
the compatibility of our prediction on the fraction of realized large losses of the assets that occur
simultaneously with the large losses of the market.
The results of our analysis are reported below in terms of the returns rather than in terms of the
excess returns above the risk free interest rate, in apparent contradiction with the prescription of the
CAPM. However, for daily returns, the difference between returns and excess returns is negligible.
Indeed, we checked that neglecting the difference between the returns and the excess returns does
not affect our results by re-running all the study described below in terms of the excess returns and
found that the tail dependence did not change by more than 0.1%.
3.1 Description of the data
We study a set of twenty assets traded on the New York Stock Exchange. The criteria presiding
over the selection of the assets (see column 1 of table 1) are that (1) they are among the stocks
with the largest capitalizations, but (2) each of them should have a weight smaller than 1% in the
Standard and Poor’s 500 index, so that the dependence studied here does not stem trivially from
their overlap with the market factor (taken as the Standard and Poor’s 500 index).
The time interval we have considered ranges from July 03, 1962 to December 29, 2000, corresponding
to 9694 data points, and represents the largest set of daily data available from the Center for
Research in Security Prices (CRSP). This large time interval is important to let us collect as many
large fluctuations of the returns as is possible in order to sample the extreme tail dependence.
Moreover, in order to allow for a non-stationarity over the four decades of the study, to check the
stability of our results and to test the stationnarity of the tail dependence over the time, we split
this set into two subsets. The first one ranges from July 1962 to December 1979, a period with
few very large return amplitudes, while the second one ranges from January 1980 to December
2000, a period which witnessed several very large price changes (see table 1 which shows the good
stability of the standard deviation between the two sub-periods while the higher cumulants such
as the excess kurtosis often increased dramatically in the second sub-period for most assets). The
table 1 presents the main statistical properties of our set of stocks during the three time intervals.
All assets exhibit an excess kurtosis significantly different from zero over the three time interval,
which is inconsistent with the assumption of Gaussianly distributed returns. While the standard
deviations remain stable over time, the excess kurtosis increases significantly from the first to the
second period. This is in resonance with the financial community’s belief that stock price volatility
has increased over time, a still controversial result (Jones and J.W.Wilson (1989), Campbell, Lettau,
Malkiel, and Xu (2001) or Xu and Malkiel (2002)).
11

9.2. Estimation du coefficient de dépendance de queue 311
3.2 Calibration of the factor model
The determination of the parameters β and of the residues ε entering in the definition of the factor
model (6) is performed for each asset by regressing the stocks returns on the market return. The
coefficient β is thus given by the ordinary least square estimator, which is consistent as long as
the residues are weak white noise and with zero mean and finite variance. The idiosyncratic noise
ε is obtained by substracting β times the market return to the stock return. Table 2 presents
the results for the three periods we consider. For each period, we give the value of the estimated
coefficient β (first columns of table 2 for each time interval). We then calculate the correlation
coefficient between the market returns and the estimated idiosyncratic noise. All of them are less
than 10 −8 , so that none of them is significantly different from zero, which allows us to conclude
that there is no linear correlation between the factor and the residues. To check one step further
the independence hypothesis, we have estimated the correlation coefficient between the square of
the factor and the square of the error-terms. In table 2, their values are given in the second of the
pair of columns presented for each period. A Fisher’s test shows that, at the 95% confidence level,
all these correlation coefficients are significantly different from zero. This result is not surprising
and shows the existence of small but significant correlations between the market volatility and
the idiosyncratic volatility. However, this will not invalidate the empirical tests of our theoretical
results, since they hold even in presence of weakly dependent factor and noise.
The coefficient β’s we obtain by regressing each asset returns on the Standard & Poor’s 500 returns
are very close to within their uncertainties to the β’s given by the CRSP database, which are
estimated by regressing the assets returns on the value-weighted market portfolio. Thus, the choice
of the Standard and Poor’s 500 index to represent the whole market portfolio is reasonable.
3.3 Estimation of the tail indexes
Assuming that the distributions of stocks and market returns are asymptotically power laws (Longin
(1996), Lux (1996), Pagan (1996) or Gopikrishnan, Meyer, Amaral, and Stanley (1998)), we now
estimate the tail index of the distribution of each stock and their corresponding residue by the factor
model, both for the positive and negative tails. Each tail index α is given by Hill’s estimator:
ˆα =
⎡
⎣ 1
k
⎤
k
log xj,N − log xk,N⎦
j=1
−1
, (21)
where x1,N ≥ x2,N ≥ · · · ≥ xN,N denotes the ordered statistics of the sample containing N independent
and identically distributed realizations of the variable X.
Hill’s estimator is asymptotically normally distributed with mean α and variance α 2 /k. But,
for finite k, it is known that the estimator is biased. As the range k increases, the variance of the
estimator decreases while its bias increases. The competition between these two effects implies that
there is an optimal choice for k = k ∗ which minimizes the mean squared error of the estimator.
To select this value k ∗ , one can apply the Danielsson and de Vries (1997)’s algorithm which is an
improvement over the Hall (1990)’s subsample bootstrap procedure. One can also prefer the more
recent Danielsson, de Haan, Peng, and de Vries (2001)’s algorithm for the sake of parsimony. We
have tested all three algorithms to determine the optimal k ∗ . It turns out that the Danielsson,
de Haan, Peng, and de Vries (2001)’s algorithm developed for high frequency data is not well
adapted to samples containing less than 100,000 data points, as is the case here. Thus, we have
focused on the two other algorithms. An accurate determination of k ∗ is rather difficult with any of
12

312 9. Mesure de la dépendance extrême entre deux actifs financiers
them, but in every case, we found that the relevant range for the tail index estimation was between
the 1% and 5% quantiles. Tables 3 and 4 give the estimated tail index for each asset and residues
at the 1%, 2.5% and 5% quantile, for both the positive and the negative tails for the two time
sub-intervals. The second time interval from January 1980 to December 2000 is characterized by
values of the tail indexes that are homogeneous over the various quantiles and range between 3 and
4 for the negative tails and between 3 and 5 for the positive tails. There is slightly more dispersions
in the first time interval from July 1962 to December 1979.
For each asset and their residue of the regression on the market factor, we tested whether the
hypothesis, according to which the tail index measured for each asset and each residue is the same
as the tail index of the Standard & Poor’s 500 index, can be rejected at the 95% confidence level,
for a given quantile. Before proceding with the presentation of our tests, two caveats have to
be accounted for. First, due to the phenomenon of volatility clustering in financial time series,
extremes are more likely to occur together. In this situation, Hill’s estimator is no more normally
distributed with variance α 2 /k. In fact, for weakly dependent time series, it can only be asserted
that the estimator remains consistent (see Rootzén and de Haan (1998)). Moreover, as shown by
Kearns and Pagan (1997) for heteroskedastic time series, the variance of the estimated tail index
can be seven times larger than the variance given by the asymptic normality assumption. Second,
the idiosyncratic noise is estimated by substracting β times the factor from the asset return. Thus,
even when the factor and the error-term are independent, the empirically estimated residues depend
on the realizations of the factor. As a consequence, the tail index estimators for the factor and for
the idiosyncratic noise are correlated. This correlation obviously depends on the exact form of the
distributions of the factor and the indiosyncratic noise. Even without the knowledge of the true
test statistics, for both problematic points, we can assert that the fluctuations of the estimators
are larger than those given by the asymptotically normal statisics for i.i.d realizations. Thus,
performing the test under the asymptotic normality assumption is more constraining than under
the true (but unknown) test statistics, so that the non-rejection of the equality hypothesis under
the assumption of a normally distributed estimator ensures that we would not be able to reject this
hypothesis under the real statistics of the estimator.
The values which reject the equality hypothesis are indicated by a star in the tables 3 and 4. During
the second time interval from January 1980 to December 2000, only four residues have a tail index
significantly different from that of that Standard & Poor’s 500, and only in the negative tail. The
situation is not as good during the first time interval, especially for the negative tail, for which
no less than 13 assets and 10 residues out of 20 have a tail index significantly different from the
Standard & Poor’s 500 ones, for the 5% quantile. Recall that the the equality tests have been
performed under the assumption of a normally distributed estimator with variance α 2 /k which, as
explained above, is too strong an hypothesis. As a consequence, a rejection under the normality
hypothesis does not imply necessarily that the equality hypothesis would have been rejected under
the true statistics. While providing a note of caution, this statement is nevertheless not very useful
from a practical point of view. More importantly, we stress that the equality of the tail indices of
the distribution of the factor and of the idiosyncratic noise is not crucial. Indeed, we shall propose
below two different estimators for the coefficient of tail dependence. One of them does not rely on
the equality of these two tail indices and thus remains operational even when they are different
and in particular when the tail index of the idiosyncratic noise appears larger than the tail of the
factor.
To summarize, our tests confirm that the tail indexes of most stock return distributions range
between three and four, even though no better precision can be given with good significance.
Moreover, in most cases, we can assume that both the asset, the factor and the residue have the
13

9.2. Estimation du coefficient de dépendance de queue 313
same tail index. We can also add that, as asserted by Loretan and Phillips (1994) or Longin (1996),
we cannot reject the hypothesis that the tail index remains the same over time. Nevertheless, it
seems that during the first period from July 1962 to December 1979, the tail indexes were sightly
larger than during the second period from January 1980 to december 2000.
3.4 Determination of the coefficient of tail dependence
Using the just established empirical fact that we cannot reject the hypothesis that the assets, the
market and the residues have the same tail index, we can use the theorem of Appendix A and
its second corollary stated in section 2. This allows us to conclude that one cannot reject the
hypothesis of a non-vanishing tail dependence between the assets and the market.
In addition, the coefficient of tail dependence is given by equations (12) and (13). These equations
provide two ways for estimating the coefficient of tail dependence: non-parametric with (12) and
parametric with (13). The first one is more general since it only requires the hypothesis of a regular
variation, while the second one explicitly assumes that the factor and the residues have distributions
with power law tails.
To estimate the tail dependence according equation (12), we need only to determine the constant l
defined (8). Consider N sorted realizations of X and Y denoted by x1,N ≥ x2,N ≥ · · · ≥ xN,N and
y1,N ≥ y2,N ≥ · · · ≥ yN,N, the quantile of F −1
−1
X (u) and of FY (u) are estimated by
−1 −1
FX
ˆ (u) = x[(1−u)·N],N and FY
ˆ (u) = y[(1−u)·N],N, (22)
where [·] denotes the integer part. Thus, the constant l is non-parametrically estimated by
ˆ lk = xk,N
yk,N
as k → 0 or N. (23)
As u goes to zero or one (or k goes to zero or N), the number of observations decreases dramatically.
However, we observe a large interval of small or large k’s such that the ratio of the empirical
quantiles remains remarkably stable and thus allows for an accurate estimation of l. A more
precise estimation could be performed with a kernel-based quantile estimator (see Shealter and
Marron (1990) or Pagan and Ullah (1999) for instance). A non-parametric estimator for λ is then
obtained by replacing l by its estimated value in equation (12)
ˆλNP =
1
max 1, ˆ 1
α =
l
ˆβ max 1, xk,N
α . (24)
ˆβ·yk,N
It can also be advantageous to follow a parametric approach, which generally allows for a more
accurate estimation of (the ratio of) the quantiles, provided that the assumed parametric form of
the distributions is not too far from the true one. For this purpose, we will use formula (13) which
requires the estimation of the scale factors for the different assets. To get the scale factors, we
proceed as follows. Consider a variable X which asymptotically follows a power law distribution
Pr{X > x} ∼ C · x −α . Given a rank ordered sample x1,N ≥ x2,N ≥ · · · ≥ xN,N, the scale factor C
can be consistently estimated from the k largest realizations by
Ĉ = k
N · (xk,N) α . (25)
The estimated value of the scale factor must not depends on the rank k for k large enough, in order
for the parameterization of the distribution in terms of a power law to hold true. Thus, denoting
14

314 9. Mesure de la dépendance extrême entre deux actifs financiers
by ĈY and Ĉε the estimated scale factors of the factor Y and of the noise ε defined in equation (6),
the estimator of the coefficient of tail dependence is
ˆλ =
1
1 + ˆ β −α · ĈY
Ĉε
=
1 +
1
εk,N
ˆβ·yk,N
α , (26)
where ˆ β denotes the estimated coefficient β. Since the estimators ĈY , Ĉε and ˆ β are consistent and
using the continuous mapping theorem, we can assert that the estimator ˆ λ is also consistent.
Since the tail indices α are impossible to determine with sufficient accuracy other than saying that
the α probably fall in the interval 3 − 4 as we have seen above, our strategy is to determine the
coefficient of tail dependence using (24) and (26) for three different common values α = 3, 3.5 and
4. This procedure allows us to test for the sensitivity of the scale factor and therefore of the tail
coefficient with respect to the uncertain value of the tail index.
Tables 5 and 6 give the values of the coefficients of lower tail dependence over the whole time
interval from July 1962 to December 2000, under the assumption that the tail index α equals 3,
for the non-parametric estimator (table 5) and the parametric one (table 6). For each table, the
coefficient of tail dependence is estimated over the first centile, the first quintile and the first decile
to also test for any possible sensitivity on the tail asymptotics. For each of these quantiles, the
mean values, their standard deviations and their minimum and maximum values are given. We
first remark that the standard deviation of the tail dependence coefficient remains small compared
with its average value and that the minimum and maximum values cluster closely around its mean
value. This shows that the coefficient of tail dependence is well-estimated by its mean over a given
quantile. Secondly, we find that these estimated coefficients of tail dependence exhibit a good
stability over the various quantiles. These two observations enable us to conclude that the average
coefficient of tail dependence over the first centile is sufficient to provide a good estimate of the
true coefficient of tail dependence.
Note that the two estimators yield essentially equivalent results, even if the coefficients of tail
dependence given by the non-parametric estimator exhibit a systematic tendency to be slightly
smaller than the estimates provided by the parametric estimator. Since the results given by these
two estimators are very close to each other, we choose to present below only those given by the
parametric one. This choice has also been guided by the lower sensibility of this last estimator
to small changes of the tail exponent α. Indeed, since the evaluation of the scale factors ĈY and
Ĉε by the formula (25) involves the tail exponent α, the small deviations from its true value are
compensated by the estimated scale factors. This explains the observation that the parametric
estimator appears more robust than the non-parametric one with respect to small changes in α.
Tables 7, 8 and 9 summarize the different values of the coefficient of tail dependence for both the
positive and the negative tails, under the assumptions that the tail index α equals 3, 3.5 and 4
respectively, over the three considered time intervals. Overall, we find that the coefficients of tail
dependence are almost equal for both the negative and the positive tail and that they are not very
sensitive to the value of the tail index in the interval considered. More precisely, during the first
time interval from July 1962 to December 1979 (table 7), the tail dependence is symmetric in both
the upper and the lower tail. During the second time interval from January 1980 to December 2000
and over the whole time interval (tables 8 and 9), the coefficient of lower tail dependence is slightly
but systematically larger than the upper one. Moreover, since these coefficients of tail dependence
are all less than 1/2, they decrease when the tail index α increases and the smaller the coefficient
of tail dependence, the larger the decay.
During the first time interval, most of the coefficients of tail dependence range between 0.15 and
15

9.2. Estimation du coefficient de dépendance de queue 315
0.35 in both tails, while during the second time interval, almost all range between 0.10 and 0.25 in
the lower tail and between 0.10 and 0.20 in the upper one. Thus, the tail dependence is smaller
during the last period compared to the first one. This result is interesting because it is in agreement
and confirms the recent studies by Campbell, Lettau, Malkiel, and Xu (2001) and Xu and Malkiel
(2002), showing that the idiosyncratic volatility of each stocks have increased relative to the market
volatility. And, as already discussed, the coefficient of tail dependence given by equation (10) must
decrease when the idiosynchratic volatility of the stocks increases relative to the market volatility.
The strong similarity of the tail dependencies in the upper and lower tails is an interesting empirical
finding which suggests that extreme co-movements reflect behaviors of agents which are more
sensitive to large amplitudes rather than to a specific direction (loss or gain). Pictorially, the
specific mechanism triggering co-movements of extreme amplitudes may well be different for losses
compared to gains, such as fear for the former and greed for the latter, but the resulting large
co-movements have similar frequencies of occurrence.
The observed lack of stationarity of the coefficient of tail dependence in the two time sub-intervals
suggests that it could be necessary to have a model where the tail dependence index is not constant
but varies as a function of past shocks (just as the volatility varies with time in a GARCH model)
in order to investigate whether large recent common shocks lead to higher future tail dependence.
This point is beyond the scope of the present study, but we will provide in our concluding remarks
some ways for explicitely accounting for this lack of stationarity.
3.5 Comparison with the historical extremes
Our determination of the coefficients of tail dependence provides predictions on the probability
that future large moves of stocks may be simultaneous to large moves of the market. This begs for
a check over the available historical period to determine whether our estimated coefficients of tail
dependence are compatible with the realized historical extremes.
For this, we consider the ten largest losses of the Standard & Poor’s 500 index during the two time
sub-intervals5 . Since λ− is by definition equal to the probability that a given asset incurs a large
loss (say, one of its ten largest losses) conditional on the occurrence of one of the ten largest losses
of the Standard & Poor’s 500 index, the probability, for this asset, to undergo n of its ten largest
losses simultaneously with any of the ten largest losses of the Standard & Poor’s 500 index is given
by the binomial law with parameter λ−:
Pλ− (n) =
10
λ−
n
n (1 − λ−) (10−n) . (27)
We stress that our consideration of only the ten largest drops ensures that the present test is not
embodied in the determination of the tail dependence coefficient, which has been determined on
a robust procedure over the 1%, 5% and 10% quantiles. We checked that removing these then
largest drops does not modify the determination of λ−. Our present test can thus be considered as
“out-of-sample,” in this sense.
Table 10 presents, for the two time sub-intervals, the number of extreme losses among the ten
largest losses incured by a given asset which occured simultaneously with one of the ten largest
losses of the standard & Poor’s 500 index. For each asset, we give the probability of occurence of
such a realisation, according to (27). We notice that during the first time interval, only two assets
5 We do not consider the whole time interval since the ten largest losses over the whole period coincide with the
ten largest ones over the second time subinterval, which would bias the statistics towards the second time interval.
16

316 9. Mesure de la dépendance extrême entre deux actifs financiers
are incompatible, at the 95% confidence level, with the value of λ− previously determined: Du Pont
(E.I.) de Nemours & Co. and Texas Instruments Inc. In contrast, during the second time interval,
four assets reject the value of λ−: Coca Cola Corp., Pepsico Inc., Pharmicia Corp. and Texaco Inc.
These results are very encouraging. However, there is a noticeable systematic bias. Indeed, during
the first time interval, 17 out of the 20 assets have a realized number of large losses lower than
their expected number (according to the estimated λ−), while during the second time interval, 19
out of the 20 assets have a realized number of large losses larger than their expected one. Thus, it
seems that during the first time interval the number of large losses is overestimated by λ− while it
is underestimated during the second time interval.
We propose to explain the underestimation of the number of large losses between January 1980
and December 2000 by a possible comonitonicity that occurred during the October 1987 crash.
Indeed, on October 19, 1987, 12 out of the 20 considered assets incurred their most severe loss,
which strongly suggests a comonotonic effect. Table 11 shows the same results as in table 10 but
corrected by substracting this comonotonic effect to the number of large losses. The compatibility
between the number of large losses and the estimated λ− becomes significantly better since only
Pepsico Inc. and Pharmicia Corp. are still rejected, and only 16 assets out of 20 are underestimated,
representing a slight decrease of the bias.
Previous works have shown that, in period of crashes, the market conditions change, herding effects
may become more important and almost dominant, so that the market enters an unusual regime,
which can be characterized by outliers present in the distribution of drawdowns Johansen and
Sornette (2002). Our detection of an anomalous comonotonicity can thus be considered as an
independent confirmation of the existence of this abnormal regime.
Another explaination for this slight discrepancy may be ascribed to a limitation of the CAPM.
Indeed, the CAPM is known to explain the relation between the expected return on an asset and
its amount of systematic risk. But, it is questionable whether extreme systematic risks as those
measured by the coefficient of tail dependence are really accounted for by the economic agents and
then effectively priced.
Concerning the overestimation of the number of large losses during the first time interval, it can
obviously not be ascribed to the comonotonicity of very large events, which in fact only occurred
once for the Coca-Cola Corp. This overestimation is probably linked with the low “volatility” of
the market during this period, which can have two effects. The first one is to lead to a less accurate
estimation of the scale factor of the power-law distribution of the assets. The second one is that
a market with smaller volatility produces fewer large losses. As a consequence, the asymptotic
regime for which the relation Pr{X < FX −1 (u)|Y < FY −1 (u)} λ− holds may not be reached
in the sample, and the number of recorded large losses remain lower than that asymptotically
expected.
4 Concluding remarks
We have used the framework offered by factor models in order to derive a general theoretical
expression for the coefficient of tail dependence between an asset and any of its explanatory factor
or between any two assets. The coefficient of tail dependence represents the probability that a given
asset incurs a large loss (say), assuming that the market (or another asset) has also undergone a
large loss. We find that factors characterized by rapidly varying distributions, such as Normal
or exponential distributions, always lead to a vanishing coefficient of tail dependence with other
17

9.2. Estimation du coefficient de dépendance de queue 317
stocks. In contrast, factors with regularly varying distributions, such as power-law distributions,
can exhibit tail dependence with other stocks, provided that the idiosyncratic noise distributions
of the corresponding stocks are not fatter-tailed than the factor.
Applying this general result to individual daily stock returns, we have been able to estimate the
coefficient of tail dependence between the returns of each stock and those of the market. This
determination of the tail dependence relies only on the simple estimation of the parameters of the
underlying factor model and on the tail parameters of the distribution of the factor and of the
idiosyncratic noise of each stock. As a consequence, the two strong advantages of our approach are
the following.
- The coefficients of tail dependence are estimated non-parametrically. Indeed, we never specify
any explicit expression of the dependence structure, contrary to most previous works (see
Longin and Solnik (2001), Malevergne and Sornette (2001) or Patton (2001) for instance);
- Our theoretical result enables us to estimate an extreme parameter, not accessible by a direct
statistical inference. This is achieved by the measurement of parameters whose estimation
involves a significant part of the data with sufficient statistics.
Having performed this estimation, we have checked the comptatibility of these estimated coefficients
of tail dependence with the historically realized extreme losses observed in the empirical time
series. A good agreement is found, notwithstanding a slight bias which leads to an overestimate
of the occurence of large events during the period from July 1962 to December 1979 and to an
underestimate during the time interval from January 1980 to December 2000.
This bias can be explained by the low volatility of the market during the first period and by a
comonotonicity effect, due to the October 1987 crach, during the second period. Indeed, from july
1962 to December 1979, the volatility was so low that the distributions of returns have probably not
sampled their tails sufficiently for the probability of large conditional losses to be represented by
its asymptotic expression given by the coefficient of tail dependence. The situation is very different
for the period from january 1980 to December 2000. On October 19, 1987, many assets incurred
their largest loss ever. This is presumably the manifestation of an ‘abnormal’ regime probably
due to herding effects and irrational behaviors and has been previously characterized as yielding
signatures in the form of outliers in the distribution of drawdowns.
Finally, the observed lack of stationarity exhibited by the coefficient of tail dependence across the
two time sub-intervals suggests the importance of going beyond a stationary view of tail dependence
and of studying its dynamics. This question, which could be of great interest in the context of the
contagion problem, could be easily treated with the new conditional quantile dynamics proposed
by Engle and Manganelli (1999). Moreover, it should be interesting to account for the change of
the β’s with incoming bad or good news, as shown by Cho and Engle (2000), for instance. These
points are left for a future work.
From a practical point of view, we stress that the coefficient λ studied here can be seen as a
generalization or a tool complementary to the CAPM’s β. These two coefficients have in common
that they probe the dependence between a given stock and the market. However, the coefficient β
quantifies only the correlations between moderate movements of both an asset and the market. In
contrast, the coefficient λ offers a measure of extreme co-movements, which is particularly useful
in period of high market volatility. In such periods, a prudent fund manager should overweight its
portofolio with assets whose λ is very small such as Texaco or Walgreen, for instance.
Moreover, the observed decrease of the tail dependence during the last year concomitant with the
18

318 9. Mesure de la dépendance extrême entre deux actifs financiers
increase of the idiosyncratic volatility suggests that the main source of risk in such a period does
not consist in the dependence between assets but rather in their instrinsic fluctuations measured
by the idiosyncratic volality.
Our study has focused on the dependence between different risks. In fact, our theorem can obviously
be applied to extreme temporal dependences, when the variable follows an autoregressive process.
This should provide an estimate of the probability that a large loss (respectively gain) is followed
by another large loss (resp. gain) in the following period. Such information is very interesting in
investment and hedging strategies.
19

9.2. Estimation du coefficient de dépendance de queue 319
A Proof of the theorem
A.1 Tail dependence between an asset and the factor
A.1.1 Statement
We consider two random variables X and Y , related by the relation
X = β · Y + ε, (28)
where ε is a random variable independent of Y and β a non-random positive coefficient.
Let PY and FY denote respectively the density with respect to the Lebesgue measure and the
distribution function of the variable Y . Let FX denotes the distribution function of X and Fε the
distribution function of ε. We state the following theorem:
Theorem 1
Assuming that
H0: The variables Y and ε have distribution functions with infinite support,
H1: For all x ∈ [1, ∞),
lim
t→∞
t PY (tx)
¯
FY (t)
= f(x), (29)
H2: There are real numbers t0 > 0, δ > 0 and A > 0, such that for all t ≥ t0 and all x ≥ 1
H3: There is a constant l ∈ R+, such that
¯FY (tx)
¯FY (t)
then, the coefficient of (upper) tail dependence of (X, Y ) is given by
A.1.2 Proof
λ =
A
≤ , (30)
xδ FX
lim
u→1
−1 (u)
FY −1 = l, (31)
(u)
∞
max{1, l
β }
dx f(x). (32)
We first give a general expression for the probability for X to be larger than F −1
X (u) knowing that
(u) :
Y is larger than F −1
Y
Lemma 1
The probability that X is larger than F −1
−1
X (u) knowing that Y is larger than FY (u) is given by :
Pr X > F −1
−1
X (u)|Y > FY (u) =
F −1
Y (u)
1 − u
∞
dx PY
1
20
−1
FY (u) x · ¯
Fε FX −1 (u) − βF −1
Y (u) x .
(33)

320 9. Mesure de la dépendance extrême entre deux actifs financiers
Proof :
Pr{X > FX −1 (u), Y > FY −1 (u)} =
E 1 {X>FX −1 (u)} · 1 {Y >FY −1
(u)}
= E E 1 {X>FX −1 (u)} · 1 {Y >FY −1 (u)} |Y
= E 1 {Y >FY −1
(u)} · E 1 {X>FX −1 (u)} |Y
= E 1 {Y >FY −1
(u)} · E 1 {ε>FX −1
=
(u)−βY }
E 1 {Y >FY −1 (u)} · ¯ Fε(FX −1
(u) − βY )
Assuming that the variable Y admits a density PY with respect to the Lebesgue measure, this
yields
Pr{X > FX −1 (u), Y > FY −1 ∞
(u)} =
F −1
Y (u)
(34)
(35)
(36)
(37)
(38)
dy PY (y) · ¯ Fε[FX −1 (u) − βy] . (39)
Performing the change of variable y = FY −1 (u) · x, in the equation above, we obtain
Pr{X > FX −1 (u), Y > FY −1 (u)} = F −1
Y (u)
∞
1
dx PY (F −1
and, dividing by ¯
FY FY −1 (u) = 1 − u, this concludes the proof.
Let us now define the function
fu(x) =
We can state the following result
−1
FY (u)
1 − u PY (F −1
Y (u) x) · ¯ Fε[FX −1 (u) − βF −1
Y
Lemma 2
Under assumption H1 and H3, for all x ∈ [1, ∞),
almost everywhere, as u goes to 1.
fu(x) −→ 1x> l
Proof: Let us apply the assumption H1. We have
lim
u→1
F −1
Y (u)
Applying now the assumption H3, we have
lim
u→1 FX −1 (u) − βF −1
Y
Y (u) x) · ¯ Fε[FX −1 (u) − βF −1
Y
(u) x] ,
(40)
(u) x] . (41)
β · f(x), (42)
1 − u PY (F −1
Y (u) x) =
t PY (t x)
lim
t→∞ FY
¯
,
(t)
(43)
= f(x). (44)
(u) x = lim
=
21
u→1
βF −1
Y (u)
−∞ if x > l
β ,
∞ if x < l
β ,
FX −1 (u)
βF −1 − x
Y (u)
(45)
(46)
(47)

9.2. Estimation du coefficient de dépendance de queue 321
which gives
and finally
lim
u→1
lim
u→1 fu(x) = lim
u→1
= 1x> l
which concludes the proof.
¯Fε[FX −1 (u) − βF −1
β Y (u) x] = 1x>
l , (48)
F −1
Y (u)
1 − u PY (F −1
Y (u) x) · lim ¯Fε[FX
u→1
−1 (u) − βF −1
Y (u) x], (49)
β · f(x), (50)
Let us now prove that there exists an integrable function g(x) such that, for all t ≥ t0 and all x ≥ 1,
we have ft(x) ≤ g(x). Indeed, let us write
t PY (tx)
¯FY (t) = t PY (tx)
¯FY (tx) · ¯ FY (tx)
¯FY (t)
For the leftmost factor in the right-hand-side of equation (51), we easily obtain
∀t, ∀x ≥ 1,
t PY (tx)
¯FY (tx) ≤ x∗ PY (x ∗ )
¯FY (x ∗ )
. (51)
· 1
, (52)
x
where x∗ denotes the point where the function x PY (x)
¯FY
reaches its maximum. The rightmost factor
(x)
in the right-hand-side of (51) is smaller than A/xδ by assumption H2, so that
Posing
∀t ≥ t0, ∀x ≥ 1,
t PY (tx)
¯FY (t) ≤ x∗ PY (x ∗ )
¯FY (x ∗ )
g(x) = x∗ PY (x ∗ )
¯FY (x ∗ )
· A
. (53)
x1+δ · A
, (54)
x1+δ and recalling that, for all ε ∈ R, ¯ Fε(ε) ≤ 1, we have found an integrable function such that for
some u0 ≥ 0, we have
∀u ∈ [u0, 1), ∀x ≥ 1, fu(x) ≤ g(x) . (55)
Thus, applying Lebesgue’s theorem of dominated convergence, we can assert that
∞
∞
lim dx fu(x) = dx 1x> u→1 1
1
l β · f(x). (56)
Since
lim
u→1
∞
the proof of theorem 1 is concluded.
1
dx fu(x) = lim Pr
u→1 X > F −1
−1
X (u)|Y > FY (u) , (57)
= λ, (58)
Remark: This result still holds in the presence of dependence between the factor and the idiosyncratic
noise. Indeed, denoting by ¯ Fε|Y the survival distribution of ε conditional on Y , lemma 1 can
easily be generalized:
Pr X > F −1
−1
X (u)|Y > FY (u) =
F −1
Y (u)
1 − u
∞
dx PY
1
22
F −1
Y (u) x · ¯ F −1
ε|Y =FY (u) x
FX −1 (u) − βF −1
Y (u) x ,
(59)

322 9. Mesure de la dépendance extrême entre deux actifs financiers
where the only change in (59) compared to (33) is to replace ¯ Fε(·) by ¯ F −1
ε|Y =FY (u) x(·). Let us
now assume that the function ¯ Fε|Y =y(x) admits a uniform limit when x and y tend to ±∞. Then,
equation (48) still holds and lemma 2 remains true.
As an example, let F denote any one-dimensional distribution fonction. Then, one can easily check
that, for any conditional distribution whose form is
¯F ε|Y =y(x) = ¯
y2 F x , (60)
+ y2
the uniform limit condition is satisfied and theorem 1 and lemma 2 still hold. In contrast, conditional
distributions of the form
¯F ε|Y =y(x) = ¯ F (x − ρy) (61)
do not fulfill the uniform limit condition, so that the result given by theorem 1 does not hold.
The full understanding of the impact of more general dependences between the factor and the
idiosyncratic noise on the coefficient of tail dependence requires a full-fledge investigation that we
defer to a future work. Our goal here has been to show that one can reasonably expect our results
to survice in the presence of weak dependence.
A.2 Tail dependence between two assets
A.2.1 Statement
We consider three random variables X1, X2 and Y , related by the relations
y 2 0
X1 = β1 · Y + ε1 (62)
X2 = β2 · Y + ε2, (63)
where ε1 and ε2 are two random variables independent of Y and β1, β2 two non-random positive
coefficients.
Let PY and FY denote respectively the density with respect to the Lebesgue measure and the
distribution function of the variable Y . Let F1, (resp. F2) denotes the distribution function of
X1 (resp. X2) and Fε1 (resp. Fε2 ) the marginal distribution function of ε1 (resp. ε2). Let Fε1,ε2
denotes the joined distribution of (ε1, ε2). We state the following theorem:
Theorem 2
Assuming that
H0: The variables Y , ε1 and ε2 have distribution functions with infinite support,
H1: For all x ∈ [1, ∞),
lim
t→∞
t PY (tx)
¯
FY (t)
= f(x), (64)
H2: There are real numbers t0 > 0, δ > 0 and A > 0, such that for all t ≥ t0 and all x ≥ 1
¯FY (tx)
¯FY (t)
23
A
≤ , (65)
xδ

9.2. Estimation du coefficient de dépendance de queue 323
H3: There is two constant (l1, l2) ∈ R+ × R+, such that
F1
lim
u→1
−1 (u)
FY −1 (u) = l1,
F2
and lim
u→1
−1 (u)
FY −1 (u) = l2, (66)
then, the coefficient of (upper) tail dependence of (X, Y ) is given by
A.2.2 Proof
λ =
∞
maxl 1
β1 , l 2
β2 dx f(x). (67)
We first give a general expression for the probability for X to be larger than F −1
X (u) knowing that
(u) :
Y is larger than F −1
Y
Lemma 3
The probability that X is larger than F −1
X
F −1
Y (u)
1 − u
dx PY
−1
(u) knowing that Y is larger than F (u) is given by :
Pr X > F −1
−1
X (u)|Y > FY (u) =
−1
FY (u) x · ¯
Fε1,ε2 F1 −1 (u) − β1F −1
Y (u) x, F2 −1 (u) − β1F −1
Y (u) x . (68)
Proof : The proof is the same for lemma 1.
Let us now define the function
F −1
Y (u)
fu(x) =
1 − u PY (F −1
We can state the following result
Lemma 4
Under assumption H1 and H3, for all x ∈ [1, ∞),
almost everywhere, as u goes to 1.
Y (u) x) · ¯ Fε1,ε2 [F1 −1 (u) − β1F −1
Y (u) x, F2 −1 (u) − β2F −1
Y
fu(x) −→ 1x>maxl 1
β1 , l 2
Proof: Applying the assumption H3, we have
and
lim
u→1 F1 −1 (u) − β1F −1
Y
lim
u→1 F2 −1 (u) − β2F −1
Y
(u) x = lim
=
u→1
(u) x = lim
=
Y
(u) x] . (69)
β2 · f(x), (70)
β1F −1
Y (u)
−∞ if x > l1
β1 ,
∞ if x < l1
β1 ,
u→1
β2F −1
Y (u)
−∞ if x > l2
β2 ,
∞ if x < l2
β2 ,
24
F1 −1 (u)
β1F −1 − x
Y (u)
F2 −1 (u)
β2F −1 − x
Y (u)
(71)
(72)
(73)
(74)
(75)
(76)

324 9. Mesure de la dépendance extrême entre deux actifs financiers
which give
lim ¯Fε1,ε2
u→1
[F1 −1 (u) − β1F −1
Y (u) x, F2 −1 (u) − β2F −1
Y (u) x] = 1x>maxl 1 , β1 l2 and following the same calculations as in part A.1, it concludes the proof.
β2 , (77)
We can now apply Lebesgue’s theorem of dominated convergence (see part A.1 for the justification),
which allows us to assert that
lim dx fu(x) =
u→1
Since
lim
u→1
the proof of theorem 2 is concluded.
dx 1x>maxl 1 , β1 l β2 2 · f(x). (78)
dx fu(x) = lim
u→1 Pr X1 > F −1
1 (u)|X2 > F −1
2 (u) , (79)
= λ, (80)
25

9.2. Estimation du coefficient de dépendance de queue 325
B Proofs of the corollaries
B.1 First corollary
Corollary 1
If the random variable Y has a rapidly varying distribution function, then λ = 0.
Proof : Let us write
For a rapidly varying function ¯ FY , we have
t PY (tx)
FY
¯ (t) = t PY (tx)
¯
FY (tx)
∀x > 1, lim
t→∞
FY (tx)
FY (t)
¯
·
¯
¯FY (tx)
¯FY (t)
. (81)
= 0, (82)
while the leftmost factor of the right-hand-side of equation (81) remains bounded as t goes to
infinity, so that
t PY (tx) FY
¯ (tx)
lim
t→∞ FY
¯
·
(tx) FY
¯
= f(x) = 0 . (83)
(t)
Since f(x) = 0, we can apply lemma 2 without the hypothesis H3, which concludes the proof.
B.2 Second corollary
Corollary 2
Let Y be regularly varying with index (−α), and assume that hypothesis H3 is satisfied. Then, the
coefficient of (upper) tail dependence is
λ =
1
max
1, l
β
where l denotes the limit, when u → 1, of the ratio FX −1 (u)/FY −1 (u).
α , (84)
Proof : Karamata’s theorem (see Embrechts, Kluppelberg, and Mikosh (1997, p 567)) ensures that
H1 is satisfied with f(x) = α
xα+1 , which is sufficient to prove the corollary. To go one step further,
let us define
where L1(·) and L2(·) are slowly varying functions.
¯Fy(y) = y −α · L1(y), (85)
¯Fε(ε) = ε −α · L2(ε), (86)
Using the proposition stated in Feller (1971, p 278), we obtain, for the distribution of the variable
X
¯FX(x) ∼ x −α
β α for large x.
x
· L1 + L2(x) ,
β
(87)
Assuming now, for simplicity, that L1 (resp. L2) goes to a constant C1 (resp. C2), this implies that
H3 is satistified, since
FX
l = lim
u→1
−1 (u)
FY −1
= β 1 +
(u) C2
βα 1
α
C1
This allows us to obtain the equations (10) and (13).
26
. (88)

326 9. Mesure de la dépendance extrême entre deux actifs financiers
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29

9.2. Estimation du coefficient de dépendance de queue 329
July 1962 - December 1979 January 1980 - December 2000 July 1962 - December 2000
Mean Std. Skew. Kurt. Mean Std. Skew. Kurt. Mean Std. Skew. Kurt.
Abbott Labs 0.6677 0.0154 0.2235 2.192 0.9217 0.0174 -0.0434 2.248 0.8066 0.0165 0.0570 2.300
American Home Products Corp. 0.4755 0.0136 0.2985 3.632 0.8486 0.0166 0.1007 8.519 0.6803 0.0154 0.1717 7.557
Boeing Co. 0.8460 0.0228 0.6753 4.629 0.7752 0.0193 0.1311 4.785 0.8068 0.0209 0.4495 4.901
Bristol-Myers Squibb Co. 0.5342 0.0152 -0.0811 2.808 0.9353 0.0175 -0.3437 16.733 0.7546 0.0165 -0.2485 12.573
Chevron Corp. 0.4916 0.0134 0.2144 2.442 0.6693 0.0169 0.0491 4.355 0.5885 0.0154 0.1033 4.209
Du Pont (E.I.) de Nemours & Co. 0.2193 0.0126 0.3493 2.754 0.6792 0.0172 -0.1021 4.731 0.4715 0.0153 0.0231 4.937
Disney (Walt) Co. 0.9272 0.0215 0.2420 2.762 0.8759 0.0195 -0.6661 17.655 0.8997 0.0204 -0.1881 9.568
General Motors Corp. 0.3547 0.0126 0.4138 4.302 0.5338 0.0183 -0.0128 5.373 0.4538 0.0160 0.0872 6.164
Hewlett-Packard Co. 0.7823 0.0199 0.0212 3.063 0.8913 0.0238 0.0254 4.921 0.8420 0.0221 0.0256 4.624
Coca-Cola Co. 0.4829 0.0138 0.0342 5.436 0.9674 0.0170 -0.1012 14.377 0.7483 0.0157 -0.0513 12.611
Minnesota Mining & MFG Co. 0.3459 0.0139 0.3016 2.997 0.6885 0.0150 -0.7861 20.609 0.5333 0.0145 -0.3550 14.066
Philip Morris Cos Inc. 0.7930 0.0153 0.2751 2.799 0.9664 0.0180 -0.2602 10.954 0.8863 0.0169 -0.0784 8.790
Pepsico Inc. 0.4982 0.0147 0.2380 2.867 0.9443 0.0180 0.1372 4.594 0.7431 0.0166 0.1786 4.413
Procter & Gamble Co. 0.3569 0.0115 0.3911 4.343 0.7916 0.0164 -1.6610 46.916 0.5947 0.0144 -1.2408 44.363
Pharmacia Corp. 0.3801 0.0145 0.2699 3.508 0.9027 0.0191 -0.6133 13.587 0.6666 0.0172 -0.3773 12.378
Schering-Plough Corp. 0.6328 0.0163 0.2619 3.112 1.0663 0.0192 0.1781 7.9979 0.8703 0.0179 0.2139 6.757
Texaco Inc. 0.3416 0.0134 0.2656 2.596 0.6644 0.0166 0.1192 6.477 0.5197 0.0152 0.1725 5.829
Texas Instruments Inc. 0.6839 0.0198 0.2076 3.174 1.0299 0.0268 0.1595 7.848 0.8726 0.0239 0.1831 7.737
United Technologies Corp 0.5801 0.0185 0.3397 2.826 0.7752 0.0170 0.0396 3.190 0.6876 0.0177 0.1933 3.034
Walgreen Co. 0.5851 0.0165 0.3530 3.030 1.1996 0.0185 0.1412 3.316 0.9217 0.0176 0.2260 3.295
30
Standart & Poor’s 500 0.1783 0.0075 0.2554 3.131 0.5237 0.0101 -1.6974 36.657 0.3674 0.0090 -1.2236 32.406
Table 1: This table gives the main statistical features of the three samples we have considered. The columns Mean, Std., Skew. and Kurt.
respectively give the average return multiplied by one thousand, the standard deviation, the skewness and the excess kurtosis of each asset
over the time intervals form July 1962 to December 1979, January 1980 to Decemeber 2000 and July 1962 to December 2000. The excess
kurtosis is given as indicative of the relative weight of large return amplitudes, and can always be calculated over a finite time series even if
it may not be asymptotically defined for power tails with exponents less than 4.

330 9. Mesure de la dépendance extrême entre deux actifs financiers
July 1962 - December 1979 January 1979 - December 2000 July 1962 - December 2000
β ρY 2 ,ε2 β ρY 2 ,ε2 β ρY 2 ,ε2 Abbott Labs 0.8994 0.0879 0.9122 0.1879 0.9081 0.1597
American Home Products Corp. 0.9855 0.1253 0.8102 0.0587 0.8652 0.0736
Boeing Co. 1.4416 0.1196 0.9036 0.0928 1.0715 0.1279
Bristol-Myers Squibb Co. 1.0832 0.1056 1.0435 0.0457 1.0559 0.0481
Chevron Corp. 1.0062 0.1191 0.8333 0.0776 0.8873 0.0906
Du Pont (E.I.) de Nemours & Co. 1.0818 0.0960 0.9451 0.0433 0.9880 0.0595
Disney (Walt) Co. 1.5530 0.0960 1.0016 0.1304 1.1736 0.0641
General Motors Corp. 1.0945 0.1531 1.0112 0.0400 1.0371 0.0563
Hewlett-Packard Co. 1.3910 0.1023 1.3074 0.0739 1.3332 0.0832
Coca-Cola Co. 1.0347 0.2146 0.9833 0.1254 0.9995 0.1238
Minnesota Mining & MFG Co. 1.1339 0.1203 0.8756 0.2605 0.9564 0.1706
Philip Morris Cos Inc. 1.0894 0.0723 0.8598 0.0340 0.9314 0.0545
Pepsico Inc. 0.9587 0.1233 0.9004 0.3294 0.9187 0.3169
Procter & Gamble Co. 0.8293 0.1873 0.8938 0.1188 0.8738 0.1287
Pharmacia Corp. 1.0750 0.0783 0.8824 0.0373 0.9429 0.0357
Schering-Plough Corp. 1.1244 0.1284 1.0480 0.0494 1.0720 0.0540
Texaco Inc. 1.4578 0.1410 1.3811 0.0674 1.4049 0.0766
Texas Instruments Inc. 0.9414 0.1354 0.6600 0.0823 0.7481 0.1053
United Technologies Corp 1.1336 0.1243 0.9049 0.1175 0.9763 0.1098
Walgreen Co. 0.6354 0.1052 0.8554 0.1087 0.7869 0.0798
31
Table 2: This table presents the estimated coefficient β for the factor model (6) and the correlation coefficient ρY 2 ,ε2 between the square of
factor and the square of the estimated idiosyncratic noise, for the different time intervals we have considered. A Fisher’s test shows that
these correlation coefficients are all significantly different from zero.

9.2. Estimation du coefficient de dépendance de queue 331
Negative Tail Positive Tail
q = 1% q = 2.5% q = 5% q = 1% q = 2.5% q = 5%
Asset ε Asset ε Asset ε Asset ε Asset ε Asset ε
Abbott Labs 5.54 5.31 3.94 4.02 3.27 3.31 5.10 4.50 4.09 3.71 3.53∗ 3.14
American Home Products Corp. 4.58 5.11 3.89 3.81 3.02∗ 3.21∗ 3.64 4.66 3.60 3.81 3.11 3.15
Boeing Co. 6.07 4.90 4.57 3.74 3.32 3.49 4.04 4.27 3.95 4.19 3.35∗ 2.93
Bristol-Myers Squibb Co. 4.32 4.27 3.31 3.95 2.99∗ 3.16∗ 5.96∗ 5.19 3.94 4.82∗ 3.62∗ 4.03∗ Chevron Corp. 5.24 4.78 3.75 3.29 2.91 3.12∗ 5.21 5.15 3.90 4.26 3.25∗ 3.07
Du Pont (E.I.) de Nemours & Co. 5.26 4.36 3.69 3.76 3.17∗ 3.23∗ 5.35 5.15 4.00 3.37 3.13 3.04
Disney (Walt) Co. 3.59 4.23 3.59 3.84 3.08∗ 3.22∗ 4.90 4.34 4.26 3.73 3.33∗ 3.29∗ General Motors Corp. 4.82 3.50 3.72 3.66 2.94∗ 3.36 3.91 4.78 3.64 3.86 2.94 3.07
Hewlett-Packard Co. 3.76 3.89 3.12∗ 3.05∗ 2.81∗ 3.00∗ 4.64 5.08 4.08 4.20 3.41∗ 3.42∗ Coca-Cola Co. 3.45 3.45 3.05∗ 3.71 2.75∗ 3.17∗ 3.91 4.26 3.16 3.61 2.81 3.16
Minnesota Mining & MFG Co. 5.16 4.86 4.06 4.35 3.43 3.71 4.35 4.47 3.96 3.31 3.14 3.06
Philip Morris Cos Inc. 4.63 3.79 3.82 3.90 3.38 3.48 4.10 4.64 3.59 3.85 3.03 3.06
Pepsico Inc. 4.89 5.35 3.93 4.49 3.02∗ 3.27 4.07 4.67 3.49 3.86 3.15 3.21∗ Procter & Gamble Co. 4.42 3.77 3.77 3.74 3.13∗ 3.42 4.14 5.39 3.59 3.73 2.97 3.40∗ Pharmacia Corp. 4.73 4.24 4.05 3.45 2.88∗ 3.34 4.46 3.72 3.95 3.90 3.14 2.99
Schering-Plough Corp. 4.59 4.70 4.20 3.87 3.37 3.33 4.60 5.88∗ 3.50 3.91 3.07 3.22∗ Texaco Inc. 5.34 4.59 3.99 3.84 3.07∗ 3.19∗ 3.83 4.10 3.94 3.67 3.14 2.98
Texas Instruments Inc. 4.08 4.54 3.36 3.13∗ 3.22∗ 2.87∗ 4.52 4.20 3.67 3.79 3.16 3.07
United Technologies Corp 4.00 4.49 3.52 3.92 3.27 3.46 4.78 4.97 3.73 3.98 3.26∗ 3.49∗ Walgreen Co. 4.63 6.50 3.85 4.26 2.94∗ 3.18∗ 5.16 4.56 3.47 3.30 3.15 2.82
32
Standart & Poor’s 500 5.17 - 4.16 - 3.91 - 3.74 - 3.34 - 2.64 -
Table 3: This table gives the estimated value of the tail index for the twenty considered assets, the Standard & Poor’s 500 index and the
residues ε obtained by regressing each asset on the Standard & Poor’s 500 index, for both the negative and the positive tails, during the
time interval from July 1962 to December 1979. The tail indexes are estimed by Hill’s estimator at the quantile 1%, 2.5% and 5% which are
the optimal quantiles given by the Hall (1990) and Danielsson and de Vries (1997)’s algorithms. The values decorrated with stars represent
the tail indexes which cannot be considered equal to the Standard & Poor’s 500 index’s tail index at the 95% confidence level.

332 9. Mesure de la dépendance extrême entre deux actifs financiers
Negative Tail Positive Tail
q = 1% q = 2.5% q = 5% q = 1% q = 2.5% q = 5%
Asset ε Asset ε Asset ε Asset ε Asset ε Asset ε
Abbott Labs 3.59 3.60 3.35 3.62 3.22 3.39 5.14 4.60 4.16 3.76 3.77 3.07
American Home Products Corp. 3.03 3.07 3.11 2.78 2.73 2.49∗ 4.01 3.47 3.28 3.02 2.87 2.79
Boeing Co. 3.39 3.97 3.23 3.53 3.02 3.21 4.86 3.65 3.45 3.16 3.13 3.23
Bristol-Myers Squibb Co. 3.21 3.15 2.90 3.41 2.80 3.16 2.98 3.74 3.35 3.12 3.20 2.75
Chevron Corp. 4.13 4.48 3.99 3.91 3.30 3.45 5.16 4.53 3.88 3.81 3.01 3.06
Du Pont (E.I.) de Nemours & Co. 3.99 3.49 3.76 3.23 3.02 3.04 5.36 4.33 4.31 3.35 3.44 2.76
Disney (Walt) Co. 2.83 3.24 2.76 2.97 2.85 2.83 3.97 3.70 3.68 3.33 3.15 2.87
General Motors Corp. 4.44 4.79 3.88 4.27∗ 3.44 3.56 5.76 5.32 4.45 3.86 3.43 3.22
Hewlett-Packard Co. 3.73 3.45 3.52 3.12 3.00 2.73 4.31 3.40 3.47 3.29 3.24 2.99
Coca-Cola Co. 3.01 3.76 3.14 3.48 2.99 2.86 4.06 3.47 3.45 3.16 3.37 2.87
Minnesota Mining & MFG Co. 3.52 3.38 3.21 3.39 2.88 3.04 3.76 3.46 3.95 3.22 3.10 2.76
Philip Morris Cos Inc. 3.58 3.34 3.33 3.12 2.68 2.53∗ 3.42 3.16 3.70 3.07 2.85 2.81
Pepsico Inc. 4.14 4.46 3.39 3.60 2.99 3.27 4.00 3.87 3.61 3.34 3.44 3.31
Procter & Gamble Co. 2.65 2.46 3.29 3.19 3.19 2.87 4.35 3.90 3.48 3.20 3.14 2.91
Pharmacia Corp. 2.96 3.20 3.09 2.79 2.80 2.70 4.12 4.70 3.44 3.50 3.31 2.89
Schering-Plough Corp. 4.22 5.20∗ 3.29 3.68 3.11 3.05 3.23 3.51 3.45 3.08 3.06 2.87
Texaco Inc. 3.09 3.20 3.10 3.15 2.88 2.84 3.65 3.36 3.20 3.04 2.86 2.70
Texas Instruments Inc. 3.49 3.53 3.35 3.31 2.89 2.99 4.00 3.42 3.36 3.30 2.97 3.06
United Technologies Corp 4.21 3.98 3.82 3.46 3.34 3.18 5.39 4.50 4.00 3.80 3.51 3.26
Walgreen Co. 4.06 4.35 3.81 4.04 3.20 3.40 4.60 5.12 3.79 3.54 3.20 3.07
33
Standart & Poor’s 500 3.16 - 3.17 - 3.16 - 4.00 - 3.65 - 3.19 -
Table 4: This table gives the estimated value of the tail index for the twenty considered assets, the Standard & Poor’s 500 index and the
residues ε obtained by regressing each asset on the Standard & Poor’s 500 index, for both the negative and the positive tails, during the time
interval from January 1980 to December 2000. The tail indexes are estimed by Hill’s estimator at the quantile 1%, 2.5% and 5% which are
the optimal quantiles given by the Hall (1990) and Danielsson and de Vries (1997)’s algorithms. The values decorrated with stars represent
the tail indexes whose value cannot be considered equal to the Standard & Poor’s 500 index’s tail index at the 95% confidence level.

9.2. Estimation du coefficient de dépendance de queue 333
First Centile First Quintile First Decile
mean std. min. max. mean std. min. max. mean std. min. max.
Abbott Labs 0.1264 0.0106 0.1039 0.1745 0.1232 0.0058 0.1039 0.1745 0.1185 0.0069 0.1039 0.1745
American Home Products Corp. 0.1181 0.0091 0.0796 0.1288 0.1349 0.0111 0.0796 0.1494 0.1401 0.0096 0.0796 0.1505
Boeing Co. 0.1116 0.0128 0.0954 0.1653 0.1090 0.0074 0.0954 0.1653 0.1066 0.0066 0.0954 0.1653
Bristol-Myers Squibb Co. 0.1927 0.0171 0.1407 0.2175 0.2220 0.0220 0.1407 0.2474 0.2180 0.0180 0.1407 0.2474
Chevron Corp. 0.1566 0.0194 0.1334 0.2368 0.1407 0.0121 0.1265 0.2368 0.1365 0.0101 0.1257 0.2368
Du Pont (E.I.) de Nemours & Co. 0.2089 0.0142 0.1785 0.2424 0.2067 0.0105 0.1785 0.2424 0.2021 0.0106 0.1785 0.2424
Disney (Walt) Co. 0.1317 0.0160 0.0930 0.1613 0.1587 0.0161 0.0930 0.1754 0.1566 0.0123 0.0930 0.1754
General Motors Corp. 0.2210 0.0149 0.2045 0.2996 0.2109 0.0096 0.1947 0.2996 0.2020 0.0120 0.1808 0.2996
Hewlett-Packard Co. 0.1455 0.0103 0.1188 0.1769 0.1615 0.0116 0.1188 0.1776 0.1603 0.0095 0.1188 0.1776
Coca-Cola Co. 0.1870 0.0259 0.1199 0.2522 0.2159 0.0204 0.1199 0.2522 0.2160 0.0164 0.1199 0.2522
Minnesota Mining & MGF Co. 0.2311 0.0254 0.1851 0.3268 0.2262 0.0126 0.1851 0.3268 0.2218 0.0115 0.1851 0.3268
Philip Morris Cos Inc. 0.1251 0.0078 0.1050 0.1526 0.1334 0.0089 0.1050 0.1526 0.1340 0.0070 0.1050 0.1526
Pepsico Inc. 0.1263 0.0111 0.1014 0.1706 0.1242 0.0071 0.1014 0.1706 0.1239 0.0057 0.1014 0.1706
Procter & Gamble Co. 0.1980 0.0325 0.1131 0.2351 0.2017 0.0153 0.1131 0.2351 0.1972 0.0131 0.1131 0.2351
Pharmacia Corp. 0.1180 0.0172 0.0596 0.1378 0.1291 0.0109 0.0596 0.1432 0.1337 0.0091 0.0596 0.1434
Schering-Plough Corp. 0.1759 0.0143 0.1320 0.2403 0.1733 0.0078 0.1320 0.2403 0.1663 0.0102 0.1320 0.2403
Texaco Inc. 0.0214 0.0017 0.0170 0.0269 0.0246 0.0023 0.0170 0.0278 0.0248 0.0018 0.0170 0.0278
Texas Instruments Inc. 0.5657 0.0687 0.3897 0.6572 0.6162 0.0438 0.3897 0.6623 0.6168 0.0368 0.3897 0.6761
United Technologies Corp 0.1300 0.0123 0.0990 0.1872 0.1254 0.0073 0.0990 0.1872 0.1192 0.0086 0.0990 0.1872
Walgreen Co. 0.0739 0.0109 0.0664 0.1185 0.0682 0.0058 0.0629 0.1185 0.0674 0.0044 0.0629 0.1185
34
Table 5: This table gives the average (mean), the standard deviation (std.), the minimum (min.) and the maximum (max.) values of the
coefficient of lower tail dependence estimated over the first centile, quintile and decile during the entire time interval from July 1962 to
December 2000, under the assumption that the tail of the distributions of the assets and the market are regularly varying with a index equal
to three.

334 9. Mesure de la dépendance extrême entre deux actifs financiers
First Centile First Quintile First Decile
mean std. min. max. mean std. min. max. mean std. min. max.
Abbott Labs 0.1670 0.0127 0.1442 0.2137 0.1633 0.0071 0.1442 0.2137 0.1540 0.0120 0.1331 0.2137
American Home Products Corp. 0.1423 0.0207 0.0910 0.1720 0.1728 0.0205 0.091 0.1963 0.1823 0.0175 0.0910 0.2020
Boeing Co. 0.1372 0.0127 0.1101 0.1804 0.1349 0.0064 0.1101 0.1804 0.1289 0.0078 0.1101 0.1804
Bristol-Myers Squibb Co. 0.2720 0.0231 0.1878 0.3052 0.2751 0.0115 0.1878 0.3052 0.2696 0.0110 0.1878 0.3052
Chevron Corp. 0.1853 0.0188 0.1656 0.2564 0.1790 0.0105 0.1634 0.2564 0.1748 0.0096 0.1606 0.2564
Du Pont (E.I.) de Nemours & Co. 0.2547 0.0148 0.2127 0.2871 0.2695 0.0117 0.2127 0.2876 0.2685 0.0103 0.2127 0.2876
Disney (Walt) Co. 0.1772 0.0149 0.1368 0.1957 0.1938 0.0123 0.1368 0.2094 0.1900 0.0109 0.1368 0.2094
General Motors Corp. 0.2641 0.0259 0.2393 0.3652 0.2565 0.0138 0.2349 0.3652 0.2545 0.0108 0.2349 0.3652
Hewlett-Packard Co. 0.1701 0.0096 0.1389 0.1914 0.2018 0.0230 0.1389 0.2303 0.2039 0.0176 0.1389 0.2303
Coca-Cola Co. 0.2343 0.0223 0.1686 0.2719 0.2576 0.0163 0.1686 0.2731 0.2579 0.0123 0.1686 0.2731
Minnesota Mining & MFG Co. 0.2844 0.0196 0.2399 0.3407 0.2873 0.0099 0.2399 0.3407 0.2802 0.0117 0.2399 0.3407
Philip Morris Cos Inc. 0.1369 0.0168 0.0983 0.1673 0.1700 0.0206 0.0983 0.1919 0.1729 0.0155 0.0983 0.1919
Pepsico Inc. 0.1634 0.0132 0.1483 0.2106 0.1535 0.0083 0.1448 0.2106 0.1512 0.0067 0.1434 0.2106
Procter & Gamble Co. 0.2284 0.0292 0.1434 0.2673 0.2461 0.0169 0.1434 0.2673 0.2413 0.0141 0.1434 0.2673
Pharmacia Corp. 0.1279 0.0104 0.0863 0.1432 0.1588 0.0192 0.0863 0.1822 0.1643 0.0149 0.0863 0.1822
Schering-Plough Corp. 0.2195 0.0190 0.1920 0.2863 0.2179 0.0103 0.1920 0.2863 0.2107 0.0123 0.1877 0.2863
Texaco Inc. 0.0327 0.0027 0.0243 0.0369 0.0369 0.0033 0.0243 0.0414 0.0371 0.0027 0.0243 0.0414
Texas Instruments Inc. 0.4355 0.0195 0.3389 0.4906 0.4500 0.0142 0.3389 0.4906 0.4515 0.011 0.3389 0.4906
United Technologies Corp 0.1570 0.0153 0.1298 0.2182 0.1562 0.0075 0.1298 0.2182 0.1511 0.0084 0.1298 0.2182
Walgreen Co. 0.0937 0.0112 0.0808 0.1384 0.0837 0.0071 0.0776 0.1384 0.0786 0.0078 0.0669 0.1384
35
Table 6: This table gives the average (mean), the standard deviation (std.), the minimum (min.) and the maximum (max.) values of the
coefficient of lower tail dependence estimated over the first centile, quintile and decile during the entire time interval from July 1962 to
December 2000, under the assumption that the tail of the distributions of the assets and the market are power laws with an exponent equal
to three.

9.2. Estimation du coefficient de dépendance de queue 335
Negative Tail Positive Tail
α = 3 α = 3.5 α = 4 α = 3 α = 3.5 α = 4
Abbott Labs 0.12 0.09 0.06 0.11 0.08 0.06
American Home Products Corp. 0.22 0.18 0.15 0.25 0.22 0.19
Boeing Co. 0.16 0.13 0.10 0.13 0.10 0.07
Bristol-Myers Squibb Co. 0.22 0.19 0.16 0.28 0.25 0.23
Chevron Corp. 0.21 0.17 0.14 0.26 0.23 0.20
Du Pont (E.I.) de Nemours & Co. 0.38 0.37 0.35 0.37 0.35 0.33
Disney (Walt) Co. 0.24 0.20 0.17 0.23 0.19 0.16
General Motors Corp. 0.39 0.37 0.35 0.48 0.47 0.47
Hewlett-Packard Co. 0.15 0.12 0.09 0.23 0.20 0.17
Coca-Cola Co. 0.26 0.22 0.19 0.26 0.23 0.20
Minnesota Mining & MFG Co. 0.35 0.32 0.30 0.35 0.33 0.31
Philip Morris Cos Inc. 0.25 0.22 0.19 0.20 0.17 0.14
Pepsico Inc. 0.15 0.12 0.09 0.17 0.14 0.11
Procter & Gamble Co. 0.23 0.19 0.16 0.24 0.21 0.18
Pharmacia Corp. 0.23 0.19 0.16 0.26 0.23 0.20
Schering-Plough Corp. 0.21 0.18 0.15 0.20 0.17 0.14
Texaco Inc. 0.06 0.04 0.03 0.07 0.05 0.03
Texas Instruments Inc. 0.47 0.46 0.46 0.49 0.49 0.49
United Technologies Corp 0.13 0.10 0.07 0.13 0.10 0.07
Walgreen Co. 0.03 0.02 0.01 0.02 0.01 0.01
Table 7: This table summarizes the mean values over the first centile of the distribution of the
coefficients of (upper or lower) tail dependence for the positive and negative tails during the time
interval from July 1962 to December 1979, for three values of the tail index α = 3, 3.5, 4.
36

336 9. Mesure de la dépendance extrême entre deux actifs financiers
Negative Tail Positive Tail
α = 3 α = 3.5 α = 4 α = 3 α = 3.5 α = 4
Abbott Labs 0.20 0.17 0.14 0.16 0.13 0.10
American Home Products Corp. 0.12 0.09 0.06 0.10 0.08 0.05
Boeing Co. 0.14 0.11 0.08 0.10 0.07 0.05
Bristol-Myers Squibb Co. 0.32 0.29 0.26 0.25 0.21 0.19
Chevron Corp. 0.18 0.14 0.11 0.13 0.09 0.07
Du Pont (E.I.) de Nemours & Co. 0.23 0.20 0.17 0.16 0.13 0.10
Disney (Walt) Co. 0.16 0.13 0.10 0.15 0.12 0.09
General Motors Corp. 0.26 0.22 0.19 0.20 0.16 0.13
Hewlett-Packard Co. 0.19 0.15 0.13 0.21 0.18 0.15
Coca-Cola Co. 0.24 0.20 0.18 0.20 0.17 0.14
Minnesota Mining & MFG Co. 0.26 0.23 0.20 0.20 0.17 0.14
Philip Morris Cos Inc. 0.11 0.08 0.06 0.11 0.08 0.06
Pepsico Inc. 0.17 0.14 0.11 0.14 0.11 0.09
Procter & Gamble Co. 0.24 0.21 0.18 0.20 0.16 0.13
Pharmacia Corp. 0.10 0.08 0.05 0.10 0.07 0.05
Schering-Plough Corp. 0.23 0.20 0.17 0.16 0.13 0.10
Texaco Inc. 0.02 0.01 0.01 0.02 0.01 0.01
Texas Instruments Inc. 0.43 0.42 0.41 0.31 0.28 0.26
United Technologies Corp 0.20 0.16 0.14 0.18 0.14 0.11
Walgreen Co. 0.15 0.12 0.09 0.09 0.07 0.05
Table 8: This table summarizes the mean values over the first centile of the distribution of the
coefficients of (upper or lower) tail dependence for the positive and negative tails during the time
interval from January 1980 to December 2000, for three values of the tail index α = 3, 3.5, 4.
37

9.2. Estimation du coefficient de dépendance de queue 337
Negative Tail Positive Tail
α = 3 α = 3.5 α = 4 α = 3 α = 3.5 α = 4
Abbott Labs 0.17 0.13 0.11 0.15 0.12 0.09
American Home Products Corp. 0.14 0.11 0.08 0.15 0.11 0.09
Boeing Co. 0.14 0.10 0.08 0.10 0.07 0.05
Bristol-Myers Squibb Co. 0.27 0.24 0.21 0.27 0.24 0.21
Chevron Corp. 0.19 0.15 0.12 0.17 0.13 0.10
Du Pont (E.I.) de Nemours & Co. 0.25 0.22 0.19 0.23 0.19 0.16
Disney (Walt) Co. 0.18 0.14 0.11 0.17 0.13 0.11
General Motors Corp. 0.26 0.23 0.20 0.24 0.21 0.18
Hewlett-Packard Co. 0.17 0.14 0.11 0.23 0.19 0.16
Coca-Cola Co. 0.23 0.20 0.17 0.23 0.20 0.17
Minnesota Mining & MFG Co. 0.28 0.25 0.23 0.25 0.22 0.19
Philip Morris Cos Inc. 0.14 0.10 0.08 0.14 0.11 0.08
Pepsico Inc. 0.16 0.13 0.10 0.16 0.12 0.10
Procter & Gamble Co. 0.23 0.20 0.17 0.22 0.18 0.15
Pharmacia Corp. 0.13 0.10 0.07 0.14 0.10 0.08
Schering-Plough Corp. 0.22 0.19 0.16 0.19 0.15 0.12
Texaco Inc. 0.03 0.02 0.01 0.03 0.02 0.01
Texas Instruments Inc. 0.44 0.42 0.41 0.37 0.35 0.33
United Technologies Corp 0.16 0.12 0.10 0.15 0.12 0.09
Walgreen Co. 0.09 0.07 0.05 0.06 0.04 0.03
Table 9: This table summarizes the mean values over the first centile of the distribution of the
coefficients of (upper or lower) tail dependence for the positive and negative tails during the time
interval from July 1962 to December 2000, for three values of the tail index α = 3, 3.5, 4.
38

338 9. Mesure de la dépendance extrême entre deux actifs financiers
July 1962 - Dec. 1979 Jan.1980 - Dec. 2000
Extremes λ− p-value Extremes λ− p-value
Abbott Labs 0 0.12 0.2937 4 0.20 0.0904
American Home Products Corp. 1 0.22 0.2432 2 0.12 0.2247
Boeing Co. 0 0.16 0.1667 3 0.14 0.1176
Bristol-Myers Squibb Co. 2 0.22 0.2987 4 0.32 0.2144
Chevron Corp. 3 0.21 0.2112 4 0.18 0.0644
Du Pont (E.I.) de Nemours & Co. 0 0.38 0.0078 4 0.23 0.1224
Disney (Walt) Co. 2 0.24 0.2901 2 0.16 0.2873
General Motors Corp. 2 0.39 0.1345 4 0.26 0.1522
Hewlett-Packard Co. 0 0.15 0.1909 2 0.19 0.3007
Coca-Cola Co. 2 0.26 0.2765 5 0.24 0.0494
Minnesota Mining & MFG Co. 2 0.35 0.1784 4 0.26 0.1571
Philip Morris Cos Inc. 1 0.25 0.1841 2 0.11 0.2142
Pepsico Inc. 2 0.15 0.2795 5 0.17 0.0141
Procter & Gamble Co. 1 0.23 0.2245 3 0.24 0.2447
Pharmacia Corp. 2 0.23 0.2956 4 0.10 0.0128
Schering-Plough Corp. 0 0.21 0.0946 4 0.23 0.1224
Texaco Inc. 0 0.06 0.5222 2 0.02 0.0212
Texas Instruments Inc. 1 0.47 0.0161 3 0.43 0.1862
United Technologies Corp 1 0.13 0.3728 4 0.20 0.0870
Walgreen Co. 1 0.03 0.2303 3 0.15 0.1373
Table 10: This table gives, for the time intervals from July 1962 to December 1979 and from
January 1980 to December 2000, the number of losses within the ten largest losses incured by an
asset which have occured together with one of the ten largest losses of the Standard & Poor’s 500
index during the same time interval. The probabilty of occurence of such a realisation is given by
the p-value derived from the binomial law (27) with parameter λ−.
39

9.2. Estimation du coefficient de dépendance de queue 339
July 1962 - Dec. 1979 Jan.1980 - Dec. 2000
Extremes λ− p-value Extremes λ− p-value
Abbott Labs 0 0.12 0.2937 4 0.20 0.0904
American Home Products Corp. 1 0.22 0.2432 1 0.12 0.3828
Boeing Co. 0 0.16 0.1667 3 0.14 0.1176
Bristol-Myers Squibb Co. 2 0.22 0.2987 3 0.32 0.2653
Chevron Corp. 3 0.21 0.2112 3 0.18 0.1708
Du Pont (E.I.) de Nemours & Co. 0 0.38 0.0078 3 0.23 0.2342
Disney (Walt) Co. 2 0.24 0.2901 1 0.16 0.3300
General Motors Corp. 2 0.39 0.1345 3 0.26 0.2536
Hewlett-Packard Co. 0 0.15 0.1909 1 0.19 0.2880
Coca-Cola Co. 1 0.26 0.1782 4 0.24 0.1318
Minnesota Mining & MFG Co. 2 0.35 0.1784 3 0.26 0.2561
Philip Morris Cos Inc. 1 0.25 0.1841 2 0.11 0.2142
Pepsico Inc. 2 0.15 0.2795 5 0.17 0.0141
Procter & Gamble Co. 1 0.23 0.2245 3 0.24 0.2447
Pharmacia Corp. 2 0.23 0.2956 4 0.10 0.0128
Schering-Plough Corp. 0 0.21 0.0946 3 0.23 0.2342
Texaco Inc. 0 0.06 0.5222 1 0.02 0.1922
Texas Instruments Inc. 1 0.47 0.0161 3 0.43 0.1862
United Technologies Corp 1 0.13 0.3728 3 0.20 0.2001
Walgreen Co. 1 0.03 0.2303 3 0.15 0.1373
Table 11: This table gives, for the time intervals from July 1962 to December 1979 and from
January 1980 to December 2000, the number of losses within the ten largest losses incured by an
asset which have occured together with one of the ten largest losses of the Standard & Poor’s 500
index during the same time interval, provided that the losses are not both the largest of each series.
The probabilty of occurence of such a realisation is given by the p-value derived from the binomial
law (27) with parameter λ−.
40

340 9. Mesure de la dépendance extrême entre deux actifs financiers
9.3 Synthèse de la description de la dépendance entre actifs financiers
Les valeurs de dépendance de queue présentées dans la partie précédente entre un actif et le facteur de
marché (l’indice) permettent aisément d’en déduire la dépendance de queue entre deux actifs, comme
étant le minimum de la dépendance de queue entre chacun des actifs et l’indice. On en déduit que si l’indice
à une distribution régulièrement variable, les actifs présentent une dépendance de queue non nulle.
En conséquence, l’hypothèse de copule gaussienne faite au chapitre 7 ne peut être considérée comme
une approximation valable, puisque nous rappelons qu’elle n’admet pas de dépendance de queue. Cependant,
comme nous l’avons montré au chapitre 3, la distribution de rendements des actifs n’est peutêtre
pas régulièrement variable, mais rapidement variable, si l’on considère les distributions exponentielles
étirées. Dans ce cas, et pour autant que l’indice de marché reste le facteur principal, les actifs ne
présentent pas de dépendance de queue, et la description de la dépendance en terme de copule gaussienne
redevient acceptable.
Ceci étant, l’examen direct de la répartition des rendements des actifs en fonction des rendements de
l’indice, au cours de la période 1980-2000, suggère clairement l’existence d’une dépendance de queue
comme le montre la figure 9.1. De plus, cette représentation confirme que la dépendance de queue pour
Texaco (panneau de gauche) est beaucoup plus faible que pour United Technologies (panneau de droite)
par exemple. En effet, on observe que pour les rendements négatifs, les extrêmes sont regroupés en fuseau
pour United Technologies alors qu’ils sont beaucoup plus dispersés pour Texaco. Ceci est parfaitement
conforme aux résultats énoncés au paragraphe précédent où les valeurs de dépendance de queue mesurées
étaient respectivement de 2% et 20%.
Texaco
4
3
2
1
0
−1
−2
−3
−4
−4 −3 −2 −1 0
S&P 500
1 2 3 4
United Technologies
4
3
2
1
0
−1
−2
−3
−4
−4 −3 −2 −1 0
S&P 500
1 2 3 4
FIG. 9.1 – Rendements de Texaco (panneau de gauche) et de United Technologies (panneau de droite)
en fonction des rendements du Standard & Poor’s 500 durant la période 1980-2000. Les distributions
marginales ont été projetées sur des distributions gaussiennes pour permettre une meilleure comparaison
et mettre en lumière l’effet de la copule.
Pour compléter cette étude, nous avons utilisé la méthode d’estimation non paramétrique de Coles, Heffernan
et Tawn (1999) mise en œuvre par Poon, Rockinger et Tawn (2001) que nous avons présentée au
paragraphe 9.1. Cette méthode est très délicate et nous semble assez peu précise puisqu’elle repose sur
l’estimation d’un exposant de queue et d’un facteur d’échelle. Nous avons déjà évoqué ces problèmes
au chapitre 1, nous n’y revenons donc pas. Malgré (ou plutôt à cause de) ces imprécisions, nous n’avons
jamais pu rejeter, à 95% de confiance, l’hypothèse de l’existence de dépendance de queue entre les actifs.
En outre, même si les valeurs numériques obtenues sont entachées d’une grande incertitude, elles

9.3. Synthèse de la description de la dépendance entre actifs financiers 341
Modèle à facteur
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Modèle elliptique
FIG. 9.2 – Coefficient de dépendance de queue estimé à l’aide du modèle à facteur en fonction du
coefficient de dépendance de queue estimé sous hypothèse de copule elliptique. L’indice de variation
régulière du facteur est de 3 et celui de la copule elliptique de 4. Ce couple de valeurs est celui qui donne
le meilleur accord entre les résultats des deux modèles.
demeurent du même ordre de grandeur que celles trouvées précédemment à l’aide du modèle à facteur.
Nous avons également estimé la dépendance de queue sous l’hypothèse de copule elliptique régulièrement
variable, dont l’expression a été dérivée par Hult et Lindskog (2001) et qui n’est fonction que de
l’indice de variation régulière et du coefficient de corrélation. Ce dernier a été estimé de manière non
paramétrique par l’intermédiaire du τ du Kendall, puis obtenu grâce à la relation
ρ = sin
π · τ
2
, (9.1)
bien connue pour la distribution gaussienne mais aussi valable pour la classe des distributions elliptiques
comme l’ont récemment montré Lindskog, McNeil et Schmock (2001). Là encore, les valeurs
numériques sont qualitativement en accord avec les résultats précédents. Plus précisément, on peut toujours
trouver un indice de variation régulière ν de la copule elliptique tel que les deux modèles donnent
exactement la même valeur de dépendance de queue, ce qui revient à accepter que la copule elliptique
soit différente d’un couple d’actifs à l’autre, mais ceci nous semble guère satisfaisant. Si par contre l’on
considère des copules elliptiques de même indice de queue pour toutes les paires, des différences importantes
sont observées par rapport aux estimations données par le modèle à facteur, comme le montre
la figure 9.2, où sont représentés les coefficients de dépendance de queue estimés à l’aide du modèle à
facteur et sous hypothèse de copule elliptique.
En conclusion, nous pensons pouvoir affirmer que les actifs auxquels nous nous sommes intéressés
présentent une dépendance de queue, et que celle-ci est de l’ordre de 5% à 30% selon les cas. Ceci
à d’importantes conséquences pratiques du point de vue de la gestion des risques. En effet, reprenant
le petit calcul présenté au point 1.2 du paragraphe 9.2, nous considérons la probabilité que deux actifs
d’un portefeuille chutent ensemble au-delà de leur VaR journalière à 99% par exemple. Négligeant
la dépendance de queue, un tel événement à une probabilité extrêmement faible de se produire (temps
de récurrence typique de 40 ans), alors que si l’on considère une dépendance de queue de 30% un tel
événement se produit typiquement tous les 16 mois (contre huit ans pour λ = 5%).

342 9. Mesure de la dépendance extrême entre deux actifs financiers
Pour ce qui est des implications théoriques, les résultats que nous venons d’exposer rendent l’approximation
de copule gaussienne non valable, à strictement parler. Toutefois, pour des actifs dont la dépendance
de queue n’est que de l’ordre de 5%, ce type d’approximation peut être considéré comme non déraisonnable,
comme vient de le montrer le petit calcul ci-dessus et dans la mesure où bien d’autres sources
d’erreurs - notamment sur la description des marginales - sont à prendre en compte dans l’estimation
globale de l’incertitude associée à l’estimation de la distribution jointe dans son ensemble.
Par ailleurs, dans les cas où la dépendance de queue est trop importante pour que la copule gaussienne
puisse être retenue, il convient aussi de s’interroger sur la pertinence de la modélisation en terme de
copule elliptique que nous avions initialement choisie. En effet, nous venons de montrer que l’accord
entre ce type de représentation et le modèle à facteur, qui a de solides points d’ancrage dans la théorie
financière, n’est pas très satisfaisant. De plus, les résultats du paragraphe 9.2 montrent que durant la
période 1980-2000, la dépendance de queue est légèrement plus importante dans la queue inférieure que
dans la queue supérieure - ce qui a été confirmé par l’approche non paramétrique - même si dans la
majeure partie des cas, cette différence n’est pas statistiquement significative. Cela dit, cette différence
systématiquement en faveur de la queue négative ne peut être négligée et est dès lors en contradiction
avec l’hypothèse de copules elliptiques qui sont des copules symétriques dans leurs parties positive et
négative et présentent donc une même dépendance de queue supérieure et inférieure.

Troisième partie
Mesures des risques extrêmes et
application à la gestion de portefeuille
343

Chapitre 10
La mesure du risque
Dans ce chapitre, nous souhaitons présenter quelques théories ayant permis la quantification du risque
associé à un actif financier ou a un portefeuille. Les sources de risque sont en fait aussi diverses que
variées. Nous pouvons citer par exemple le risque de défaut lié au fait qu’une contrepartie ne puisse
faire face à ses obligations ou bien le risque de liquidité lié à la capacité limitée des marchés à pouvoir
absorber un afflux massif de titres à la vente ou à l’achat, mais il nous semble que la source principale du
risque est le risque de marché, c’est-à-dire le risque associé aux fluctuations des actifs financiers. C’est
pourquoi nous nous bornerons à décrire comment quantifier ce type de risque.
Depuis le milieu du vingtième siècle, plusieurs théories ont vu le jour pour tenter de cerner le comportement
des individus face à l’incertain en vue d’en déduire des règles de décisions et des mesures de risque.
On peut citer tout d’abord les travaux de von Neumann et Morgenstern (1947) et Savage (1954) sur la
formalisation de la théorie de l’utilité espérée, visant à décrire les préférences des agents économiques,
et l’introduction de la notion d’aversion au risque. Puis, au vu des incompatibilités de cette théorie avec
certains comportements observés par Allais (1953) ou Ellsberg (1961), de nouvelles approches ont vu le
jour, telle la “théorie de la perspective” de Kahneman et Tversky (1979), qui permet de rendre compte
d’une part du fait que les agents s’attachent plus aux perspectives d’évolution de leur richesse qu’à leur
richesse elle-même, et d’autre part qu’ils traitent leurs gains et leurs pertes de manière disymétriques : ils
sont risquophobes face à des gains potentiels mais risquophiles vis-à-vis de pertes à venir. Une autre alternative
plus générale est ensuite apparue (Quiggin 1982, Gilboa et Schmeidler 1989) et s’est développée
autour des modèles dits non-additifs, c’est-à-dire des modèles où les probabilités ne jouissent plus de
la propriété d’additivité, et sont donc remplacées par des capacités, ce qui permet notamment de rendre
compte du phénomène de distorsion des probabilités souvent observé chez la plupart des agents.
Récemment, dans un cadre plus restreint que celui de la théorie de la décision, Artzner, Delbaen, Eber
et Heath (1999) ont proposé une nouvelle approche de la notion de risque, reposant sur les propriétés
minimales que l’on est en droit d’attendre d’une mesure de risque. Ceci a donné naissance à la notion de
mesures de risque cohérentes, qui a ensuite été étendue par Heath (2000) puis Föllmer et Schied (2002a)
à la notion de mesures de risque convexes. Cependant, comme nous le verrons, ce type de mesure de
risque ne semble pas toujours être le mieux adapté à la quantification des risques d’un portefeuille. En
effet, il nous semble qu’il convient de bien différencier deux types de risques :
– premièrement, ce que l’on peut appeler la mesure du risque en terme de capital économique, c’est-àdire
la somme d’argent dont doit disposer un gestionnaire de portefeuille / une institution pour pouvoir
faire face à ses obligations et ainsi éviter la ruine,
– et deuxièmement le risque lié aux fluctuations statistiques de la richesse ou du rendement d’un portefeuille
autour de l’objectif de rentabilité préalablement fixé.
345

346 10. La mesure du risque
C’est pour rendre compte de cette deuxième catégorie de risques que nous proposons une autre approche
dans laquelle il est fait usage des moments de la distribution des rendements d’un actif pour quantifier le
risque associé aux fluctuations de cet actif.
10.1 La théorie de l’utilité
Notre objectif dans ce paragraphe est de passer en revue les avancées récentes de la théorie de la décision,
dont on sait qu’elle joue un rôle très important en économie et en finance au travers de la notion d’utilité.
Notre présentation ne fait qu’effleurer la surface de ce vaste problème et nous renvoyons le lecteur à
l’article de Cohen et Tallon (2000) notamment pour de plus amples détails.
10.1.1 Théorie de l’utilité en environnement certain
La théorie de l’utilité trouve ses fondements dans le courant de pensée sociale développé de la fin du
18ème siècle au milieu du 19ème par J. Bentham et J.S. Mill, fondateurs de la philosophie de l’utilitarisme
1 et selon laquelle on doit juger une action à ses résultats et conséquences sur le bien-être des
individus. En cela, l’utilitarisme s’oppose farouchement au rigorisme moral prôné, à la même époque,
par Kant, pour qui la valeur d’une action ne peut se juger qu’à l’aune des principes et intentions qui en
sont à l’origine.
Loin de ces querelles philosophiques, les économistes se sont rapidement emparés du principe selon lequel
les individus agissent en vue de maximiser leur bien-être et en ont fait le moteur des comportements
individuels des agents économique. On peut d’ailleurs faire remonter à Adam Smith (1776) l’introduction
de la notion d’utilité en économie du fait de la distinction qu’il introduit entre les concepts de “valeur
à l’échange” (le prix) et de “valeur à l’usage” (l’utilité) d’un bien et qui sont à la base de la loi de l’offre
et de la demande et de la notion d’équilibre de marchés. La formalisation mathématique de la théorie de
l’utilité n’interviendra que près de deux siècles plus tard et repose (en univers certain) sur deux axiomes
simples décrivant les capacités des agents à déterminer leurs préférences.
Considérons à partir de maintenant, l’ensemble B des actifs financiers (ou plus généralement des biens)
accessibles aux agents économiques, et postulons deux axiomes simples, un axiome de comparaison et
un axiome de continuité :
AXIOME 1 (COMPARABILITÉ)
Un agent économique est capable d’établir une préférence entre tous les actifs de B. Cela revient à dire
qu’il existe un préordre complet “l’actif X est préféré à l’actif Y ”, notée X Y , entre tous les actifs
X, Y ∈ B.
AXIOME 2 (CONTINUITÉ)
La relation d’ordre est continue. Cela signifie qu’étant donné trois actifs X, Y, Z ∈ B il existe toujours
deux reéls α, β ∈]0, 1[ tels que d’une part, le portefeuille composé d’une proportion α de l’actif X et
(1 − α) de l’actif Y est strictement préféré à l’actif Z et d’autre part, l’actif Z est strictement préféré au
portefeuille composé d’une proportion β de l’actif X et (1 − β) de l’actif Y :
α X + (1 − α) Y ≻ Z et Z ≻ β X + (1 − β) Y. (10.1)
1 En toute rigueur les prémices de la philosophie de l’utilité remontent plutôt à la fin du 17ème siècle avec Hobbes puis avec
Helvétius au début 18ème (voire, si l’on veut aller aussi loin dans le temps à Epicure).

10.1. La théorie de l’utilité 347
Moyennant ces deux axiomes et quelques hypothèses à caractère purement technique que nous omettons,
il est possible de montrer qu’il existe une fonction dite fonction d’utilité telle que :
DÉFINITION 5 (FONCTION D’UTILITÉ)
La relation de préférence sur l’ensemble des actifs B peut être représentée par une fonction d’utilité
U : B → R, telle que :
∀(X, Y ) ∈ B 2 , X Y ⇐⇒ U(X) ≥ U(Y ). (10.2)
Cette définition signifie simplement que l’actif X est préféré à l’actif Y si et seulement si l’utilité (ou
valeur à l’usage) de l’actif X est supérieure à l’utilité de l’actif Y . Bien évidemment, la fonction d’utilité
n’est pas unique, puisque pour toute fonction g : R → R strictement croissante, la fonction V = g ◦ U
est aussi une fonction d’utilité. La fonction U ainsi définie n’a donc qu’une valeur ordinale. Si l’on
souhaite pouvoir considérer qu’un accroissement de la fonction d’utilité mesure une augmentation de
la satisfaction de l’agent économique, il faut admettre que la fonction d’utilité a une valeur cardinale,
et elle n’est alors plus définie qu’à une tranformation affine croissante près. Dans toute la suite nous
considérerons uniquement des fonctions d’utilité cardinales.
Si l’on s’intéresse, comme le plus fréquemment, à l’utilité de la richesse W d’un individu, on déduit
aisément du comportement des agents économiques que la fonction U(W ) est croissante, ce qui traduit
l’appât du gain ou insatiabilité, et généralement concave ce qui exprime la décroissance marginale de
l’utilité de la richesse : cent euros ne représentent pas la même utilité pour un agent possédant en tout et
pour tout mille euros ou un million d’euros.
10.1.2 Théorie de la décision face au risque
Considérons maintenant que l’on s’intéresse au comportement des agents économiques à l’égard d’actifs
dont la valeur future n’est pas parfaitement connue et dépend, à l’instant futur T , de l’état de la nature
dans lequel se trouvera l’univers en T . Nous sommes alors confrontés à un problème de décision en
univers risqué. Le premier exemple de résolution d’un tel problème remonte à Daniel Bernoulli (1738)
qui apparaît comme le précurseur de l’introduction de l’utilité espérée, dont il fit usage pour résoudre le
célèbre paradoxe de Saint-Pétersbourg.
Dans ce paradoxe, un individu se voit offrir la possibilité de jouer au jeu suivant : on lance une pièce
parfaitement équilibrée autant de fois que nécessaire pour voir le coté pile apparaître. A ce momentlà,
le jeu s’arrête et le joueur reçoit 2 n euros, n étant le nombre de fois que la pièce a été lancée. La
question est alors de savoir combien est prêt à payer l’individu pour pouvoir participer à ce jeu. Un
simple calcul d’espérance mathématique montre qu’en moyenne ce jeu offre un gain infini 2 . Donc, pour
être équitable, le joueur devrait accepter de payer une mise sinon infinie, du moins colossale. Or en
réalité, on observe que les joueurs n’acceptent guère de payer plus de quelques euros pour participer au
jeu, d’où le paradoxe.
La solution proposée par Bernoulli (1738) consiste à supposer que les joueurs ne s’intéressent pas à la
valeur moyenne des gains espérés mais plutôt à l’espérance du logarithme des gains, et l’on obtient alors :
∞
2 −n ln (2 n ) = 2 ln 2. (10.3)
n=1
Donc, pour reprendre la terminologie d’Adam Smith, les joueurs ne s’intéresse pas à la “valeur à l’échange”
- ici les gains espérés - du jeu, mais plutôt à sa “valeur à l’usage” et donc à l’espérance du logarithme
2 n 1 n.
La probabilité de gagner 2 euros est égale à la probabilité d’obtenir n fois de suite le coté face de la pièce, soit 2

348 10. La mesure du risque
des gains. Cette approche coïncide exactement avec la théorie de l’utilité espérée dont von Neumann
et Morgenstern (1947) poseront les bases plus de deux siècles plus tard et que nous allons maintenant
exposer.
Pour cela considérons l’ensemble Ω des états de la nature et F une tribu sur Ω de sorte que l’espace
(Ω, F) soit mesurable. A chaque actif X ∈ B est associé une loi de probabilité PX sur (Ω, F)
représentant la distribution de la valeur future de l’actif X. Par abus de langage et pour alléger les notations,
la variable aléatoire donnant la valeur future de l’actif X ∈ B sera elle même notée X (mais cette
fois, X ∈ (Ω, F, PX)). Comme précédemment en environnement certain, nous supposons que les agents
économiques sont capables d’établir un préordre total sur l’ensemble des actifs ou des valeurs futures
des actifs et que ces préférences sont continues. Nous admettons de plus que
AXIOME 3 (INDÉPENDENCE)
Pour tout actif X, Y, Z ∈ B et tout reél α ∈]0, 1],
X Y ⇐⇒ α X + (1 − α) Z α Y + (1 − α) Z, (10.4)
ce qui suppose que l’adjonction d’un même actif ne modifie pas l’ordre des préférences. Cet axiome est
central dans la théorie de l’utilité espérée de von Neumann et Morgenstern (1947). C’est en effet grâce à
lui que l’on peut montrer que l’utilité U d’un actif X dont la valeur future est risquée s’exprime comme
U(X) = E[u(X)], (10.5)
où u(·) est une fonction continue, croissante et définie à une transformation affine croissante près. La
fonction u(·) est donc elle même une fonction d’utilité, de sorte que l’utilité U d’un bien risqué apparait
comme la moyenne de l’utilité u de ce même bien dans chacun des états de la nature.
La fonction u permet de définir la notion d’aversion pour le risque. Selon Rotschild et Stiglitz (1970),
un actif Y est plus risqué qu’un actif X si pour toute fonction croissante et concave u(·), E[u(X)] ≥
E[u(Y )]. Ceci est en fait équivalent (Levy 1998) au fait que X domine Y au sens de la dominance
stochastique d’ordre deux :
∀t ∈ R,
t
FY (y) dy ≥
−∞
t
−∞
FX(x) dx. (10.6)
Ainsi donc, un individu qui préfère l’actif X à l’actif Y présente de l’aversion pour le risque, et une
fonction d’utilité u(·) concave caractérise un individu risquophobe. Le coefficient absolu d’aversion
pour le risque de cet individu est alors défini par
a = − u′′
. (10.7)
u ′
La notion de dominance stochastique d’ordre deux peut être étendue à un ordre n quelconque, et un
individu présentant une aversion pour le risque au sens de la dominance stochastique d’ordre n possède
une fonction d’utilité u(·), telle que pour tout x, et tout k ≤ n :
(−1) k u (k) (x) ≤ 0, (10.8)
(Levy 1998). De telles fonctions d’utilité caractèrisent une aversion pour le risque dite standard selon
la terminologie de Kimball (1993) et pour n = 4 par exemple, un individu ayant une fonction d’utilité
vérifiant (10.8) est dit insatiable (u ′ > 0), risquophobe (u ′′ < 0), prudent (u (3) > 0) et tempéré (u (4) <
0).

10.1. La théorie de l’utilité 349
Cette formulation simple et parcimonieuse de la théorie de la décision face au rique a contribué à rendre
l’approche de von Neumann et Morgenstern (1947) très populaire. Cependant, cette simplicité ne va
pas sans soulever quelques difficultés et incohérences, au premier rang desquelles se trouve la violation
du postulat d’indépendance, clé de voute de cette théorie. En effet, Allais (1953) a montré par des tests
simples, consistant à proposer une série d’alternatives à des sujets, que la majorité de ceux-ci émettent des
choix en contradiction avec cet axiome d’independance. En particulier, les agents semblent très sensibles
aux petites variations de probabilités au voisinage du certain : ils accordent beaucoup d’importance au
passage d’une probabilité de 0 à 0.01 ou de 0.99 à 1, alors qu’un changement de 5 à 10 pour-cent au
voisinage d’un niveau de probabilité de 0.50 les laissent bien souvent indifférents. Ainsi, les agents sont
sujets à une distorsion de leur perception des probabilités. D’autre part, en plus de cette contradiction
empirique vient s’ajouter une limitation théorique : la fonction u(·) joue un double rôle. Elle quantifie en
même temps l’aversion pour le risque du décideur et la décroissance marginale de l’utilité de la richesse,
de sorte qu’il est impossible de modéliser un agent qui est à la fois risquophile et dont l’utilité marginale
décroit.
En fait, ces deux contradictions peuvent être levées en affaiblissant le postulat d’indépendance, qui sera
remplacé par :
AXIOME 4 (CHOSE SÛRE COMONOTONE DANS LE RISQUE)
Soit deux actifs X, Y ∈ B dont les valeurs futures sont données par les variables aléatoires (supposées
discrètes pour simplifier) : X = (x1, p1; · · · ; xk, pk; · · · ; xn, pn) et Y = (y1, p1; · · · ; yk, pk; · · · ; yn, pn),
telles que x1 ≤ · · · ≤ xk ≤ · · · ≤ xn et y1 ≤ · · · ≤ yk ≤ · · · ≤ yn avec xk = yk. Soient alors les actifs
X ′ , Y ′ ∈ B obtenus en remplaçant xk par x ′ k dans les actifs X et Y de sorte que xk−1 ≤ x ′ k ≤ xk+1 et
yk−1 ≤ x ′ k = y′ k ≤ yk+1. Alors
X Y ⇐⇒ X ′ Y ′ . (10.9)
Cela veut simplement dire que l’on ne modifie pas l’ordre de préférence de deux actifs lorsque l’on
modifie leur commune valeur future sans changer le rang de celle-ci, ce qui fait toute la différence avec
l’axiome d’indépendance. Moyennant cela, les paradoxes d’Allais (1953) sont levés et l’on est amené
à généraliser la théorie de l’utilité espérée par la théorie de l’utilité dépendante du rang originellement
développée par Quiggin (1982). En effet, il est alors possible de montrer que l’on peut caractériser le
comportement de tout agent économique par deux fonctions croissantes. La première, u : B → R,
définie à une transformation affine croissante près, joue le rôle de fonction d’utilité dans le certain. La
seconde, ϕ : [0, 1] → [0, 1] est unique et représente la fonction de transformation (ou de distorsion) des
probabilités. Ainsi, l’utilité de l’actif X est :
U(X) = − u(x) dϕ(Pr{X > x}) = Eϕ◦PX [u(x)]. (10.10)
Cette intégrale est en fait une intégrale de Choquet, c’est-à-dire une intégrale par rapport à une mesure
généralement non-additive. Dans le cas particulier où ϕ(x) = x, on retrouve bien évidemment l’expression
de l’utilité espérée de von Neumann et Morgenstern (1947), qui est donc englobée par la théorie de
l’utilité espérée dépendante du rang.
L’intérêt de cette nouvelle formulation est de complètement découpler les notions de décroissance marginale
de l’utilité, mesurée par la concavité de la fonction u(·), et d’aversion pour le risque, entièrement
caractérisée par la fonction de transformation des probabilités ϕ(x) : un agent dont la fonction de transformation
des probabilités est telle que ϕ(x) ≤ x sera dit pessimiste dans le risque. En effet, reprenant
l’exemple discret de l’axiome 4, l’équation (10.10) devient :
U(X) = u(x1) + ϕ(p2 + · · · + pn) · [u(x2) − u(x1)] + · · · + ϕ(pn) · [u(xn) − u(xn−1)], (10.11)

350 10. La mesure du risque
ce qui montre qu’un tel agent commence par calculer l’utilité minimale que peut lui procurer l’actif X,
soit u(x1), puis il ajoute les accroissements possibles de l’utilité u(xk) − u(xk−1) qu’il peut recevoir en
les pondérant non pas par leur probabilités d’occurence mais par sa fonction de distorsion des probabiltés.
Ainsi, lorsque ϕ(x) ≤ x, il sous-estime la probabilité d’événements favorables et sous-pondère les
accroissements d’utilité qu’il peut en retirer.
10.1.3 Théorie de la décision face à l’incertain
Nous venons d’exposer les bases de la théorie de la décision face au risque, c’est-à-dire lorsque le
décideur connait de façon objective les probabilités associées aux différents états de la nature Ω. Cependant,
dans la plupart des situations économiques et financières, celles-ci ne sont que partiellement
révélées voire totalement inconnues. Il convient donc de s’intéresser à cette situation, qualifiée de théorie
de la décision dans l’incertain, par opposition à la théorie de la décision face au risque, où l’on suppose
données les probabilités sur les états de la nature.
L’approche classique (ou bayésienne) de ce problème est celle de Savage (1954) qui consiste à réduire
le problème de décision dans l’incertain à un problème de décision face au risque, à l’aide de la notion
de probabilités subjectives. Ces probabilités, dites subjectives, diffèrent des probabilités objectives de
la même manière que les courses de chevaux diffèrent du jeu de roulette au casino : à la roulette, la
table étant parfaitement équilibrée, tous les joueurs connaissent la probabilité que sorte le trois, rouge,
impair et passe, alors qu’au tiercé, nul ne connaît avec exactitude la probabilité que tel ou tel cheval a de
l’emporter.
De plus, les probabilités objectives ont une interprétation très simple dans la mesure où elles sont reliées
à la fréquence typique d’occurrence d’un événement. En effet, la probabilité d’obtenir face en jetant une
pièce parfaitement équilibrée est de un demi, tout simplement parce ce qu’en répétant un grand nombre
de fois ce lancer de pièce, on observe que celle-ci tombe sur face la moitié du temps, et ce quelle que
soit la personne effectuant les lancers. Donc, la probabilité objective est une propriété intrinsèque de
l’objet (ici, la pièce) ou de l’événement (ici, tomber sur face) considéré. Au contraire, les probabilités
subjectives mesurent un degré de croyance en la vraisemblance d’un événement. Quelle est la probabilité
qu’existe une vie extra-terrestre ? Nous ne pouvons pas faire d’expériences à ce sujet. Donc, la probabilité
accordée à ce type d’événement ne peut être que fonction de l’opinion de chacun sur la question.
Ainsi une probabilité subjective n’a pas une valeur unique et dépend de chaque individu. Pour autant, ces
probabilités subjectives obéissent aux mêmes règles que les probabilités objectives en vertu du théorème
du “ dutch book” (de Finetti 1937). Selon ce théorème, tout pari basé sur un ensemble de probabilités
subjectives est équitable (et ne peut donc conduire à un gain certain) si et seulement si la probabilité subjective
attribuée à un événement certain vaut un, ainsi que la somme des probabilités de deux événements
complémentaires.
Moyennant cette nouvelle interprétation des probabilités, les problèmes de décision dans l’incertain
peuvent être ramenés à de “simples” problèmes de décision face au risque, ce qui, outre l’axiome de
préordre total, repose selon Savage (1954), sur l’axiome suivant :
AXIOME 5 (PRINCIPE DE LA CHOSE SÛRE)
Etant donné un sous-ensemble ˜ Ω ∈ Ω de l’ensemble des états de la nature et des actifs X, X ′ , Y, Y ′ ∈ B
tels que ∀ω ∈ ˜ Ω, X(ω) = X ′ (ω), Y (ω) = Y ′ (ω) et ∀ω ∈ ˜ Ω, X(ω) = Y (ω), X ′ (ω) = Y ′ (ω), alors
X Y ⇐⇒ X ′ Y ′ . (10.12)
Ceci signifie qu’une modification commune de la partie commune de deux actifs ne modifie pas l’ordre

10.1. La théorie de l’utilité 351
des préférences. Ajouté aux axiomes de comparabilité et de continuité, cet axiome permet d’affirmer
qu’il existe une unique probabilité P sur (Ω, F) et une fonction u(·) continue et croissante (définie à une
fonction affine croissante près) telle que l’utilité U de l’actif X est donnée par
U(X) = EP[u(X)], (10.13)
où u(·) joue comme d’habitude le rôle de fonction d’utilité dans le certain. Cette expression est analogue
à celle obtenue par la théorie de l’utilité espérée de von Neumann et Morgenstern (1947), à la différence
notoire qu’ici, la mesure de probabilité P est subjective et non pas objective.
L’axiome de la chose sûre est extrêmement fort car il permet de traiter tout problème de décision dans
l’incertain comme un problème de décision face au risque. Ceci n’est en fait pas très réaliste, si bien
que cette approche est rejetée sur le plan théorique comme sur le plan pratique. En effet, de même
qu’Allais (1953) avait prouvé que l’axiome d’indépendance était contredit empiriquement par la majorité
des agents, Ellsberg (1961) a pu montrer que dans des situations très simple de choix dans l’incertain
l’axiome de la chose sûre ne résistait pas non plus à l’expérience. En fait, la plupart des agents présentent
une aversion pour l’ambiguïté, dans le sens où, pour une même mise, ils préférent parier pour ou contre
un événement de probabilité P connue plutôt que pour ou contre un événement dont ils savent seulement
que sa probabilité est comprise P − ε et P + ε. On peut alors montrer que les agents satisfaisant au
modèle de Savage (1954) sont indifférents à l’ambiguité, puisqu’ils ne peuvent pas faire la différence
entre ces deux paris.
Pour dépasser cette contradiction empirique il convient d’affaiblir l’axiome de la chose sûre. Pour cela,
Schmeidler (1989) a proposé l’alternative suivante :
AXIOME 6 (CHOSE SÛRE COMONOTONE)
Soit une partition {Ωk} n k=1
de l’ensemble des états de la nature Ω et deux actifs X, Y ∈ B dont les
valeurs futures sont données par les variables aléatoires : X = (x1, Ω1; · · · ; xk, Ωk; · · · ; xn, Ωn) et
Y = (y1, Ω1; · · · ; yk, Ωk; · · · ; yn, Ωn), telles que x1 ≤ · · · ≤ xk ≤ · · · ≤ xn et y1 ≤ · · · ≤ yk ≤ · · · ≤
dans les actifs X
yn avec xk = yk. Soient alors les actifs X ′ , Y ′ ∈ B obtenus en remplaçant xk par x ′ k
et Y de sorte que xk−1 ≤ x ′ k ≤ xk+1 et yk−1 ≤ x ′ k = y′ k ≤ yk+1. Alors
X Y ⇐⇒ X ′ Y ′ . (10.14)
Cet axiome est très proche de l’axiome de la chose sûre comonotone dans le risque, si ce n’est que cette
fois, les probabilités pi des états Ωi ne sont pas connues.
A l’aide de cela, on montre qu’il existe non plus une unique probabilité P sur {Ω, F} mais une unique
capacité 3 v sur {Ω, F} et une fonction u(·) croissante et continue (définie à une transformation affine
3
Une capacité v est une fonction d’ensemble de {Ω, F} dans [0, 1] telle que :
– v(∅) = 0,
– v(Ω) = 1,
– ∀A, B ∈ F, A ⊂ B =⇒ v(A) ≤ v(B).
Rappelons qu’une mesure de probabilité P (addititive) vérifierait en plus
Une capacité est dite convexe si
∀A, B ∈ F, P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
∀A, B ∈ F, v(A) + v(B) ≤ v(A ∪ B) + v(A ∩ B).
Toute capacité convexe a un noyau non vide, où le noyau de v est
core(v) = {P ∈ P | ∀A ∈ F, P (A) ≥ v(A)} ,
et P est l’ensemble des mesures de probabilités additives sur {Ω, F}.

352 10. La mesure du risque
croissante près) telles que l’utilité de l’actif X est
U(X) =
u(X) dv, (10.15)
qui est une intégrale de Choquet par rapport à la mesure non-additive (capacité) v. On peut noter la
très forte ressemblance de cette expression avec celle obtenue pour l’utilité dépendante du rang (cf
équation (10.10)). En fait, dans (10.10), l’expression ϕ◦P est une capacité. De plus, si dans le modèle de
Schmeidler (1989), il existe une probabilité objective P sur {Ω, F}, la capacité v peut s’exprimer comme
v = ϕ ◦ P, où ϕ est unique et v est convexe si et seulement si ϕ l’est aussi.
Dans le cadre de ce modèle, Montessano et Giovannoni (1996) définissent la notion d’aversion pour l’incertitude,
à savoir qu’un agent présente de l’aversion pour l’incertitude, s’il existe une loi de probabilité
P telle que quelque soit X ∈ B, u(X) dv ≤ EP[u(X)], ce qui implique que le noyau de la capacité
v contient P et est donc non vide. Réciproqement, on peut donc affirmer que tout agent caracterisé par
une capacité convexe (donc à noyau non vide) a de l’aversion pour l’incertitude. Intuitivement, un agent
averse à l’incertitude affectera toujours un événement de la “probabilité” la moins favorable parmi toutes
les probabilités attribuées à cet événement par l’ensemble des lois présentes dans le noyau de la capacité.
Schmeidler (1986) fournit une interpretation de ce modèle en terme de croyances. En effet, sous l’hypothèse
que la capacité v est convexe, son noyau est non vide et
∀X ∈ B,
u(X) dv = min
P∈core(v) EP[u(X)], (10.16)
donc l’utilité U(X) est donnée par l’espérance minimale de u(X) calculée sur un ensemble de scénarii.
Ceci a conduit Gilboa et Schmeidler (1989), Nakamura (1990), Chateauneuf (1991) ou encore Casadesus-
Masanell, Klibanoff et Ozdenoren (2000) à développer des modèles dits multi-prior, où les agents se
donnent, a priori, un ensemble de distributions de probabilités P (ou scénarii) et définissent l’utilité
comme
∀X ∈ B, U(X) = min
P∈P EP[u(X)]. (10.17)
Il faut bien remarquer que les deux approches ne sont pas équivalentes, car tout ensemble (fermé et
convexe) P n’est pas nécessairement le noyau d’une capacité v. De plus, cette dernière approche peut
paraitre excessivement pessimiste puisqu’elle ne retient que la plus petite utilité possible, vu l’ensemble
de scénarii considérés. En tout cas, elle est beaucoup plus pessimiste que l’utilité dérivée du modèle de
Schmeidler (1986) puisque Jaffray et Philippe (1997) ont montré que l’intégrale de Choquet pouvait toujours
s’exprimer comme la somme pondérée de deux termes : le minimum et le maximum de l’espérance
d’utilité par rapport à un ensemble de distributions de probabilité, le poids relatif de ces deux termes
permettant de définir un indice de pessimisme de l’agent.
Cependant, malgré les récentes avancées de la théorie de la décision que nous venons de présenter, il
faut reconnaitre qu’une de ses limitations fondamentales demeure, à savoir comment mettre en œuvre
concrètement et pratiquement cette théorie. En effet, dans la mesure où chaque agent possède une fonction
d’utilité différente, il est très difficile de décider de manière objective laquelle employer. Donc, il va
s’avérer utile de considérer d’autres outils de décision et d’autres moyens de mesurer les risques.

10.2. Les mesures de risque cohérentes 353
10.2 Les mesures de risque cohérentes
10.2.1 Définition
Selon Artzner et al. (1999), le risque associé aux variations de la valeur d’une position est mesuré par la
somme d’argent qui doit être investie dans un actif sans risque pour que dans le futur, cette position reste
acceptable, c’est-à-dire pour que les pertes éventuelles liées à la valeur future de la position ne mettent pas
en péril les projets du gestionnaire de fond, de l’entreprise ou plus généralement la personne / l’organisme
qui garantit la position. En ce sens, une mesure de risque constitue pour Artzner et al. (1999) une mesure
de capital économique. Donc, la mesure de risque, notée ρ dans toute la suite, pourra être positive ou
négative selon que, respectivement, il faille augmenter la somme investie dans l’actif sans risque pour
garantir la position risquée ou bien que l’on puisse réduire cette somme tout en maintenant cette garantie.
Une mesure de risque sera dite cohérente au sens de Artzner et al. (1999) si elle vérifie les quatres
propriétés ou axiomes que nous allons exposer ci-dessous. Décidons tout d’abord d’appeler G l’espace
des risques. Si l’espace des états de la nature Ω est supposé fini (hypothèse faite par Artzner et al. (1999)),
G est isomorphe à R N et une position risquée X n’est alors rien d’autre qu’un vecteur de R N . Une mesure
de risque ρ est alors une application de R N dans R. Une généralisation à d’autres espaces de risque G a
été proposée par Delbaen (2000).
Soit donc une position risquée X et une somme d’argent α investie dans l’actif sans risque en début de
période, de sorte qu’en fin de période, le montant investi dans l’actif sans risque est α · (1 + r), où r
représente le taux d’intérêt sans risque, alors :
AXIOME 7 (INVARIANCE PAR TRANSLATION)
∀X ∈ G et ∀α ∈ R, ρ(X + α · (1 + r)) = ρ(X) − α. (10.18)
Ceci signifie simplement qu’investir une somme α dans l’actif sans risque, diminue le risque de la même
quantité α. En particulier, pour toute position risquée X, ρ(X + ρ(X) · (1 + r)) = 0.
Considérons maintenant deux positions risquées X1 et X2, représentant par exemple les positions de
deux traders dans une salle de marché. Il est commode pour le superviseur de cette salle de marché que
le risque agrégé de tous les traders soit inférieur ou égal à la somme des risques de chaque trader, donc
en particulier il est souhaitable que le risque associé à la position (X1 + X2) soit inférieur ou égal à la
somme des risques associés aux positions X1 et X2 séparément :
AXIOME 8 (SOUS-ADDITIVITÉ)
∀(X1, X2) ∈ G × G, ρ(X1 + X2) ≤ ρ(X1) + ρ(X2). (10.19)
De plus, la sous-additivité garantit qu’un gestionnaire de portefeuille a interêt à agréger ses diverses
positions afin d’en diminuer le risque par diversification.
Le troisième axiome est un axiome d’homogénéité (ou d’extensivité pour reprendre le language des
physiciens) :
AXIOME 9 (POSITIVE HOMOGÉNÉITÉ)
∀X ∈ G et ∀λ ≥ 0, ρ(λ · X) = λ · ρ(X), (10.20)

354 10. La mesure du risque
ce qui signifie simplement que le risque d’une position croît avec la taille de cette position, et plus
précisément, le risque est ici supposé proportionnel à la taille de la position risquée. Nous reviendrons
un peu plus loin sur l’hypothèse sous-jacente que suggère un tel axiome.
Enfin, sachant que dans tous les états de la nature, le risque X conduit à une perte supérieure à l’actif Y
(c’est-à-dire que toutes les composantes du vecteur X de R N sont toujours inférieures ou égales à celles
du vecteur Y ), la mesure de risque ρ(X) doit être supérieure ou égale à ρ(Y ) :
AXIOME 10 (MONOTONIE)
∀X, Y ∈ G tel que X ≤ Y, ρ(X) ≥ ρ(Y ). (10.21)
Ainsi posés, ces quatre axiomes définissent ce que l’on appelle les mesures de risques cohérentes.
10.2.2 Quelques exemples de mesures de risque cohérentes
Beaucoup de mesures de risque communément utilisées aussi bien dans la recherche académique que par
les professionels s’avèrent être non cohérentes au sens de Artzner et al. (1999). En effet, il est évident
que la variance - dont l’utilisation comme mesure de risque remonte à Markovitz (1959) - ne satisfait pas
à l’axiome de monotonie. De même, il est aisé de montrer que la Value-at-Risk n’est généralement pas
sous-additive. Nous rappelons que la Value-at-Risk, calculée au seuil de confiance α est définie par
DÉFINITION 6 (VALUE-AT-RISK)
Soit X ∈ G supposé à distribution continue. La Value-at-Risk, calculée au seuil de confiance α ∈ [0, 1]
et notée VaRα est
Pr[X + (1 + r) · VaRα ≥ 0] = α. (10.22)
A ce sujet, la classe des actifs dont la distribution jointe est elliptique constitue une exception notable
pour laquelle Embrechts et al. (2002) ont montré que la VaR est sous-additive, et demeure donc une
mesure cohérente de risque.
Vue l’utilisation très répandue de la VaR dans le milieu professionel, il était souhaitable d’essayer de
construire une mesure de risque cohérente se rapprochant le plus possible de la VaR et qui de plus
complète l’information fournie par celle-ci, à savoir : conditionnée au fait de subir une perte dépassant
la VaR, quelle est, en moyenne, la perte observée. Ceci a conduit à la définition de l’Expected Shortfall :
DÉFINITION 7 (EXPECTED SHORTFALL)
Soit X ∈ G supposé à distribution continue. L’Expected Shortfall, calculée au niveau de confiance α est
X
ESα = −E
X
1 + r ≤ −V aRα . (10.23)
1 + r
Dans le cas où la distribution de X n’est pas supposée continue, l’expression de l’Expected Shortfall est
un peu plus compliquée (voir Acerbi et Tasche (2002) ou Tasche (2002) par exemple).
Lorsque l’on souhaite tenir compte des grands risques, une autre approche, sur laquelle nous reviendrons
en détail dans la section 10.3, consiste à prendre en compte l’effet des moments d’ordres supérieurs
(à deux). La question concernant la cohérence de ce type de mesures de risques contruites à partir des
moments est abordée par Delbaen (2000) et plus particulièrement par Fisher (2001). Ce dernier montre
que toute mesure de risque
ρ(X) = −E[X] + a · σp, (10.24)

10.2. Les mesures de risque cohérentes 355
où 0 ≤ a ≤ 1 et
σp = (E [max{(E[X] − X) p , 0}]) 1/p
(10.25)
est le semi-moment centré (inférieur) d’ordre p, est une mesure cohérente de risque. Plus généralement,
du fait que toute somme convexe de mesures cohérentes de risque est une mesure cohérente de risque, la
mesure
∞
ρ(X) = −E[X] + ap · σp, (10.26)
avec
est cohérente.
∞
p=1
p=1
ap ≤ 1 et ap ≥ 0, (10.27)
Ceci permet d’obtenir de façon simple les équivalents cohérents de certaines mesures de risque ou fonctions
d’utilité. Par exemple, la mesure de risque ρ(X) = −E[X] + a · σ2 peut être considérée comme la
généralisation cohérente de la fonction d’utilité moyenne-variance.
10.2.3 Représentation des mesures de risque cohérentes
Les axiomes présentés au paragraphe 10.2.1 ainsi que les quelques exemples que nous venons de donner
montrent qu’il existe une grande variété de mesures cohérentes : les axiomes ne sont pas assez contraignants
pour spécifier complétement une (unique) mesure de risque cohérente. Aussi est-il intéressant de
rechercher une représentation de ce type de mesures. Artzner et al. (1999) ont montré que
THÉORÈME 2 (REPRÉSENTATION DES MESURES COHÉRENTES DE RISQUE)
Une mesure de risque ρ est cohérente si et seulement si il existe une famille P de mesures de probabilité
sur l’espace des états de la nature telle que :
ρ(X) = sup
P∈P
EP
− X
. (10.28)
1 + r
Ainsi, une mesure cohérente de risque apparait comme l’espérance de perte maximale sur un ensemble
de scénarii réalisables. Il est alors évident que plus l’ensemble de scénarii considérés sera grand, plus
ρ(X) sera grand lui aussi, toutes choses égales par ailleurs. Ainsi, plus l’ensemble de scénarii considérés
est grand, plus la mesure de risque est conservatrice.
Cette expression mathématique n’est pas sans rappeler certaines formules que nous avons rencontrées
concernant la théorie de la décision dans l’incertain (voir les équations 10.16 et 10.17). Ceci est en
fait très naturel car les axiomes choisis par Artzner et al. (1999) pour définir les mesures de risques
cohérentes sont tout-à-fait similaires à ceux dont découlent le modèle d’utilité de Schmeidler (1986). La
légère différence entre les expressions (10.16) et (10.28), c’est-à-dire le changement du min en sup et le
passage d’une fonction d’utilité u(·) croissante à la fonction décroissante − ·
1+r
vient simplement du fait
que le décideur ou le gestionnaire de risques tend à maximiser son utilité alors qu’il cherche à minimiser
sa prise de risque. Ceci a comme conséquence que dans le modèle multi-prior, l’utilité est une quantité
super-additive et non pas sous-additive comme les mesures de risque cohérentes. De plus, la spécification
de la fonction − ·
1+r apparaissant dans (10.28) vient de l’axiome d’invariance par translation qui impose
que pour un investissement α dans l’actif sans risque ρ(α (1 + r)) = −α.
Aussi générale soit-elle, la représentation des mesures de risque cohérentes fournie par le théorème 2
n’est pas d’une utilisation ou d’une mise en oeuvre très simple. En effet, s’il est facile, dans la pratique,

356 10. La mesure du risque
d’utiliser l’Expected Shortfall comme mesure cohérente de risque ou toute autre mesure cohérente ayant
une expression analytique simple, comment doit-on choisir l’ensemble des scénarii réalisables si l’on
veut en rester au degré de généralité proposé par le théorème de représentation ? Cette question n’admet
guère de réponse satisfaisante, et dans l’optique d’une mise en œuvre pratique, le point fondamental est
plutôt de savoir si la mesure de risque que l’on souhaite utiliser peut être estimée à partir des données
empiriques. De telles mesure de risques sont dites law-invariant, et nous nous restreindrons désormais à
la seule étude de ce type de mesures cohérentes. Si de plus, on ne s’intéresse qu’aux mesures comonotoniquement
additives 4 , on définit alors ce qu’Acerbi (2002) qualifie de mesures spectrales, pour lesquelles
Kusuoka (2001) puis Tasche (2002) ont démontré le théorème de représentation suivant :
THÉORÈME 3 (REPRÉSENTATION DES MESURES SPECTRALES)
Soit F une fonction de distribution continue et convexe et un réel p ∈ [0, 1]. La mesure de risque ρ est une
mesure spectrale, c’est-à-dire cohérente, law-invariant et comonotoniquement additive, si et seulement
si elle admet la représentation
ρ(X) = p
1
0
VaRu(X) F (du) + (1 − p)VaR1(X). (10.29)
Si de plus F admet une densité φ par rapport à la mesure de Lebesgue, (et en supposant p = 1 pour
simplifier) alors
ρ(X) =
1
0
VaRu(X) φ(u) du, (10.30)
et ρ(X) apparait comme la somme pondérée par φ(u) des VaRu, ce qui justifie, suivant Acerbi (2002),
que l’on puisse qualifier φ de “fonction d’aversion pour le risque”, puisque φ quantifie l’importance
accordée aux différents niveaux de risque quantifiés par le seuil de confiance u. Dans le cas où φ(u) =
α −1 · 1 (u

10.2. Les mesures de risque cohérentes 357
Le premier point assure l’absence d’encaisse oisive. Le second est un principe de diversification : mieux
vaut agréger les risques. Le troisième point stipule que seul le risque est un paramètre pertinent pour
l’allocation de capital. Enfin, le quatrième point exprime simplement que le capital investi dans l’actif
sans risque ne peut l’être ailleurs.
Considérant maintenant que la mesure de risque ρ est cohérente, Denault (2001) a prouvé, par analogie
avec la théorie des jeux, l’existence, et parfois l’unicité, d’allocations cohérentes. Ces résultats étant
avant tout théorique et pour l’heure sans application pratique directe, nous n’irons pas plus avant sur
cette voie.
En vue d’une mise en oeuvre pratique, mais aussi sur le plan théorique, il est intéressant de noter que de
par les axiomes de sous-additivité et de positive homogénéité, il est évident que les mesures de risque
cohérentes sont convexes, ce qui assure aux problèmes d’optimisation de portefeuille l’existence d’une
unique solution optimale. Cependant, malgré le bon comportement mathématique de la fonction à minimiser,
la mise en oeuvre pratique se révèle parfois délicate. Dans le cas particulier de l’Expected
Shortfall, Pflug (2000) et Rockafellar et Uryasev (2000) ont établi un algorithme d’optimisation efficace
par l’introdution de variables auxiliaires qui permettent de rendre le problème linéaire par morceau. Cet
algorithme a ensuite été généralisé au cas des mesures spectrales par Acerbi et Simonetti (2002).
Enfin, il peut être utile de calculer le risque marginal associé à chaque actif au sein d’un portefeuille. Etant
donné N actifs X1, · · · , XN et w1, · · · , wN le nombre (ou le poids) de chaque actif dans le portefeuille,
le risque marginal de l’actif i est la contribution d’une unité de cet actif au risque du portefeuille. D’après
l’axiome d’homogénéité et en conséquence du théorème d’Euler sur les fonctions homogènes, le risque
ρw = ρ(w1 · X1 + · · · + wN · XN) du portefeuille peut s’écrire
ρw =
N
i=1
wi · ∂ρw
, (10.31)
∂wi
sous l’hypothèse que la mesure de risque est différentiable par rapport aux wi. En conséquence, il apparait
clairement que la contribution marginale de l’actif i au risque du portefeuille est
ρi = ∂ρw
. (10.32)
∂wi
Ceci permet en outre, généralisant l’approche de Gouriéroux, Laurent et Scaillet (2000) concernant la
Value-at-Risk et de Scaillet (2000a) pour ce qui est de l’Expected-Shortfall, d’étudier la sensibilité du
risque du portefeuille par rapport à l’allocation des différents actifs.
10.2.5 Critique des mesures cohérentes de risque
Le mérite de l’axiomatique des mesures de risque cohérentes est de proposer une approche mathématique
rigoureuse de la problèmatique associée à la notion de risque. Cependant, comme toute approche axiomatique,
certaines des hypothèses de base peuvent être soumises à discussion. Si les axiomes d’invariance
par translation et de monotonie ne semblent pas souffrir la critique, on peut cependant légitimement
s’interroger sur la validité et les limites des deux autres axiomes.
Commencons par discuter l’axiome d’homogénéité. Comme nous l’avons précédemment écrit, cet axiome
n’est rien d’autre qu’une propriété d’extensivité de la mesure du risque : le risque associé à une position
est proportionel à la taille de cette position. Cependant, cela suppose qu’à tout moment l’on soit capable
de liquider cette position, quelle qu’en soit sa taille. Sous l’hypothèse d’un marché parfaitement liquide,
cette hypothèse est tout-à-fait raisonnable. Mais, dès que l’on souhaite considérer un marché réel, et donc

358 10. La mesure du risque
à liquidité limitée, il est alors bien évident que la taille de la position devient un facteur de risque, aucun
marché n’étant en mesure d’absorber, sans fluctuation de cours, n’importe quelle taille d’ordre. Il peut
donc sembler a priori raisonnable de reformuler l’axiome d’homogénéité de la façon suivante :
AXIOME 11 (POSITIVE HOMOGÉNÉITÉ EN MARCHÉ ILLIQUIDE)
Il existe une constante β > 1, telle que
∀X ∈ G et ∀λ ≥ 0, ρ(λ · X) = λ β · ρ(X) , (10.33)
et plus la constante β est grande, plus les positions de grandes tailles sont pénalisées.
Cet axiome suppose en fait seulement que l’impact de la liquidité limitée du marché est la même pour
tous les actifs. Ceci n’est peut-être pas rigoureusement vrai, mais demeure une très bonne approximation
pour des compagnies de taille comparable (Lillo et al. 2002).
Il faut cependant noter qu’en tant que tel, cet axiome n’est pas compatible avec l’axiome d’invariance
par translation. En effet, considérons le risque ρ(λ(X +α·(1+r))), avec X ∈ G et α et λ deux réels. En
appliquant tout d’abord l’axiome d’invariance par translation, puis l’axiome d’homogénéité en marché
illiquide, on obtient :
ρ(λ(X + α · (1 + r))) = ρ(λX + λα · (1 + r)), (10.34)
= ρ(λX) − λα, (10.35)
= λ β · ρ(X) − λα. (10.36)
Si maintenant on fait usage de ces deux axiomes dans l’ordre inverse, il vient :
ρ(λ(X + α · (1 + r))) = λ β · ρ(X + α · (1 + r)), (10.37)
= λ β · ρ(X) − λ β α, (10.38)
ce qui est en contradiction avec le résultat précédent donné par l’équation (10.36)
Donc, si l’on veut restaurer la compatiblité entre l’axiome d’invariance par translation et l’axiome d’homogénéité
en marche illiquide, il faut restreindre ce dernier aux positions purement risquées, ce qui
revient à admettre la parfaite liquidité de l’actif sans risque. Une autre alternative consiste à modifier
l’axiome d’invariance par translation de sorte que pour un investissement de taille α dans l’actif sans
risque ρ(α · (1 + r)) = −α β , auquel cas, on tient aussi compte du risque d’illiquidité pour l’actif sans
risque.
Cette approche n’est donc pas très satisfaisante. Une meilleure solution a tout d’abord été proposée par
Heath (2000) puis par Föllmer et Schied (2002a). Leur idée consiste pour commencer à remarquer que
l’axiome de sous-additivité peut être remplacé par un axiome de convexité :
AXIOME 12 (CONVEXITÉ)
∀(X1, X2) ∈ G × G et ∀λ ∈ [0, 1], ρ(λ X1 + (1 − λ) X2) ≤ λ ρ(X1) + (1 − λ) ρ(X2). (10.39)
Cette substitution est parfaitement légitime puisque pour les fonctions homogènes, convexité et sousadditivité
sont équivalentes. Notons que comme l’axiome de sous-additivité, l’axiome de convexité garantit
que l’agrégation de positions risquées assure leur diversification.
Ainsi, les mesures de risque cohérentes peuvent être définies par un ensemble d’axiomes équivalents à
ceux énoncés en section 10.2.1 et qui sont les axiomes d’invariance par translation, d’homogénéité, de
convexité et de monotonie.

10.3. Les mesures de fluctuations 359
Pour prendre en compte le risque de liquidité, Heath (2000) et Föllmer et Schied (2002a) proposent
de rejeter l’axiome d’homogénéité. Moyennant cela, ils définissent un nouvel ensemble de mesures de
risque dites convexes, qui englobent les mesures de risque cohérentes, et pour lesquelles ils donnent un
théorème de représentation :
THÉORÈME 4 (REPRÉSENTATION DES MESURES DE RISQUE CONVEXES)
Une mesure de risque ρ est convexe si et seulement si il existe une famille Q de mesures de probabilité
sur l’espace des états de la nature et une fonctionnelle α sur Q telle que :
où la fonctionnelle α est donnée par
et
ρ(X) = sup
Q∈Q
α(Q) = sup
X∈Aρ
(EQ [−X] − α(Q)) , (10.40)
EQ[−X], (10.41)
Aρ = {X ∈ G | ρ(X) ≤ 0} . (10.42)
Dans l’énoncé du théorème, nous avons omis le facteur d’actualisation, afin d’alléger l’écriture, mais il
peut être réintroduit de manière évidente.
Ici encore, comme souligné par Föllmer et Schied (2002b), le lien avec la théorie de la décision dans
l’incertain est immédiat, et permet d’ancrer ces mesures de risque convexes dans la théorie de l’utilité et
ainsi de leur donner un sens économique très net. Cependant, nous avons vu que cette théorie conduisait
à prendre des décisions extrêmement pessismistes puisqu’elle ne retient que l’utilité minimale que peut
retirer un agent étant donné l’ensemble des situations qu’il considère. Ce même pessimisme excessif
affecte les mesures de risque convexes (et donc cohérentes) puisque là aussi, le gestionnaire n’est sensible
qu’à la plus grande perte qu’il peut subir.
Enfin, il convient de signaler que la mesure du risque en terme de capital économique demeure insuffisante.
Certes elle garantit, avec un certain niveau de confiance déterminé, que le portefeuille ou l’entreprise
évitera la ruine, ce qui est fondamental du point de vue du régulateur, mais si l’on se place du
point de vue du gestionnaire de fond ou d’un éventuel investisseur, cela ne suffit pas. Il faut aussi être
capable de mesurer les fluctuations autour de l’objectif de rentabilité fixé, c’est-à-dire de la richesse du
portefeuille autour de la richesse espérée (ou richesse moyenne). En effet, la qualité d’un portefeuille se
juge aussi à la régularité de ses performances.
10.3 Les mesures de fluctuations
Comme nous venons de l’exposer, la mesure du risque en terme de capital économique, pour nécessaire
qu’elle soit - elle constitue en fait la première des exigences - ne suffit cependant pas. Il semble en
effet tout-à-fait souhaitable de pouvoir mesurer les fluctuations d’un actif ou d’un portfeuille autour
de sa valeur moyenne ou plus généralement autour d’un objectif de rentabilité préalablement établi.
Les qualités du portefeuille seront alors d’autant meilleures que les fluctuations seront plus petites. Il
convient donc de rechercher les propriétés minimales que doit satisfaire une mesure de fluctuation ρ.
Ceci est exposé dans Malevergne et Sornette (2002c) que nous présenterons au chapitre 14 section 1.2,
et dont nous donnons ici un résumé.
En premier lieu, nous requèrons qu’une mesure de fluctuation soit positive :

360 10. La mesure du risque
AXIOME 13 (POSITIVITÉ)
Soit X ∈ G une grandeur risquée, alors ρ(X) ≥ 0. De plus, ρ(X) = 0 si et seulement si X est non
risquée (ou certain).
En particulier, tout actif sans risque a une mesure de fluctuation égale à zéro, ce qui est bien naturel. De
plus, l’ajout d’une quantité certaine à une valeur risquée ne modifiant en rien les fluctuations de cette
dernière, nous devons avoir :
AXIOME 14 (INVARIANCE PAR TRANSLATION)
∀X ∈ G et ∀µ ∈ R, ρ(X + α) = ρ(X). (10.43)
Enfin, nous demandons que la mesure de fluctuation soit une fonction croissante, et plus spécifiquement
homogéne, de la taille de la position :
AXIOME 15 (POSITIVE HOMOGÉNÉITÉ)
Il existe une constante β ≥ 1, telle que
∀X ∈ G et ∀λ ≥ 0, ρ(λ · X) = λ β · ρ(X). (10.44)
Dans le cas où β égale un, la mesure de fluctuations est extensive par rapport à la taille de la position
mais ne prend alors pas en compte le risque de liquidité.
Les mesures de fluctuations satisfaisant les axiomes 14 et 15 sont connues sous le nom de semi-invariants.
Ils en existent de très nombreux, parmi lesquels on peut citer par exemples les moments centrés
ou les cumulants
µn(X) = E [(X − E[X]) n ] , (10.45)
Cn(X) = 1
i n · n!
d E eikX
dk
k=0
. (10.46)
L’utilisation des moments centrés comme mesure du risque associé aux fluctuations d’un actif n’est pas
nouvelle. Elle remonte au moins à Markovitz (1959) qui choisit d’utiliser la variance (moment centré
d’ordre deux) comme mesure du risque des actifs financiers. Plus tard, Rubinstein (1973) montrera que
les moments centrés d’ordre supérieur à deux - pour autant qu’ils existent (cf. chapitres 1 et 3) - interviennent
de façon naturelle pour quantifier des risques plus grands que ceux pris en compte par la
variance, en les mettant en relation avec la théorie de l’utilité espérée de von Neumann et Morgenstern
(1947). Dans le cas général, c’est-à-dire pour les cumulants ou pour toute autre mesure de fluctuations,
il ne semble cependant pas que l’on puisse trouver de lien avec la théorie de l’utilité.
L’axiome de positivité permet de restreindre les semi-invariants acceptables. En effet, par définition, les
moments centrés d’ordre pairssont positifs, mais ce n’est pas nécessairement le cas pour ceux d’ordre
impair. La situation est beaucoup plus floue pour ce qui est des cumulants, puisqu’aucun résultat général
ne peut être donné concernant leur positivité. En fait tout dépend de la distribution de la variable aléatoire
X.
Partant des moments centrés, il est en fait facile de construire une mesure de fluctuation qui satisfait les
trois axiomes, quelque soit la valeur de β. En effet, il suffit de considérer les moments absolus centrés :
¯µβ(X) = E |X − E[X]| β
, (10.47)

10.4. Conclusion 361
et de manière plus générale : ¯µp β/p . Ceci permet d’ailleurs de construire de façon très simple d’autres
mesures de fluctuations. En effet, il est aisé de montrer que toute somme (positive mais non nécessairement
convexe) de mesures de fluctuations de même degré d’homogénéité β est une mesure de fluctuation de
degré d’homogénéité β. Donc, dans l’esprit des mesures spectrales d’Acerbi (2002), nous pouvons définir
ρ(X) = dα φ(α) E [|X − E[X]| α ] β/α , (10.48)
pourvu que l’intégrale existe. Là encore, la fonction φ permet de quantifier l’aversion du gestionnaire de
risque vis-à-vis des grandes fluctuations.
10.4 Conclusion
Les avancées récentes en matière de théorie de la décision et de mesure de risque, que nous venons
d’exposer, nous permettrons au chapitre suivant de montrer comment mettre en œuvre une gestion de
portefeuille efficace vis-à-vis des grands risques. Toutefois, il convient de garder à l’esprit qu’une des
limites les plus importantes des mesures de risque que nous venons de présenter est de se restreindre à une
prise en compte mono-périodique des risques et de négliger l’approche inter-temporelle, qui est pourtant
fondamentale dans la mesure où les contraintes de risque ont souvent pour objectif d’être intégrées dans
un problème d’optimisation de portefeuille qui généralement est un problème dynamique.
Malheureusement, l’étude dynamique des risques est beaucoup moins avancée car beaucoup plus délicate
à formaliser que l’étude statique. On peut cependant présenter quelques tentatives ayant permis d’appréhender
ce problème. Tout d’abord citons les approches de Dacorogna, Gençay, Müller et Pictet
(2001) et Muzy, Sornette, Delour et Arnéodo (2001) notamment qui, partant de mesures de risques
mono-périodiques, montrent qu’il est possible de construire des mesures “multi-échelles” en moyennant
les mesures de risque mono-périodiques calculées à différentes échelles de temps. Ceci peut paraître
une méthode purement ad hoc, mais a d’une part révélé d’intéressants résultats et d’autre part repose
tout de même sur l’existence d’une cascade causale entre les différentes échelles temporelles (Arnéodo
et al. 1998, Muzy et al. 2001). Plus généralement, Wang (1999) a proposé un ensemble d’axiomes que
les mesures de risques dynamiques doivent satisfaire et a complété ainsi les travaux précédents de Shapiro
et Basak (2000) concernant la maximisation de l’utilité en temps continu ou de Ahn, Boudoukh,
Richardson et Whitelaw (1999) sur l’optimisation de la VaR en temps continu. Enfin, signalons certaines
mesures de risques telles les drawdowns, qui mesurent les pertes cumulées indépendamment de
leur durée, et qui ont suscité récemment un certain regain d’intérêt (Grossman et Zhou 1993, Cvitanic et
Karatzas 1995, Chekhlov, Uryasev et Zabarankin 2000, Johansen et Sornette 2002).

362 10. La mesure du risque

Chapitre 11
Portefeuilles optimaux et équilibre de
marché
On doit à Markovitz (1959) la première formalisation mathématique de la gestion de portefeuille. Celle-ci
est basée sur la nécessité d’accepter un compromis entre l’obtention d’un rendement espéré le plus élevé
possible et d’une quantité de risques encourus la plus faible possible, ce qui conduit tout naturellement
à s’intéresser aux portefeuilles dits optimaux, c’est-à-dire les portefeuilles tels que pour un niveau de
risque donné et un jeu de contraintes à satisfaire - telle que l’absence de vente à découvert, par exemple
- il n’existe pas de portefeuilles de rendement supérieur au rendement du portefeuille optimal. La courbe
représentant l’ensemble des portefeuilles optimaux dans le plan risque / rendement définit la frontière
efficiente au-dessus de laquelle aucun couple risque / rendement ne peut être atteint.
Dans l’approche initiale de Markovitz (1959), le risque est mesuré par la variance (ou déviation standard)
du rendement des actifs. Cela revient à admettre soit que leur distribution (multivariée) est gaussienne
- puisque c’est le seul cas où la variance caractérise complètement les fluctuations des actifs - soit que
la fonction d’utilité des agents est quadratique, auquel cas les décisions des agents ne sont régies que
par les deux premiers moments de la distribution des actifs. Moyennant ces hypothèses, il est possible
de dériver analytiquement la composition des portefeuilles efficients en fonction du seul vecteur des
rendements espérés des actifs et de leur matrice de covariance (Elton et Gruber 1995, par exemple).
Cependant, il est désormais bien admis que la variance ne saurait suffire à quantifier convenablement les
risques puisqu’elle ne rend compte que des petites fluctuations du rendement des actifs autour de leur
valeur moyenne, négligeant ainsi totalement les grands risques dont l’impact est généralement le plus
conséquent. Aussi, est-il important de se tourner vers d’autres mesures de risques ou d’autres critères
d’optimisation.
L’une des premières alternatives proposées à l’analyse moyenne-variance a été de considérer les portefeuilles
dont la moyenne non pas arithmétique mais géométrique des rendements est la plus grande,
car ce sont eux qui ont la probabilité la plus élevée de dépasser un niveau donné de rentabilité, quelque
soit l’intervalle de temps considéré (Brieman 1960, Hakansson 1971, Roll 1973). Cette approche n’est
cependant compatible avec la théorie de la décision que pour des fonctions d’utilité logarithmiques. Une
seconde alternative, dite “Safety First”, consiste à mettre l’accent sur les pertes qu’encourt le portefeuille.
De nombreux critères ont vu le jour, tel le critère de Roy (1952) qui consiste à minimiser la probabilité
de subir une perte supérieure à une valeur prédéterminée, ou encore les critères de Kataoka ou de Telser
(voir Elton et Gruber (1995)) qui sont en fait très proches de critères d’optimisation sous contrainte de
VaR. De manière générale, toute optimisation sous contrainte de capital économique et donc utilisant notamment
les mesures de risques cohérentes s’apparente à cette approche. Enfin, une troisième alternative
363

364 11. Portefeuilles optimaux et équilibre de marché
consiste, pour palier directement aux limitations de l’approche moyenne-variance (Samuelson 1958), à
prendre en compte l’effet des moments d’ordre supérieur, tels la skewness (Arditti 1967, Krauss et Lintzenburger
1976) ou plus généralement les mesures de fluctuations que nous avons présentées au chapitre
précédent.
Outre l’intérêt pratique des méthodes d’optimisation de portefeuilles, celles-ci ont aussi un intérêt théorique,
dans la mesure où la composition des portefeuilles optimaux permet de déduire des relations entre les
prix des actifs et le prix du portefeuille de marché à l’équilibre. Ceci permet alors de généraliser le CAPM
de Sharpe (1964), Lintner (1965) et Mossin (1966) dérivé dans un univers de Markovitz et donc soumis
aux mêmes limites quant à sa validité.
11.1 Les limites de l’approche moyenne - variance
Dans l’approche de Markovitz (1959), le vecteur des rendements espérés et la matrice de corrélation
jouent un rôle crucial. Or, si l’estimation des rendements moyens peut être réalisée avec une assez bonne
précision dans la mesure où les distributions des actifs décroissent plus vite qu’une loi de puissance d’exposant
deux (cf chapitre 1), l’estimation de la matrice de covariance pose beaucoup plus de problèmes,
car son estimation correcte nécessite que la distribution de rendement décroisse plus rapidement qu’une
loi de puissance d’exposant quatre dans la région intermédiaire, ce qui n’est pas le cas.
En effet, lorsque l’on s’intéresse à des portefeuilles de grande taille, la matrice de covariance empirique
est à telle point bruitée, que ses propriétés sont très proches des propriétés universelles de certains
ensembles de matrices aléatoires. Plus précisément, Laloux, Cizeau, Bouchaud et Potters (1999), Laloux,
Cizeau, Bouchaud et Potters (2000) ou encore Plerou, Gopikrishnan, Rosenow, Amaral et Stanley
(1999) ont montré que les distributions de valeurs propres et vecteurs propres de ces matrices, ainsi
que la distribution des écarts entre valeurs propres, étaient très proches de celles des matrices de l’ensemble
de Wishart (1928), à l’exception des quelques plus grandes valeurs propres qui semblent pouvoir
être associées à des facteurs comme le marché ou certains secteurs d’activité. Cependant, comme
suggéré par les résultats de Meerschaert et Scheffler (2001), l’ensemble de Whishart n’est peut-être pas
le plus adapté. En effet, l’ensemble de Whishart est l’ensemble des matrices de corrélations empiriques
dérivées d’échantillons gaussiens. Or, si l’on admet que les queues de distributions des rendements sont
en lois de puissance (ou régulièrement variables) d’exposant de queue inférieur à quatre, les ensembles
de matrice de Lévy (Burda, Janik, Jurkiewicz, Nowak, Papp et Zahed 2002) semblent plus indiqués. Les
résultats de Burda, Jurkiewicz, Nowak, Papp et Zahed (2001a) et Burda, Jurkiewicz, Nowak, Papp et Zahed
(2001b) montrent alors que les matrices de covariances estimées sont encore plus bruitées que prévu
par comparaison à l’ensemble de Whishart, puisque même les plus grandes valeurs ne paraissent pas significatives.
En conséquence, le contenu des matrices de covariances estimées de grandes tailles semble
peu informatif. En fait, nous montrons en annexe de ce chapitre que les plus grandes valeurs propres
peuvent être estimées avec une bonne précision tandis que le cœur de la distribution de valeurs propres
s’écarte sensiblement de la distribution de Wishart. En outre, nous justifions, à l’aide de la théorie des
matrices aléatoires, comment émerge de tout système de grande taille, dont la corrélation moyenne entre
éléments est non nulle, un facteur (ou valeur propre) dominant qui permet à lui seul de rendre compte
de la plus grande partie des interactions (ou corrélations) entre les éléments du système (Malevergne et
Sornette 2002a).
Ceci a d’importantes conséquences quant à la composition et la stabilité dans le temps des portefeuilles
optimaux obtenus à l’aide de ces matrices de corrélations empiriques (Rosenow, Plerou, Gopikrishnan
et Stanley 2001). En fait, l’importance du bruit dépend du contexte. D’une part, elle semble assez faible
pour les portefeuilles optimisés sous contraintes linéaires plutôt que sous contraintes non linéaires (Pafka

11.2. Prise en compte des grands risques 365
et Kondor 2001). D’autre part, cette influence dépend du rapport r = N/T , où N est le nombre d’actifs
dans le portefeuille et T la taille des séries temporelles ayant servi à estimer la matrice de covariance.
Pafka et Kondor (2002) ont montré que pour un rapport supérieur ou de l’ordre de 0.6, le bruit à une
influence primordiale, alors que pour r inférieur à 0.2, son impact devient négligeable. Ceci implique
qu’il est nécessaire de disposer de séries temporelles dont la longueur est au moins cinq fois supérieure à
la taille du portefeuille considéré. Ainsi, pour un portefeuille d’une centaine d’actifs gérés sur la base de
données journalières, cela requièrt des échantillons de cinq cents points, soit deux années de cotations, ce
qui reste très raisonnable. En revanche, pour un portefeuille d’un millier d’actifs, il faut alors des séries
temporelles de cinq mille points, ce qui représente une vingtaine d’années de cotations, et se posent alors
d’autres problèmes tels que celui de la stationnarité de ces données.
Quand bien même ces problèmes pratiques ne se poseraient pas, il faut garder à l’esprit que l’approche
moyenne - variance de Markovitz (1959) ne se révèle adaptée que dans l’hypothèse où les actifs sont
conjointement gaussiens ou dans la mesure où les agents forment des préférences quadratiques dans leur
richesse, ce qui dans un cas comme dans l’autre traduit que l’on ne s’intéresse qu’à des risques de faibles
amplitudes. Dès lors que l’on souhaite intégrer d’autre dimension du risque, c’est-à-dire des risques
associés à de grandes pertes ou de grandes fluctuations, il convient de se tourner vers d’autres mesures
de risques que la variance.
11.2 Prise en compte des grands risques
L’utilisation de nouvelles mesures de risques permettant de prendre en compte les grands risques est
nécessaire à l’obtention de portefeuilles moins sensibles que les portefeuilles moyenne - variance aux
grandes variations de cours. Pour cela, nous devons nous intéresser à des mesures de risques accordant
plus d’importance aux événements rares et de grandes amplitudes. De telles mesures ont été présentées
au chapitre précédent, et selon Tasche (2000) et Malevergne et Sornette (2002c) peuvent être divisées en
deux classes :
– premièrement les mesures de risques associées au capital économique pour lesquelles peuvent être
requises des conditions de cohérence (Artzner et al. 1999) ou de convexité (Heath 2000, Föllmer et
Schied 2002a),
– et deuxièmement les mesures des fluctuations du rendement autour de sa valeur espérée (Malevergne et
Sornette 2002c), ce qui historiquement est la première approche à avoir vu le jour puisque la variance
est une mesure de fluctuation et pas une mesure de capital économique.
En fait, ces deux classes ne suffisent pas à englober toutes les mesures de grands risques : en restant dans
un cadre strictement mono - périodique, on peut au moins citer le coefficient de dépendance de queue qui
permet de quantifier les co-mouvements extrêmes entre actifs (Malevergne et Sornette 2002b), ou encore
la “covariance de queue” utilisée par Bouchaud, Sornette, Walter et Aguilar (1998) pour quantifier le
risque d’un portefeuille d’actifs dont les rendements suivent des lois de puissances, généralisant ainsi
l’approche de Fama (1965b) valable uniquement pour des actifs distribués selon des lois stables. Si l’on
s’autorise à considérer des mesures inter - temporelles, les “drawdowns” (Chekhlov et al. 2000), par
exemple peuvent être pris en compte.
11.2.1 Optimisation sous contrainte de capital économique
L’optimisation d’un portefeuille par rapport à des contraintes portant sur le capital économique conduit
naturellement à s’intéresser aux portefeuilles VaR-efficients (Consigli 2002, Huisman, Koedijk et Pownall
2001, Kaplanski et Kroll 2001a) ou Expected Shortfall-efficients (Frey et McNeil 2002). En effet,

366 11. Portefeuilles optimaux et équilibre de marché
ces deux mesures de risques sont, à l’heure actuelle, les plus employées. Comme il est légitime de le
penser, l’allocation optimale selon des critères moyenne - VaR ou moyenne - ES est très différente de
celle obtenue selon le critère moyenne variance (Alexander et Baptista 2002), ce dernier n’étant pas à
même de tenir compte des grands risques.
D’un point de vue pratique, l’optimisation selon des critères de VaR est délicate pour deux raisons.
D’une part du fait de sa non convexité, qui impose d’avoir recours à des algorithmes de minimisation
non standards (algorithmes génétiques, par exemple) et d’autre part, son estimation dans un cadre non
- paramétrique est généralement difficile car sensible à la méthode utilisée et grosse consommatrice
de temps de calcul 1 . Aussi des méthodes de calculs approximatifs (Tasche et Tibiletti 2001) basées
notamment sur l’application de la théorie des valeurs extrêmes (Longin 2000, Danielson et de Vries
2000, Consigli, Frascella et Sartorelli 2001) ou des approches paramétriques (Malevergne et Sornette
2002d), prennent tout leur sens. L’Expected-Shortfall présente quant à elle l’avantage de satisfaire aux
contraintes de cohérences 2 de Artzner et al. (1999). Ainsi, elle conduit à des problèmes d’optimisation
bien conditionnés pour lesquels des algorithmes de minimisation particulièrement simples et efficaces
existent (Rockafellar et Uryasev 2002).
11.2.2 Optimisation sous contrainte de fluctuations autour du rendement espéré
Le capital économique n’est pas la seule quantité à minimiser et les fluctuations du portefeuille autour de
son rendement moyen ou de tout autre objectif de rentabilité sont aussi à prendre en compte. La variance
réalise cela, mais elle se focalise uniquement sur les écarts à la moyenne de faibles amplitudes et néglige
donc complètement les grands risques. C’est pourquoi il convient d’utiliser d’autres quantités partageant
certaines des propriétés de la variance mais mettant l’emphase sur les grandes fluctuations. Rubinstein
(1973) fut l’un des premiers à s’intéresser à cette approche, suggérant que les moments centrés d’ordre
supérieur à deux ne devaient être négligés puisqu’ils apparaissent naturellement dans le développement
en série de la fonction d’utilité. Plus récemment, Sornette, Andersen et Simonetti (2000) et Sornette,
Simonetti et Andersen (2000) ont émis l’idée que les cumulants pouvaient aussi fournir d’utiles mesures
de fluctuations, permettant notamment de rendre compte du comportement de certains agents globalement
risquophobes dans le sens où ils cherchent à éviter les grands risques mais sont près à accepter un
certain niveau de petits risques (Andersen et Sornette 2002a, Malevergne et Sornette 2002c). Dans le cas
où les distributions marginales des actifs suivent des lois exponentielles étirées (cf chapitre 3) et où la
copule décrivant leur dépendance est gaussienne (cf chapitres 7 et 8), des expressions analytiques ont pu
être dérivées pour l’expression des moments et cumulants des portefeuilles constitués de tels actifs (voir
chapitre 14).
Ces quelques exemples sont des cas particuliers des mesures de fluctuations définies au chapitre précédent
et permettent de dériver simplement la plupart des propriétés générales des portefeuilles optimaux. Ces
propriétés sont en fait des généralisations immédiates, aux cas des grands risques, des propriétés dont
jouissent les portefeuilles moyenne - variance efficients.
11.2.3 Optimisation sous d’autres contraintes
Lorsque l’on souhaite considérer les risques extrêmes systématiques, c’est-à-dire les mouvements extrêmes
que subissent les actifs conjointement avec le marché, le coefficient de dépendance de queue λ peut
1 Ce point est discuté en détail dans Chabaane, Duclos, Laurent, Malevergne et Turpin (2002)
2 Voir Acerbi et Tasche (2002) pour une discussion des propriétés de cohérence de l’Expected-Shortfall selon la définition
adoptée.

11.3. Equilibre de marché 367
s’avérer utile. En effet, comme nous le montrons dans Malevergne et Sornette (2002b), les portefeuilles
constitués d’actifs de faibles λ présentent globalement beaucoup moins de dépendance dans les extrêmes
que les portefeuilles d’actifs ayant individuellement de grands λ.
Lorsque l’on abandonne le strict cadre mono - périodique, les choses se compliquent. L’approche statique
de Markovitz (1959) peut certes être généralisée dans un cadre dynamique (Merton 1992), mais cela nous
ramène à la seule prise en compte des petits risques. Quelques tentatives ont été menées pour tenter de
concilier grand risques et approche inter - temporelle telle par exemple la minimisation des “drawdowns”
(Grossman et Zhou 1993, Cvitanic et Karatzas 1995, Chekhlov et al. 2000), ou encore l’utilisation des
cumulants à différentes échelles temporelles (Muzy et al. 2001).
11.3 Equilibre de marché
L’allocation de capital et le choix des actifs effectués par les agents a bien évidemment une influence sur
le prix de marché de ces actifs. Sous les hypothèses que le marché est parfaitement liquide et en l’absence
de taxe de quelque sorte que ce soit, il est possible de dériver des équations reliant les rendements
espérés de chaque actif au rendement du marché (à l’équilibre). Le premier modèle d’équilibre, ancré
dans l’univers de Markovitz, est le CAPM dérivé par Sharpe (1964), Lintner (1965) et Mossin (1966).
Très tôt, de nombreuses généralisations ont été obtenues pour tenir compte notamment de l’effet des
moments d’ordre supérieur à deux (Jurczenko et Maillet 2002, et les références citées dans cet article),
et ainsi tenter de résoudre l’ “equity premium puzzle”. Les résultats n’étant pas très concluants, d’autres
voies ont été explorées pour rendre compte notamment des implications économiques de l’optimisation
de portefeuille sous d’autres contraintes que la variance, telle la VaR par exemple (Alexander et Baptista
2002, Kaplanski et Kroll 2001b), sans malheureusement beaucoup plus de succès.
Pour notre part, nous avons montré que la relation standard du CAPM reste valable pour les mesures de
fluctuations que nous avons considérées ainsi que dans le cas où les agents n’utilisent pas tous la même
mesure de risque, et donc lorsque le marché est hétérogène (Malevergne et Sornette 2002c), reprenant et
généralisant certains travaux antérieurs de Lintner (1969) ou Gonedes (1976) notamment.
Toutes ces approches demeurent dans le cadre de l’étude mono-périodique de l’équilibre des marchés.
Mais, de même que le problème de choix de portefeuille a reçu certaines extensions multi-périodiques,
des généralisations inter-temporelles du CAPM ont vu le jour dont celles proposées par Fama (1970) ou
Merton (1973) pour ne citer que les plus célèbres, que nous ne faisons que mentionner puisque nous ne
nous y sommes pas du tout intéressés. Comme pour le problème de sélection de portefeuille, nous nous
sommes restreints au cas mono-périodiques.
11.4 Conclusion
Nous avons synthétisé dans ce chapitre les résultats antérieurs obtenus dans le domaine de la gestion de
portefeuille quantitative et les conséquences théoriques qui en découlaient concernant les équilibres de
marchés. Ceci nous a permis de situer les résultats que nous avons obtenus sur ce sujet par rapport à ceux
déjà établis.
Ces résultat seront présentés en détail dans les chapitres suivants, en commençant par les conséquences de
la prise en compte des risques grands et extrêmes, notamment au travers du coefficient de dépendance de
queue (chapitre 12), puis les implications et difficultés de l’optimisation de portefeuille sous contraintes
de mesures de risques (cohérentes ou non) associées au capital économique (chapitre 13) et enfin nous

368 11. Portefeuilles optimaux et équilibre de marché
étudierons les portefeuilles efficients et équilibres de marchés qui en découlent sous contraintes de fluctuations
autour d’un objectif de rentabilité sur le rendement espéré (chapitre 14).
11.5 Annexe
A l’aide de calculs simples et de simulations numériques, nous démontrons l’existence générique d’un
état macroscopique auto-organisé dans tout système de grande taille présentant une corrélation moyenne
non-nulle entre une fraction finie de toutes ces paires d’éléments. Nous montrons que la coexistence
d’un spectre de valeurs propres, prédit par la théorie des matrices aléatoires, et de quelques très grandes
valeurs propres dans les matrices de corrélation empiriques de grande taille résulte d’un effet collectif
des séries temporelles sous-jacentes plutôt que de l’impact de facteurs. Nos résultats, en excellent accord
avec de précédentes études menées sur les matrices de corrélation financières, montrent également que le
cœur du spectre de valeurs propres contient une part significative d’information et rationalise la présence
de facteurs de marchés jusqu’ici introduits de manière ad hoc.

11.5. Annexe 369
Collective Origin of the Coexistence of Apparent RMT Noise
and Factors in Large Sample Correlation Matrices
Y. Malevergne1, 2 1, 3
and D. Sornette
1 Laboratoire de Physique de la Matière Condensée CNRS UMR 6622
Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France
2 Institut de Science Financière et d’Assurances - Université Lyon I
43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
3 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science
University of California, Los Angeles, California 90095, USA
Through simple analytical calculations and numerical simulations, we demonstrate the generic
existence of a self-organized macroscopic state in any large multivariate system possessing nonvanishing
average correlations between a finite fraction of all pairs of elements. The coexistence of
an eigenvalue spectrum predicted by random matrix theory (RMT) and a few very large eigenvalues
in large empirical correlation matrices is shown to result from a bottom-up collective effect of the
underlying time series rather than a top-down impact of factors. Our results, in excellent agreement
with previous results obtained on large financial correlation matrices, show that there is relevant
information also in the bulk of the eigenvalue spectrum and rationalize the presence of market factors
previously introduced in an ad hoc manner.
Since Wigner’s seminal idea to apply random matrix
theory (RMT) to interpret the complex spectrum of energy
levels in nuclear physics [1], RMT has made enormous
progress [2] with many applications in physical sciences
and elsewhere such as in meteorology [3] and image
processing [4]. A new application was proposed a
few years ago to the problem of correlations between financial
assets and to the portfolio optimization problem.
It was shown that, among the eigenvalues and principal
components of the empirical correlation matrix of the
returns of hundreds of asset on the New York Stock Exchange
(NYSE), apart from the few highest eigenvalues,
the marginal distribution of the other eigenvalues and
eigenvectors closely resembles the spectral distribution
of a positive symmetric random matrix with maximum
entropy, suggesting that the correlation matrix does not
contain any specific information beyond these few largest
eigenvalues and eigenvectors [5]. These results apparently
invalidate the standard mean-variance portfolio optimization
theory [6] consecrated by the financial industry
[7] and seemingly support the rationale behind factor
models such as the capital asset pricing model (CAPM)
[8] and the arbitrage pricing theory (APT) [9], where the
correlations between a large number of assets are represented
through a small number of so-called market factors.
Indeed, if the spectrum of eigenvalues of the empirical
covariance or correlation matrices are predicted
by RMT, it seems natural to conclude that there is no
usable information in these matrices and that empirical
covariance matrices should not be used for portfolio optimization.
In contrast, if one detects deviations between
the universal – and therefore non-informative – part of
the spectral properties of empirically estimated covariance
and correlation matrices and those of the relevant
ensemble of random matrices [10], this may quantify the
amount of real information that can be used in portfolio
optimization from the “noise” that should be discarded.
More generally, in many different scientific fields, one
needs to determine the nature and amount of information
contained in large covariance and correlation matrices.
This occurs as soon as one attempts to estimate
very large covariance and correlation matrices in multivariate
dynamics of systems exhibiting non-Gaussian
fluctuations with fat tails and/or long-range time correlations
with intermittency. In such cases, the convergence
of the estimators of the large covariance and correlation
matrices is often too slow for all practical purposes. The
problem becomes even more complex with time-varying
variances and covariances as occurs in systems with heteroskedasticity
[11] or with regime-switching [12]. A
prominent example where such difficulties arise is the
data-assimilation problem in engineering and in meteorology
where forecasting is combined with observations
iteratively through the Kalman filter, based on the estimation
and forward prediction of large covariance matrices
[13].
As we said in the context of financial time series, the
rescuing strategy is to invoke the existence of a few dominant
factors, such as an overall market factor and the
factors related to firm size, firm industry and book-tomarket
equity, thought to embody most of the relevant
dependence structure between the studied time series
[14]. Indeed, there is no doubt that observed equity prices
respond to a wide variety of unanticipated factors, but
there is much weaker evidence that expected returns are
higher for equities that are more sensitive to these factors,
as required by Markowitz’s mean-variance theory,
by the CAPM and the APT [15]. This severe failure of
the most fundamental finance theories could conceivably
be attributable to an inappropriate proxy for the market
portfolio, but nobody has been able to show that this is
really the correct explanation. This remark constitutes

370 11. Portefeuilles optimaux et équilibre de marché
the crux of the problem: the factors invoked to model the
cross-sectional dependence between assets are not known
in general and are either postulated based on economic
intuition in financial studies or obtained as black box results
in the recent analyses using RMT [5].
Here, we show that the existence of factors results from
a collective effect of the assets, similar to the emergence
of a macroscopic self-organization of interacting microscopic
constituents. For this, we unravel the general
physical origin of the large eigenvalues of large covariance
and correlation matrices and provide a complete
understanding of the coexistence of features resembling
properties of random matrices and of large “anomalous”
eigenvalues. Through simple analytical calculations and
numerical simulations, we demonstrate the generic existence
of a self-organized macroscopic state in any large
system possessing non-vanishing average correlations between
a finite fraction of all pairs of elements.
Let us first consider a large system of size N with correlation
matrix C in which every non-diagonal pairs of
elements exhibits the same correlation coefficient Cij = ρ
for i = j and Cii = 1. Its eigenvalues are
λ1 = 1 + (N − 1)ρ and λi≥2 = 1 − ρ (1)
with multiplicity N − 1 and with ρ ∈ (0, 1) in order for
the correlation matrix to remain positive definite. Thus,
in the thermodynamics limit N → ∞, even for a weak
positive correlation ρ → 0 (with ρN ≫ 1), a very large
eigenvalue appears, associated with the delocalized eigenvector
v1 = (1/ √ N)(1, 1, · · · , 1), which dominates completely
the correlation structure of the system. This trivial
example stresses that the key point for the emergence
of a large eigenvalue is not the strength of the correlations,
provided that they do not vanish, but the large size
N of the system.
This result (1) still holds qualitatively when the correlation
coefficients are all distinct. To see this, it is convenient
to use a perturbation approach. We thus add a
small random component to each correlation coefficient:
Cij = ρ + ɛ · aij for i = j , (2)
where the coefficients aij = aji have zero mean, variance
σ 2 and are independently distributed (There are additional
constraints on the support of the distribution of
the aij’s in order for the matrix Cij to remain positive
definite with probability one). The determination of the
eigenvalues and eigenfunctions of Cij is performed using
the perturbation theory developed in quantum mechanics
[16] up to the second order in ɛ. We find that the
largest eigenvalue becomes
E[λ1] = (N −1)ρ+1+
(N − 1)(N − 2)
N 2 · ɛ2σ2 ρ +O(ɛ3 ) (3)
while, at the same order, the corresponding eigenvector
v1 remains unchanged. The degeneracy of the eigenvalue
12
10
8
6
4
2
0
0 0.5 1 1.5 2 2.5 3
12
10
8
6
4
2
largest
eigenvalue
0
0 20 40 60
FIG. 1: Spectrum of eigenvalues of a random correlation matrix
with average correlation coefficient ρ = 0.14 and standard
deviation of the correlation coefficients σ = 0.345/ √ N. The
size N = 406 of the matrix is the same as in previous studies
[5] for the sake of comparison. The continuous curve is
the theoretical translated semi-circle distribution of eigenvalues
describing the bulk of the distribution which passes the
Kolmogorov test. The center value λ = 1 − ρ ensures the
conservation of the trace equal to N. There is no adjustable
parameter. The inset represents the whole spectrum with the
largest eigenvalue whose size is in agreement with the prediction
ρN = 56.8.
λ = 1 − ρ is broken and leads to a complex set of smaller
eigenvalues described below.
In fact, this result (3) can be generalized to the nonperturbative
domain of any correlation matrix with independent
random coefficients Cij, provided that they have
the same mean value ρ and variance σ 2 . Indeed, it has
been shown [17] that, in such a case, the expectations of
the largest and second largest eigenvalues are
E[λ1] = (N − 1) · ρ + 1 + σ 2 /ρ + o(1) , (4)
E[λ2] ≤ 2σ √ N + O(N 1/3 log N) . (5)
Moreover, the statistical fluctuations of these two largest
eigenvalues are asymptotically (for large fluctuations t >
O( √ N)) bounded by a Gaussian distribution according
to the following large deviation theorem
Pr{|λ1,2 − E[λ1,2]| ≥ t} ≤ e −c1,2t2
2
, (6)
for some positive constant c1,2 [18].
This result is very different from that obtained when
the mean value ρ vanishes. In such a case, the distribution
of eigenvalues of the random matrix C is given by
the semi-circle law [2]. However, due to the presence of
the ones on the main diagonal of the correlation matrix
C, the center of the circle is not at the origin but at the

11.5. Annexe 371
point λ = 1. Thus, the distribution of the eigenvalues of
random correlation matrices with zero mean correlation
coefficients is a semi-circle of radius 2σ √ N centered at
λ = 1.
The result (4) is deeply related to the so-called “friendship
theorem” in mathematical graph theory, which
states that, in any finite graph such that any two vertices
have exactly one common neighbor, there is one
and only one vertex adjacent to all other vertices [19].
A more heuristic but equivalent statement is that, in a
group of people such that any pair of persons have exactly
one common friend, there is always one person (the
“politician”) who is the friend of everybody. The connection
is established by taking the non-diagonal entries Cij
(i = j) equal to Bernouilli random variable with parameter
ρ, that is, P r[Cij = 1] = ρ and P r[Cij = 0] = 1 − ρ.
Then, the matrix Cij − I, where I is the unit matrix,
becomes nothing but the adjacency matrix of the random
graph G(N, ρ) [18]. The proof of [19] of the “friendship
theorem” indeed relies on the N-dependence of the
largest eigenvalue and on the √ N-dependence of the second
largest eigenvalue of Cij as given by (4) and (5).
Figure 1 shows the distribution of eigenvalues of a
random correlation matrix. The inset shows the largest
eigenvalue lying at the predicting size ρN = 56.8, while
the bulk of the eigenvalues are much smaller and are described
by a modified semi-circle law centered on λ =
1 − ρ, in the limit of large N. The result on the largest
eigenvalue emerging from the collective effect of the crosscorrelation
between all N(N −1)/2 pairs provides a novel
perspective to the observation [20] that the only reasonable
explanation for the simultaneous crash of 23 stock
markets worldwide in October 1987 is the impact of a
world market factor: according to our demonstration,
the simultaneous occurrence of significant correlations
between the markets worldwide is bound to lead to the
existence of an extremely large eigenvalue, the world market
factor constructed by ... a linear combination of the
23 stock markets! What our result shows is that invoking
factors to explain the cross-sectional structure of stock returns
is cursed by the chicken-and-egg problem: factors
exist because stocks are correlated; stocks are correlated
because of common factors impacting them [24].
Figure 2 shows the eigenvalues distribution of the sample
correlation matrix reconstructed by sampling N =
406 time series of length T = 1309 generated with a given
correlation matrix C with theoretical spectrum shown in
figure 1. The largest eigenvalue is again very close to
the prediction ρN = 56.8 while the bulk of the distribution
departs very strongly from the semi-circle law and
is not far from the Wishart prediction, as expected from
the definition of the Wishart ensemble as the ensemble of
sample covariance matrices of Gaussian distributed time
series with unit variance and zero mean. A Kolmogorov
test shows however that the bulk of the spectrum (renormalized
so as to take into account the presence of the
frequency
14
12
10
8
6
4
2
0
0 20 40 60
λ
0
0 0.5 1 1.5 2 2.5
λ
3 3.5 4 4.5 5
frequency
14
12
10
8
6
4
2
largest
eigenvalue
FIG. 2: Spectrum estimated from the sample correlation matrix
obtained from N = 406 time series of length T = 1309
(the same length as in [5]) with the same theoretical correlation
matrix as that presented in figure 1.
outlier eigenvalue) is not in the Wishart class, in contradiction
with previous claims lacking formal statistical
tests [5]. This result holds for different simulations of the
sample correlation matrix and different realizations of the
theoretical correlation matrix with the same parameters
(ρ, σ). The statistically significant departure from the
Wishart prediction implies that there is actually some
information in the bulk of the spectrum of eigenvalues,
which can be retrieved using Marsili’s procedure [10]. We
have also checked that these results remain robust for
non-Gaussian distribution of returns as long as the second
moments exist. Indeed, correlated time series with
multivariate Gaussian or Student distributions with three
degrees of freedom (which provide more acceptable proxies
for financial time series [21]) give no discernible differences
in the spectrum of eigenvalues. This is surprising
as the estimator of a correlation coefficient is asymptotically
Gaussian for time series with finite fourth moment
and Lévy stable otherwise [22].
Empirically [5], a few other eigenvalues below the
largest one have an amplitude of the order of 5 − 10
that deviate significantly from the bulk of the distribution.
Our analysis provides a very simple constructive
mechanism for them, justifying the postulated model of
Ref.[23]. The solution consists in considering, as a first
approximation, the block diagonal matrix C ′ with diagonal
elements made of the matrices A1, · · · , Ap of sizes
N1, · · · , Np with Ni = N, constructed according to
(2) such that each matrix Ai has the average correlation
coefficient ρi. When the coefficients of the matrix C ′
outside the matrices Ai are zero, the spectrum of C ′ is
given by the union of all the spectra of the Ai’s, which
3

372 11. Portefeuilles optimaux et équilibre de marché
frequency
14
12
10
8
6
4
2
0
0 20 40 60
λ
0
0 0.5 1 1.5
λ
2 2.5 3
frequency
15
10
5
largest
eigenvalue
FIG. 3: Spectrum of eigenvalues estimated from the sample
correlation matrix of N = 406 time series of length T = 1309.
The times series have been constructed from a multivariate
Gaussian distribution with a correlation matrix made of three
block-diagonal matrices of sizes respectively equal to 130, 140
and 136 and mean correlation coefficients equal to 0.18 for all
of them. The off-diagonal elements are all equal to 0.1. The
same results hold if the off-diagonal elements are random.
are each dominated by a large eigenvalue λ1,i ρi · Ni.
The spectrum of C ′ then exhibits p large eigenvalues.
Each block Ai can be interpreted as a sector of the
economy, including all the companies belonging to a same
industrial branch and the eigenvector associated with
each largest eigenvalue represents the main factor driving
this sector of activity [25]. For similar sector sizes Ni and
average correlation coefficients ρi, the largest eigenvalues
are of the same order of magnitude. In order to recover
a very large unique eigenvalue, we reintroduce some coupling
constants outside the block diagonal matrices. A
well-known result of perturbation theory in quantum mechanics
states that such coupling leads to a repulsion between
the eigenstates, which can be observed in figure 3
where C ′ has been constructed with three block matrices
A1, A2 and A3 and non-zero off-diagonal coupling described
in the figure caption. These values allow us to
quantitatively replicate the empirical finding of Laloux
et al. in [5], where the three first eigenvalues are approximately
λ1 57, λ2 10 and λ3 8. The bulk of the
spectrum (which excludes the three largest eigenvalues)
is similar to the Wishart distribution but again statistically
different from it as tested with a Kolmogorov test.
As a final remark, expressions (3,4) and our numerical
tests for a large variety of correlation matrices show that
the delocalized eigenvector v1 = (1/ √ N)(1, 1, · · · , 1), associated
with the largest eigenvalue is extremely robust
and remains (on average) the same for any large system.
Thus, even for time-varying correlation matrices
(see Drozdz et al. in [5]) – as in finance with important
heteroskedastic effects – the composition of the main
factor remains almost the same. This can be seen as a
generalized limit theorem reflecting the bottom-up organization
of broadly correlated time series.
[1] Wigner, E.P., Ann. Math. 53, 36 (1951).
[2] Mehta, M.L., Random matrices, 2nd ed. (Boston: Academic
Press, 1991).
[3] Santhanam, M.S. and P. K. Patra, Phys. Rev. E 64,
016102 (2001).
[4] Setpunga, A.M. and P.P. Mitra, Phys. Rev E 60, 3389
(1999).
[5] Laloux, L. et al., Phys. Rev. Lett. 83, 1467 (1999);
Plerou, V., et al., Phys. Rev. Lett. 83, 1471 (1999);
Maslov, S., Physica A 301, 397 (2001); Drozdz, S. et al.,
Physica A 287, 440 (2000); Physica A 294, 226 (2001);
Plerou, V. et al., Phys. Rev E 6506 066126 (2002).
[6] Markowitz, H., Portfolio selection (John Wiley and Sons,
New York, 1959).
[7] RiskMetrics Group, RiskMetrics (Technical Document,
NewYork: J.P. Morgan/Reuters, 1996).
[8] Sharpe, W.F., J. Finance (September), 425 (1964); Lintner,
J., Rev. Econ. Stat. (February), 13 (1965); Mossin,
J., Econometrica (October), 768 (1966); Black, F., J.
Business (July), 444 (1972).
[9] Ross, S.A., J. Economic Theory (December), 341 (1976).
[10] Marsili, M., cond-mat/0003241; Giada, L. and Marsili,
M., Phys. Rev. E art. no. 061101, 6306 N6 PT1:1101,U17-
U23 (2001); T. Guhr and B. Kalber, cond-mat/0206577.
[11] Engle, R.F. and K. Sheppard, NBER Working Paper No.
W8554 (2001).
[12] Schaller, H. and van Norden, S., Appl. Financial Econ.
7, 177 (1997).
[13] Brammer, K., Kalman-Bucy filters (Gerhard Siffling,
Norwood, MA: Artech House, 1989).
[14] Fama, E.F. and Kenneth R., J. Finance 51, 55 (1996); J.
Financial Econ. 33, 3 (1993); Fama, E.F. et al., Financial
Analysts J. 49, 37 (1993).
[15] Roll, R., Financial Management 23, 69 (1994).
[16] Cohen-Tannoudji, C., B. Diu and F. Laloe, Quantum
mechanics (New York: Wiley, 1977).
[17] F¨redi Z. and J. Komlós, Combinatorica 1, 233-241 (1981).
[18] Krivelevich, M. and V. H. Vu, math-ph/0009032 (2000).
[19] Erdos, P. et al., Studia Sci. Math. 1, 215 (1966).
[20] R. Roll, Financial Analysts J. 44, 19 (1988).
[21] Gopikrishnan, P. et al., Eur. Phys. Journal B 3 ¯ , 139
(1998); Guillaume, D.M., et al., Finance and Stochastics
1, 95 (1997); Lux, L., Appl. Financial Economics 6,
463 (1996); Pagan, A., J. Emp. Fin. 3, 15 (1996).
[22] Davis, R.A. and J.E. Marengo, Commun. Statist.-
Stochastic Models 6, 483 (1990); Meerschaert, M.M. and
H.P. Scheffler, J. Time Series Anal. 22, 481 (2001).
[23] Noh, J.D., Phys.Rev.E 61, 5981 (2000).
[24] see D. Sornette et al., in press in Risk, condmat/0204626,
for a mechanism and empirical results contrasting
the endogenous character of the October 1987
crash and other large endogenous market moves.
[25] Mantegna, R.N., Eur. Phys. J. 11, 193 (1999); Marsili,
M., Quant. Fin. 2, 297 (2002).
4

Chapitre 12
Gestion des risques grands et extrêmes
Dans la première partie de ce chapitre, nous exposons nos idées concernant la gestion des risques
extrêmes mais aussi des risques “intermédiaires”, si l’on peut dire, que nous qualifions de grands risques
dans le sens où leur impact est bien plus important que celui associé aux risques quantifiés par la variance
mais reste très inférieur de part leur conséquences aux risques extrêmes. Cela nous permet donc
de discuter des moyens d’appréhender toute la gamme des risques : des plus petits aux plus extrêmes.
La deuxième partie du chapitre est exclusivement consacrée aux risques extrêmes et nous nous interrogeons
sur la façon de s’en prémunir. En fait, à cause de l’existence d’une dépendance de queue
entre les actifs, il n’est pas possible de diversifier parfaitement les risques extrêmes par agrégation. On
peut néanmoins construire des portefeuilles dont les actifs ont chacun de très faibles coefficients de
dépendance de queue, ce qui assure au portefeuille une assez faible sensibilité aux grands mouvements
de ses constituants. Pour autant, l’absence des risques extrêmes est un objectif qui semble hors d’atteinte.
Nous nous sommes focalisés sur des portefeuilles “traditionnels”, c’est-à-dire où les ventes à découvert
ne sont pas autorisées. Il est cependant intéressant de se demander si des stratégies mixtes consistant à
détenir des positions longues sur certains actifs et courtes sur d’autres ne permettraient pas de se prémunir
contre les risques extrêmes dans la mesure où les très grandes baisses des uns seraient compensées par
les très grandes baisses (concomitantes) des autres. Cette stratégie fonctionne très bien dans cette configuration,
mais fait apparaître une autre situation critique : celle où les actifs en position longue baissent
et où ceux en position courte montent. Or, des pertes extrêmes concomitantes de hausses extrêmes sont
tout à fait envisageables : que l’on se remémore la figure 7.1 page 187 et la figure 1 page 214 concernant
la copule de Student et son coefficient de dépendance de queue. On y observe que les grandes
hausses concomitantes de grandes pertes ont une probabilité d’occurrence non nulle, même si elle est
systématiquement plus faible que la probabilité d’occurrence simultanée de deux pertes (ou hausses)
extrêmes.
Ainsi, contrairement à l’approche moyenne-variance où les stratégies mixtes permettent de complètement
décorréler le portefeuille du marché, par exemple, elles ne peuvent apporter de réelle solution face au
problème des risques extrêmes, même s’il semble qu’elles présentent une amélioration par rapport aux
stratégies exclusivement en positions longues.
373

374 12. Gestion des risques grands et extrêmes
12.1 Comprendre et gérer les risques grands et extrêmes
L’impact des risques grands et extrêmes sur l’activité financière et le secteur de l’assurance est devenu
si important qu’il ne peut plus être ignoré des gestionnaires de portefeuille. C’est pourquoi nous nous
proposons de synthétiser ici les étapes successives conduisant à une gestion de portefeuille rigoureuse
visant à prendre en compte (1) le comportement sous-exponentiel des distributions de rendement, (2)
les dépendances non-gaussiennes entre actifs et (3) les dépendances temporelles intermittentes amenant
les grandes pertes tout en déployant (4) le concept de risque sur ses différentes dimensions allant des
“petits” risques jusqu’aux risques “extrêmes” afin de définir des fonctions de décision cohérentes et
pratiques permettant d’établir des portefeuilles optimaux.
Reprint from : J.V Andersen, Y. Malevergne et D.Sornette, 2002, Comprendre et gérer les risques grands
et extêmes, Risque 49, 105-110.

12.1. Comprendre et gérer les risques grands et extrêmes 375
COMPRENDRE ET GÉRER
LES RISQUES GRANDS ET EXTRÊMES
Jorgen V. Andersen
Chargé de recherche à l’université Paris X-Nanterre (Thema)
et à l’université de Nice Sophia-Antipolis
Yannick Malevergne
Doctorant à l’université de Nice Sophia-Antipolis
et à l’université Lyon I (ISFA)
Didier Sornette
Directeur de recherche à l’université de Nice Sophia-Antipolis
et professeur à l’université de Californie, Los Angeles (Ucla)
L’impact des risques grands et extrêmes sur l’activité financière et le secteur
de l’assurance est devenu si important qu’il ne peut plus être ignoré des gestionnaires
de portefeuille. C’est pourquoi nous nous proposons de synthétiser
ici les étapes successives conduisant à une gestion de portefeuille rigoureuse
visant à prendre en compte (1) le comportement sous-exponentiel des distributions
de rendement, (2) les dépendances non-gaussiennes entre actifs et (3)
les dépendances temporelles intermittentes amenant les grandes pertes tout en
déployant (4) le concept de risque sur ses différentes dimensions allant des
“petits” risques jusqu’aux risques “extrêmes” afin de définir des fonctions de
décision cohérentes et pratiques permettant d’établir des portefeuilles optimaux.
La capitalisation totale des marchés financiers à
travers le monde a considérablement augmenté
depuis le début des années 1980. En effet, alors
qu’elle ne représente que 3380 milliards de dollars
en 1983, soit 4 fois le budget annuel des
États-Unis d’Amérique, elle atteint, en 1999, le
chiffre de 38700 milliards de dollars, soit 22 fois
budget annuel des États-Unis, pour cette annéelà.
Ainsi, en moins de vingt ans, la capitalisation
boursière mondiale est passée de 4 fois à 22
fois le budget des Etats-Unis ! Rien qu’en ce qui
concerne la dernière décennie, la capitalisation
boursière et les volumes échangés ont triplé, alors
que le volume d’actions émises a été multiplié par
six. De plus, la volatilité a connu une croissance
significative depuis le début des années 1990,
surtout pour les marchés intégrant des sociétés
fortement centrées sur le secteur des technologies
de l’information (indice américain Nasdaq ou
finlandais Helsinki General (Hex) par exemple).
La même tendance, certes moins prononcée, est
également visible sur des marchés plus traditionnels
comme par exemple le Cac40 (bourse de Paris),
le Dow Jones (Bourse de New York) ou le
Ftse100 (bourse de Londres).
Cette intense et lucrative activité financière est
cependant tempérée par quelques rares mais très
violentes secousses. En effet, l’éclatement des
bulles spéculatives de la fin des années 1990,

376 12. Gestion des risques grands et extrêmes
ainsi que deux années de tourmentes sur les
marchés financiers, ont fait fondre la capitalisation
boursière mondiale de plus de 30% par rapport
à son niveau de 1999, pour la ramener à
un montant de 25100 milliards de dollars. Un
autre crash d’une telle ampleur, se déclenchant
simultanément (comme en octobre 1987) dans la
plupart des bourses mondiales, amènerait encore
une perte quasi-instantanée de près de 7500 milliards
de dollars. Ainsi, de par les sommes astronomiques
qu’ils engloutissent, les crashs financiers
peuvent anéantir en quelques instants les
plus gros fonds d’investissement, ruinant, par la
même, des années d’épargne et de financement
de retraite. Ce pourrait-il même qu’ils soient,
comme en 1929-33 après le grand crash d’octobre
1929, les précurseurs ou les déclencheurs de
récessions majeures ? Voire, qu’ils puissent mener
à un écroulement général des systèmes financiers
et bancaires qui semblent y avoir échappé de justesse
déjà quelques fois dans le passé ?
Les grandes crises et les crashs financiers sont
également fascinants parce-qu’ils personnifient
une classe de phénomènes appelés “phénomènes
extrêmes”. Des recherches récentes en physique,
psychologie, en théorie de jeux ou
encore en sciences cognitives au sens large
suggèrent qu’ils sont les caractéristiques incontournables
de systèmes complexes autoorganisés.
Marchés turbulents, crises, crashs exposent
donc un investisseur à de grands risques
dont la compréhension précise devient essentielle.
Compte-tenu de l’évolution des marchés et de
leurs caractéristiques citées plus haut, il est plus
que jamais dans l’intérêt des gestionnaires de portefeuilles
et des investisseurs en général de comprendre
et de gérer les risques extrêmes.
Distributions
des rendements
à queues épaisses
Le premier pas vers une quantification des grands
risques est d’admettre que les statistiques des
risques – que ce soient les risques de marchés associés
aux fluctuations des actions ou la distri-
bution des remboursements survenant à la suite
de sinistres ou des sinistres eux-mêmes – suivent
des distributions à queues épaisses. Ainsi, le paradigme
gaussien, en vogue en finance jusqu’à une
période relativement récente, n’est plus de mise
aujourd’hui. Sa disparition a laissé le champ libre
à diverses modélisations possibles des risques,
par exemple la modélisation Parétienne, très appliquée
en finance, et la modélisation à l’aide des
distributions dites “exponentielles étirées” que
nous avons développée ces dernières années.
Ces deux classes de distributions sont qualifiées
de sous-exponentielles, c’est-à-dire que la probabilité
d’occurrence d’événements extrêmes est
plus probable qu’avec une distribution exponentielle.
Cela a pour conséquence immédiate que
de telles distributions n’admettent pas de moment
exponentiel, ou pour adopter le langage de la
théorie de la ruine, ces distributions ne satisfont
pas à la condition de Cramér-Lundberg.
Sous l’hypothèse que les rendements sont distribués
de manière identique et indépendante ou
ne possèdent qu’une faible dépendance, ces distributions
caractérisent complètement les risques.
Cela a l’énorme avantage de permettre d’établir
des lois de comportement universelles, liées à
certains théorèmes de convergence mathématique
tels que la loi des grands nombres, le théorème de
la limite centrale, la théorie des valeurs extrêmes
et des grandes déviations. L’immense majorité des
théories sur la gestion des risques, établies aussi
bien en finance qu’en assurance, est fondée sur
cette hypothèse d’indépendance.
Dépendance temporelle
intermittente à l’origine
des grandes pertes
Ce premier pas s’avère en fait très insuffisant
pour apprécier toute la dimension des risques
réels encourus. En effet, des études récentes
indiquent que l’hypothèse d’indépendance des
rendements sur des périodes successives (par
exemple journalières) tombe en défaut lors de
grands mouvements qui s’avèrent persistants :

12.1. Comprendre et gérer les risques grands et extrêmes 377
la distribution des drawdowns (ou somme de
pertes quotidiennes successives) d’un actif, que
ce soit d’un indice financier, du taux de change
entre deux monnaies ou de la côte d’une action,
présente un comportement anormal pour les très
grands drawdowns. Autrement dit, les très grands
drawdowns n’appartiennent pas à la même population
que le reste de la statistique observée et font
apparaître d’importantes corrélations sérielles qui
les rend beaucoup plus probables.
Les distributions Parétiennes ou en exponentielles
étirées sont insuffisantes pour quantifier
ces grands risques intermittents, que nous appelons
“outliers”, pour faire référence au vocable
statistique désignant des occurrences anormales,
distinctes du reste de la population. La
nature outlier des évènements extrêmes semble
ne pas se confiner aux systèmes financiers
mais a été proposée également pour la rupture
catastrophique de matériaux et de structures
industrielles, les tremblements de terre,
les catastrophes météorologiques et enfin divers
phénomènes biologiques et sociaux. Les crises
extrêmes semblent donc résulter de mécanismes
amplificateurs spécifiques signalant probablement
des phénomènes coopératifs. Ainsi, des études
comportementales, dans lesquelles l’économie
dite “cognitive” tient un grand rôle, permettent
d’associer ces corrélations sérielles intermittentes
concomitantes des grandes pertes à certains comportements
des acteurs économiques, tels que des
effets de paniques et/ou d’imitations.
Dans ce contexte, les mesures de risques réalisées
à partir d’outils standards comme la VaR
(Value-at-Risk) peuvent se révéler totalement
inadéquates. En effet, une perte journalière de 2 %
ou 3 % sur les marchés financiers n’est pas rare,
et ne constitue pas un événement extrême. Mais si
une perte d’une telle ampleur vient à se reproduire
plusieurs jours de suite, qui plus est en s’amplifiant,
la perte cumulée peut alors atteindre 10%,
20% ou même beaucoup plus, ce qui entraîne des
conséquences bien plus dramatiques que ne l’indique
la VaR à l’échelle quotidienne. La prise
en compte de ce type de dépendances sérielles
intermittentes nécessite impérativement le calcul
d’indicateurs de risques à plusieurs échelles temporelles
permettant de couvrir la distribution des
durées de drawdowns, comme le suggère le Comité
de Basle quand il recommande de calculer la
VaR sur un intervalle de dix jours. La distribution
des drawdowns fournit un tel indicateur parmi
d’autres. Notons aussi que de nombreux investisseurs
professionnels attachent une grande importance
aux drawdowns pour caractériser leurs
risques et la qualité d’une stratégie ou d’un portefeuille.
Dépendance de queue
et contagion
Cependant, une gestion des risques digne de ce
nom ne peut se réduire à une étude individuelle de
chaque actif. En effet, l’épine dorsale de la gestion
des risques est la diversification par la constitution
de portefeuilles de risques, aussi bien en assurance
qu’en finance. De la même manière que
le paradigme gaussien est inadéquat pour quantifier
les grands risques des distributions marginales
des rendements, la covariance intervenant dans la
théorie standard du portefeuille ne donne qu’une
idée très limitée des grands risques collectifs. Ces
grands risques de portefeuille résultent en effet de
la conjonction d’effets non-gaussien dans les distributions
marginales à queue épaisse et dans les
dépendances entre actifs.
On peut donner une idée de l’importance des effets
de dépendance non-gaussienne en étudiant la
“dépendance de queue” λ c’est-à-dire la probabilité
pour que l’actif X subisse une perte plus
grande que Xq, associée au quantile q tendant vers
zéro, conditionnée à la réalisation d’une perte de
l’actif Y plus grande que Yq associée au même
quantile q. Il se trouve que λ est nul pour les actifs
à dépendance gaussienne ! Par contre, dans le
cadre des modèles à facteurs, nous avons montré
que seules les distributions sous-exponentielles
(et plus particulièrement Parétiennes) présentent
une dépendance de queue asymptotique (pour q
tendant vers 0). Il est possible d’accéder au paramètre
de dépendance de queue entre deux actifs
par des méthodes qui ne font pas appel à une
détermination statistique directe. Nos tests empiriques
trouvent alors un bon accord entre le ca-

378 12. Gestion des risques grands et extrêmes
librage du coefficient de dépendance de queue et
les grandes pertes réalisées entre les années 1962
et 2000 pour des actions principales et de grands
indices de marché. Conditionné à un grand mouvement
du marché, on peut ainsi déduire la probabilité
que tel ou tel actif subisse une perte du
même ordre.
Nature multidimensionnelle
des risques
Le Graal est de conjuguer la description des
distributions marginales sous-exponentielles avec
les dépendances inter-actifs non-gaussiennes et
idéalement les dépendances temporelles intermittentes
amenant les grandes pertes pour établir un
portefeuille optimal. Le problème est alors que la
notion d’“optimalité” n’est pas évidente à définir
en pratique : si la théorie économique nous dit
de maximiser la fonction d’utilité de l’investisseur,
en réalité nous ne la connaissons pas avec
précision. Le problème se complique par les multiples
dimensions du risque introduites par la nature
non-gaussienne des distributions de rendement
et des dépendances .
Dans une série d’articles, nous avons développé
une théorie du portefeuille reposant sur la caractérisation
des risques par les cumulants de
la distribution des rendements du portefeuille.
Les cumulants, notés cn, s’expriment comme des
combinaisons de moments, et quantifient notamment
l’écart à la gaussienne. De même que les
moments, les cumulants n’existent pas tous pour
les distributions Parétiennes mais sont définis à
tout ordre pour les distributions exponentielles
étirées. En particulier, les cumulants d’ordres un
et deux sont respectivement la moyenne et la variance
des rendements, tandis que les cumulants
d’ordre trois et quatre (après normalisation par
l’écart type) permettent de définir la skewness
et la kurtosis. De façon générale, les cumulants
d’ordres pairs quantifient des risques d’autant plus
grands que l’ordre du cumulant considéré est
élevé, tandis que les cumulants d’ordres impairs
caractérisent la dissymétrie entre les queues positives
et négatives de la distribution des rende-
ments. Plus l’ordre n du cumulant cn considéré
est grand, plus celui-ci accorde d’importance aux
événements extrêmes. L’ordre n des cumulants allant
de 1 à l’infini, varier n revient à étaler ou
développer toutes les dimensions du risque : les
“petits” risques quantifiés par c2 et les “grands”
risques quantifiés par c4 et les cumulants d’ordres
plus élevés.
Notre théorie du portefeuille utilise les distributions
marginales de la famille des exponentielles
étirées et la dépendance entre actif est décrite par
la copule gaussienne. Si on souhaite créer un portefeuille
qui évite les grands risques, on choisit le
poids des actifs de telle manière que les cumulants
c4, c6, c8, etc. soient tous proches de leur
minimum, tout en laissant libre la variance c2 (petits
risques). Cette approche est très différente de
l’approche standard de Markowitz qui se focalise
sur c2 et de plus construit une frontière efficiente
dans l’espace (rendement-variance).
Petits risques, grands
risques et rendement
Pour illustrer l’impact de la décomposition
des risques en “petits” et “grands” risques,
considérons le cas simple d’un portefeuille avec
seulement deux actifs : l’action Chevron et la devise
malaise : le Ringgit. Ces deux actifs ont des
caractéristiques très différentes et illustrent admirablement
un effet surprenant a priori. La figure 1a
montre le rendement quotidien d’un portefeuille
dont la proportion w1, investie dans l’action Chevron,
a été obtenue en minimisant la variance. La
figure 1b donne la solution de la minimisation de
cn vis-à-vis du poids w1. Les lignes des points
horizontaux sont les valeurs maximales du rendement
quotidien dans le cas où l’on optimise c2n,
pour n>1. Les rendements quotidiens pour le portefeuille
de la figure 1b surpassent ces limites,
i.e. le portefeuille de la figure 1a subit plus de
fluctuations de grandes amplitudes.Ces deux figures
illustrent clairement le fait que minimiser
des petits risques peut faire augmenter les grands
risques ! De plus, le gain cumulatif de la figure 1c
montre que le portefeuille de la figure 1b voit son

12.1. Comprendre et gérer les risques grands et extrêmes 379
gain s’accroître considérablement par rapport au
portefeuille standard à la Markovitz. Autrement
dit “on peut avoir le beurre et l’argent du beur-
re” : diminuer les grands risques et augmenter le
profit !
Rendements quotidiens annualisés (en pourcentage) et gain cumulé pour les deux portefeuilles
correspondants au minimum de la variance (poids Chevron w1 = 0,095) et au minimum des cumulants
c2n, d’ordre 2n > 2 (poids Chevron w1 = 0,38).
Rendement quotiedien
Rendement quotiedien
Richesse cummulée
10
¤
20
20
10
0
¤
20
10
¤
20
10
0
£
6
¦
4
¤
2
0
0
0
0
¡
1000 2000 3000
¤
1000 2000
¡
3000
¡
1000 2000 3000
Le mécanisme de cet effet remarquable est
simple : le Ringgit malais contribue le plus au
cumulant d’ordres élevés (grands risques) et a de
plus un rendement très faible par rapport à celui de
Chevron. Par contre, sa distribution étroite dans la
zone des petits rendements lui donne une faible
variance. L’optimisation à la Markowitz mettra
donc plus de poids sur le Ringgit qui semble apporter
une diversification intéressante du point de
vue de la variance. Mais cela est une illusion dangereuse,
car le risque réel du Ringgit est beaucoup
plus grand que ne le fait croire sa variance. Le cumulant
c4, par exemple, le quantifie clairement et
sa minimisation conduit en conséquence à réduire
le poids de la devise malaisienne dans le porte-
¥
w1=0.095
w1=0.38
¢
4000 5000
¦
4000
¢
5000
¢
4000 5000
£
6000
£
6000
w1=0.38
w1=0.095
£
6000
7000
§
7000
7000
feuille. Du coup, les grands risques sont réduits.
Comme le Ringgit n’a que peu de rendement, on
gagne alors sur les deux tableaux car alors le rendement
augmente nettement.
Bibliographie
Andersen, J. V. and D. Sornette (2001) Have your
cake and eat it too : increasing returns while lowering
large risks ! Journal of Risk Finance 2 (3),
70-82.
Embrechts P., C. Klüppelberg et T. Mikosch
(1997), Modelling Extremal Events for Insurance
(a)
(b)
(c)

380 12. Gestion des risques grands et extrêmes
and Finance (Springer, New York).
Malevergne, Y. and D. Sornette (2001) General
framework for a portfolio theory with non-
Gaussian risks and non-linear correlations, paper
presented at the 18th International conference in
Finance, June 2001, Namur, Belgium (e-print at
http : //arXiv.org/abs/cond−mat/0103020).
Malevergne, Y. and D. Sornette (2002) Tail Dependence
of Factor Models, working paper (http :
//arXiv.org/abs/cond − mat/0202356).
Johansen, A. et D. Sornette (2002), “Large
Price Drawdowns Are Outliers”, Journal of
Risk 4 (2), (http : //arXiv.org/abs/cond −
mat/0010050).
Sornette, D. (1999) Complexity, catastrophe and
physics, Physics World 12 (N12), 57-57.
Sornette, D. (2002) Predictability of catastrophic
events : material rupture, earthquakes, turbulence,
financial crashes and human birth, Proceedings
of the National Academy of Sciences USA, vol.
4, (e-print at http : //arXiv.org/abs/cond −
mat/0107173).
Sornette, D., P. Simonetti and J. V. Andersen
(2000) φ q -field theory for Portfolio optimization :
``fat tails” and non-linear correlations, Physics
Report 335 (2), 19-92.
Zajdenweber, D., Economie des extrêmes (Flammarion,
2000).

12.2. Minimiser l’impact des grands co-mouvements 381
12.2 Minimiser l’impact des grands co-mouvements
A l’aide des résultats exposés au chapitre 9 section 9.2, nous montrons comment le coefficient de
dépendance de queue entre un actif et un de ces facteurs explicatifs (le marché par exemple) ou entre
deux actifs, peut facilement être calibré. Nous construisons ensuite des portefeuilles composés d’actifs
ayant de très faibles coefficients de dépendance de queue avec le marché et montrons qu’ils présentent
une corrélation remarquablement plus faible que des portefeuilles composés d’actifs ayant une plus forte
dépendance de queue avec le marché, et ce sans dégradation de la performance mesurée par le ratio de
Sharpe.
Reprint from : Y. Malevergne et D.Sornette (2002), Minimizing extremes, RISK 15(11), 129-134.

382 12. Gestion des risques grands et extrêmes
M
ore than 100 years ago, Vilfred Pareto discovered a statistical relationship,
now known as the 80-20 rule, that manifests itself over
and over in large systems: “In any series of elements to be controlled,
a selected small fraction, in terms of numbers of elements, always
accounts for a large fraction in terms of effect.” The stock market is no exception:
events occurring over a very small fraction of the total invested
time may account for most of the gains and/or losses. Diversifying away
such large risks requires novel approaches to portfolio management, which
must take into account the non-Gaussian fat tail structure of distributions
of returns and their dependence. Recent economic shocks and crashes
have shown that standard portfolio diversification works well in normal
times but may break down in stressful times, precisely when diversification
is most important. One could say that diversification works when one
does not really need it and may fail severely when it is most needed.
Technically, the question boils down to whether large price movements
occur mainly in an isolated manner or in a co-ordinated way. This question
is vital for fund managers who take advantage of the diversification
to minimise their risks. Here, we introduce a new technique to quantify
and empirically estimate the propensity for assets to exhibit extreme comovements,
through the use of the so-called coefficient of tail dependence.
Using a factor model framework and tools from extreme value
theory, we provide novel analytical formulas for the coefficient of tail dependence
between arbitrary assets, which yields an efficient non-parametric
estimator. We then construct portfolios of stocks with minimal tail
dependence with the market represented by the S&P 500, and show that
their superior behaviour in stressed times comes together with qualities
in terms of Sharpe ratio and standard quality measures that are at least as
good as standard portfolios.
Assessing large co-movements
Standard estimators of the dependence between assets include the correlation
coefficient and the Spearman’s rank correlation. However, as stressed
by Embrechts, McNeil & Straumann (1999), these kind of dependence measures
suffer from many deficiencies. Moreover, their values are mostly controlled
by relatively small moves of the asset prices around their mean. To
solve this problem, it has been proposed to use the correlation coefficients
conditioned on large movements of the assets. But Boyer, Gibson & Lauretan
(1997) have emphasised that this approach suffers also from a severe
systematic bias leading to spurious strategies: the conditional
correlation in general evolves with time even when the true non-conditional
correlation remains constant. In fact, Malevergne & Sornette (2002a)
have shown that any approach based on conditional dependence measures
implies a spurious change of the intrinsic value of the dependence,
measured for instance by copulas. Recall that the copula of several random
variables is the (unique) function (for continuous marginals) that completely
embodies the dependence between these variables, irrespective of
their marginal behaviour (see Nelsen, 1998, for a mathematical description
of the notion of copula).
In view of these limitations of the standard statistical tools, it is natural
to turn to extreme value theory. In the univariate case, extreme value theory
is very useful and provides many tools for investigating the extreme
Portfolio tail risk l
tails of distributions of assets’ returns. These new developments rest on
the existence of a few fundamental results on extremes, such as the Gnedenko-Pickands-Balkema-de
Haan theorem, which gives a general expression
for the conditional distribution of exceedance over a large
threshold. In this framework, the study of large and extreme co-movements
requires the multivariate extreme values theory, which, in contrast with the
univariate case, cannot be used to constrain accurately the distribution of
large co-movements since the class of limiting extreme-value distributions
is too broad.
In the spirit of the mean-variance portfolio or of utility theory, which
establish an investment decision on a unique risk measure, we use the coefficient
of tail dependence, which, to our knowledge, was first introduced
in a financial context by Embrechts, McNeil & Straumann (2002). The coefficient
of tail dependence between assets Xi and Xj is a very natural and
easily comprehensible measure of extreme co-movements. It is defined as
the probability that the asset Xi incurs a large loss (or gain) assuming that
the asset Xj also undergoes a large loss (or gain) at the same probability
level, in the limit where this probability level explores the extreme tails of
the distribution of returns of the two assets. Mathematically speaking, the
coefficient of lower tail dependence between the two assets Xi and Xj , denoted
by λ _
ij , is defined by:
−
λij u→0
−1 −1
{ Xi Fi u Xj Fj u }
= lim Pr < ( ) < ( )
where F i –1 (u) and Fj –1 (u) represent the quantiles of assets Xi and X j at the
level u. Similarly, the coefficient of upper tail dependence is:
{ }
+
−1 −1
= > ( ) > ( )
Cutting edge
Minimising extremes
Portfolio diversification often breaks down in stressed market environments, but the comovement
of asset prices in a tail risk regime may be modelled using a coefficient of tail
dependence. Here, Yannick Malevergne and Didier Sornette show how such coefficients can
be estimated analytically using the parameters of factor models, while avoiding the problem
of under-sampling of extreme values
(2)
λ _
ij (respectively λ+ ij ) is of concern to investors with long (respectively
short) positions. We refer to Coles, Heffernan & Tawn (1999) and references
therein for a survey of the properties of the coefficient of tail dependence.
Let us stress that the use of quantiles in the definition of λ _
ij
and λ + ij makes them independent of the marginal distribution of the asset
returns. As a consequence, the tail dependence parameters are intrinsic
dependence measures. The obvious gain is an ‘orthogonal’ decomposition
of the risks into (1) individual risks carried by the marginal distribution
of each asset and (2) their collective risk described by their
dependence structure or copula.
Being a probability, the coefficient of tail dependence varies between
zero and one. A large value of λ _
ij means that large losses are more likely
to occur together. Then, large risks cannot be diversified away and the assets
crash together. This investor and portfolio manager nightmare is further
amplified in real life situations by the limited liquidity of markets.
When λ _
λij lim Pr Xi Fi u Xj Fj u
u→1
ij vanishes, these assets are said to be asymptotically independent,
but this term hides the subtlety that the assets can still present a non-zero
dependence in their tails. For instance, two assets with a bivariate normal
distribution with correlation coefficient less than one can be shown to have
a vanishing coefficient of tail dependence. Nevertheless, unless their correlation
coefficient is zero, these assets are never independent. Thus, asymptotic
independence must be understood as the weakest dependence
(1)
WWW.RISK.NET ● NOVEMBER 2002 RISK 129

12.2. Minimiser l’impact des grands co-mouvements 383
Cutting edge l
Portfolio tail risk
1. Tail dependence versus correlation
λ
λ
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
–1.0 –0.8 –0.6 –0.4 –0.2 0
ρ
0.2 0.4 0.6 0.8 1.0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Tail index ν = 3
Tail index ν = 10
0
–1.0 –0.8 –0.6 –0.4 –0.2 0
ρ
0.2 0.4 0.6 0.8 1.0
Evolution as a function of the correlation coefficient ρ of the coefficient
of tail dependence for an elliptical bivariate Student distribution (solid
line) and for the additive factor model with Student factor and noise
(dashed line)
that can be quantified by the coefficient of tail dependence (for other details,
the reader is referred to Ledford & Tawn, 1998).
For practical implementations, a direct application of the definitions (1)
and (2) fails to provide reasonable estimations due to the double curse of
dimensionality and under-sampling of extreme values, so that a fully nonparametric
approach is not reliable. It turns out to be possible to circumvent
this fundamental difficulty by considering the general class of factor
models, which are among the most widespread and versatile models in finance.
They come in two classes: multiplicative and additive factor models
respectively. The multiplicative factor models are generally used to
model asset fluctuations due to an underlying stochastic volatility (see, for
example, Hull & White, 1987, and Taylor, 1994, for a survey of the properties
of these models). The additive factor models are made to relate asset
fluctuations to market fluctuations, as in the capital asset pricing model
and its generalisations (see, for example, Sharpe, 1964, and Rubinstein,
1973), or to any set of common factors as in Ross’ (1976) arbitrage pricing
theory. The coefficient of tail dependence is known in closed form for both
classes of factor models, which allows, as we shall see, for an efficient empirical
estimation.
Tail dependence generated by factor models
We first examine multiplicative factor models, which account for most of
the stylised facts observed on financial time series. A multivariate stochastic
volatility model with a common stochastic volatility factor can be
written as:
130 RISK NOVEMBER 2002 ● WWW.RISK.NET
where σ is a positive random variable modelling the volatility, Y is a Gaussian
random vector, independent of σ, and X is the vector of asset returns.
In this framework, the multivariate distribution of asset returns X is an elliptical
multivariate distribution. For instance, if the inverse of the square
of the volatility 1/σ2 is a constant times a χ2-distributed random variable
with ν degrees of freedom, the distribution of asset returns will be the Student
distribution with ν degrees of freedom. When the volatility follows
Arch or Garch processes, the asset returns are also elliptically distributed
with fat-tailed marginal distributions. Thus, any asset Xi is asymptotically
distributed according to a regularly varying distribution1 : Pr{|Xi | > x} ~ L(x)
× x –ν , where L(⋅) denotes a slowly varying function, with the same exponent
ν for all assets, due to the ellipticity of their multivariate distribution.
Hult & Lindskog (2002) have shown that the necessary and sufficient
condition for any two assets Xi and Xj with an elliptical multivariate distribution
to have a non-vanishing coefficient of tail dependence is that their
distribution be regularly varying. Denoting by ρij the correlation coefficient
between the assets Xi and Xj and by ν the tail index of their distributions,
they obtain:
π / 2
ν
∫
dt cos t
± ( π/ 2−arcsin ρij
) / 2
ν + 1
λij
= = 2I
1+
ρ ,
π / 2
ij
(4)
ν
dt cos t
2 2
1 ⎛ ⎞
⎝
⎜
2⎠
⎟
where the function:
∫
0
X =σY
1
Ix( z, w)=
Bzw ( , )
( )
x z−1
w−1
dt t 1 − t
0
denotes the incomplete beta function. This expression holds for any regularly
varying elliptical distribution, irrespective of the exact shape of the
distribution. Only the tail index is important in the determination of the
coefficient of tail dependence because λ ± ij probes the extreme end of the
tail of the distributions, which all have, roughly speaking, the same behaviour
for regularly varying distributions. In contrast, when the marginal
distributions decay faster than any power law, such as the Gaussian distribution,
the coefficient of tail dependence is zero.
Let us now turn to the second class of additive factor models, whose
introduction in finance goes back at least to the arbitrage pricing theory
(Ross, 1976). They are now widely used in many branches of finance, including
to model stock returns, interest rates and credit risks. Here, we
shall only consider the effect of a single factor, which may represent the
market, for example. This factor will be denoted by Y and its cumulative
distribution by FY . As previously, the vector X is the vector of asset returns
and ε will denote the vector of idiosyncratic noises assumed independent2 of Y. β is the vector whose components are the regression coefficients of
the Xi on the factor Y. Thus, the factor model reads:
(6)
In contrast with multiplicative factor models, the multivariate distribution
of X cannot be obtained in an analytical form, in the general case. In
the particular situation when Y and ε are normally distributed, the multivariate
distribution of X is also normal but this case is not very interesting.
In a sense, additive factor models are richer than the multiplicative ones,
since they give birth to a larger set of distributions of asset returns.
Notwithstanding these difficulties, it turns out to be possible to obtain
the coefficient of tail dependence for any pair of assets Xi and Xj . In a first
step, let us consider the coefficient of tail dependence λ ± i between any
asset Xi and the factor Y itself. Malevergne & Sornette (2002b) have shown
that λ ± X = βY+ ε
i is also identically zero for all rapidly varying factors, that is, for all
factors whose distribution decays faster than any power law, such as the
Gaussian, exponential or gamma laws. When the factor Y has a distribution
that is regularly varying with tail index ν, we have:
1 See Bingham, Goldie & Teugel (1987) for details on regular variations
2 In fact, ε and Y can be weakly dependent (see Malevergne & Sornette, 2002b, for details)
∫
(3)
(5)

384 12. Gestion des risques grands et extrêmes
where:
λ
+
i =
l =
A similar expression holds for λ – i , which is obtained by simply replacing
the limit u → 1 by u → 0 in the definition of l. λ ± i is non-zero as long
as l remains finite, that is, when the tail of the distribution of the factor is
not thinner than the tail of the idiosyncratic noise εi . Therefore, two conditions
must hold for the coefficient of tail dependence to be non-zero:
the factor must be intrinsically ‘wild’ (to use the terminology of Mandelbrot,
1997) so that its distribution is regularly varying; and
the factor must be sufficiently ‘wild’ in its intrinsic variability, so that its
influence is not dominated by the idiosyncratic component of the asset.
Then, the amplitude of λ ± i is determined by the trade-off between the relative
tail behaviours of the factor and the idiosyncratic noise.
As an example, let us consider that the factor and the idiosyncratic noise
follow Student distribution with νY and ν degrees of freedom and scale
εi
factor σY and σ respectively. Expression (7) leads to:
εi
λi = 0 if νY > νεi
λi =
1
1
σεi
ν
if νY = νε= ν
i
+ ( βσ i Y)
−1
lim
u→1
Y
ν
l { 1 β } i
( )
( )
FXu −1
F u
λ = 1 if ν < ν
i Y
The tail dependence decreases when the idiosyncratic volatility increases
relative to the factor volatility. Therefore, λi decreases in periods
of high idiosyncratic volatility and increases in periods of high market
volatility. From the viewpoint of the tail dependence, the volatility of an
asset is not relevant. What is governing extreme co-movement is the relative
weights of the different components of the volatility of the asset.
Figure 1 compares the coefficient of tail dependence as a function of
the correlation coefficient for the bivariate Student distribution (expression
(4)) and for the factor model with the factor and the idiosyncratic noise
following Student distributions (equation (8)). Contrary to the coefficient
of tail dependence of the Student factor model, the tail dependence of the
(elliptical) Student distribution does not vanish for negative correlation coefficients.
For large values of the correlation coefficient, the former is always
larger than the latter.
Once the coefficients of tail dependence between the assets and the
common factor are known, the coefficient of tail dependence between any
two assets Xi and Xj with a common factor Y is simply equal to the weakest
tail dependence between the assets and their common factor:
λ min λ , λ
(9)
= { }
ij i j
This result is very intuitive: since the dependence between the two assets
is due to their common factor, this dependence cannot be stronger than
the weakest dependence between each of the assets and the factor.
Practical implementation and consequences
The two mathematical results (4) and (7) have a very important practical
effect for estimating the coefficient of tail dependence. As we have already
pointed out, its direct estimation is essentially impossible since, by definition,
the number of observations goes to zero as the probability level of
the quantile goes to zero (or one). In contrast, the formulas (4) and (7–9)
tell us that one has just to estimate a tail index and a correlation coefficient.
These estimations can be reasonably accurate because they make
use of a significant part of the data beyond the few extremes targeted by
λ. Moreover, equation (7) does not explicitly assume a power law behaviour,
but only a regularly varying behaviour, which is far more general. In
1
max ,
εi
(7)
(8)
A. Coefficients of tail dependence
Lower tail dependence Upper tail dependence
Bristol-Myers Squibb 0.16 (0.03) 0.14 (0.01)
Chevron 0.05 (0.01) 0.03 (0.01)
Hewlett-Packard 0.13 (0.01) 0.12 (0.01)
Coca-Cola 0.12 (0.01) 0.09 (0.01)
Minnesota Mining & MFG 0.07 (0.01) 0.06 (0.01)
Philip Morris 0.04 (0.01) 0.04 (0.01)
Procter & Gamble 0.12 (0.02) 0.09 (0.01)
Pharmacia 0.06 (0.01) 0.04 (0.01)
Schering-Plough 0.12 (0.01) 0.11 (0.01)
Texaco 0.04 (0.01) 0.03 (0.01)
Texas Instruments 0.17 (0.02) 0.12 (0.01)
Walgreen 0.11 (0.01) 0.09 (0.01)
This table presents the coefficients of lower and upper tail dependence with the S&P
500 index for a set of 12 major stocks traded on the New York Stock Exchange from
January 1991 to December 2000. The numbers in brackets give the estimated standard
deviation of the empirical coefficients of tail dependence
2. Quantile ratio
I = X k,N /Y k,N
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0 0.05 0.10 0.15 0.20 0.25
k/N
^
Empirical estimate l of the quantile ratio l in (7) versus the empirical
^
quantile k/N. We observe a very good stability of l for quantiles
ranging between 0.005 and 0.05
such a case, the empirical quantile ratio l in (7) turns out to be stable
enough for its accurate non-parametric estimation, as shown in figure 2.
As an example, table A presents the results obtained both for the upper
and lower coefficients of tail dependence between several major stocks
and the market factor represented here by the S&P 500 index, over the
past decade. The estimation has been performed under the assumption
that equation (6) holds, rather than under the ellipticality assumption yielding
equation (4). In the present context of the dependence between stocks
and an index (not between two stocks), favouring the factor model is very
reasonable since, according to the financial theory, the market’s return is
well known to be the most important explanatory factor for each individual
asset return. 3 The technical aspects of the method are given in the Appendix.
The coefficient of tail dependence between any two assets is easily
derived from (9). It is interesting to observe that the coefficients of tail dependence
seem almost identical in the lower and the upper tail. Nonetheless,
the coefficient of lower tail dependence is always slightly larger than
the upper one, showing that large losses are more likely to occur together
than large gains.
Two clusters of assets stand out: those with a tail dependence of about
3 In a situation where the common factor cannot be easily identified or estimated, the
ellipticality assumption may provide a useful alternative
WWW.RISK.NET ● NOVEMBER 2002 RISK 131

12.2. Minimiser l’impact des grands co-mouvements 385
Cutting edge l
Portfolio tail risk
3. Portfolios versus market
Portfolio daily return
0.10
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
Portfolio 1
Linear regression of
the portfolio 1 on the
S&P 500 index
Portfolio 2
Linear regression of
the portfolio 2 on the
S&P 500 index
0.10
0.08 0.06 0.04 0.02 0 0.02 0.04 0.06
S&P 500 daily return
Daily returns of two equally weighted portfolios P 1 (made of four stocks
with small λ ≤ 0.06) and P 2 (made of four stocks with large λ ≥ 0.12) as a
function of the daily returns of the S&P 500 from Jan 1991–Dec 2000
10% (or more) and those with a tail dependence of about 5%. Since the
estimation of the tail dependence coefficients has been performed under
the assumption that equation (6) holds, it is interesting to compare the
values with the tail dependence coefficient under the assumption of joint
ellipticality leading to (4). To get a reliable correlation coefficient ρ, we
calculate the Kendall’s tau coefficient τ, use the relation r = sin(πτ/2) and
derive λ from (4), assuming ν = 3 or 4. For ν = 3 (respectively ν = 4), all
λ’s are in the range 0.25–0.30 (respectively 0.20–0.25). Thus, assuming
joint ellipticality, the tail-dependence coefficients between stocks and the
index are much more homogeneous than found with the factor model.
As we show in figure 3, it is clear that the portfolio with stocks with small
tail-dependence coefficients with the index (measured with the factor
model) exhibits significantly less dependence than the portfolio constructed
with stocks with large λ’s (measured with the factor model). This
132 RISK NOVEMBER 2002 ● WWW.RISK.NET
should not be observed if all λ’s are the same as predicted under the assumption
of joint ellipticality.
We now explore some consequences of the existence of stocks with
drastically different tail-dependence coefficients with the index. These
stocks offer the interesting possibility of devising a prudential portfolio that
can be significantly less sensitive to the large market moves. Figure 3 compares
the daily returns of the S&P 500 index with those of two portfolios
P 1 and P 2 . P 1 comprises the four stocks (Chevron, Philip Morris, Pharmacia
and Texaco) with the smallest λ’s while P 2 comprises the four stocks
(Bristol-Meyer Squibb, Hewlett-Packard, Schering-Plough and Texas Instruments)
with the largest λ’s. For each set of stocks, we have constructed
two portfolios, one in which each stock has the same weight 1/4 and
the other with asset weights chosen to minimise the variance of the resulting
portfolio. We find that the results are almost the same between the
equally weighted and minimum-variance portfolios. This makes sense since
the tail-dependence coefficient of a bivariate random vector does not depend
on the variances of the components, which only account for price
moves of moderate amplitudes.
Figure 3 shows the results for the equally weighted portfolios generated
from the two groups of assets. Observe that only one large drop occurs
simultaneously for P 1 and for the S&P 500 index, in contrast with P 2 ,
for which several large drops are associated with the largest drops of the
index and only a few occur desynchronised. The figure clearly shows an
almost circular scatter plot for the large moves of P 1 and the index compared
with a rather narrow ellipse, whose long axis is approximately along
the first diagonal, for the the large returns of P 2 and the index, illustrating
that the small tail dependence between the index and the four stocks in
P 1 automatically implies that their mutual tail dependence is also very small,
according to (9). As a consequence, P 1 offers a better diversification with
respect to large drops than P 2 . This effect, already quite significant for such
small portfolios, should be overwhelming for large ones. The most interesting
result stressed in figure 3 is that optimising for minimum tail dependence
automatically diversifies away the large risks.
These advantages of portfolio P 1 with small tail dependence compared
with portfolio P 2 with large tail dependence with the S&P 500
index come at almost no cost in terms of the daily Sharpe ratio, which
is equal respectively to 0.058 and 0.061 for the equally weighted and
minimum variance P 1 and to 0.069 and 0.071 for the equally weighted
and minimum variance P 2 .
The straight lines represent the linear regression of the two portfolios’

386 12. Gestion des risques grands et extrêmes
returns on the index returns, which shows that there is significantly less
linear correlation between P 1 and the index (correlation coefficient of 0.52
for both the equally weighted and the minimum variance P 1 ) compared
with P 2 and the index (correlation coefficient of 0.73 for the equally weighted
P 2 and of 0.70 for the minimum variance P 2 ). Theoretically, it is possible
to construct two random variables with small correlation coefficient
and large λ and vice versa. Recall that the correlation coefficient and the
tail-dependence coefficient are two opposite end-members of dependence
measures. The correlation coefficient quantifies the dependence between
relatively small moves while the tail-dependence coefficient measures the
dependence during extreme events. The finding that P 1 comes with both
the smallest correlation and the smallest tail-dependence coefficients suggests
that they are not independent properties of assets. This intuition is
in fact explained and encompassed by the factor model since the larger
β is, the larger the correlation coefficient and the larger the tail dependence.
Diversifying away extreme shocks may provide a useful diversification
tool for less extreme dependences, thus improving the potential
usefulness of a strategy of portfolio management based on tail dependence
proposed here.
As a final remark, the almost identical values of the coefficients of tail
dependence for negative and positive tails show that assets that are the
most likely to suffer from the large losses of the market factor are also
those that are the most likely to take advantage of its large gains. This
has the following consequence: minimising the large concomitant losses
between the stocks and the market means renouncing the potential
concomitant large gains. This point is well exemplified by our two portfolios
(see figure 3): P 2 obviously underwent severe negative co-movements
but it also enjoyed large gains with the large positive movements
of the index. In contrast, P 1 is almost completely decoupled from the
large negative movements of the market but is also insensitive to the large
positive movements of the index. Thus, a good dynamic strategy seems
to be: invest in P 1 during bearish or trend-less market phases and prefer
P 2 in a bullish market. ■
Yannick Malevergne is a PhD student at the University of Nice-Sophia
Antipolis and at the ISFA Actuarial School – University of Lyon. Didier
Bingham N, C Goldie and J Teugel, 1987
Regular variation
Cambridge University Press, Cambridge
Boyer B, M Gibson and M Lauretan, 1997
Pitfalls in tests for changes in correlations
International Finance Discussion Paper 597, Board of the Governors of the Federal
Reserve System
Coles S, J Heffernan and J Tawn, 1999
Dependence measures for extreme value analysis
Extremes 2, pages 339–365
Embrechts P, A McNeil and D Straumann, 1999
Correlation: pitfalls and alternatives
Risk May, pages 69–71
Embrechts P, A McNeil and D Straumann, 2002
Correlation and dependence in risk management: properties and pitfalls
In Risk Management: Value at Risk and Beyond, edited by M Dempster, pages
176–223, Cambridge University Press, Cambridge
Hull J and A White, 1987
The option pricing on assets with stochastic volatilities
Journal of Finance 42, pages 281–300
Hult H and F Lindskog, 2002
Multivariate extremes, aggregation and dependence in elliptical distributions
Forthcoming in Advances in Applied Probability 34(3)
Ledford A and J Tawn, 1998
Concomitant tail behaviour for extremes
Advances in Applied Probability 30, pages 197–215
REFERENCES
Appendix: empirical estimation of the
coefficient of tail dependence
We show how to estimate the coefficient of tail dependence
between an asset X and the market factor Y related by the relation
(6) where ε is an idiosyncratic noise uncorrelated with X.
Given a sample of N realisations {X 1 , X 2 , ... , X N } and
{Y 1 , Y 2 , ... , Y N } of X and Y, we first estimate the coefficient β using
the ordinary least square estimator. Let β ^ denote its estimate. Then,
using Hill’s estimator, we obtain the tail index ν^ of the factor Y:
⎡ k 1
ˆ = ⎢ ∑ logY −logY
⎣⎢
k j=
1
νk j, N k, N
where Y 1, N ≥ Y 2, N ≥ ... ≥ Y N, N are the order statistics of the N realisations
of Y. The constant l is non-parametrically estimated with
the formula:
FXu X
l = lim ≃
u→1
−1
F u Y
for k = o(N), which means that k must remain very small with
respect to N but large enough to ensure an accurate determination
of l. Figure 2 presents l ^ as a function of k/N.
Finally, using equation (7), the estimated λ ^ is:
ˆ
max , ˆ
+ 1
λ =
νˆ
{ 1
βˆ
} l
Sornette is a CNRS research director at the University of Nice-Sophia
Antipolis and professor of geophysics at the University of California at
Los Angeles. They acknowledge helpful discussions with Jean-Paul
Laurent. This work was partially supported by the James S McDonnell
Foundation twenty-first century scientist award/studying complex
system. e-mail: Yannick.Malevergne@unice.fr and sornette@unice.fr
Malevergne Y and D Sornette, 2002a
Investigating extreme dependences: concepts and tools
Working paper, available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id =
303465
Malevergne Y and D Sornette, 2002b
Tail dependence of factor models
Working paper, available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id =
301266
Mandelbrot B, 1997
Fractals and scaling in finance: discontinuity, concentration
Springer-Verlag, New York
Nelsen R, 1998
An introduction to copulas
Lectures Notes in Statistics 139, Springer Verlag, New York
Ross S, 1976
The arbitrage theory of capital asset pricing
Journal of Economic Theory 17, pages 254–286
Rubinstein M, 1973
The fundamental theorem of parameter-preference security valuation
Journal of Financial and Quantitative Analysis 8, pages 61–69
Sharpe W, 1964
Capital asset pricing: a theory of market equilibrium under conditions of risk
Journal of Finance 19, pages 425–442
Taylor S, 1994
Modeling stochastic volatility
Mathematical Finance 4, pages 183–204
−1
Y
( )
( )
kN ,
kN ,
⎤
⎥
⎦⎥
WWW.RISK.NET ● NOVEMBER 2002 RISK 133

Chapitre 13
Gestion de portefeuille sous contraintes de
capital économique : l’exemple de la
Value-at-Risk et de l’Expected-Shortfall
A l’aide d’une famille de distributions de Weibull modifiées, englobant à la fois des distributions super
et sous-exponentielles (dont l’interêt a été souligné au chapitre 3), nous paramétrons les distributions
de rendements des actifs financiers. L’hypothèse de copule gaussienne est utilisée pour modéliser la
dépendance entre les actifs. Cela nous permet d’obtenir les expressions analytiques des queues de la
distribution P (S) du rendement S d’un portefeuille composé de tels actifs. Nous montrons que les queues
de la distribution P (S) demeurent asymptotiquement des distributions de Weibull modifiées avec un
facteur d’échelle χ fonction du poids des actifs dans le portefeuille et dont l’expression différe selon
le comportement super ou sous-exponentiel des actifs. Nos traitons alors en détails le problème de la
minimisation du risque pour de tels portefeuilles, les mesures de risque considérées étant la Value-at-
Risk et l’Expected-Shortfall, que nous montrons être asymptotiquement équivalentes dans le cadre de la
représentaion adoptée.
387

388 13. Gestion de portefeuille sous contraintes de capital économique

VaR-Efficient Portfolios for a Class of Super- and
Sub-Exponentially Decaying Assets Return Distributions
Y. Malevergne 1,2 and D. Sornette 1,3
1 Laboratoire de Physique de la Matière Condensée CNRS UMR 6622
Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France
2 Institut de Science Financière et d’Assurances - Université Lyon I
43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex
3 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science
University of California, Los Angeles, California 90095, USA
email: Yannick.Malevergne@unice.fr and sornette@unice.fr
fax: (33) 4 92 07 67 54
Abstract
Using a family of modified Weibull distributions, encompassing both sub-exponentials and super-exponentials,
to parameterize the marginal distributions of asset returns and their multivariate generalizations
with Gaussian copulas, we offer exact formulas for the tails of the distribution P (S) of returns S of a
portfolio of arbitrary composition of these assets. We find that the tail of P (S) is also asymptotically
a modified Weibull distribution with a characteristic scale χ function of the asset weights with different
functional forms depending on the super- or sub-exponential behavior of the marginals and on the
strength of the dependence between the assets. We then treat in details the problem of risk minimization
using the Value-at-Risk and Expected-Shortfall which are shown to be (asymptotically) equivalent in this
framework.
Introduction
In recent years, the Value-at-Risk has become one of the most popular risk assessment tool (Duffie and
Pan 1997, Jorion 1997). The infatuation for this particular risk measure probably comes from a variety of
factors, the most prominent ones being its conceptual simplicity and relevance in addressing the ubiquitous
large risks often inadequately accounted for by the standard volatility, and from its prominent role in the
recommendations of the international banking authorities (Basle Commitee on Banking Supervision 1996,
2001). Moreover, down-side risk measures such as the Value-at-risk seem more in accordance with observed
behavior of economic agents. For instance, according to prospect theory (Kahneman and Tversky 1979), the
perception of downward market movements is not the same as upward movements. This may be reflected
in the so-called leverage effect, first discussed by (Black 1976), who observed that the volatility of a stock
tends to increase when its price drops (see (Fouque et al. 2000, Campbell, Lo and McKinley 1997, Bekaert
and Wu 2000, Bouchaud et al. 2001) for reviews and recent works). Thus, it should be more natural to
consider down-side risk measures like the VaR than the variance traditionally used in portfolio management
(Markowitz 1959) which does not differentiate between positive and negative change in future wealth.
1
389

390 13. Gestion de portefeuille sous contraintes de capital économique
However, the choice of the Value-at-Risk has recently been criticized (Szergö 1999, Danielsson et al. 2001)
due to its lack of coherence in the sense of Artzner et al. (1999), among other reasons. This deficiency
leads to several theoretical and practical problems. Indeed, other than the class of elliptical distributions,
the VaR is not sub-additive (Embrechts et al. 2002a), and may lead to inefficient risk diversification policies
and to severe problems in the practical implementation of portfolio optimization algorithms (see (Chabaane
et al. 2002) for a discussion). Alternative have been proposed in terms of Conditional-VaR or Expected-
Shortfall (Artzner et al. 1999, Acerbi and Tasche 2002, for instance), which enjoy the property of subadditivity.
This ensures that they yield coherent portfolio allocations which can be obtained by the simple
linear optimization algorithm proposed by Rockafellar and Uryasev (2000).
From a practical standpoint, the estimation of the VaR of a portfolio is a strenuous task, requiring large computational
time leading sometimes to disappointing results lacking accuracy and stability. As a consequence,
many approximation methods have been proposed (Tasche and Tilibetti 2001, Embrechts et al. 2002b, for
instance). Empirical models constitute another widely used approach, since they provide a good trade-off
between speed and accuracy.
From a general point of view, the parametric determination of the risks and returns associated with a given
portfolio constituted of N assets is completely embedded in the knowledge of their multivariate distribution
of returns. Indeed, the dependence between random variables is completely described by their joint
distribution. This remark entails the two major problems of portfolio theory: 1) the determination of the
multivariate distribution function of asset returns; 2) the derivation from it of a useful measure of portfolio
risks, in the goal of analyzing and optimizing portfolios. These objective can be easily reached if one can
derive an analytical expression of the portfolio returns distribution from the multivariate distribution of asset
returns.
In the standard Gaussian framework, the multivariate distribution takes the form of an exponential of minus
a quadratic form X ′ Ω −1 X, where X is the uni-column of asset returns and Ω is their covariance matrix.
The beauty and simplicity of the Gaussian case is that the essentially impossible task of determining a
large multidimensional function is collapsed onto the very much simpler one of calculating the N(N +
1)/2 elements of the symmetric covariance matrix. And, by the statibility of the Gaussian distribution,
the risk is then uniquely and completely embodied by the variance of the portfolio return, which is easily
determined from the covariance matrix. This is the basis of Markowitz (1959)’s portfolio theory and of the
CAPM (Sharpe 1964, Lintner 1965, Mossin 1966). The same phenomenon occurs in the stable Paretian
portfolio analysis derived by (Fama 1965) and generalized to separate positive and negative power law tails
(Bouchaud et al. 1998). The stability of the distribution of returns is essentiel to bypass the difficult problem
of determining the decision rules (utility function) of the economic agents since all the risk measures are
equivalent to a single parameter (the variance in the case of a Gaussian universe).
However, it is well-known that the empirical distributions of returns are neither Gaussian nor Lévy Stable
(Lux 1996, Gopikrishnan et al. 1998, Gouriéroux and Jasiak 1998) and the dependences between assets are
only imperfectly accounted for by the covariance matrix (Litterman and Winkelmann 1998). It is thus desirable
to find alternative parameterizations of multivariate distributions of returns which provide reasonably
good approximations of the asset returns distribution and which enjoy asymptotic stability properties in the
tails so as to be relevant for the VaR.
To this aim, section 1 presents a specific parameterization of the marginal distributions in terms of so-called
modified Weibull distributions introduced by Sornette et al. (2000b), which are essentially exponential of
minus a power law. This family of distributions contains both sub-exponential and super-exponentials,
including the Gaussian law as a special case. It is shown that this parameterization is relevant for modeling
the distribution of asset returns in both an unconditional and a conditional framework. The dependence
structure between the asset is described by a Gaussian copula which allows us to describe several degrees of
2

dependence: from independence to comonotonicity. The relevance of the Gaussian copula has been put in
light by several recent studies (Sornette et al. 2000a, Sornette et al. 2000b, Malevergne and Sornette 2001,
Malevergne and Sornette 2002c).
In section 2, we use the multivariate construction based on (i) the modified Weibull marginal distributions
and (ii) the Gaussian copula to derive the asymptotic analytical form of the tail of the distribution of returns
of a portfolio composed of an arbitrary combination of these assets. In the case where individual asset
returns have modified-Weibull distributions, we show that the tail of the distribution of portfolio returns S
is asymptotically of the same form but with a characteristic scale χ function of the asset weights taking
different functional forms depending on the super- or sub-exponential behavior of the marginals and on the
strength of the dependence between the assets. Thus, this particular class of modified-Weibull distributions
enjoys (asymptotically) the same stability properties as the Gaussian or Lévy distributions. The dependence
properties are shown to be embodied in the N(N + 1)/2 elements of a non-linear covariance matrix and the
individual risk of each assets are quantified by the sub- or super-exponential behavior of the marginals.
Section 3 then uses this non-Gaussian nonlinear dependence framework to estimate the Value-at-Risk (VaR)
and the Expected-Shortfall. As in the Gaussian framework, the VaR and the Expected-Shortfall are (asymptotically)
controlled only by the non-linear covariance matrix, leading to their equivalence. More generally,
any risk measure based on the (sufficiently far) tail of the distribution of the portfolio returns are equivalent
since they can be expressed as a function of the non-linear covariance matrix and the weights of the assets
only.
Section 4 uses this set of results to offer an approach to portfolio optimization based on the asymptotic
form of the tail of the distribution of portfolio returns. When possible, we give the analytical formulas of
the explicit composition of the optimal portfolio or suggest the use of reliable algorithms when numerical
calculation is needed.
Section 5 concludes.
Before proceeding with the presentation of our results, we set the notations to derive the basic problem
addressed in this paper, namely to study the distribution of the sum of weighted random variables with given
marginal distributions and dependence. Consider a portfolio with ni shares of asset i of price pi(0) at time
t = 0 whose initial wealth is
N
W (0) = nipi(0) . (1)
A time τ later, the wealth has become W (τ) = N
i=1 nipi(τ) and the wealth variation is
where
δτ W ≡ W (τ) − W (0) =
N
i=1
wi =
i=1
nipi(0) pi(τ) − pi(0)
pi(0)
nipi(0)
N
j=1 njpj(0)
= W (0)
391
N
wixi(t, τ), (2)
is the fraction in capital invested in the ith asset at time 0 and the return xi(t, τ) between time t − τ and t of
asset i is defined as:
xi(t, τ) = pi(t) − pi(t − τ)
.
pi(t − τ)
(4)
Using the definition (4), this justifies us to write the return Sτ of the portfolio over a time interval τ as the
3
i=1
(3)

392 13. Gestion de portefeuille sous contraintes de capital économique
weighted sum of the returns ri(τ) of the assets i = 1, ..., N over the time interval τ
Sτ = δτ W
W (0) =
N
wi xi(τ) . (5)
In the sequel, we shall thus consider the asset returns Xi as the fundamental variables and study their
aggregation properties, namely how the distribution of portfolio return equal to their weighted sum derives
for their multivariable distribution. We shall consider a single time scale τ which can be chosen arbitrarily,
say equal to one day. We shall thus drop the dependence on τ, understanding implicitly that all our results
hold for returns estimated over time step τ.
1 Definitions and important concepts
1.1 The modified Weibull distributions
We will consider a class of distributions with fat tails but decaying faster than any power law. Such possible
behavior for assets returns distributions have been suggested to be relevant by several empirical works
(Mantegna and Stanley 1995, Gouriéroux and Jasiak 1998, Malevergne et al. 2002) and has also been asserted
to provide a convenient and flexible parameterization of many phenomena found in nature and in
the social sciences (Lahèrre and Sornette 1998). In all the following, we will use the parameterization
introduced by Sornette et al. (2000b) and define the modified-Weibull distributions:
DEFINITION 1 (MODIFIED WEIBULL DISTRIBUTION)
A random variable X will be said to follow a modified Weibull distribution with exponent c and scale
parameter χ, denoted in the sequel X ∼ W(c, χ), if and only if the random variable
follows a Normal distribution.
i=1
Y = sgn(X) √ 2
c
|X| 2
χ
These so-called modified-Weibull distributions can be seen to be general forms of the extreme tails of product
of random variables (Frisch and Sornette 1997), and using the theorem of change of variable, we can
assert that the density of such distributions is
where c and χ are the two key parameters.
p(x) = 1
2 √ c
π χ c |x|
2
c
2 −1 e −|x|
χc
, (7)
These expressions are close to the Weibull distribution, with the addition of a power law prefactor to the
exponential such that the Gaussian law is retrieved for c = 2. Following Sornette et al. (2000b), Sornette
et al. (2000a) and Andersen and Sornette (2001), we call (7) the modified Weibull distribution. For c < 1,
the pdf is a stretched exponential, which belongs to the class of sub-exponential. The exponent c determines
the shape of the distribution, fatter than an exponential if c < 1. The parameter χ controls the scale or
characteristic width of the distribution. It plays a role analogous to the standard deviation of the Gaussian
law.
The interest of these family of distributions for financial purposes have also been recently underlined by
Brummelhuis and Guégan (2000) and Brummelhuis et al. (2002). Indeed these authors have shown that
4
(6)

given a series of return {rt}t following a GARCH(1,1) process, the large deviations of the returns rt+k
and of the aggregated returns rt + · · · + rt+k conditional on the return at time t are distributed according
to a modified-Weibull distribution, where the exponent c is related to the number of step forward k by the
formula c = 2/k .
A more general parameterization taking into account a possible asymmetry between negative and positive
values (thus leading to possible non-zero mean) is
p(x) =
p(x) =
1
2 √ π
1
2 √ π
c+
c +
2
+
χ
c−
c− 2
−
χ
|x| c +
2 −1 e −|x| +
χ +c
|x| c− 2 −1 e −|x| −
χ−c 393
if x ≥ 0 (8)
if x < 0 . (9)
In what follows, we will assume that the marginal probability distributions of returns follow modified
Weibull distributions. Figure 1 shows the (negative) “Gaussianized” returns Y defined in (6) of the Standard
and Poor’s 500 index versus the raw returns X over the time interval from January 03, 1995 to December
29, 2000. With such a representation, the modified-Weibull distributions are qualified by a power law of
exponent c/2, by definition 1. The double logarithmic scales of figure 1 clearly shows a straight line over an
extended range of data, qualifying a power law relationship. An accurate determination of the parameters
(χ, c) can be performed by maximum likelihood estimation (Sornette 2000, pp 160-162). However, note
that, in the tail, the six most extreme points significantly deviate from the modified-Weibull description.
Such an anomalous behavior of the most extreme returns can be probably be associated with the notion
of “outliers” introduced by Johansen and Sornette (1998, 2002) and associated with behavioral and crowd
phenomena during turbulent market phases.
The modified Weibull distributions defined here are of interest for financial purposes and specifically for
portfolio and risk management, since they offer a flexible parametric representation of asset returns distribution
either in a conditional or an unconditional framework, depending on the standpoint prefered by
manager. The rest of the paper uses this family of distributions.
1.2 Tail equivalence for distribution functions
An interesting feature of the modified Weibull distributions, as we will see in the next section, is to enjoy the
property of asymptotic stability. Asymptotic stability means that, in the regime of large deviations, a sum
of independent and identically distributed modified Weibull variables follows the same modified Weibull
distribution, up to a rescaling.
DEFINITION 2 (TAIL EQUIVALENCE)
Let X and Y be two random variables with distribution function F and G respectively.
X and Y are said to be equivalent in the upper tail if and only if there exists λ+ ∈ (0, ∞) such that
1 − F (x)
lim
x→+∞ 1 − G(x) = λ+. (10)
Similarly, X and Y are said equivalent in the lower tail if and only if there exists λ− ∈ (0, ∞) such that
F (x)
lim
x→−∞ G(x) = λ−. (11)
5

394 13. Gestion de portefeuille sous contraintes de capital économique
Applying l’Hospital’s rule, this gives immediately the following corollary:
COROLLARY 1
Let X and Y be two random variables with densities functions f and g respectively. X and Y are equivalent
in the upper (lower) tail if and only if
1.3 The Gaussian copula
f(x)
lim
x→±∞ g(x) = λ±, λ± ∈ (0, ∞). (12)
We recall only the basic properties about copulas and refer the interested reader to (Nelsen 1998), for instance,
for more information. Let us first give the definition of a copula of n random variables.
DEFINITION 3 (COPULA)
A function C : [0, 1] n −→ [0, 1] is a n-copula if it enjoys the following properties :
• ∀u ∈ [0, 1], C(1, · · · , 1, u, 1 · · · , 1) = u ,
• ∀ui ∈ [0, 1], C(u1, · · · , un) = 0 if at least one of the ui equals zero ,
• C is grounded and n-increasing, i.e., the C-volume of every boxes whose vertices lie in [0, 1] n is
positive.
The fact that such copulas can be very useful for representing multivariate distributions with arbitrary
marginals is seen from the following result.
THEOREM 1 (SKLAR’S THEOREM)
Given an n-dimensional distribution function F with continuous marginal distributions F1, · · · , Fn, there
exists a unique n-copula C : [0, 1] n −→ [0, 1] such that :
F (x1, · · · , xn) = C(F1(x1), · · · , Fn(xn)) . (13)
This theorem provides both a parameterization of multivariate distributions and a construction scheme for
copulas. Indeed, given a multivariate distribution F with margins F1, · · · , Fn, the function
C(u1, · · · , un) = F F −1
1 (u1), · · · , F −1
n (un)
(14)
is automatically a n-copula. Applying this theorem to the multivariate Gaussian distribution, we can derive
the so-called Gaussian copula.
DEFINITION 4 (GAUSSIAN COPULA)
Let Φ denote the standard Normal distribution and ΦV,n the n-dimensional Gaussian distribution with cor-
relation matrix V. Then, the Gaussian n-copula with correlation matrix V is
−1
CV (u1, · · · , un) = ΦV,n Φ (u1), · · · , Φ −1 (un) , (15)
whose density
cV (u1, · · · , un) = ∂CV (u1, · · · , un)
∂u1 · · · ∂un
6
(16)

eads
cV (u1, · · · , un) =
1
√ exp −
det V 1
2 yt (u) (V−1
− Id)y (u)
with yk(u) = Φ −1 (uk). Note that theorem 1 and equation (14) ensure that CV (u1, · · · , un) in equation (15)
is a copula.
It can be shown that the Gaussian copula naturally arises when one tries to determine the dependence between
random variables using the principle of entropy maximization (Rao 1973, Sornette et al. 2000b, for
instance). Its pertinence and limitations for modeling the dependence between assets returns has been tested
by Malevergne and Sornette (2001), who show that in most cases, this description of the dependence can
be considered satisfying, specially for stocks, provided that one does not consider too extreme realizations
(Malevergne and Sornette 2002a, Malevergne and Sornette 2002b, Mashal and Zeevi 2002).
2 Portfolio wealth distribution for several dependence structures
2.1 Portfolio wealth distribution for independent assets
Let us first consider the case of a portfolio made of independent assets. This limiting (and unrealistic) case
is a mathematical idealization which provides a first natural benchmark of the class of portolio return distributions
to be expected. Moreover, it is generally the only case for which the calculations are analytically
tractable. For such independepent assets distributed with the modified Weibull distributions, the following
results prove the asymptotic stability of this set of distributions:
THEOREM 2 (TAIL EQUIVALENCE FOR I.I.D MODIFIED WEIBULL RANDOM VARIABLES)
Let X1, X2, · · · , XN be N independent and identically W(c, χ)-distributed random variables. Then, the
variable
SN = X1 + X2 + · · · + XN
(18)
is equivalent in the lower and upper tail to Z ∼ W(c, ˆχ), with
395
(17)
ˆχ = N c−1
c χ, c > 1, (19)
ˆχ = χ, c ≤ 1. (20)
This theorem is a direct consequence of the theorem stated below and is based on the result given by Frisch
and Sornette (1997) for c > 1 and on general properties of sub-exponential distributions when c ≤ 1.
THEOREM 3 (TAIL EQUIVALENCE FOR WEIGHTED SUMS OF INDEPENDENT VARIABLES)
Let X1, X2, · · · , XN be N independent and identically W(c, χ)-distributed random variables. Let w1,
w2, · · · , wN be N non-random real coefficients. Then, the variable
SN = w1X1 + w2X2 + · · · + wNXN
is equivalent in the upper and the lower tail to Z ∼ W(c, ˆχ) with
ˆχ =
N
|wi| c
c−1
c−1
c
· χ, c > 1, (22)
i=1
(21)
ˆχ = max
i {|w1|, |w2|, · · · , |wN|}, c ≤ 1. (23)
7

396 13. Gestion de portefeuille sous contraintes de capital économique
The proof of this theorem is given in appendix A.
COROLLARY 2
Let X1, X2, · · · , XN be N independent random variables such that Xi ∼ W(c, χi). Let w1, w2, · · · , wN
be N non-random real coefficients. Then, the variable
SN = w1X1 + w2X2 + · · · + wNXN
is equivalent in the upper and the lower tail to Z ∼ W(c, ˆχ) with
ˆχ =
N
i=1
|wiχi| c
c−1
c−1
c
(24)
, c > 1, (25)
ˆχ = max
i {|w1χ1|, |w2χ2|, · · · , |wNχN|}, c ≤ 1. (26)
The proof of the corollary is a straightforward application of theorem 3. Indeed, let Y1, Y2, · · · , YN be N
independent and identically W(c, 1)-distributed random variables. Then,
which yields
(X1, X2, · · · , XN) d = (χ1Y1, χ2Y2, · · · , χNYN), (27)
SN d = w1χ1 · Y1 + w2χ2 · Y2 + · · · + wNχN · YN . (28)
Thus, applying theorem 3 to the i.i.d variables Yi’s with weights wiχi leads to corollary 2.
2.2 Portfolio wealth distribution for comonotonic assets
The case of comonotonic assets is of interest as the limiting case of the strongest possible dependence
between random variables. By definition,
DEFINITION 5 (COMONOTONICITY)
the variables X1, X2, · · · , XN are comonotonic if and only if there exits a random variable U and nondecreasing
functions f1, f2, · · · , fN such that
(X1, X2, · · · , XN) d = (f1(U), f2(U), · · · , fN(U)). (29)
In terms of copulas, the comonotonicity can be expressed by the following form of the copula
C(u1, u2, · · · , uN) = min(u1, u2, · · · , uN) . (30)
This expression is known as the Fréchet-Hoeffding upper bound for copulas (Nelsen 1998, for instance). It
would be appealing to think that estimating the Value-at-Risk under the comonotonicity assumption could
provide an upper bound for the Value-at-Risk. However, it turns out to be wrong, due –as we shall see in the
sequel– to the lack of coherence (in the sense of Artzner et al. (1999)) of the Value-at-Risk, in the general
case. Notwithstanding, an upper and lower bound can always be derived for the Value-at-Risk (Embrechts
et al. 2002b). But in the present situation, where we are only interested in the class of modified Weibull
distributions with a Gaussian copula, the VaR derived under the comonoticity assumption will actually
represent the upper bound (at least for the VaR calculated at sufficiently hight confidence levels).
8

THEOREM 4 (TAIL EQUIVALENCE FOR A SUM OF COMONOTONIC RANDOM VARIABLES)
Let X1, X2, · · · , XN be N comonotonic random variables such that Xi ∼ W(c, χi). Let w1, w2, · · · , wN
be N non-random real coefficients. Then, the variable
SN = w1X1 + w2X2 + · · · + wNXN
is equivalent in the upper and the lower tail to Z ∼ W(c, ˆχ) with
ˆχ =
wiχi . (32)
The proof is obvious since, under the assumption of comonotonicity, the portfolio wealth S is given by
S =
and for modified Weibull distributions, we have
i
i
wi · Xi d =
fi(·) = sgn(·) χi
397
(31)
N
wi · fi(U), (33)
i=1
2/ci | · |
√2 , (34)
in the symmetric case while U is a Gaussian random variable. If, in addition, we assume that all assets have
the same exponent ci = c, it is clear that S ∼ W(c, ˆχ) with
ˆχ =
wiχi. (35)
i
It is important to note that this relation is exact and not asymptotic as in the case of independent variables.
When the exponents ci’s are different from an asset to another, a similar result holds, since we can still write
the inverse cumulative function of S as
F −1
S (p) =
N
i=1
wiF −1
(p), p ∈ (0, 1), (36)
Xi
which is the property of additive comonotonicity of the Value-at-Risk1 . Let us then sort the Xi’s such that
c1 = c2 = · · · = cp < cp+1 ≤ · · · ≤ cN. We immediately obtain that S is equivalent in the tail to
Z ∼ W(c1, ˆχ), where
p
ˆχ = wiχi. (37)
i=1
In such a case, only the assets with the fatest tails contributes to the behavior of the sum in the large deviation
regime.
1 This relation shows that, in general, the VaR calculated for comonotonic assets does not provide an upper bound of the VaR,
whatever the dependence structure the portfolio may be. Indeed, in such a case, we have VaR(X1 + X2) = VaR(X1) + VaR(X2)
while, by lack of coherence, we may have VaR(X1 + X2) ≥ VaR(X1) + VaR(X2) for some dependence structure between X1
and X2.
9

398 13. Gestion de portefeuille sous contraintes de capital économique
2.3 Portfolio wealth under the Gaussian copula hypothesis
2.3.1 Derivation of the multivariate distribution with a Gaussian copula and modified Weibull margins
An advantage of the class of modified Weibull distributions (7) is that the transformation into a Gaussian,
and thus the calculation of the vector y introduced in definition 1, is particularly simple. It takes the form
yk = sgn(xk) √
|xk|
2
χk
c k 2
, (38)
where yk is normally distributed . These variables Yi then allow us to obtain the covariance matrix V of the
Gaussian copula :
|xi|
Vij = 2 · E sgn(xixj)
χi
c i
2 |xj|
χj
cj
2
, (39)
which always exists and can be efficiently estimated. The multivariate density P (x) is thus given by:
P (x1, · · · , xN) = cV (x1, x2, · · · , xN)
=
1
2 N π N/2√ V
N
pi(xi) (40)
i=1
N ci|xi| c/2−1
⎡
exp ⎣−
i=1
χ c/2
i
i,j
V −1
ij
Obviously, similar transforms hold, mutatis mutandis, for the asymmetric case (8,9).
⎤
c/2 c/2 |xi| |xj|
⎦ . (41)
2.3.2 Asymptotic distribution of a sum of modified Weibull variables with the same exponent c > 1
We now consider a portfolio made of dependent assets with pdf given by equation (41) or its asymmetric
generalization. For such distributions of asset returns, we obtain the following result
THEOREM 5 (TAIL EQUIVALENCE FOR A SUM OF DEPENDENT RANDOM VARIABLES)
Let X1, X2, · · · , XN be N random variables with a dependence structure described by the Gaussian copula
with correlation matrix V and such that each Xi ∼ W(c, χi). Let w1, w2, · · · , wN be N (positive) nonrandom
real coefficients. Then, the variable
SN = w1X1 + w2X2 + · · · + wNXN
is equivalent in the upper and the lower tail to Z ∼ W(c, ˆχ) with
ˆχ =
where the σi’s are the unique (positive) solution of
i
i
wiχiσi
c−1
c
χi
χj
(42)
, (43)
V −1
ik σi c/2 = wkχk σ 1−c/2
k , ∀k . (44)
10

Independent Assets
Comonotonic Assets
N c
i=1 |wiχi| c−1
ˆχ λ−
c−1
c
, c > 1
N−1
c 2
2(c−1)
max{|w1χ1|, · · · , |wNχN|}, c ≤ 1 Card {|wiχi| = maxj{ |wjχj|}}
N
i=1 wiχi
Gaussian copula (
c−1
i wiχiσi) c , c > 1 see appendix B
Table 1: Summary of the various scale factors obtained for different distribution of asset returns.
The proof of this theorem follows the same lines as the proof of theorem 3. We thus only provide a heuristic
derivation of this result in appendix B. Equation (44) is equivalent to
Vik wkχk σk 1−c/2 = σi c/2 , ∀i . (45)
k
which seems more attractive since it does not require the inversion of the correlation matrix. In the special
case where V is the identity matrix, the variables Xi’s are independent so that equation (43) must yield the
same result as equation (22). This results from the expression of σk = (wkχk) 1
c−1 valid in the independent
case. Moreover, in the limit where all entries of V equal one, we retrieve the case of comonotonic assets.
Obviously, V−1 does not exist for comonotonic assets and the derivation given in appendix B does not hold,
but equation (45) remains well-defined and still has a unique solution σk = ( wkχk) 1
c−1 which yields the
scale factor given in theorem 4.
2.4 Summary
In the previous sections, we have shown that the wealth distribution FS(x) of a portfolio made of assets with
modified Weibull distributions with the same exponent c remains equivalent in the tail to a modified Weibull
distribution W(c, ˆχ). Specifically,
FS(x) ∼ λ− FZ(x) , (46)
when x → −∞, and where Z ∼ W(c, ˆχ). Expression (46) defines the proportionality factor or weight λ−
of the negative tail of the portfolio wealth distribution FS(x). Table 1 summarizes the value of the scale
parameter ˆχ for the different types of dependence we have studied. In addition, we give the value of the
coefficient λ−, which may also depend on the weights of the assets in the portfolio in the case of dependent
assets.
11
1
399

400 13. Gestion de portefeuille sous contraintes de capital économique
3 Value-at-Risk
3.1 Calculation of the VaR
We consider a portfolio made of N assets with all the same exponent c and scale parameters χi, i ∈
{1, 2, · · · , N}. The weight of the i th asset in the portfolio is denoted by wi. By definition, the Valueat-Risk
at the loss probability α, denoted by VaRα, is given , for a continuous distribution of profit and loss,
by
Pr{W (τ) − W (0) < −VaRα} = α, (47)
which can be rewritten as
Pr S < − VaRα
= α. (48)
W (0)
In this expression, we have assumed that all the wealth is invested in risky assets and that the risk-free
interest rate equals zero, but it is easy to reintroduce it, if necessary. It just leads to discount VaRα by the
discount factor 1/(1 + µ0), where µ0 denotes the risk-free interest rate.
Now, using the fact that FS(x) ∼ λ− FZ(x), when x → −∞, and where Z ∼ W(c, ˆχ), we have
1
Pr S < − VaRα
√2
c/2
VaRα
1 − Φ
, (49)
W (0)
W (0) ˆχ
λ−
as VaRα goes to infinity, which allows us to obtain a closed expression for the asymptotic Value-at-Risk
with a loss probability α:
VaRα W (0) ˆχ
21/c
Φ −1
1 − α
2/c , (50)
λ−
ξ(α) 2/c W (0) · ˆχ, (51)
where the function Φ(·) denotes the cumulative Normal distribution function and
ξ(α) ≡ 1
2 Φ−1
1 − α
. (52)
λ−
In the case where a fraction w0 of the total wealth is invested in the risk-free asset with interest rate µ0, the
previous equation simply becomes
VaRα ξ(α) 2/c (1 − w0) · W (0) · ˆχ − w0W (0)µ0. (53)
Due to the convexity of the scale parameter ˆχ, the VaR is itself convex and therefore sub-additive. Thus, for
this set of distributions, the VaR becomes coherent when the considered quantiles are sufficiently small.
The Expected-Shortfall ESα, which gives the average loss beyond the VaR at probability level α, is also
very easily computable:
α
ESα = 1
VaRu du (54)
α 0
= ζ(α)(1 − w0) · W (0) · ˆχ − w0W (0)µ0, (55)
where ζ(α) = 1
α
α 0 ξ(u)2/c du . Thus, the Value-at-Risk, the Expected-Shortfall and in fact any downside
risk measure involving only the far tail of the distribution of returns are entirely controlled by the scale
parameter ˆχ. We see that our set of multivariate modified Weibull distributions enjoy, in the tail, exactly the
same properties as the Gaussian distributions, for which, all the risk measures are controlled by the standard
deviation.
12

3.2 Typical recurrence time of large losses
Let us translate these formulas in intuitive form. For this, we define a Value-at-Risk VaR ∗ which is such that
its typical frequency is 1/T0. T0 is by definition the typical recurrence time of a loss larger than VaR ∗ . In
our present example, we take T0 equals 1 year for example, i.e., VaR ∗ is the typical annual shock or crash.
Expression (49) then allows us to predict the recurrence time T of a loss of amplitude VaR equal to β times
this reference value VaR ∗ :
401
T
ln (β
T0
c ∗ c VaR
− 1)
+ O(ln β) . (56)
W (0) ˆχ
Figure 2 shows ln T versus β. Observe that T increases all the more slowly with β, the smaller is the
T0
exponent c. This quantifies our expectation that large losses occur more frequently for the “wilder” subexponential
distributions than for super-exponential ones.
4 Optimal portfolios
In this section, we present our results on the problem of the efficient portfolio allocation for asset distributed
according to modified Weibull distributions with the different dependence structures studied in the previous
sections. We focus on the case when all asset modified Weibull distributions have the same exponent c, as
it provides the richest and more varied situation. When this is not the case and the assets have different
exponents ci, i = 1, ..., N, the asymptotic tail of the portfolio return distribution is dominated by the asset
with the heaviest tail. The largest risks of the portfolio are thus controlled by the single most risky asset
characterized by the smallest exponent c. Such extreme risk cannot be diversified away. In such a case, for
a risk-averse investor, the best strategy focused on minimizing the extreme risks consists in holding only the
asset with the thinnest tail, i.e., with the largest exponent c.
4.1 Portfolios with minimum risk
Let us consider first the problem of finding the composition of the portfolio with minimum risks, where
the risks are measured by the Value-at-Risk. We consider that short sales are not allowed, that the risk free
interest rate equals zero and that all the wealth is invested in stocks. This last condition is indeed the only
interesting one since allowing to invest in a risk-free asset would automatically give the trivial solution in
which the minimum risk portfolio is completely invested in the risk-free asset.
The problem to solve reads:
VaR ∗ α = min VaRα = ξ(α) 2/c W (0) · min ˆχ (57)
N
i=1 wi = 1 (58)
wi ≥ 0 ∀i. (59)
In some cases (see table 1), the prefactor ξ(α) defined in (52) also depends on the weight wi’s through λ−
defined in (46). But, its contribution remains subdominant for the large losses. This allows to restrict the
minimization to ˆχ instead of ξ(α) 2/c · ˆχ.
13

402 13. Gestion de portefeuille sous contraintes de capital économique
4.1.1 Case of independent assets
“Super-exponential” portfolio (c > 1)
Consider assets distributed according to modified Weibull distributions with the same exponent c > 1.
The Value-at-Risk is given by
VaRα = ξ(α) 2/c W (0) ·
N
i=1
|wiχi| c
c−1
c−1
2
Introducing the Lagrange multiplier λ, the first order condition yields
∂ ˆχ
∂wi
and the composition of the minimal risk portfolio is
=
λ
ξ(α) W (0)
wi ∗ = χ−c
i
j χ−c
j
which satistifies the positivity of the Hessian matrix Hjk = ∂2 ˆχ
∂wj∂wk {w∗ i }
The minimal risk portfolio is such that
VaR ∗ α = ξ(α)2/c W (0)
i χ−c
1
c
j
, (60)
∀i, (61)
, µ ∗
=
i χ−c
i µi
j χ−c
j
where µi is the return of asset i and µ ∗ is the return of the minimum risk portfolio.
(second order condition).
(62)
, (63)
sub-exponential portfolio (c ≤ 1)
Consider assets distributed according to modified Weibull distributions with the same exponent c < 1.
The Value-at-Risk is now given by
VaRα = ξ(α) c/2 W (0) · max{|w1χ1|, · · · , |wNχN|}. (64)
Since the weights wi are positive, the modulus appearing in the argument of the max() function can be
removed. It is easy to see that the minimum of VaRα is obtained when all the wiχi’s are equal, provided
that the constraint wi = 1 can be satisfied. Indeed, let us start with the situation where
w1χ1 = w2χ2 = · · · = wNχN . (65)
Let us decrease the weight w1. Then, w1χ1 decreases with respect to the initial maximum situation (65) but,
in order to satisfy the constraint
i wi = 1, at least one of the other weights wj, j ≥ 2 has to increase, so
that wjχj increases, leading to a maximum for the set of the wiχi’s greater than in the initial situation where
(65) holds. Therefore,
and the constraint
i wi = 1 yields
w ∗ i = A
A =
χi
, ∀i, (66)
1
i χ−1
i
14
, (67)

and finally
w ∗ i = χ−1
i
j χ−1
j
, VaR ∗ α = ξ(α)c/2 W (0)
, µ ∗
=
i χ−1
i
i χ−1
i µi
j χ−1
j
403
. (68)
The composition of the optimal portfolio is continuous in c at the value c = 1. This is the consequence
of the continuity as a function of c at c = 1 of the scale factor ˆχ for a sum of independent variables. In
this regime c ≤ 1, the Value-at-Risk increases as c decreases only through its dependence on the prefactor
ξ(α) 2/c since the scale factor ˆχ remains constant.
4.1.2 Case of comonotonic assets
For comonotonic assets, the Value-at-Risk is
VaRα = ξ(α) c/2 W (0) ·
which leads to a very simple linear optimization problem. Indeed, denoting χ1 = min{χ1, χ2, · · · , χN},
we have
wi = χ1, (70)
i
wiχi ≥ χ1
i
i
wiχi
which proves that the composition of the optimal portfolio is w ∗ 1 = 1, w∗ i
= 0 i ≥ 2 leading to
(69)
VaR ∗ α = ξ(α) c/2 W (0)χ1, µ ∗ = µ1. (71)
This result is not surprising since all assets move together. Thus, the portfolio with minimum Value-at-Risk
is obtained when only the less risky asset, i.e., with the smallest scale factor χi, is held. In the case where
there is a degeneracy in the smallest χ of order p (χ1 = χ2 = ... = χp = min{χ1, χ2, · · · , χN}), the
optimal choice lead to invest all the wealth in the asset with the larger expected return µj, j ∈ {1, · · · , p}.
However, in an efficient market with rational agents, such an opportunity should not exist since the same
risk embodied by χ1 = χ2 = ... = χp should be remunerated by the same return µ1 = µ2 = ... = µp.
4.1.3 Case of assets with a Gaussian copula
In this situation, we cannot solve the problem analytically. We can only assert that the miminization problem
has a unique solution, since the function VaRα({wi}) is convex. In order to obtain the composition of the
optimal portfolio, we need to perform the following numerical analysis.
It is first needed to solve the set of equations −1
i Vij σc/2
1−c/2
i = wjχjσj or the equivalent set of equations
given
by (45), which can be performed by Newton’s algorithm. Then one have the minimize the quantity
wiχiσi({wi}). To this aim, one can use the gradient algorithm, which requires the calculation of the
derivatives of the σi’s with respect to the wk’s. These quantities are easily obtained by solving the linear set
of equations
c
· V
2 −1
c
2
ij σ −1
i σ c
2 −1
∂σi c
1 ∂σj
j + − 1 wjχj = χj · δjk. (72)
∂wk 2 σj ∂wk
i
Then, the analytical solution for independent assets or comonotonic assets can be used to initialize the
minimization algorithm with respect to the weights of the assets in the portfolio.
15

404 13. Gestion de portefeuille sous contraintes de capital économique
4.2 VaR-efficient portfolios
We are now interested in portfolios with minimum Value-at-risk, but with a given expected return µ =
i wiµi. We will first consider the case where, as previously, all the wealth is invested in risky assets and
we then will discuss the consequences of the introduction of a risk-free asset in the portfolio.
4.2.1 Portfolios without risky asset
When the investors have to select risky assets only, they have to solve the following minimization problem:
VaR ∗ α = min VaRα = ξ(α) W (0) · min ˆχ (73)
N
i=1 wiµi = µ (74)
N
i=1 wi = 1 (75)
wi ≥ 0 ∀i. (76)
In contrast with the research of the minimum risk portfolios where analytical results have been derived, we
need here to use numerical methods in every situation. In the case of super-exponential portfolios, with or
without dependence between assets, the gradient method provides a fast and easy to implement algorithm,
while for sub-exponential portfolios or portfolios made of comonotonic assets, one has to use the simplex
method since the minimization problem is then linear.
Thus, although not as convenient to handle as analytical results, these optimization problems remain easy to
manage and fast to compute even for large portfolios.
4.2.2 Portfolios with risky asset
When a risk-free asset is introduced in the portfolio, the expression of the Value-at-Risk is given by equation
(53), the minimization problem becomes
VaR ∗ α = min ξ(α) 2/c (1 − w0) · W (0) · ˆχ − w0W (0)µ0
(77)
N
i=1 wiµi = µ (78)
N
i=1 wi = 1 (79)
wi ≥ 0 ∀i. (80)
When the risk-free interest rate µ0 is non zero, we have to use the same numerical methods as above to
solve the problem. However, if we assume that µ0 = 0, the problem becomes amenable analytically. Its
Lagrangian reads
L = ξ(α) 2/c ⎛
(1 − w0) · W (0) · ˆχ − λ1 ⎝
⎞
N
wiµi − µ ⎠ − λ2 wi − 1 , (81)
= ξ(α) 2/c
⎛
⎝
j=0
which allows us to show that the weights of the optimal portfolio are
w ∗ i = (1 − w0) ·
wi
ˆwi
N
j=1 ˆwj
i=0
⎞
⎛
⎠ · W (0) · ˆχ − λ1 ⎝
⎞
wiµi − µ ⎠ , (82)
i=0
i=0
and VaR ∗ α = ξ(α)2/c (1 − w0) · W (0)
2
16
· µ, (83)

where the ˆwi’s are solution of the set of equations
ˆχ +
N
i=1
ˆwi
∂ ˆχ
∂ ˆwi
405
= µi . (84)
Expression (83) shows that the efficient frontier is simply a straight line and that any efficient portfolio is
the sum of two portfolios: a “riskless portfolio” in which a fraction w0 of the initial wealth is invested and
a portfolio with the remaining (1 − w0) of the initial wealth invested in risky assets. This provides another
example of the two funds separation theorem. A CAPM then holds, since equation (84) together with the
market equilibrium assumption yields the proportionality between any stock return and the market return.
However, these three properties are rigorously established only for a zero risk-free interest rate and may not
remain necessarily true as soon as the risk-free interest rate becomes non zero.
Finally, for practical purpose, the set of weights w∗ i ’s obtained under the assumption of zero risk-free interest
rate µ0, can be used to initialize the optimization algorithms when µ0 does not vanish.
5 Conclusion
The aim of this work has been to show that the key properties of Gaussian asset distributions of stability
under convolution, of the equivalence between all down-side riks measures, of coherence and of simple
use also hold for a general family of distributions embodying both sub-exponential and super-exponential
behaviors, when restricted to their tail. We then used these results to compute the Value-at-Risk (VaR) and
to obtain efficient porfolios in the risk-return sense, where the risk is characterized by the Value-at-Risk.
Specifically, we have studied a family of modified Weibull distributions to parameterize the marginal distributions
of asset returns, extended to their multivariate distribution with Gaussian copulas. The relevance to
finance of the family of modified Weibull distributions has been proved in both a context of conditional and
unconditional portfolio management. We have derived exact formulas for the tails of the distribution P (S)
of returns S of a portfolio of arbitrary composition of these assets. We find that the tail of P (S) is also
asymptotically a modified Weibull distribution with a characteristic scale χ function of the asset weights
with different functional forms depending on the super- or sub-exponential behavior of the marginals and
on the strength of the dependence between the assets. The derivation of the portfolio distribution has shown
the asymptotic stability of this family of distribution with the important economic consequence that any
down-side risk measure based upon the tail of the asset returns distribution are equivalent, in so far as they
all depends on the scale factor χ and keep the same functional form whatever the number of assets in the
portfolio may be. Our analytical study of the properties of the VaR has shown the VaR to be coherent. This
justifies the use of the VaR as a coherent risk measure for the class of modified Weibull distributions and
ensures that portfolio optimization problems are always well-conditioned even when not fully analytically
solvable. The Value-at-Risk and the Expected-Shortfall have also been shown to be (asymptotically) equivalent
in this framework. In fine, using the large class of modified Weibull distributions, we have provided
a simple and fast method for calculating large down-side risks, exemplified by the Value-at-Risk, for assets
with distributions of returns which fit quite reasonably the empirical distributions.
17

406 13. Gestion de portefeuille sous contraintes de capital économique
A Proof of theorem 3 : Tail equivalence for weighted sums of modified
Weibull variables
A.1 Super-exponential case: c > 1
Let X1, X2, · · · , XN be N i.i.d random variables with density p(·). Let us denote by f(·) and g(·) two
positive functions such that p(·) = g(·) · e −f(·) . Let w1, w2, · · · , wN be N real non-random coefficients,
and S = N
i=1 wixi.
Let X = {x ∈ R N , N
i=1 wixi = S}. The density of the variable S is given by
PS(S) =
We will assume the following conditions on the function f
X
dx e −N
i=1 [f(xi)−ln g(xi)] , (85)
1. f(·) is three times continuously differentiable and four times differentiable,
2. f (2) (x) > 0, for |x| large enough,
3. limx→±∞ f (3) (x)
(f (2) (x)) 2 = 0,
4. f (3) is asymptotically monotonous,
5. there is a constant β > 1 such that f (3) (β·x)
f (3) (x)
remains bounded as x goes to infinity,
6. g(·) is ultimately a monotonous function, regularly varying at infinity with indice ν.
Let us start with the demonstration of several propositions.
PROPOSITION 1
under hypothesis 3, we have
Proof
lim
x→±∞ |x| · f ′′ (x) = 0. (86)
d 1
Hypothesis 3 can be rewritten as lim
x→±∞ dx f (2) = 0, so that
(x)
∀ɛ > 0, ∃Aɛ/x > Aɛ =⇒
d 1
dx
f (2)
(x) ≤ ɛ. (87)
Now, since f ′′ is differentiable, 1/f ′′ is also differentiable, and by the mean value theorem, we have
for some ξ ∈ (x, y).
1
f
′′ 1
−
(x) f ′′
(y) = |x − y| ·
d 1
dξ
f ′′
(ξ)
18
(88)

Choosing x > y > Aɛ, and applying equation (87) together with (88) yields
1
f
′′ 1
−
(x) f ′′
(y) ≤ ɛ · |x − y|. (89)
Now, dividing by x and letting x go to infinity gives
lim
1
x→∞ x
· f ′′
(x) ≤ ɛ, (90)
which concludes the proof.
PROPOSITION 2
Under assumption 3, we have
Proof
407
lim
x→±∞ f ′ (x) = +∞. (91)
According to assumption 3 and proposition 1, lim
x→±∞ x · f ′′ (x) = ∞, which means
This thus gives
∀α > 0, ∃Aα/x > Aɛ =⇒ x · f ′′ (x) ≥ α. (92)
∀x ≥ aα, x · f ′′ (x) ≥ α ⇐⇒ f ′′ (x) ≥ α
x
=⇒
x
Aα
f ′′ (t) dt ≥ α ·
x
Aα
dt
t
(93)
(94)
=⇒ f ′ (x) ≥ α · ln x − α · ln Aα + f ′ (Aα). (95)
The right-hand-side of this last equation goes to infinity as x goes to infinity, which concludes the proof.
PROPOSITION 3
Under assumptions 3 and 6, the function g(·) satisfies
uniformly in h, for any positive constant C.
Proof For g non-decreasing, we have
∀|h| ≤ C
f ′′ (x) ,
∀|h| ≤ C
f ′′ g(x + h)
, lim = 1, (96)
(x) x→±∞ g(x)
g
x
1 − C
x·f ′′
(x) g(x + h)
≤
g(x)
g(x) ≤
g x
1 + C
x·f ′′ (x)
g(x)
. (97)
If g is non-increasing, the same inequalities hold with the left and right terms exchanged. Therefore, the
final conclusion is easily shown to be independent of the monotocity property of g. From assumption 3 and
proposition 1, we have
∀α > 0, ∃Aα/x > Aɛ =⇒ x · f ′′ (x) ≥ α. (98)
Thus, for all x larger than Aα and all |h| ≤ C/f ′′ (x)
g x 1 − C
α g(x + h)
≤
g(x) g(x) ≤ g x 1 + C
α
g(x)
19
(99)

408 13. Gestion de portefeuille sous contraintes de capital économique
Now, letting x go to infinity,
for all α as large as we want, which concludes the proof.
1 − C
ν g(x + h)
≤ lim
α x→∞ g(x) ≤
1 + C
ν , (100)
α
PROPOSITION 4
Under assumptions 1, 3 and 4 we have, for any positive constant C:
Proof
Let us first remark that
∀|h| ≤ C
f ′′ , lim
(x) x→±∞
sup
ξ∈[x,x+h] f (3) (ξ)
f ′′ (x) 2
sup
ξ∈[x,x+h] f (3) (ξ)
f ′′ (x) 2 = 0. (101)
= sup
ξ∈[x,x+h] f (3) (ξ) f
f (3) (x)
·
(3) (x)
f ′′ . (102)
(x) 2
The rightmost factor in the right-hand-side of the equation above goes to zero as x goes to infinity by assumption
3. Therefore, we just have to show that the leftmost factor in the right-hand-side remains bounded
as x goes to infinity to prove Proposition 4.
Applying assumption 4 according to which f (3) is asymptotically monotonous, we have
sup f ξ∈[x,x+h]
(3) (ξ)
f (3) (x)
≤
f (3)
x + C
f ′′ =
(x)
f (3) (x)
f
(103)
(3)
x 1 + C
x·f ′′
(x)
f (3) (x) ≤
,
f (3)
(104)
x 1 + C
α f (3) (x)
, (105)
for every x larger than some positive constant Aα by assumption 3 and proposition 1. Now, for α large
enough, 1 + C
α is less than β (assumption 5) which shows that supξ∈[x,x+h]|f (3) (ξ)|
|f (3) (x)|
remains bounded for large
x, which conclude the proof.
We can now show that under the assumptions stated above, the leading order expansion of PS(S) for large
S and finite N > 1 is obtained by a generalization of Laplace’s method which here amounts to remark that
the set of x∗ i ’s that maximize the integrand in (85) are solution of
f ′ i(x ∗ i ) = σ(S)wi , (106)
where σ(S) is nothing but a Lagrange multiplier introduced to minimize the expression N
i=1 fi(xi) under
the constraint N
i=1 wixi = S. This constraint shows that at least one xi, for instance x1, goes to infinity
as S → ∞. Since f ′ (x1) is an increasing function by assumption 2 which goes to infinity as x1 → +∞
(proposition 2), expression (106) shows that σ(S) goes to infinity with S, as long as the weight of the asset
1 is not zero. Putting the divergence of σ(S) with S in expression (106) for i = 2, ..., N ensures that each
x∗ i increases when S increases and goes to infinity when S goes to infinity.
20

Expanding fi(xi) around x ∗ i yields
where the set of hi = xi − x∗ i obey the condition
409
f(xi) = f(x ∗ i ) + f ′ (x ∗ x∗ i +hi
i ) · hi +
x∗ t
dt
i
x∗ du f
i
′′ (u) (107)
N
wihi = 0 . (108)
i=1
Summing (106) in the presence of relation (108), we obtain
N
N
f(xi) = f(x ∗ N
i ) +
i=1
i=1
i=1
i=1
i=1
x ∗ i +hi
Thus exp(− f(xi)) can be rewritten as follows :
N
N
exp − f(xi) = exp f(x ∗ N
i ) +
x ∗ i
i=1
t
dt
x∗ du f
i
′′ (u) . (109)
x ∗ i +hi
x ∗ i
t
dt
x∗ du f
i
′′
(u) . (110)
Let us now define the compact set AC = {h ∈ RN , N i=1 f ′′ (x∗ i )2 · h2 i ≤ C2 } for any given positive
constant C and the set H = {h ∈ RN , N i=1 wihi = 0}. We can thus write
PS(S) =
=
H
dh e −N
i=1 [f(xi)−ln g(xi)] , (111)
AC∩H
dh e −N
i=1 [f(xi)−ln
g(xi)]
+
AC∩H
dh e −N
i=1 [f(xi)−ln g(xi)]
, (112)
We are now going to analyze in turn these two terms in the right-hand-side of (112).
First term of the right-hand-side of (112).
Let us start with the first term. We are going to show that
lim
S→∞
i=1x
AC∩H dh e−N
∗ i +hi x∗ dtt
x
i
∗ du f
i
′′ (u)−ln g(xi)
(2π) N−1 i
2 g(x∗ i ) N w
i=1
2 f
iN
j=1 ′′
j (x∗ j )
f ′′
i (x∗ i )
= 1, for some positive C. (113)
In order to prove this assertion, we will first consider the leftmost factor of the right-hand-side of (112):
g(xi) e −f(xi) = g(x ∗ i + hi) e −x ∗
= g(x ∗ 1
−
i + hi) e
Since for all ξ ∈ R, e −|ξ| ≤ e −ξ ≤ e |ξ| , we have
e −x ∗ x∗ i
i +h i
dtt
x∗ du [f
i
′′ (u)−f ′′ (x∗ i )]
≤ e −x ∗ x∗ i
i +h i
x ∗ i
i +h i
dtt
x∗ du f
i
′′ (u)
, (114)
2f ′′ (x∗ i )h2 i e −x ∗ i +hi x∗ i
dtt
x ∗ i
21
du [f ′′ (u)−f ′′ (x∗ i )] ≤ ex ∗ i +hi x∗ i
dtt
x∗ du [f
i
′′ (u)−f ′′ (x∗ i )] . (115)
dtt
x∗ du [f
i
′′ (u)−f ′′ (x∗ i )]
,
(116)

410 13. Gestion de portefeuille sous contraintes de capital économique
and
e −x ∗ x∗ i
i +h i
dtt
x ∗ i
du |f ′′ (u)−f ′′ (x ∗ i )| ≤ e −x ∗
x ∗ i
i +h i
since whatever the sign of hi, the quantity x ∗ i +hi
x ∗ i
dtt
x ∗ i
du [f ′′ (u)−f ′′ (x∗ i )] ≤ ex ∗ i +hi x∗ i
dt t
x∗ du |f
i
′′ (u) − f ′′ (x∗ i
dtt
x∗ du |f
i
′′ (u)−f ′′ (x∗ i )| ,
(117)
)| remains always positive.
But, |u − x∗ i | ≤ |hi| ≤ C
f ′′ (x∗ i ), which leads, by the mean value theorem and assumption 1, to
which yields
Thus
|f ′′ (u) − f ′′ (x ∗ i )| ≤ sup
ξ∈(x∗ i ,x∗ i +hi)
|f (3) (ξ)| · |u − x ∗ i |, (118)
0 ≤
1
− 2i
e sup |f (3) (ξ)|
i
≤ sup
ξ∈(x∗ i ,x∗ i +hi)
|f (3) (ξ)| C
f ′′ (x∗ i ),
≤ sup |f
ξ∈Gi
(3) (ξ)| C
f ′′ (x∗ i ), where Gi =
x ∗
i +hi
x ∗ i
t
dt
x∗ du |f
i
′′ (u) − f ′′ (x ∗ i )| ≤ 1
2
i
C
f ′′ (x∗ i ) h2 i
≤ e −x ∗ i +hi x∗ i
dtt
x ∗ i
(119)
x ∗ i − C
f ′′ (x∗ i ), x∗i + C
f ′′ (x∗ i )
, (120)
sup
ξ∈Gi
|f (3) (ξ)|
du [f ′′ (u)−f ′′ (x∗ i )] 1
2i
≤ e
sup |f (3) (ξ)|
C
f ′′ (x ∗ i )h2 i . (121)
C
f ′′ (x ∗ i ) h2 i , (122)
where sup ξ∈Gi |f (3) (ξ)|, have been denoted by sup |f (3) (ξ)| in the previous expression, in order not to
cumber the notations.
By proposition 4, we know that for all h ∈ AC and all ɛi > 0
sup
|f
(3) (ξ)
f ′′ (x∗ i )
≤ ɛi, for x ∗ i large enough, (123)
so that
∀ɛ ′ C·ɛ′
i −
> 0 and ∀h ∈ AC, e 2 h2 i ≤ e −x ∗ x∗ i
for |x| large enough.
i +h i
Moreover, from proposition 3, we have for all ɛi > 0 and x∗ i large enough:
so, for all ɛ ′′ > 0
dtt
x∗ du [f
i
′′ (u)−f ′′ (x∗ i )] ≤ e C·ɛ′
i 2 h2 i , (124)
∀h ∈ AC, (1 − ɛi) ν ≤ g(x∗i + hi)
g(x∗ i ) ≤ (1 + ɛi) ν , (125)
∀h ∈ AC, (1 − ɛ ′′ ) Nν ≤
Then for all ɛ > 0 and |x| large enough, this yields :
i
g(x ∗ i
+ hi)
g(x∗ i ) ≤ (1 + ɛ′′ ) Nν . (126)
(1 − ɛ) Nν 1
−
e 2i (f ′′ (x∗ i )+C·ɛ)·h2 i ≤ g(xi)
g(x∗ i )e−f(xi) Nν −
≤ (1 + ɛ) e 1
2i (f ′′ (x∗ i )−C·ɛ)·h2 i , (127)
i
22

for all h ∈ AC. Thus, integrating over all the h ∈ AC ∩ H and by continuity of the mapping
G(Y) = dh g(h, Y) (128)
AC∩H
1
− where g(h, Y) = e 2Yi·h 2 i , we can conclude that,
Now, we remark that
with
H
AC∩H
g(x∗ i )
1
−
dh e 2f ′′ (x∗ i )h2 i =
and
dh e
AC∩H
H
g(xi) e −f(xi)
AC∩H
AC∩H
1
−
dh e 2f ′′ (x∗ i )h2 i =
dh e− 1
2f ′′ (x ∗ i )h2 i
411
S→∞
−−−→ 1. (129)
1
−
dh e 2f ′′ (x∗ i )h2
i +
AC∩H
1
−
dh e 2f ′′ (x∗ i )h2 i , (130)
1
− 2f ′′ (x∗ i )h2 i ∼ O
(2π) N−1
2
N w
i=1
2 iN
j=1 f ′′
j (x∗ j )
f ′′
i (x∗ i )
, (131)
α
−
e f ′′ (x∗
) , α > 0, (132)
where x ∗ = max{x ∗ i } (note that 1/f ′′ (x) → ∞ with x by Proposition 1). Indeed, we clearly have
AC∩H
1
−
dh e 2f ′′ (x∗ i )h2 i ≤
=
AC
1
−
dh e 2f ′′ (x∗ i )h2 i , (133)
(2π) N/2
f ′′ (x ∗ i )
BC
du
i
1 u
− 2
e 2 i
f ′′ (x∗ i )
2π f ′′ (x∗ i )
, (134)
where we have performed the change of variable ui = f ′′ (x∗ i ) · hi and denoted by BC the set {h ∈
RN , u2 i ≤ C2 }. Now, let x∗ max = max{x∗ i } and x∗min = min{x∗i }. Expression (134) then gives
AC∩H
1
−
dh e 2f ′′ (x∗ i )h2 i ≤
SN−1
(2π) N/2
f ′′ (x∗ min )N/2
du
BC
e−
1
2 f ′′ (x∗ max )u 2 i
(2π f ′′ (x∗ min
f
= SN−1
′′ (x∗ max) N/2
f ′′ (x∗
N
Γ
min
)N 2 ,
C2 2 f ′′ (x∗ max)
N
2 −1
f ′′ (x ∗ max) N/2
f ′′ (x ∗ min )N
C 2
2 f ′′ (x ∗ max)
))N/2 , (135)
, (136)
· e −
C 2
2 f ′′ (x∗ max ) , (137)
which decays exponentially fast for large S (or large x ∗ max) as long as f ′′ goes to zero at infinity, i.e, for any
function f which goes to infinity not faster than x 2 . So, finally
AC∩H
1
−
dh e 2f ′′ (x∗ i )h2 i =
which concludes the proof of equation (113).
(2π) N−1
2
N w
i=1
2 iN
j=1 f ′′
j (x∗ j )
f ′′
i (x∗ i )
23
+ O e −
α
f ′′ (x∗ max )
, (138)

412 13. Gestion de portefeuille sous contraintes de capital économique
Second term of the right-hand-side of (112).
We now have to show that
dh e
AC∩H
−f(x ∗ i +hi)−g(x∗ i +hi)
(139)
can be neglected. This is obvious since, by assumption 2 and 6, the function f(x) − ln g(x) remains convex
for x large enough, which ensures that f(x) − ln g(x) ≥ C1|x| for some positive constant C1 and x large
enough. Thus, choosing the constant C in AC large enough, we have
dh e −N
i=1 f(xi)−ln
g(xi)
≤ dh e −C1N
i=1 |x∗ i +hi|
∼ O
AC∩H
AC∩H
Thus, for S large enough, the density PS(S) is asymptotically equal to
PS(S) =
i
g(x ∗ i )
In the case of the modified Weibull variables, we have
f(x) =
(2π) N−1
2
N w
i=1
2 iN
j=1 f ′′
j (x∗ j )
f ′′
i (x∗ i )
|x|
χ
α′
−
e f ′′ (x∗
) . (140)
. (141)
c
, (142)
and
c
g(x) =
2 √ c
· |x| 2
πχc/2 −1 , (143)
which satisfy our assumptions if and only if c > 1. In such a case, we obtain
x ∗ i = w
1
c−1
i
c
c−1
wi
which, after some simple algebraic manipulations, yield
N−1
c
2 c
P (S) ∼
2(c − 1) 2 √ π
with
as announced in theorem 3.
A.2 Sub-exponential case: c ≤ 1
ˆχ = w
c
c−1
i
· S, (144)
1 c
|S| 2
ˆχ c/2 −1 e −|S|
ˆχc
c−1
c
(145)
· χ. (146)
Let X1, X2, · · · , XN be N i.i.d sub-exponential modified Weibull random variables W(c, χ), with distribution
function F . Let us denote by GS the distribution function of the variable
where w1, w2, · · · , wN are real non-random coefficients.
SN = w1X1 + w2X2 + · · · + wNXN, (147)
Let w ∗ = max{|w1|, |w2|, · · · , |wN|}. Then, theorem 5.5 (b) of Goldie and Klüppelberg (1998) states that
GS(x/w
lim
x→∞
∗ )
F (x) = Card {i ∈ {1, 2, , N} : |wi| = w ∗ }. (148)
By definition 2, this allows us to conclude that SN is equivalent in the upper tail to Z ∼ W(c, w ∗ χ).
A similar calculation yields an analogous result for the lower tail.
24

B Asymptotic distribution of the sum of Weibull variables with a Gaussian
copula.
We assume that the marginal distributions are given by the modified Weibull distributions:
Pi(xi) = 1
2 √ π
c
χ c/2
i
|xi| c/2−1 e −|x i |
χ ic
413
(149)
and that the χi’s are all equal to one, in order not to cumber the notation. As in the proof of corollary 2, it
will be sufficient to replace wi by wiχi to reintroduce the scale factors.
Under the Gaussian copula assumption, we obtain the following form for the multivariate distribution :
c
P (x1, · · · , xN) =
N
2Nπ N/2√ N
x
det V
c/2−1
⎡
i exp ⎣−
V −1
ij xc/2
i xc/2
⎤
⎦
j . (150)
Let
i=1
f(x1, · · · , xN) =
i,j
i,j
V −1
ij xi c/2 xj c/2 . (151)
We have to minimize f under the constraint wixi = S. As for the independent case, we introduce a
Lagrange multiplier λ which leads to
c
j
V −1
jk x∗ j c/2 x ∗ k c/2−1 = λwk . (152)
The left-hand-side of this equation is a homogeneous function of degree c − 1 in the x∗ i ’s, thus necessarily
where the σi’s are solution of
j
x ∗ i =
1
λ c−1
· σi, (153)
c
V −1
jk σj c/2 σk c/2−1 = wk . (154)
The set of equations (154) has a unique solution due to the convexity of the minimization problem. This
set of equations can be easily solved by a numerical method like Newton’s algorithm. It is convenient to
simplify the problem and avoid the inversion of the matrix V , by rewritting (154) as
Using the constraint wix ∗ i
so that
k
= S, we obtain
Vjk wk σk 1−c/2 = σ c/2
j . (155)
1
λ c−1
c
x ∗ i =
=
S
, (156)
wiσi
σi
· S. (157)
wiσi
25

414 13. Gestion de portefeuille sous contraintes de capital économique
Thus
f(x1, · · · , xN) = f(x ∗ 1, · · · , x ∗ N) + ∂f
=
Sc ( 1
+
wiσi) c−1 2
i
ij
∂xi
hi + 1
2
ij
∂2f hihj + · · · (158)
∂xi∂xj
∂2f hihj + · · · , (159)
∂xi∂xj
where, as in the previous section, hi = xi − x ∗ i and the derivatives of f are expressed at x∗ 1 , ..., x∗ N .
It is easy to check that the nth-order derivative of f with respect to the xi’s evaluated at {x∗ i } is proportional
to Sc−n . In the sequel, we will use the following notation :
∂n
f
∂xi1 · · · ∂xin
We can write :
f(x1, · · · , xN) =
up to the fourth order. This leads to
P (S) ∝ e −
Sc ( Sc−2
+
wiσi) c−1 2
( w
S c
iσi ) c−1
{x ∗ i }
ij
= M (n)
i1···in Sc−n . (160)
M (2)
ij hihj + Sc−3
6
ijk
Sc−2ij
(2)
− M
dh1 · · · dhNe 2
ij hihj
δ wihi
×
⎡
⎣1 + Sc−3
6
Using the relation δ ( wihi) = dk
2π e−ikj wjhj , we obtain :
P (S) ∝ e −
or in vectorial notation :
( w
S c
iσi ) c−1
P (S) ∝ e −
( w
dk
2π
S c
iσi ) c−1
×
M (3)
ijk
M (3)
ijk hihjhk + · · · (161)
ijk hihjhk + · · ·
×
⎤
Sc−2ij
(2)
− M
dh1 · · · dhNe 2
ij hihj−ikj wjhj ×
×
dk
2π
⎡
⎡
⎣1 + Sc−3
6
⎣1 + Sc−3
6
Let us perform the following standard change of variables :
M (3)
ijk
ijk hihjhk + · · ·
Sc−2
−
dh e 2 htM (2) h−ikwth ×
⎤
M (3)
(M (2)−1 exists since f is assumed convex and thus M (2) positive) :
S c−2
ijk
ijk hihjhk + · · ·
⎦ . (162)
⎤
⎦ , (163)
⎦ . (164)
h = h ′ − ik
S c−2 M(2)−1 w . (165)
2 ht M (2) h + ikw t h = Sc−2
2 h′t M (2) h ′ + k2
2S c−2 wt M (2)−1 w . (166)
26

This yields
×
h+
Sc−2
− 2 dh e ik
Sc−2 M(2)−1 w wt
M (2)h+
ik
Sc−2 M(2)−1
S c ( wiσi ) c−1
P (S) ∝ e −
⎡
⎣1 + Sc−3
M
6
(3)
dk k2
e− 2S
2π c−2 wtM (2)−1w ×
ijk
ijk hihjhk + · · ·
⎤
415
⎦ . (167)
Denoting by 〈·〉h the average with respect to the Gaussian distribution of h and by 〈·〉k the average with
respect to the Gaussian distribution of k, we have :
×
⎡
⎣1 + Sc−3
6
ijk
P (S) ∝
M (3)
ijk 〈〈hihjhk〉h〉k + Sc−4
24
det M (2)−1
w t M (2)−1 w (2πS2−c ) N−1
2 e −
S c ( wiσi ) c−1 ×
M (4)
ijkl 〈〈hihjhkhl〉h〉k
⎤
+ · · · ⎦ . (168)
We now invoke Wick’s theorem 2 , which states that each term 〈〈hi · · · hp〉h〉k can be expressed as a product
of pairwise correlation coefficients. Evaluating the average with respect to the symmetric distribution of k,
it is obvious that odd-order terms will vanish and that the count of powers of S involved in each even-order
term shows that all are sub-dominant. So, up to the leading order :
P (S) ∝
The matrix M (2) can be calculated, which yields
M (2)
kl
and shows that
=
=
ijkl
det M (2)−1
w t M (2)−1 w (2πS2−c ) N−1
2 e −
S c ( wiσi ) c−1 . (169)
1
( wiσi) c−2
c
wk
c − 1 δkl +
2 σk
c2 c
−1 2 V
2 kl σ −1
k σ c
2 −1
l , (170)
1
( wiσi) c−2 ˜ Mkl, (171)
det M (2)−1
w t M (2)−1 w =
c
(N−1)( 2
wiσi
−1)
The inverse matrix ˜ M −1 satisfies
l ˜ Mkl · ( ˜ M −1 )lj = δkj which can be rewritten:
or equivalently
c
wk
c − 1 (
2 σk
˜ M −1 )kj + c2
2
c
c − 1 wk (
2 ˜ M −1 )kj + c2
2
V −1
kl · ( ˜ M −1 )ljσ c
k
l
V −1
kl · ( ˜ M −1 )ljσ c
2
k
l
det ˜M −1
wt ˜M −1 . (172)
w
2 −1
σ c
2 −1
l = δkj (173)
σ c
2 −1
l = δkj · σk (174)
2 See for instance (Brézin et al. 1976) for a general introduction, (Sornette 1998) for an early application to the portfolio problem
and (Sornette et al. 2000b) for a systematic utilization with the help of diagrams.
27

416 13. Gestion de portefeuille sous contraintes de capital économique
which gives
c
c − 1
2
j,k
wk ( ˜ M −1 )kj wj + c2
2
V −1
kl · ( ˜ M −1 )ljσ c c
2
k σl j,k,l
2 −1
wj =
δkj · σkwj . (175)
Summing the rightmost factor of the left-hand-side over k, and accounting for equation (154) leads to
so that
Moreover
c
c − 1
2
c N
2 N π N/2√ det V
j,k
wk ( ˜ M −1 )kj wj + c2
2
N
w t ˜M −1 w =
j,l
1
c(c − 1)
c N
x
i=1
∗ i c/2−1 =
2Nπ N/2√det V
Thus, putting together equations (169), (172), (177) and (178) yields
P (S) c(c − 1) det ˜ M −1
1
·
det V
2 √ π
with
ˆχ =
c N−1 σ c/2−1
i
2 (N−1)/2
wiχiσi
28
j,k
wl ( ˜ M −1 )ljwj =
j
σjwj . (176)
wjσj . (177)
j
c
2 σ −1
i
( c
wiσi)
N( 2
−1) SN( c
2 −1) . (178)
c
ˆχ c/2 |S|c/2−1e −|S|
ˆχc
, (179)
c−1
c
. (180)

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419

420 13. Gestion de portefeuille sous contraintes de capital économique
Gaussian Returns
10 1
10 0
10 −1
10 −4
10 −3
Standard & Poor’s 500
Raw Returns
c/2 = 0.73
Figure 1: Graph of the Gaussianized Standard & Poor’s 500 index returns versus its raw returns, during the
time interval from January 03, 1995 to December 29, 2000 for the negative tail of the distribution.
32
10 −2
10 −1

ln(T/T 0 )
T
Figure 2: Logarithm ln of the ratio of the recurrence time T to a reference time T0 for the recurrence
T0
of a given loss V aR as a function of β defined by β = VaR
VaR∗ . VaR∗ (resp. VaR) is the Value-at-Risk over a
time interval T0 (resp. T ).
α
33
c>1
c=1
c

422 13. Gestion de portefeuille sous contraintes de capital économique

Chapitre 14
Gestion de Portefeuilles multimoments et
équilibre de marché
Nous présentons un nouvel ensemble de mesures de risque qualifiées de consistantes, en termes des
semi-invariants de la distribution de rendement d’un portefeuille, tels que les moments centrés ou les
cumulants, et qui accordent plus de poids aux événements rares. Nous dérivons les frontières efficientes
basées sur cet ensemble de mesures de risques et présentons une version généralisée du CAPM dans le
cas où le marché est composé d’agents formant des anticipations homogènes ou hétérogènes.
Utilisant alors une famille de distributions de Weibull modifiées, qui englobe aussi bien des lois superexponentielles
que sous-exponentielles, pour paramétrer les distributions marginales du rendement des
actifs et une copule gaussienne pour représenter la dépendance entre les actifs, nous dérivons les expressions
analytiques des moments et cumulants de la distribution de rendement du portefeuille pour
une composition arbitraire de ses actifs. Faisant usage de fonctions hypergéométriques, nous sommes en
particulier capables d’étendre certains résultats antérieurs au cas où l’exposant de Weibull est différent
d’un actif à l’autre.
A l’aide de cette représentation paramétrique, nous traitons en détail le problème de la minimisation
du risque mesuré par les cumulants pour un portefeuille de deux actifs et comparons nos prédictions
théoriques aux estimations empiriques directes. Ces formules nous permettent en outre de déterminer
analytiquement dans quelles conditions il est possible de construire un portefeuille présentant à la fois
un rendement plus élevé et moins de grands risques.
423

424 14. Gestion de Portefeuilles multimoments et équilibre de marché

Multi-Moments Method for Portfolio Management:
Generalized Capital Asset Pricing Model
in Homogeneous and Heterogeneous markets ∗
Y. Malevergne 1,2 and D. Sornette 1,3
1 Laboratoire de Physique de la Matière Condensée
CNRS UMR6622 and Université de Nice-Sophia Antipolis
Parc Valrose, 06108 Nice Cedex 2, France
2 Institut de Science Financière et d’Assurances - Université Lyon I
43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex
3 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science
University of California, Los Angeles, California 90095
e-mails: Yannick.Malevergne@unice.fr and sornette@unice.fr
Abstract
We use a new set of consistent measures of risks, in terms of the semi-invariants of pdf’s, such that the
centered moments and the cumulants of the portfolio distribution of returns that put more emphasis on
the tail the distributions. We derive generalized efficient frontiers, based on these novel measures of
risks and present the generalized CAPM, both in the cases of homogeneous and heterogeneous markets.
Then, using a family of modified Weibull distributions, encompassing both sub-exponentials and superexponentials,
to parameterize the marginal distributions of asset returns and their natural multivariate
generalizations, we offer exact formulas for the moments and cumulants of the distribution of returns of
a portfolio made of an arbitrary composition of these assets. Using combinatorial and hypergeometric
functions, we are in particular able to extend previous results to the case where the exponents of the
Weibull distributions are different from asset to asset and in the presence of dependence between assets.
In this parameterization, we treat in details the problem of risk minimization using the cumulants as
measures of risks for a portfolio made of two assets and compare the theoretical predictions with direct
empirical data. Our extended formulas enable us to determine analytically the conditions under which
it is possible to “have your cake and eat it too”, i.e., to construct a portfolio with both larger return and
smaller “large risks”.
1 Introduction
The Capital Asset Pricing Model (CAPM) is still the most widely used approach to relative asset evaluation,
although its empirical roots are been found weaker and weaker in recent years. This asset valuation model
describing the relationship between expected risk and expected return for marketable assets is strongly
∗ We acknowledge helpful discussions and exchanges with J.V. Andersen, J.P. Laurent and V. Pisarenko. We are grateful to
participants of the workshop on “Multi-moment Capital Asset Pricing Models and Related Topics”, ESCP-EAP European School
of Management, Paris, April,19, 2002, and in particular to Philippe Spieser, for their comments. This work was partially supported
by the James S. Mc Donnell Foundation 21st century scientist award/studying complex system.
1
425

426 14. Gestion de Portefeuilles multimoments et équilibre de marché
entangled with the Mean-Variance Portfolio Model. Indeed both of them fundamentally rely on the description
of the probability distribution function (pdf) of asset returns in terms of Gaussian functions. The
Mean-Variance description is thus at the basis of Markovitz’s portfolio theory (Markovitz 1959) and of the
CAPM (see for instance (Merton 1990)).
Otherwise, the determination of the risks and returns associated with a given portfolio constituted of N
assets is completely embedded in the knowledge of their multivariate distribution of returns. Indeed, the
dependence between random variables is completely described by their joint distribution. This remark
entails the two major problems of portfolio theory: 1) determine the multivariate distribution function of
asset returns; 2) derive from it useful measures of portfolio risks and use them to analyze and optimize
portfolios.
The variance (or volatility) of portfolio returns provides the simplest way to quantify its fluctuations and
is at the fundation of the (Markovitz 1959)’s portfolio selection theory. Nonetheless, the variance of a
portfolio offers only a limited quantification of incurred risks (in terms of fluctuations), as the empirical
distributions of returns have “fat tails” (Lux 1996, Gopikrishnan et al. 1998, among many others) and the
dependences between assets are only imperfectly accounted for by the covariance matrix (Litterman and
Winkelmann 1998). It is thus essential to extend portfolio theory and the CAPM to tackle these empirical
facts.
The Value-at-Risk (Jorion 1997) and many other measures of risks (Artzner et al. 1997, Sornette 1998,
Artzner et al. 1999, Bouchaud et al. 1998, Sornette et al. 2000b) have then been developed to account for the
larger moves allowed by non-Gaussian distributions and non-linear correlations but they mainly allow for the
assessment of down-side risks. Here, we consider both-side risk and define general measures of fluctuations.
It is the first goal of this article. Indeed, considering the minimum set of properties a fluctuation measure
must fulfil, we characterize these measures. In particular, we show that any absolute central moments and
some cumulants satisfy these requirement as well as do any combination of these quantities. Moreover,
the weights involved in these combinations can be interpreted in terms of the portfolio manager’s aversion
against large fluctuations.
Once the definition of the fluctuation measures have been set, it is possible to classify the assets and portfolios
using for instance a risk adjustment method (Sharpe 1994, Dowd 2000) and to develop a portfolio
selection and optimization approach. It is the second goal of this article.
Then a new model of market equilibrium can be derived, which generalizes the usual Capital Asset Pricing
Model (CAPM). This is the third goal of our paper. This improvement is necessary since, although the use
of the CAPM is still widely spread, its empirical justification has been found less and less convincing in the
past years (Lim 1989, Harvey and Siddique 2000).
The last goal of this article is to present an efficient parametric method allowing for the estimation of the
centered moments and cumulants, based upon a maximum entropy principle. This parameterization of
the problem is necessary in order to obtain accurate estimates of the high order moment-based quantities
involved the portfolio optimization problem with our generalized measures of fluctuations.
The paper is organized as follows.
Section 2 presents a new set of consistent measures of risks, in terms of the semi-invariants of pdf’s, such as
the centered moments and the cumulants of the portfolio distribution of returns, for example.
Section 3 derives the generalized efficient frontiers, based on these novel measures of risks. Both cases with
and without risk-free asset are analyzed.
Section 4 offers a generalization of the Sharpe ratio and thus provides new tools to classify assets with
respect to their risk adjusted performance. In particular, we show that this classification may depend on the
2

choosen risk measure.
Section 5 presents the generalized CAPM based on these new measures of risks, both in the cases of homogeneous
and heterogeneous agents.
Section 6 introduces a novel general parameterization of the multivariate distribution of returns based on two
steps: (i) the projection of the empirical marginal distributions onto Gaussian laws via nonlinear mappings;
(ii) the use of an entropy maximization to construct the corresponding most parsimonious representation of
the multivariate distribution.
Section 7 offers a specific parameterization of marginal distributions in terms of so-called modified Weibull
distributions, which are essentially exponential of minus a power law. Notwithstanding their possible fat-tail
nature, all their moments and cumulants are finite and can be calculated. We present empirical calibration
of the two key parameters of the modified Weibull distribution, namely the exponent c and the characteristic
scale χ.
Section 8 provides the analytical expressions of the cumulants of the distribution of portfolio returns for the
parameterization of marginal distributions in terms of so-called modified Weibull distributions, introduced
in section 6. Empirical tests comparing the direct numerical evaluation of the cumulants of financial time
series to the values predicted from our analytical formulas find a good consistency.
Section 9 uses these two sets of results to illustrate how portfolio optimization works in this context. The
main novel result is an analytical understanding of the conditions under which it is possible to simultaneously
increase the portfolio return and decreases its large risks quantified by large-order cumulants. It thus appears
that the multidimensional nature of risks allows one to break the stalemate of no better return without more
risks, for some special kind of rational agents.
Section 10 concludes.
Before proceeding with the presentation of our results, we set the notations to derive the basic problem
addressed in this paper, namely to study the distribution of the sum of weighted random variables with
arbitrary marginal distributions and dependence. Consider a portfolio with ni shares of asset i of price pi(0)
at time t = 0 whose initial wealth is
N
W (0) = nipi(0) . (1)
A time τ later, the wealth has become W (τ) = N
i=1 nipi(τ) and the wealth variation is
where
δτ W ≡ W (τ) − W (0) =
N
i=1
wi =
i=1
nipi(0) pi(τ) − pi(0)
pi(0)
nipi(0)
N
j=1 njpj(0)
= W (0)
427
N
wiri(t, τ), (2)
is the fraction in capital invested in the ith asset at time 0 and the return ri(t, τ) between time t − τ and t of
asset i is defined as:
ri(t, τ) = pi(t) − pi(t − τ)
.
pi(t − τ)
(4)
Using the definition (4), this justifies us to write the return Sτ of the portfolio over a time interval τ as the
weighted sum of the returns ri(τ) of the assets i = 1, ..., N over the time interval τ
Sτ = δτ W
W (0) =
3
i=1
(3)
N
wi ri(τ) . (5)
i=1

428 14. Gestion de Portefeuilles multimoments et équilibre de marché
In the sequel, we shall thus consider asset returns as the fundamental variables (denoted xi or Xi in the
sequel) and study their aggregation properties, namely how the distribution of portfolio return equal to their
weighted sum derives for their multivariable distribution. We shall consider a single time scale τ which
can be chosen arbitrarily, say equal to one day. We shall thus drop the dependence on τ, understanding
implicitely that all our results hold for returns estimated over the time step τ.
2 Measuring large risks of a portfolio
The question on how to assess risk is recurrent in finance (and in many other fields) and has not yet received
a general solution. Since the middle of the twentieth century, several paths have been explored. The pioneering
work by (Von Neuman and Morgenstern 1947) has given birth to the mathematical definition of the
expected utility function which provides interesting insights on the behavior of a rational economic agent
and formalized the concept of risk aversion. Based upon the properties of the utility function, (Rothschild
and Stiglitz 1970) and (Rothschild and Stiglitz 1971) have attempted to define the notion of increasing risks.
But, as revealed by (Allais 1953, Allais 1990), empiric investigations has proven that the postulates chosen
by (Von Neuman and Morgenstern 1947) are actually often violated. Many generalizations have been
proposed for curing the so-called Allais’ Paradox, but up to now, no generally accepted procedure has been
found in this way.
Recently, a theory due to (Artzner et al. 1997, Artzner et al. 1999) and its generalization by (Föllmer and
Schied 2002a, Föllmer and Schied 2002b), have appeared. Based on a series of postulates that are quite
natural, this theory allows one to build coherent (convex) measures of risks. In fact, this theory seems welladapted
to the assessment of the needed economic capital, that is, of the fraction of capital a company must
keep as risk-free assets in order to face its commitments and thus avoid ruin. However, for the purpose of
quantifying the fluctuations of the asset returns and of developing a theory of portfolios, this approach does
not seem to be operational. Here, we shall rather revisit (Markovitz 1959)’s approach to investigate how
its extension to higher-order moments or cumulants, and any combination of these quantities, can be used
operationally to account for large risks.
2.1 Why do higher moments allow to assess larger risks?
In principle, the complete description of the fluctuations of an asset at a given time scale is given by the
knowledge of the probability distribution function (pdf) of its returns. The pdf encompasses all the risk
dimensions associated with this asset. Unfortunately, it is impossible to classify or order the risks described
by the entire pdf, except in special cases where the concept of stochastic dominance applies. Therefore, the
whole pdf can not provide an adequate measure of risk, embodied by a single variable. In order to perform
a selection among a basket of assets and construct optimal portfolios, one needs measures given as real
numbers, not functions, which can be ordered according to the natural ordering of real numbers on the line.
In this vein, (Markovitz 1959) has proposed to summarize the risk of an asset by the variance of its pdf of
returns (or equivalently by the corresponding standard deviation). It is clear that this description of risks is
fully satisfying only for assets with Gaussian pdf’s. In any other case, the variance generally provides a very
poor estimate of the real risk. Indeed, it is a well-established empirical fact that the pdf’s of asset returns
has fat tails (Lux 1996, Pagan 1996, Gopikrishnan et al. 1998), so that the Gaussian approximation underestimates
significantly the large prices movements frequently observed on stock markets. Consequently, the
variance can not be taken as a suitable measure of risks, since it only accounts for the smallest contributions
to the fluctuations of the assets returns.
4

The variance of the return X of an asset involves its second moment E[X2
] and, more precisely, is equal
to its second centered moment (or moment about the mean) E (X − E[X]) 2
. Thus, the weight of a given
fluctuation X entering in the definition of the variance of the returns is proportional to its square. Due to the
decay of the pdf of X for large X bounded from above by ∼ 1/|X| 1+α with α > 2, the largest fluctuations
do not contribute significantly to this expectation. To increase their contributions, and in this way to account
for the largest fluctuations, it is natural to invoke higher order moments of order n > 2. The large n is, the
larger is the contribution of the rare and large returns in the tail of the pdf. This phenomenon is demonstrated
in figure 1, where we can observe the evolution of the quantity x n · P (x) for n = 1, 2 and 4, where P (x), in
this example, is the standard exponential distribution e −x . The expectation E[X n ] is then simply represented
geometrically as equal to the area below the curve x n ·P (x). These curves provide an intuitive illustration of
the fact that the main contributions to the moment E[X n ] of order n come from values of X in the vicinity
of the maximum of x n · P (x) which increases fast with the order n of the moment we consider, all the more
so, the fatter is the tail of the pdf of the returns X. For the exponential distribution chosen to construct figure
1, the value of x corresponding to the maximum of x n · P (x) is exactly equal to n. Thus, increasing the
order of the moment allows one to sample larger fluctuations of the asset prices.
2.2 Quantifying the fluctuations of an asset
Let us now examine what should be the properties that coherent measures of risks adapted to the portfolio
problem must satisfy in order to best quantify the asset price fluctuations. Let us consider an asset denoted
X, and let G be the set of all the risky assets available on the market. Its profit and loss distribution is the
distribution of δX = X(τ) − X(0), while the return distribution is given by the distribution of X(τ)−X(0)
X(0) .
The risk measures will be defined for the profit and loss distributions and then shown to be equivalent to
another definition applied to the return distribution.
Our first requirement is that the risk measure ρ(·), which is a functional on G, should always remain positive
AXIOM 1 ∀X ∈ G, ρ(δX) ≥ 0 ,
where the equality holds if and only if X is certain. Let us now add to this asset a given amount a invested
in the risk free-asset whose return is µ0 (with therefore no randomness in its price trajectory) and define the
new asset Y = X +a. Since a is non-random, the fluctuations of X and Y are the same. Thus, it is desirable
that ρ enjoys the property of translational invariance, whatever the asset X and the non-random coefficient
a may be:
AXIOM 2 ∀X ∈ G, ∀a ∈ R, ρ(δX + µ · a) = ρ(δX).
We also require that our risk measure increases with the quantity of assets held in the portfolio. A priori,
one should expect that the risk of a position is proportional to its size. Indeed, the fluctuations associated
with the variable 2 · X are naturally twice larger as the fluctuations of X. This is true as long as we can
consider that a large position can be liquidated as easily as a smaller one. This is obviously not true, due
to the limited liquidity of real markets. Thus, a large position in a given asset is more risky than the sum
of the risks associated with the many smaller positions which add up to the large position. To account for
this point, we assume that ρ depends on the size of the position in the same manner for all assets. This
assumption is slightly restrictive but not unrealistic for companies with comparable properties in terms of
market capitalization or sector of activity. This requirement reads
AXIOM 3 ∀X ∈ G, ∀λ ∈ R+, ρ(λ · δX) = f(λ) · ρ(δX),
5
429

430 14. Gestion de Portefeuilles multimoments et équilibre de marché
where the function f : R+ −→ R+ is increasing and convex to account for liquidity risk. In fact, it is
straightforward to show 1 that the only functions statistying this axiom are the fonctions fα(λ) = λ α with
α ≥ 1, so that axiom 3 can be reformulated in terms of positive homogeneity of degree α:
AXIOM 4
∀X ∈ G, ∀λ ∈ R+, ρ(λ · δX) = λ α · ρ(δX). (6)
Note that the case of liquid markets is recovered by α = 1 for which the risk is directly proportionnal to the
size of the position.
These axioms, which define our risk measures for profit and loss can easily be extended to the returns of
the assets. Indeed, the return is nothing but the profit and loss divided by the initial value X(0) of the asset.
One can thus easily check that the risk defined on the profit and loss distribution is X(0) α times the risk
defined on the return distribution. In the sequel, we will only consider this later definition, and, to simplify
the notations since we will only consider the returns and not the profit and loss, the notation X will be used
to denote the asset and its return as well.
We can remark that the risk measures ρ enjoying the two properties defined by the axioms 2 and 4 are known
as the semi-invariants of the distribution of the profit and loss / returns of X (see (Stuart and Ord 1994, p
86-87)). Among the large familly of semi-invariants, we can cite the well-known centered moments and
cumulants of X.
2.3 Examples
The set of risk measures obeying axioms 1-4 is huge since it includes all the homogeneous functionals of
(X − E[X]), for instance. The centered moments (or moments about the mean) and the cumulants are two
well-known classes of semi-invariants. Then, a given value of α can be seen as nothing but a specific choice
of the order n of the centered moments or of the cumulants. In this case, our risk measure defined via these
semi-invariants fulfills the two following conditions:
ρ(X + µ) = ρ(X), (7)
ρ(λ · X) = λ n · ρ(X). (8)
In order to satisfy the positivity condition (axiom 1), we need to restrict the set of values taken by n. By
construction, the centered moments of even order are always positive while the odd order centered moments
can be negative. Thus, only the even order centered moments are acceptable risk measures. The situation
is not so clear for the cumulants, since the even order cumulants, as well as the odd order ones, can be
negative. In full generality, only the centered moments provide reasonable risk measures satifying our
axioms. However, for a large class of distributions, even order cumulants remain positive, especially for
fat tail distributions (eventhough there are simple but somewhat artificial counter-examples). Therefore,
cumulants of even order can be useful risk measures when restricted to these distributions.
Indeed, the cumulants enjoy a property which can be considered as a natural requirement for a risk measure.
It can be desirable that the risk associated with a portfolio made of independent assets is exactly the sum
of the risk associated with each individual asset. Thus, given N independent assets {X1, · · · , XN}, and the
portfolio SN = X1 + · · · + XN, we wish to have
ρn(SN) = ρn(X1) + · + ρn(XN) . (9)
1 using the trick ρ(λ1λ2 · δX) = f(λ1) · ρ(λ2 · δX) = f(λ1) · f(λ2) · ρ(δX) = f(λ1 · λ2) · ρ(δX) leading to f(λ1 · λ2) =
f(λ1) · f(λ2). The unique increasing convex solution of this functional equation is fα(λ) = λ α with α ≥ 1.
6

This property is verified for all cumulants while is not true for centered moments. In addition, as seen from
their definition in terms of the characteristic function (63), cumulants of order larger than 2 quantify deviation
from the Gaussian law, and thus large risks beyond the variance (equal to the second-order cumulant).
Thus, centered moments of even orders possess all the minimal properties required for a suitable portfolio
risk measure. Cumulants fulfill these requirement only for well behaved distributions, but have an additional
advantage compared to the centered moments, that is, they fulfill the condition (9). For these reasons, we
shall consider below both the centered moments and the cumulants.
In fact, we can be more general. Indeed, as we have written, the centered moments or the cumulants of order
n are homogeneous functions of order n, and due to the positivity requirement, we have to restrict ourselves
to even order centered moments and cumulants. Thus, only homogeneous functions of order 2n can be
considered. Actually, this restrictive constraint can be relaxed by recalling that, given any homogeneous
function f(·) of order p, the function f(·) q is also homogeneous of order p · q. This allows us to decouple
the order of the moments to consider, which quantifies the impact of the large fluctuations, from the influence
of the size of the positions held, measured by the degres of homogeneity of ρ. Thus, considering any even
order centered moments, we can build a risk measure ρ(X) = E (X − E[X]) 2n α/2n which account for
the fluctuations measured by the centered moment of order 2n but with a degree of homogeneity equal to α.
A further generalization is possible to odd-order moments. Indeed, the absolute centered moments satisfy
our three axioms for any odd or even order. We can go one step further and use non-integer order absolute
centered moments, and define the more general risk measure
where γ denotes any positve real number.
431
ρ(X) = E [|X − E[X]| γ ] α/γ , (10)
These set of risk measures are very interesting since, due to the Minkowsky inegality, they are convex for
any α and γ larger than 1 :
ρ(u · X + (1 − u) · Y ) ≤ u · ρ(X) + (1 − u) · ρ(Y ), (11)
which ensures that aggregating two risky assets lead to diversify their risk. In fact, in the special case γ = 1,
these measures enjoy the stronger sub-additivity property.
Finally, we should stress that any discrete or continuous (positive) sum of these risk measures, with the same
degree of homogeneity is again a risk measure. This allows us to define “spectral measures of fluctuations”
in the same spirit as in (Acerbi 2002):
ρ(X) = dγ φ(γ) E [(X − E[X]) γ ] α/γ , (12)
where φ is a positive real valued function defined on any subinterval of [1, ∞) such that the integral in
(12) remains finite. It is interesting to restrict oneself to the functions φ whose integral sums up to one:
dγ φ(γ) = 1, which is always possible, up to a renormalization. Indeed, in such a case, φ(γ) represents
the relative weight attributed to the fluctuations measured by a given moment order. Thus, the function φ
can be considered as a measure of the risk aversion of the risk manager with respect to the large fluctuations.
Let us stress that the variance, originally used in (Markovitz 1959)’s portfolio theory, is nothing but the
second centered moment, also equal to the second order cumulant (the three first cumulants and centered
moments are equal). Therefore, a portfolio theory based on the centered moments or on the cumulants
automatically contain (Markovitz 1959)’s theory as a special case, and thus offers a natural generalization
emcompassing large risks of this masterpiece of the financial science. It also embodies several other generalizations
where homogeneous measures of risks are considered, a for instance in (Hwang and Satchell 1999).
7

432 14. Gestion de Portefeuilles multimoments et équilibre de marché
3 The generalized efficient frontier and some of its properties
We now address the problem of the portfolio selection and optimization, based on the risk measures introduced
in the previous section. As we have already seen, there is a large choice of relevant risk measures
from which the portfolio manager is free to choose as a function of his own aversion to small versus large
risks. A strong risk aversion to large risks will lead him to choose a risk measure which puts the emphasis
on the large fluctuations. The simplest examples of such risk measures are provided by the high-order centered
moments or cumulants. Obviously, the utility function of the fund manager plays a central role in his
choice of the risk measure. The relation between the central moments and the utility function has already
been underlined by several authors such as (Rubinstein 1973) or (Jurczenko and Maillet 2002), who have
shown that an economic agent with a quartic utility function is naturally sensitive to the first four moments
of his expected wealth distribution. But, as stressed before, we do not wish to consider the expected utility
formalism since our goal, in this paper, is not to study the underlying behavior leading to the choice of any
risk measure.
The choice of the risk measure also depends upon the time horizon of investment. Indeed, as the time
scale increases, the distribution of asset returns progressively converges to the Gaussian pdf, so that only
the variance remains relevant for very long term investment horizons. However, for shorter time horizons,
say, for portfolio rebalanced at a weekly, daily or intra-day time scales, choosing a risk measure putting the
emphasis on the large fluctuations, such as the centered moments µ6 or µ8 or the cumulants C6 or C8 (or of
larger orders), may be necessary to account for the “wild” price fluctuations usually observed for such short
time scales.
Our present approach uses a single time scale over which the returns are estimated, and is thus restricted
to portfolio selection with a fixed investment horizon. Extensions to a portofolio analysis and optimization
in terms of high-order moments and cumulants performed simultaneously over different time scales can be
found in (Muzy et al. 2001).
3.1 Efficient frontier without risk-free asset
Let us consider N risky assets, denoted by X1, · · · , XN. Our goal is to find the best possible allocation, given
a set of constraints.The portfolio optimization generalizing the approach of (Sornette et al. 2000a, Andersen
and Sornette 2001) corresponds to accounting for large fluctuations of the assets through the risk measures
introduced above in the presence of a constraint on the return as well as the “no-short sells” constraint:
⎧
⎪⎨
⎪⎩
inf wi∈[0,1] ρα({wi})
i≥1 wi = 1
i≥1 wiµ(i) = µ ,
wi ≥ 0, ∀i > 0,
where wi is the weight of Xi and µ(i) its expected return. In all the sequel, the subscript α in ρα will refer
to the degree of homogeneity of the risk measure.
This problem cannot be solved analytically (except in the Markovitz’s case where the risk measure is given
by the variance). We need to perform numerical calculations to obtain the shape of the efficient frontier.
Nonetheless, when the ρα’s denotes the centered moments or any convex risk measure, we can assert that
this optimization problem is a convex optimization problem and that it admits one and only one solution
which can be easily determined by standard numerical relaxation or gradient methods.
As an example, we have represented In figure 2, the mean-ρα efficient frontier for a portfolio made of seventeen
assets (see appendix A for details) in the plane (µ, ρ 1/α
α ), where ρα represents the centered moments
8
(13)

µn=α of order n = 2, 4, 6 and 8. The efficient frontier is concave, as expected from the nature of the optimization
problem (13). For a given value of the expected return µ, we observe that the amount of risk
measured by µ 1/n
n increases with n, so that there is an additional price to pay for earning more: not only
the µ2-risk increases, as usual according to Markowitz’s theory, but the large risks increases faster, the more
so, the larger n is. This means that, in this example, the large risks increases more rapidly than the small
risks, as the required return increases. This is an important empirical result that has obvious implications for
portfolio selection and risk assessment. For instance, let us consider an efficient portfolio whose expected
(daily) return equals 0.12%, which gives an annualized return equal to 30%. We can see in table 1 that the
typical fluctuations around the expected return are about twice larger when measured by µ6 compared with
µ2 and that they are 1.5 larger when measured with µ8 compared with µ4.
3.2 Efficient frontier with a risk-free asset
Let us now assume the existence of a risk-free asset X0. The optimization problem with the same set of
constraints as previoulsy can be written as:
⎧
⎪⎨
⎪⎩
inf wi∈[0,1] ρα({wi})
i≥0 wi = 1
i≥0 wiµ(i) = µ ,
wi ≥ 0, ∀i > 0,
This optimization problem can be solved exactly. Indeed, due to existence of a risk-free asset, the normalization
condition wi = 1 is not-constraining since one can always adjust, by lending or borrowing money,
the fraction w0 to a value satisfying the normalization condition. Thus, as shown in appendix B, the efficient
frontier is a straight line in the plane (µ, ρα 1/α ), with positive slope and whose intercept is given by the
value of the risk-free interest rate:
µ = µ0 + ξ · ρα 1/α , (15)
where ξ is a coefficient given explicitely below. This result is very natural when ρα denotes the variance,
since it is then nothing but (Markovitz 1959)’s result. But in addition, it shows that the mean-variance result
can be generalized to every mean-ρα optimal portfolios.
We present in figure 3 the results given by numerical simulations. The set of assets is the same as before and
the risk-free interest rate has been set to 5% a year. The optimization procedure has been performed using
a genetic algorithm on the risk measure given by the centered moments µ2, µ4, µ6 and µ8. As expected,
we observe three increasing straight lines, whose slopes monotonically decay with the order of the centered
moment under consideration. Below, we will discuss this property in greater detail.
3.3 Two funds separation theorem
The two funds separation theorem is a well-known result associated with the mean-variance efficient portfolios.
It results from the concavity of the Markovitz’s efficient frontier for portfolios made of risky assets
only. It states that, if the investors can choose between a set of risky assets and a risk-free asset, they invest
a fraction w0 of their wealth in the risk-free asset and the fraction 1 − w0 in a portfolio composed only with
risky assets. This risky portofolio is the same for all the investors and the fraction w0 of wealth invested
in the risk-free asset depends on the risk aversion of the investor or on the amount of economic capital an
institution must keep aside due to the legal requirements insuring its solvency at a given confidence level.
We shall see that this result can be generalized to any mean-ρα efficient portfolio.
9
433
(14)

434 14. Gestion de Portefeuilles multimoments et équilibre de marché
Indeed, it can be shown (see appendix B) that the weights of the optimal portfolios that are solutions of (14)
are given by:
w ∗ 0 = w0, (16)
w ∗ i = (1 − w0) · ˜wi, i ≥ 1, (17)
where the ˜wi’s are constants such that ˜wi = 1 and whose expressions are given appendix B. Thus,
denoting by Π the portfolio only made of risky assets whose weights are the ˜wi’s, the optimal portfolios are
the linear combination of the risk-free asset, with weight w0, and of the portfolio Π, with weigth 1 − w0.
This result generalizes the mean-variance two fund theorem to any mean-ρα efficient portfolio.
To check numerically this prediction, figure 4 represents the five largest weights of assets in the portfolios
previously investigated as a function of the weight of the risk-free asset, for the four risk measures given
by the centered moments µ2, µ4, µ6 and µ8. One can observe decaying straight lines that intercept the
horizontal axis at w0 = 1, as predicted by equations (16-17).
In figure 2, the straight lines representing the efficient portfolios with a risk-free asset are also represented.
They are tangent to the efficient frontiers without risk-free asset. This is natural since the efficient portfolios
with the risk-free asset are the weighted sum of the risk-free asset and the optimal portfolio Π only made
of risky assets. Since Π also belongs to the efficient frontier without risk-free asset, the optimum is reached
when the straight line describing the efficient frontier with a risk-free asset and the (concave) curve of the
efficient frontier without risk-free asset are tangent.
3.4 Influence of the risk-free interest rate
Figure 3 has shown that the slope of the efficient frontier (with a risk-free asset) decreases when the order
n of the centered moment used to measure risks increases. This is an important qualitative properties of the
risk measures offered by the centered moments, as this means that higher and higher large risks are sampled
under increasing imposed return.
Is it possible that the largest risks captured by the high-order centered moments could increase at a slower
rate than the small risks embodied in the small-order centered cumulants? For instance, is it possible for
the slope of the mean-µ6 efficient frontier to be larger than the slope of the mean-µ4 frontier? This is an
important question as it conditions the relative costs in terms of the panel of risks under increasing specified
returns. To address this question, consider figure 2. Changing the value of the risk-free interest rate amounts
to move the intercept of the straight lines along the ordinate axis so as to keep them tangent to the efficient
frontiers without risk-free asset. Therefore, it is easy to see that, in the situation depicted in figure 2, the
slope of the four straight lines will always decay with the order of the centered moment.
In order to observe an inversion in the order of the slopes, it is necessary and sufficient that the efficient
frontiers without risk-free asset cross each other. This assertion is proved by visual inspection of figure
5. Can we observe such crossing of efficient frontiers? In the most general case of risk measure, nothing
forbids this occurence. Nonetheless, we think that this kind of behavior is not realistic in a financial context
since, as said above, it would mean that the large risks could increase at a slower rate than the small risks,
implying an irrational behavior of the economic agents.
4 Classification of the assets and of portfolios
Let us consider two assets or portfolios X1 and X2 with different expected returns µ(1), µ(2) and different
levels of risk measured by ρα(X1) and ρα(X2). An important question is then to be able to compare
10

these two assets or portfolios. The most general way to perform such a comparison is to refer to decision
theory and to calculate the utility of each of them. But, as already said, the utility function of an agent is
generally not known, so that other approaches have to be developed. The simplest solution is to consider
that the couple (expected return, risk measure) fully characterizes the behavior of the economic agent and
thus provides a sufficiently good approximation for her utility function.
In the (Markovitz 1959)’s world for instance, the preferences of the agents are summarized by the two first
moments of the distribution of assets returns. Thus, as shown by (Sharpe 1966, Sharpe 1994) a simple way
to synthetize these two parameters, in order to get a measure of the performance of the assets or portfolios,
is to build the ratio of the expected return µ (minus the risk free interest rate) over the standard deviation σ:
S =
435
µ − µ0
, (18)
σ
which is the so-called Sharpe ratio and simply represents the amount of expected return per unit of risk,
measured by the standard deviation. It is an increasing function of the expected return and a decreasing
function of the level of risk, which is natural for risk-averse or prudential agent.
4.1 The risk-adjustment approach
This approach can be generalized to any type of risk measures (see (Dowd 2000), for instance) and thus
allows for the comparison of assets whose risks are not well accounted for by the variance (or the standard
deviation). Indeed, instead of considering the variance, which only accounts for the small risks, one can build
the ratio of the expected return over any risk measure. In fact, looking at the equation (113) in appendix B,
the expression
µ − µ0
, (19)
ρα(X) 1/α
naturally arises and is constant for every efficient portfolios. In this expression, α denotes the coefficient
of homogeneity of the risk measure. It is nothing but a simple generalisation of the usual Sharpe ratio.
Indeed, when ρα is given by the variance σ 2 , the expression above recovers the Sharpe ratio. Thus, once
the portfolio manager has chosen his measure of fluctuations ρα, he can build a consistent risk-adjusted
performance measure, as shown by (19).
As just said, these generalized Sharpe ratios are constant for every efficient portfolios. In fact, they are not
only constant but also maximum for every efficient portfolios, so that looking for the portfolio with maximum
generalized Sharpe ratio yields the same optimal portfolios as those found with the whole optimization
program solved in the previous section.
As an illutration, table 2 gives the risk-adjusted performance of the set of seventeen assets already studied,
for several risk measures. We have considered the three first even order centered moments (columns 2 to 4)
and the three first even order cumulants (columns 2, 5 and 6) as fluctuation measures. Obviously the second
order centered moment and the second order cumulant are the same, and give again the usual Sharpe ratio
(18). The assets have been sorted with respect to their Sharpe Ratio.
The first point to note is that the rank of an asset in terms of risk-adjusted perfomance strongly depends on
the risk measure under consideration. The case of MCI Worldcom is very striking in this respect. Indeed,
according to the usual Sharpe ratio, it appears in the 12 th position with a value larger than 0.04 while
according to the other measures it is the last asset of our selection with a value lower than 0.02.
The second interesting point is that, for a given asset, the generalize Sharpe ratio is always a decreasing
function of the order of the considered centered moment. This is not particular to our set of assets since we
11

436 14. Gestion de Portefeuilles multimoments et équilibre de marché
can prove that
so that
∀p > q,
(E [|X| p ]) 1/p ≥ (E [|X| q ]) 1/q , (20)
µ − µ0
(E [|X| p ≤
1/p
])
µ − µ0
(E [|X| q . (21)
1/q
])
On the contrary, when the cumulants are used as risk measures, the generalized Sharpe ratios are not monotonically
decreasing, as exhibited by Procter & Gamble for instance. This can be surprising in view of our
previous remark that the larger is the order of the moments involved in a risk measure, the larger are the fluctuations
it is accounting for. Extrapolating this property to cumulants, it would mean that Procter & Gamble
presents less large risks according to C6 than according to C4, while according to the centered moments, the
reverse evolution is observed.
Thus, the question of the coherence of the cumulants as measures of fluctuations may arise. And if we accept
that such measures are coherent, what are the implications on the preferences of the agents employing such
measures ? To answer this question, it is informative to express the cumulants as a function of the moments.
For instance, let us consider the fourth order cumulant
C4 = µ4 − 3 · µ2 2 , (22)
= µ4 − 3 · C2 2 . (23)
An agent assessing the fluctuations of an asset with respect to C4 presents aversion for the fluctuations
quantified by the fourth central moment µ4 – since C4 increases with µ4 – but is attracted by the fluctuations
measured by the variance - since C4 decreases with µ2. This behavior is not irrational since it remains
globally risk-averse. Indeed, it depicts an agent which tries to avoid the larger risks but is ready to accept
the smallest ones.
This kind of behavior is characteristic of any agent using the cumulants as risk measures. It thus allows us to
understand why Procter & Gamble is more attractive for an agent sentitive to C6 than for an agent sentitive
to C4. From the expression of C6, we remark that the agent sensitive to this cumulant is risk-averse with
respect to the fluctuations mesured by µ6 and µ2 but is risk-seeker with respect to the fluctuations mesured
by µ4 and µ3. Then, is this particular case, the later ones compensate the former ones.
It also allows us to understand from a behavioral stand-point why it is possible to “have your cake and eat
it too” in the sense of (Andersen and Sornette 2001), that is, why, when the cumulants are choosen as risk
measures, it may be possible to increase the expected return of a portfolio while lowering its large risks, or in
other words, why its generalized Sharpe ratio may increase when one consider larger cumulants to measure
its risks. We will discuus this point again in section 9.
4.2 Marginal risk of an asset within a portofolio
Another important question that arises is the contribution of a given asset to the risk of the whole portfolio.
Indeed, it is crucial to know whether the risk is homogeneously shared by all the assets of the portfolio or if
it is only held by a few of them. The quality of the diversification is then at stake. Moreover, this also allows
for the sensitivity analysis of the risk of the portfolio with respect to small changes in its composition 2 ,
which is of practical interest since it can prevent us from recalculating the whole risk of the portfolio after a
small re-adjustment of its composition.
2 see (Gouriéroux et al. 2000, Scaillet 2000) for a sensitivity analysis of the Value-at-Risk and the expected shortfall.
12

Due to the homogeneity property of the fluctuation measures and to Euler’s theorem for homogeneous
functions, we can write that
ρ({w1, · · · , wN}) = 1
N
wi ·
α
∂ρ
, (24)
∂wi
provided the risk measure ρ is differentiable which will be assumed in all the sequel. In this expression, the
coefficient α again denotes the degree of homogeneity of the risk measure ρ
This relation simply shows that the amount of risk brought by one unit of the asset i in the portfolio is given
by the first derivative of the risk of the portfolio with respect to the weight wi ot this asset. Thus, α −1 · ∂ρ
∂wi
represents the marginal amount of risk of asset i in the portfolio. It is then easy to check that, in a portfolio
with minimum risk, irrespective of the expected return, the weight of each asset is such that the marginal
risks of the assets in the portfolio are equal.
5 A new equilibrum model for asset prices
Using the portfolio selection method explained in the two previous sections, we now present an equilibrium
model generalizing the original Capital Asset Pricing Model developed by (Sharpe 1964, Lintner 1965,
Mossin 1966). Many generalizations have already been proposed to account for the fat-tailness of the assets
return distributions, which led to the multi-moments CAPM. For instance (Rubinstein 1973) and (Krauss
and Lintzenberger 1976) or (Lim 1989) and (Harvey and Siddique 2000) have underlined and tested the role
of the asymmetry in the risk premium by accounting for the skewness of the distribution of returns. More
recently, (Fang and Lai 1997) and (Hwang and Satchell 1999) have introduced a four-moments CAPM to
take into account the letpokurtic behavior of the assets return distributions. Many other extentions have
been presented such as the VaR-CAPM (see (Alexander and Baptista 2002)) or the Distributional-CAPM
by (Polimenis 2002). All these generalization become more and more complicated and not do not provide
necessarily more accurate prediction of the expected returns.
Here, we will assume that the relevant risk measure is given by any measure of fluctuations previously
presented that obey the axioms I-IV of section 2. We will also relax the usual assumption of an homogeneous
market to give to the economic agents the choice of their own risk measure: some of them may choose a
risk measure which put the emphasis on the small fluctuations while others may prefer those which account
for the large ones. We will show that, in such an heterogeneous market, an equilibrium can still be reached
and that the excess returns of individual stocks remain proportional to the market excess return.
For this, we need the following assumptions about the market:
• H1: We consider a one-period market, such that all the positions held at the begining of a period are
cleared at the end of the same period.
• H2: The market is perfect, i.e., there are no transaction cost or taxes, the market is efficient and the
investors can lend and borrow at the same risk-free rate µ0.
We will now add another assumption that specifies the behavior of the agents acting on the market, which
will lead us to make the distinction between homogeneous and heterogeneous markets.
13
i1
437

438 14. Gestion de Portefeuilles multimoments et équilibre de marché
5.1 Equilibrium in a homogeneous market
The market is said to be homogeneous if all the agents acting on this market aim at fulfilling the same
objective. This means that:
• H3-1: all the agents want to maximize the expected return of their portfolio at the end of the period
under a given constraint of measured risk, using the same measure of risks ρα for all of them.
In the special case where ρα denotes the variance, all the agents follow a Markovitz’s optimization procedure,
which leads to the CAPM equilibrium, as proved by (Sharpe 1964). When ρα represents the centered
moments, we will be led to the market equilibrium described by (Rubinstein 1973). Thus, this approach
allows for a generalization of the most popular asset pricing in equilibirum market models.
When all the agents have the same risk function ρα, whatever α may be, we can assert that they have all a
fraction of their capital invested in the same portfolio Π, whose composition is given in appendix B, and the
remaining in the risk-free asset. The amount of capital invested in the risky fund only depends on their risk
aversion or on the legal margin requirement they have to fulfil.
Let us now assume that the market is at equilibrium, i.e., supply equals demand. In such a case, since the
optimal portfolios can be any linear combinations of the risk-free asset and of the risky portfolio Π, it is
straightforward to show (see appendix C) that the market portfolio, made of all traded assets in proportion
of their market capitalization, is nothing but the risky portfolio Π. Thus, as shown in appendix D, we can
state that, whatever the risk measure ρα chosen by the agents to perform their optimization, the excess return
of any asset over the risk-free interest rate is proportional to the excess return of the market portfolio Π over
the risk-free interest rate:
µ(i) − µ0 = β i α · (µΠ − µ0), (25)
where
β i α = ·
∂ ln
1
ρα α
∂wi
w ∗ 1 ,···,w ∗ N
, (26)
where w ∗ 1 , · · · , w∗ N are defined in appendix D. When ρα denotes the variance, we recover the usual β i given
by the mean-variance approach:
β i = Cov(Xi, Π)
. (27)
Var(Π)
Thus, the relations (25) and (26) generalize the usual CAPM formula, showing that the specific choice of
the risk measure is not very important, as long as it follows the axioms I-IV characterizing the fluctuations
of the distribution of asset returns.
5.2 Equilibrium in a heterogeneous market
Does this result hold in the more realistic situation of an heterogeneous market? A market will be said to be
heterogeneous if the agents seek to fulfill different objectives. We thus consider the following assumption:
• H3-2: There exists N agents. Each agent n is characterized by her choice of a risk measure ρα(n) so
that she invests only in the mean-ρα(n) efficient portfolios.
According to this hypothesis, an agent n invests a fraction of her wealth in the risk-free asset and the
remaining in Πn, the mean-ρα(n) efficient portfolio, only made of risky assets. The fraction of wealth
14

invested in the risky fund depends on the risk aversion of each agents, which may vary from an agent to
another one.
The composition of the market portfolio for such a heterogenous market is derived in appendix C. We find
that the market portfolio Π is nothing but the weighted sum of the mean-ρα(n) optimal portfolio Πn:
Π =
439
N
γnΠn, (28)
where γn is the fraction of the total wealth invested in the fund Πn by the n th agent.
n=1
Appendix D demonstrates that, for every asset i and for any mean-ρα(n) efficient portfolio Πn, for all n, the
following equation holds
µ(i) − µ0 = β i n · (µΠn − µ0) . (29)
Multiplying these equations by γn/β i n, we get
γn
βi n
for all n, and summing over the different agents, we obtain
γn
· (µ(i) − µ0) =
so that
with
n
β i n
· (µ(i) − µ0) = γn · (µΠn − µ0), (30)
n
γn · µΠn
− µ0, (31)
µ(i) − µ0 = β i · (µΠ − µ0), (32)
β i =
n
γn
βi n
−1
. (33)
This allows us to conclude that, even in a heterogeneous market, the expected excess return of each individual
stock is directly proportionnal to the expected excess return of the market portfolio, showing that
the homogeneity of the market is not a key property necessary for observing a linear relationship between
individual excess asset returns and the market excess return.
6 Estimation of the joint probability distribution of returns of several assets
A priori, one of the main practical advantage of (Markovitz 1959)’s method and its generalization presented
above is that one does not need the multivariate probability distribution function of the assets returns, as the
analysis solely relies on the coherent measures ρ(X) defined in section 2, such as the centered moments
or the cumulants of all orders that can in principle be estimated empirically. Unfortunately, this apparent
advantage maybe an illusion. Indeed, as underlined by (Stuart and Ord 1994) for instance, the error of
the empirically estimated moment of order n is proportional to the moment of order 2n, so that the error
becomes quickly of the same order as the estimated moment itself. Thus, above n = 6 (or may be n = 8) it is
not reasonable to estimate the moments and/or cumulants directly. Thus, the knowledge of the multivariate
distribution of assets returns remains necessary. In addition, there is a current of thoughts that provides
evidence that marginal distributions of returns may be regularly varying with index µ in the range 3-4
(Lux 1996, Pagan 1996, Gopikrishnan et al. 1998), suggesting the non-existence of asymptotically defined
moments and cumulants of order equal to or larger than µ.
15

440 14. Gestion de Portefeuilles multimoments et équilibre de marché
In the standard Gaussian framework, the multivariate distribution takes the form of an exponential of minus
a quadratic form X ′ Ω −1 X, where X is the unicolumn of asset returns and Ω is their covariance matrix. The
beauty and simplicity of the Gaussian case is that the essentially impossible task of determining a large multidimensional
function is reduced into the very much simpler one of calculating the N(N + 1)/2 elements
of the symmetric covariance matrix. Risk is then uniquely and completely embodied by the variance of the
portfolio return, which is easily determined from the covariance matrix. This is the basis of Markovitz’s
portfolio theory (Markovitz 1959) and of the CAPM (see for instance (Merton 1990)).
However, as is well-known, the variance (volatility) of portfolio returns provides at best a limited quantification
of incurred risks, as the empirical distributions of returns have “fat tails” (Lux 1996, Gopikrishnan et
al. 1998) and the dependences between assets are only imperfectly accounted for by the covariance matrix
(Litterman and Winkelmann 1998).
In this section, we present a novel approach based on (Sornette et al. 2000b) to attack this problem in terms
of the parameterization of the multivariate distribution of returns involving two steps: (i) the projection of
the empirical marginal distributions onto Gaussian laws via nonlinear mappings; (ii) the use of an entropy
maximization to construct the corresponding most parsimonious representation of the multivariate distribution.
6.1 A brief exposition and justification of the method
We will use the method of determination of multivariate distributions introduced by (Karlen 1998) and
(Sornette et al. 2000b). This method consists in two steps: (i) transform each return x into a Gaussian
variable y by a nonlinear monotonous increasing mapping; (ii) use the principle of entropy maximization to
construct the corresponding multivariate distribution of the transformed variables y.
The first concern to address before going any further is whether the nonlinear transformation, which is in
principle different for each asset return, conserves the structure of the dependence. In what sense is the
dependence between the transformed variables y the same as the dependence between the asset returns x? It
turns out that the notion of “copulas” provides a general and rigorous answer which justifies the procedure
of (Sornette et al. 2000b).
For completeness and use later on, we briefly recall the definition of a copula (for further details about
the concept of copula see (Nelsen 1998)). A function C : [0, 1] n −→ [0, 1] is a n-copula if it enjoys the
following properties :
• ∀u ∈ [0, 1], C(1, · · · , 1, u, 1 · · · , 1) = u ,
• ∀ui ∈ [0, 1], C(u1, · · · , un) = 0 if at least one of the ui equals zero ,
• C is grounded and n-increasing, i.e., the C-volume of every boxes whose vertices lie in [0, 1] n is
positive.
Skar’s Theorem then states that, given an n-dimensional distribution function F with continuous marginal
distributions F1, · · · , Fn, there exists a unique n-copula C : [0, 1] n −→ [0, 1] such that :
F (x1, · · · , xn) = C(F1(x1), · · · , Fn(xn)) . (34)
This elegant result shows that the study of the dependence of random variables can be performed independently
of the behavior of the marginal distributions. Moreover, the following result shows that copulas
are intrinsic measures of dependence. Consider n continuous random variables X1, · · · , Xn with copula
16

C. Then, if g1(X1), · · · , gn(Xn) are strictly increasing on the ranges of X1, · · · , Xn, the random variables
Y1 = g1(X1), · · · , Yn = gn(Xn) have exactly the same copula C (Lindskog 2000). The copula is thus
invariant under strictly increasing tranformation of the variables. This provides a powerful way of studying
scale-invariant measures of associations. It is also a natural starting point for construction of multivariate
distributions and provides the theoretical justification of the method of determination of mutivariate distributions
that we will use in the sequel.
6.2 Transformation of an arbitrary random variable into a Gaussian variable
Let us consider the return X, taken as a random variable characterized by the probability density p(x). The
transformation y(x) which obtains a standard normal variable y from x is determined by the conservation
of probability:
Integrating this equation from −∞ and x, we obtain:
441
p(x)dx = 1 y2
− √ e 2 dy . (35)
2π
F (x) = 1
2
where F (x) is the cumulative distribution of X:
F (x) =
This leads to the following transformation y(x):
1 + erf
y√2
, (36)
x
dx
−∞
′ p(x ′ ) . (37)
y = √ 2 erf −1 (2F (x) − 1) , (38)
which is obvously an increasing function of X as required for the application of the invariance property of
the copula stated in the previous section. An illustration of the nonlinear transformation (38) is shown in
figure 6. Note that it does not require any special hypothesis on the probability density X, apart from being
non-degenerate.
In the case where the pdf of X has only one maximum, we may use a simpler expression equivalent to (38).
Such a pdf can be written under the so-called Von Mises parametrization (Embrechts et al. 1997) :
p(x) = C f ′ (x) 1
− e 2
|f(x)| f(x) , (39)
where C is a constant of normalization. For f(x)/x 2 → 0 when |x| → +∞, the pdf has a “fat tail,” i.e., it
decays slower than a Gaussian at large |x|.
Let us now define the change of variable
Using the relationship p(y) = p(x) dx
dy , we get:
y = sgn(x) |f(x)| . (40)
p(y) = 1 y2
− √ e 2 . (41)
2π
It is important to stress the presence of the sign function sgn(x) in equation (40), which is essential in order
to correctly quantify dependences between random variables. This transformation (40) is equivalent to (38)
but of a simpler implementation and will be used in the sequel.
17

442 14. Gestion de Portefeuilles multimoments et équilibre de marché
6.3 Determination of the joint distribution : maximum entropy and Gaussian copula
Let us now consider N random variables Xi with marginal distributions pi(xi). Using the transformation
(38), we define N standard normal variables Yi. If these variables were independent, their joint distribution
would simply be the product of the marginal distributions. In many situations, the variables are not
independent and it is necessary to study their dependence.
The simplest approach is to construct their covariance matrix. Applied to the variables Yi, we are certain that
the covariance matrix exists and is well-defined since their marginal distributions are Gaussian. In contrast,
this is not ensured for the variables Xi. Indeed, in many situations in nature, in economy, finance and in
social sciences, pdf’s are found to have power law tails ∼ A
x1+µ for large |x|. If µ ≤ 2, the variance and the
covariances can not be defined. If 2 < µ ≤ 4, the variance and the covariances exit in principle but their
sample estimators converge poorly.
We thus define the covariance matrix:
V = E[yy t ] , (42)
where y is the vector of variables Yi and the operator E[·] represents the mathematical expectation. A
classical result of information theory (Rao 1973) tells us that, given the covariance matrix V , the best joint
distribution (in the sense of entropy maximization) of the N variables Yi is the multivariate Gaussian:
1
P (y) =
(2π) N/2det(V ) exp
− 1
2 ytV −1
y . (43)
Indeed, this distribution implies the minimum additional information or assumption, given the covariance
matrix.
Using the joint distribution of the variables Yi, we obtain the joint distribution of the variables Xi:
∂yi
P (x) = P (y)
∂xj
, (44)
where ∂yi
is the Jacobian of the transformation. Since
we get
∂xj
This finally yields
P (x) =
∂yi
∂xj
= √ 2πpj(xj)e 1
2 y2 i δij , (45)
∂yi
N
∂xj
= (2π)N/2
i=1
pi(xi)e 1
2 y2 i . (46)
1
exp −
det(V ) 1
2 yt (x) (V −1
N
− I)y (x) pi(xi) . (47)
i=1
As expected, if the variables are independent, V = I, and P (x) becomes the product of the marginal
distributions of the variables Xi.
Let F (x) denote the cumulative distribution function of the vector x and Fi(xi), i = 1, ..., N the N corresponding
marginal distributions. The copula C is then such that
F (x1, · · · , xn) = C(F1(x1), · · · , Fn(xn)) . (48)
18

Differentiating with respect to x1, · · · , xN leads to
where
is the density of the copula C.
P (x1, · · · , xn) = ∂F (x1, · · · , xn)
∂x1 · · · ∂xn
= c(F1(x1), · · · , Fn(xn))
c(u1, · · · , uN) = ∂C(u1, · · · , uN)
∂u1 · · · ∂uN
443
N
pi(xi) , (49)
Comparing (50) with (47), the density of the copula is given in the present case by
1
c(u1, · · · , uN) = exp
det(V )
, (51)
i=1
− 1
2 yt (u) (V −1 − I)y (u)
which is the “Gaussian copula” with covariance matrix V. This result clarifies and justifies the method
of (Sornette et al. 2000b) by showing that it essentially amounts to assume arbitrary marginal distributions
with Gaussian copulas. Note that the Gaussian copula results directly from the transformation to Gaussian
marginals together with the choice of maximizing the Shannon entropy under the constraint of a fixed covariance
matrix. Under differents constraint, we would have found another maximum entropy copula. This
is not unexpected in analogy with the standard result that the Gaussian law is maximizing the Shannon entropy
at fixed given variance. If we were to extend this formulation by considering more general expressions
of the entropy, such that Tsallis entropy (Tsallis 1998), we would have found other copulas.
6.4 Empirical test of the Gaussian copula assumption
We now present some tests of the hypothesis of Gaussian copulas between returns of financial assets. This
presentation is only for illustration purposes, since testing the gaussian copula hypothesis is a delicate task
which has been addressed elsewhere (see (Malevergne and Sornette 2001)). Here, as an example, we propose
two simple standard methods.
The first one consists in using the property that Gaussian variables are stable in distribution under addition.
Thus, a (quantile-quantile or Q − Q) plot of the cumulative distribution of the sum y1 + · · · + yp versus
the cumulative Normal distribution with the same estimated variance should give a straight line in order to
qualify a multivariate Gaussian distribution (for the transformed y variables). Such tests on empirical data
are presented in figures 7-9.
The second test amounts to estimating the covariance matrix V of the sample we consider. This step is
simple since, for fast decaying pdf’s, robust estimators of the covariance matrix are available. We can then
estimate the distribution of the variable z 2 = y t V −1 y. It is well known that z 2 follows a χ 2 distribution
if y is a Gaussian random vector. Again, the empirical cumulative distribution of z 2 versus the χ 2 cumulative
distribution should give a straight line in order to qualify a multivariate Gaussian distribution (for the
transformed y variables). Such tests on empirical data are presented in figures 10-12.
First, one can observe that the Gaussian copula hypothesis appears better for stocks than for currencies.
As discussed in (Malevergne and Sornette 2001), this result is quite general. A plausible explanation lies
in the stronger dependence between the currencies compared with that between stocks, which is due to
the monetary policies limiting the fluctuations between the currencies of a group of countries, such as was
the case in the European Monetary System before the unique Euro currency. Note also that the test of
aggregation seems systematically more in favor of the Gaussian copula hypothesis than is the χ 2 test, maybe
due to its smaller sensitivity. Nonetheless, the very good performance of the Gaussian hypothesis under the
19
(50)

444 14. Gestion de Portefeuilles multimoments et équilibre de marché
aggregation test bears good news for a porfolio theory based on it, since by definition a portfolio corresponds
to asset aggregation. Even if sums of the transformed returns are not equivalent to sums of returns (as we
shall see in the sequel), such sums qualify the collective behavior whose properties are controlled by the
copula.
Notwithstanding some deviations from linearity in figures 7-12, it appears that, for our purpose of developing
a generalized portfolio theory, the Gaussian copula hypothesis is a good approximation. A more systematic
test of this goodness of fit requires the quantification of a confidence level, for instance using the Kolmogorov
test, that would allow us to accept or reject the Gaussian copula hypothesis. Such a test has been performed
in (Malevergne and Sornette 2001), where it is shown that this test is sensitive enough only in the bulk of the
distribution, and that an Anderson-Darling test is preferable for the tails of the distributions. Nonetheless,
the quantitative conclusions of these tests are identical to the qualitative results presented here. Some other
tests would be useful, such as the multivariate Gaussianity test presented by (Richardson and Smith 1993).
7 Choice of an exponential family to parameterize the marginal distributions
7.1 The modified Weibull distributions
We now apply these constructions to a class of distributions with fat tails, that have been found to provide
a convenient and flexible parameterization of many phenomena found in nature and in the social sciences
(Laherrère and Sornette 1998). These so-called stretched exponential distributions can be seen to be general
forms of the extreme tails of product of random variables (Frisch and Sornette 1997).
Following (Sornette et al. 2000b), we postulate the following marginal probability distributions of returns:
p(x) = 1
2 √ c
π χ c |x|
2
c
2 −1 e −|x|
χc
, (52)
where c and χ are the two key parameters. A more general parameterization taking into account a possible
asymmetry between negative and positive returns (thus leading to possible non-zero average return) is
p(x) = Q √
π
χ
p(x) =
1 − Q
√ π
c+
c +
2
+
|x| c +
2 −1 e −|x| +
χ +c
c−
c− 2
−
χ
|x| c− 2 −1 e −|x| −
χ−c if x ≥ 0 (53)
if x < 0 , (54)
where Q (respectively 1 − Q) is the fraction of positive (respectively negative) returns. In the sequel, we
will only consider the case Q = 1
2 , which is the only analytically tractable case. Thus the pdf’s asymmetry
will be only accounted for by the exponents c+, c− and the scale factors χ+, χ−.
We can note that these expressions are close to the Weibull distribution, with the addition of a power law
prefactor to the exponential such that the Gaussian law is retrieved for c = 2. Following (Sornette et
al. 2000b, Sornette et al. 2000a, Andersen and Sornette 2001), we call (52) the modified Weibull distribution.
For c < 1, the pdf is a stretched exponential, also called sub-exponential. The exponent c determines the
shape of the distribution, which is fatter than an exponential if c < 1. The parameter χ controls the scale or
characteristic width of the distribution. It plays a role analogous to the standard deviation of the Gaussian
law. See chapter 6 of (?) for a recent review on maximum likelihood and other estimators of such generalized
Weibull distributions.
20

7.2 Transformation of the modified Weibull pdf into a Gaussian Law
One advantage of the class of distributions (52) is that the transformation into a Gaussian is particularly
simple. Indeed, the expression (52) is of the form (39) with
c |x|
f(x) = 2 . (55)
χ
Applying the change of variable (40) which reads
leads automatically to a Gaussian distribution.
yi = sgn(xi) √
|xi|
2
χi
c i
2
These variables Yi then allow us to obtain the covariance matrix V :
Vij = 2
T
2 N π N/2√ V exp
T
|xi|
sgn(xixj)
n=1
i,j
V −1
ij
χi
χi
c i
2 |xj|
χj
445
, (56)
χj
c j
2
and thus the multivariate distributions P (y) and P (x) :
P (x1, · · · , xN) =
1
⎡
⎣−
⎤
c/2 c/2 |xi| |xj|
⎦
, (57)
N
ci|xi| c/2−1
i=1
χ c/2
i
e −|x i |
χc
.
(58)
Similar transforms hold, mutatis mutandis, for the asymmetric case. Indeed, for asymmetric assets of interest
for financial risk managers, the equations (53) and (54) yields the following change of variable:
yi = √ 2
xi
c+ i
2
χ + i
yi = − √
|xi|
2
χ − i
c− i 2
and xi ≥ 0, (59)
and xi < 0 . (60)
This allows us to define the correlation matrix V and to obtain the multivariate distribution P (x), generalizing
equation (58) for asymmetric assets. Since this expression is rather cumbersome and nothing but a
straightforward generalization of (58), we do not write it here.
7.3 Empirical tests and estimated parameters
In order to test the validity of our assumption, we have studied a large basket of financial assets including
currencies and stocks. As an example, we present in figures 13 to 17 typical log-log plot of the transformed
return variable Y versus the return variable X for a certain number of assets. If our assumption was right,
we should observe a single straight line whose slope is given by c/2. In contrast, we observe in general
two approximately linear regimes separated by a cross-over. This means that the marginal distribution of
returns can be approximated by two modified Weibull distributions, one for small returns which is close to a
Gaussian law and one for large returns with a fat tail. Each regime is depicted by its corresponding straight
line in the graphs. The exponents c and the scale factors χ for the different assets we have studied are given
in tables 3 for currencies and 4 for stocks. The coefficients within brackets are the coefficients estimated for
small returns while the non-bracketed coefficients correspond to the second fat tail regime.
21

446 14. Gestion de Portefeuilles multimoments et équilibre de marché
The first point to note is the difference between currencies and stocks. For small as well as for large returns,
the exponents c− and c+ for currencies (excepted Poland and Thailand) are all close to each other. Additional
tests are required to establish whether their relatively small differences are statistically significant. Similarly,
the scale factors are also comparable. In contrast, many stocks exhibit a large asymmetric behavior for large
returns with c+ −c− 0.5 in about one-half of the investigated stocks. This means that the tails of the large
negative returns (“crashes”) are often much fatter than those of the large positive returns (“rallies”).
The second important point is that, for small returns, many stocks have an exponent 〈c+〉 ≈ 〈c−〉 2 and
thus have a behavior not far from a pure Gaussian in the bulk of the distribution, while the average exponent
for currencies is about 1.5 in the same “small return” regime. Therefore, even for small returns, currencies
exhibit a strong departure from Gaussian behavior.
In conclusion, this empirical study shows that the modified Weibull parameterization, although not exact on
the entire range of variation of the returns X, remains consistent within each of the two regimes of small
versus large returns, with a sharp transition between them. It seems especially relevant in the tails of the
return distributions, on which we shall focus our attention next.
8 Cumulant expansion of the portfolio return distribution
8.1 link between moments and cumulants
Before deriving the main result of this section, we recall a standard relation between moments and cumulants
that we need below.
The moments Mn of the distribution P are defined by
ˆP (k) =
+∞
n=0
(ik) n
where ˆ P is the characteristic function, i.e., the Fourier transform of P :
Similarly, the cumulants Cn are given by
Differentiating n times the equation
ln
ˆP (k) =
ˆP (k) = exp
+∞
n=0
(ik) n
+∞
−∞
n! Mn
+∞
n! Mn , (61)
dS P (S)e ikS . (62)
n=1
=
(ik) n
n! Cn
+∞
n=1
(ik) n
. (63)
n! Cn , (64)
we obtain the following recurrence relations between the moments and the cumulants :
Mn =
n−1
n − 1
MpCn−p ,
p
(65)
p=0
n−1
n − 1
Cn = Mn −
CpMn−p . (66)
n − p
p=1
22

In the sequel, we will first evaluate the moments, which turns out to be easier, and then using eq (66) we
will be able to calculate the cumulants.
8.2 Symmetric assets
We start with the expression of the distribution of the weighted sum of N assets :
PS(s) =
R N
447
N
dx P (x)δ( wixi − s) , (67)
where δ(·) is the Dirac distribution. Using the change of variable (40), allowing us to go from the asset
returns Xi’s to the transformed returns Yi’s, we get
1
PS(s) =
(2π) N/2
det(V )
R N
i=1
1
−
dy e 2 ytV −1y δ(
Taking its Fourier transform ˆ PS(k) = dsPS(s)eiks , we obtain
ˆPS(k)
1
=
(2π) N/2
det(V )
where ˆ PS is the characteristic function of PS.
R N
N
wisgn(yi)f −1 (y 2 i ) − s) . (68)
i=1
1
−
dy e 2 ytV −1y+ikN i=1 wisgn(yi)f −1 (y2 i ) , (69)
In the particular case of interest here where the marginal distributions of the variables Xi’s are the modified
Weibull pdf,
f −1 (yi) = χi| yi
√2 | qi (70)
with
the equation (69) becomes
ˆPS(k)
1
=
(2π) N/2
det(V )
R N
qi = 2/ci , (71)
1
−
dy e 2 ytV −1y+ikN i=1 wisgn(yi)χi| y √2 i | qi . (72)
The task in front of us is to evaluate this expression through the determination of the moments and/or
cumulants.
8.2.1 Case of independent assets
In this case, the cumulants can be obtained explicitely (Sornette et al. 2000b). Indeed, the expression (72)
can be expressed as a product of integrals of the form
We obtain
+∞
0
C2n =
u2
− 2 du e +ikwiχiu i
√2q
. (73)
N
c(n, qi)(χiwi) 2n , (74)
i=1
23

448 14. Gestion de Portefeuilles multimoments et équilibre de marché
and
⎧
⎨n−2
c(n, qi) = (2n)! (−1)
⎩
n Γ qi(n − p) + 1
2
(2n − 2p)!π1/2 p=0
Γ qi + 1
2
2!π 1/2
p
− (−1)n
n
Γ qi + 1
2
2!π
1/2
⎫
n⎬ .
⎭
(75)
Note that the coefficient c(n, qi) is the cumulant of order n of the marginal distribution (52) with c = 2/qi
and χ = 1. The equation (74) expresses simply the fact that the cumulants of the sum of independent variables
is the sum of the cumulants of each variable. The odd-order cumulants are zero due to the symmetry
of the distributions.
8.2.2 Case of dependent assets
Here, we restrict our exposition to the case of two random variables. The case with N arbitrary can be
treated in a similar way but involves rather complex formulas. The equation (72) reads
ˆPS(k)
1
=
2π 1 − ρ2
dy1dy2 exp − 1
2 ytV −1
q1
y1
y + ik χ1w1sgn(y1)
√
2
+
q2
y2
+χ2w2sgn(y2)
√
2
, (76)
and we can show (see appendix E) that the moments read
n
n
Mn =
p
with
γq1q2
γq1q2 (2n, 2p) = χ2p
1 χ2(n−p) 2
(2n, 2p + 1) = 2χ2p+1
1
p=0
Γ q1p + 1
2
χ 2(n−p)−1
2
, −q2(n − p) + q2 + 1
2
where 2F1 is an hypergeometric function.
w p
1wn−pγq1q2 2 (n, p) , (77)
Γ q2(n − p) + 1
2
2F1 −q1p, −q2(n − p);
π
1
; ρ2 , (78)
2
Γ q2(n − p) + 1 − q2
2 ρ 2F1 −q1p −
π
q1 − 1
,
2
; 3
; ρ2 , (79)
2
Γ q1p + 1 + q1
2
These two relations allow us to calculate the moments and cumulants for any possible values of q1 = 2/c1
and q2 = 2/c2. If one of the qi’s is an integer, a simplification occurs and the coefficients γ(n, p) reduce to
polynomials. In the simpler case where all the qi’s are odd integer the expression of moments becomes :
with
Mn =
n
p=0
n
(w1χ1)
p
p (w2χ2) n−p
min{q1p,q2(n−p)}
a (2n)
2p = (2p)!
a (2n)
2p+1
s=0
2(n − p)
(2(n − p) − 1)!! =
2n
ρ s s! a (q1p)
s a (q2(n−p))
s , (80)
(2n)!
2n−p , (81)
(n − p)!
= 0 , (82)
a (2n+1)
2p = 0 , (83)
a (2n+1)
2(n − p)
(2n + 1)!
2p+1 = (2p + 1)!
(2(n − p) − 1)!! =
2n + 1
2n−p . (84)
(n − p)!
24

8.3 Non-symmetric assets
In the case of asymmetric assets, we have to consider the formula (53-54), and using the same notation as in
the previous section, the moments are again given by (77) with the coefficient γ(n, p) now equal to :
Γ
4π
−
q1 p
+2Γ + 1 Γ
2
Γ
4π
−
q1 p
−2Γ + 1 Γ
2
Γ
4π
+
q 1 p
−2Γ + 1 Γ
2
Γ
4π
+
q 1 p
+2Γ + 1 Γ
2
γ(n, p) = (−1)n (χ − 1 )p (χ − 2 )n−p
+ (−1)p (χ − 1 )p (χ + 2 )n−p
+ (−1)n−p (χ + 1 )p (χ − 2 )n−p
+ (χ+ 1 )p (χ + 2 )n−p
− −
q1 p + 1 q2 (n − p) + 1
Γ
2F1
2
2
−
q2 (n − p)
+ 1 ρ 2F1 −
2
q− 1 p − 1
2
− +
q1 p + 1 q 2 (n − p) + 1
Γ
2F1
2
2
+
q 2 (n − p)
+ 1 ρ 2F1 −
2
q− 1 p − 1
2
+ −
q 1 p + 1 q2 (n − p) + 1
Γ
2F1
2
2
−
q2 (n − p)
+ 1 ρ 2F1 −
2
q+ 1 p − 1
2
+ +
q 1 p + 1 q 2 (n − p) + 1
Γ
2F1
2
2
+
q 2 (n − p)
+ 1 ρ 2F1 −
2
q+ 1
− q− 1 p
2 , −q− 2
, − q− 2
− q− 1 p
2 , −q+ 2
, − q+ 2
− q+ 1 p
2 , −q− 2
, − q− 2
− q+ 1 p
2 , −q+ 2
p − 1
, −
2
q+ 2
(n − p)
2
449
; 1
; ρ2 +
2
(n − p) − 1
;
2
3
; ρ2 +
2
(n − p)
;
2
1
; ρ2 +
2
(n − p) − 1
;
2
3
; ρ2 +
2
(n − p)
;
2
1
; ρ2 +
2
(n − p) − 1
;
2
3
; ρ2 +
2
(n − p)
;
2
1
; ρ2 +
2
(n − p) − 1
;
2
3
; ρ2 .
2
(85)
This formula is obtained in the same way as for the formulas given in the symmetric case. We retrieve the
formula (78) as it should if the coefficients with index ’+’ are equal to the coefficients with index ’-’.
8.4 Empirical tests
Extensive tests have been performed for currencies under the assumption that the distributions of asset
returns are symmetric (Sornette et al. 2000b).
As an exemple, let us consider the Swiss franc and the Japanese Yen against the US dollar. The calibration
of the modified Weibull distribution to the tail of the empirical histogram of daily returns give (qCHF =
1.75, cCHF = 1.14, χCHF = 2.13) and (qJP Y = 2.50, cJP Y = 0.8, χJP Y = 1.25) and their correlation
coefficient is ρ = 0.43.
Figure 18 plots the excess kurtosis of the sum wCHF xCHF + wJP Y xJP Y as a function of wCHF , with
the constraint wCHF + wJP Y = 1. The thick solid line is determined empirically, by direct calculation of
the kurtosis from the data. The thin solid line is the theoretical prediction using our theoretical formulas
with the empirically determined exponents c and characteristic scales χ given above. While there is a nonnegligible
difference, the empirical and theoretical excess kurtosis have essentially the same behavior with
their minimum reached almost at the same value of wCHF .
Three origins of the discrepancy between theory and empirical data can be invoked. First, as already pointed
out in the preceding section, the modified Weibull distribution with constant exponent and scale parameters
describes accurately only the tail of the empirical distributions while, for small returns, the empirical
distributions are close to a Gaussian law. While putting a strong emphasis on large fluctuations, cumulants
of order 4 are still significantly sensitive to the bulk of the distributions. Moreover, the excess kurtosis is
25

450 14. Gestion de Portefeuilles multimoments et équilibre de marché
normalized by the square second-order cumulant, which is almost exclusively sensitive to the bulk of the
distribution. Cumulants of higher order should thus be better described by the modified Weibull distribution.
However, a careful comparison between theory and data would then be hindered by the difficulty in estimating
reliable empirical cumulants of high order. This estimation problem is often invoked as a criticism
against using high-order moments or cumulants. Our approach suggests that this problem can be in large
part circumvented by focusing on the estimation of a reasonable parametric expression for the probability
density or distribution function of the assets returns. The second possible origin of the discrepancy between
theory and data is the existence of a weak asymmetry of the empirical distributions, particularly of the Swiss
franc, which has not been taken into account. The figure also suggests that an error in the determination of
the exponents c can also contribute to the discrepancy.
In order to investigate the sensitivity with respect to the choice of the parameters q and ρ, we have also
constructed the dashed line corresponding to the theoretical curve with ρ = 0 (instead of ρ = 0.43) and
the dotted line corresponding to the theoretical curve with qCHF = 2 rather than 1.75. Finally, the dasheddotted
line corresponds to the theoretical curve with qCHF = 1.5. We observe that the dashed line remains
rather close to the thin solid line while the dotted line departs significantly when wCHF increases. Therefore,
the most sensitive parameter is q, which is natural because it controls directly the extend of the fat tail of the
distributions.
In order to account for the effect of asymmetry, we have plotted the fourth cumulant of a portfolio composed
of Swiss Francs and British Pounds. On figure 19, the solid line represents the empirical cumulant while
the dashed line shows the theoretical cumulant. The agreement between the two curves is better than under
the symmetric asumption. Note once again that an accurate determination of the parameters is the key point
to obtain a good agreement between empirical data and theoretical prediction. As we can see in figure 19,
the paramaters of the Swiss Franc seem well adjusted since the theoretical and empirical cumulants are both
very close when wCHF 1, i.e., when the Swiss Franc is almost the sole asset in the portfolio, while when
wCHF 0, the theoretical cumulant is far from the empirical one, i.e., the parameters of the Bristish Pound
are not sufficiently well-adjusted.
9 Can you have your cake and eat it too ?
Now that we have shown how to accurately estimate the multivariate distribution fonction of the assets
return, let us come back to the portfolio selection problem. In figure 2, we can see that the expected return
of the portfolios with minimum risk according to Cn decreases when n increases. But, this is not the general
situation.
Figure 20 and 21 show the generalized efficient frontiers using C2 (Markovitz case), C4 or C6 as relevant
measures of risks, for two portfolios composed of two stocks : IBM and Hewlett-Packard in the first case
and IBM and Coca-Cola in the second case.
Obviously, given a certain amount of risk, the mean return of the portfolio changes when the cumulant
considered changes. It is interesting to note that, in figure 20, the minimisation of large risks, i.e., with
respect to C6, increases the average return while, in figure 21, the minimisation of large risks lead to decrease
the average return.
This allows us to make precise and quantitative the previously reported empirical observation that it is
possible to “have your cake and eat it too” (Andersen and Sornette 2001). We can indeed give a general
criterion to determine under which values of the parameters (exponents c and characteristic scales χ of
the distributions of the asset returns) the average return of the portfolio may increase while the large risks
decrease at the same time, thus allowing one to gain on both account (of course, the small risks quantified
26

y the variance will then increase). For two independent assets, assuming that the cumulants of order n and
n + k of the portfolio admit a minimum in the interval ]0, 1[, we can show that
if and only if
(µ(1) − µ(2)) ·
µ ∗ n < µ ∗ n+k
Cn(1) 1
n−1
−
Cn(2)
451
(86)
1
Cn+k(1) n+k−1
> 0 , (87)
Cn+k(2)
where µ ∗ n denotes the return of the portfolio evaluated with respect to the minimum of the cumulant of order
n and Cn(i) is the cumulant of order n for the asset i.
The proof of this result and its generalisation to N > 2 are given in appendix F. In fact, we have observed
that when the exponent c of the assets remains sufficiently different, this result still holds in presence of
dependence between assets. This last empirical observation in the presence of dependence between assets
has not been proved mathematically. It seems reasonable for assets with moderate dependence while it may
fail when the dependence becomes too strong as occurs for comonotonic assets.
For the assets considered above, we have found µIBM = 0.13, µHW P = 0.07, µKO = 0.05 and
C2(IBM)
C2(HW P )
C2(IBM)
C2(KO)
= 1.76 >
= 0.96 <
1
C4(IBM) 3
C4(HW P )
1
C4(IBM) 3
C4(KO)
C6(IBM)
= 1.03 >
C6(HW P )
C6(IBM)
= 1.01 <
C6(KO)
1
5
1
5
= 0.89 (88)
= 1.06 , (89)
which shows that, for the portfolio IBM / Hewlett-Packard, the efficient return is an increasing function of
the order of the cumulants while, for the portfolio IBM / Coca-Cola, the inverse phenomenon occurs. This
is exactly what is shown on figures 20 and 21.
The underlying intuitive mechanism is the following: if a portfolio contains an asset with a rather fat tail
(many “large” risks) but narrow waist (few “small” risks) with very little return to gain from it, minimizing
the variance C2 of the return portfolio will overweight this asset which is wrongly perceived as having little
risk due to its small variance (small waist). In contrast, controlling for the larger risks quantified by C4 or
C6 leads to decrease the weight of this asset in the portfolio, and correspondingly to increase the weight
of the more profitable assets. We thus see that the effect of “both decreasing large risks and increasing
profit” appears when the asset(s) with the fatter tails, and therefore the narrower central part, has(ve) the
smaller overall return(s). A mean-variance approach will weight them more than deemed appropriate from
a prudential consideration of large risks and consideration of profits.
From a behavioral point of view, this phenomenon is very interesting and can probably be linked with the
fact that the main risk measure considered by the agents is the volatility (or the variance), so that the other
dimensions of the risk, measured by higher moments, are often neglected. This may sometimes offer the
opportunity of increasing the expected return while lowering large risks.
10 Conclusion
We have introduced three axioms that define a consistent set of risk measures, in the spirit of (Artzner et
al. 1997, Artzner et al. 1999). Contrarily to the risk measures of (Artzner et al. 1997, Artzner et al. 1999),
our consistent risk measures may account for both-side risks and not only for down-side risks. Thus, they
supplement the notion of coherent measures of risk and are well adapted to the problem of portfolio risk
27

452 14. Gestion de Portefeuilles multimoments et équilibre de marché
assessment and optimization. We have shown that these risk measures, which contain centered moments
(and cumulants with some restriction) as particular examples, generalize them significantly. We have presented
a generalization of previous generalizations of the efficient frontiers and of the CAPM based on these
risk measures in the cases of homogeneous and heterogeneous agents. We have then proposed a simple but
powerful specific von Mises representation of multivariate distribution of returns that allowed us to obtain
new analytical results on and empirical tests of a general framework for a portfolio theory of non-Gaussian
risks with non-linear correlations. Quantitative tests have been presented on a basket of seventeen stocks
among the largest capitalization on the NYSE.
This work opens several novel interesting avenues for research. One consists in extending the Gaussian copula
assumption, for instance by using the maximum-entropy principle with non-extensive Tsallis entropies,
known to be the correct mathematical information-theoretical representation of power laws. A second line
of research would be to extend the present framework to encompass simultaneously different time scales τ
in the spirit of (Muzy et al. 2001) in the case of a cascade model of volatilities.
28

A Description of the data set
We have considered a set of seventeen assets traded on the New York Stock Exchange: Applied Material,
Coca-Cola, EMC, Exxon-Mobil, General Electric, General Motors, Hewlett Packard, IBM, Intel, MCI
WorldCom, Medtronic, Merck, Pfizer, Procter & Gambel, SBC Communication, Texas Instrument, Wall
Mart. These assets have been choosen since they are among the largest capitalizations of the NYSE at the
time of writing.
The dataset comes from the Center for Research in Security Prices (CRSP) database and covers the time
interval from the end of January 1995 to the end of December 2000, which represents exactly 1500 trading
days. The main statistical features of the compagnies composing the dataset are presented in the table 5.
Note the high kurtosis of each distribution of returns as well as the large values of the observed minimum and
maximum returns compared with the standard deviations, that clearly underlines the non-Gaussian behavior
of these assets.
29
453

454 14. Gestion de Portefeuilles multimoments et équilibre de marché
B Generalized efficient frontier and two funds separation theorem
Let us consider a set of N risky assets X1, · · · , XN and a risk-free asset X0. The problem is to find the
optimal allocation of these assets in the following sense:
⎧
⎨
⎩
inf wi∈[0,1] ρα({wi})
i≥0 wi = 1
i≥0 wiµ(i) = µ ,
In other words, we search for the portfolio P with minimum risk as measured by any risk measure ρα
obeying axioms I-IV of section 2 for a given amount of expected return µ and normalized weights wi.
Short-sells are forbidden except for the risk-free asset which can be lent and borrowed at the same interest
rate µ0. Thus, the weights wi’s are assumed positive for all i ≥ 1.
B.1 Case of independent assets when the risk is measured by the cumulants
To start with a simple example, let us assume that the risky assets are independent and that we choose to
measure the risk with the cumulants of their distributions of returns. The case when the assets are dependent
and/or when the risk is measured by any ρα will be considered later. Since the assets are assumed
independent, the cumulant of order n of the pdf of returns of the portfolio is simply given by
Cn =
(90)
N
wi n Cn(i), (91)
i=1
where Cn(i) denotes the marginal nth order cumulant of the pdf of returns of the asset i. In order to solve
this problem, let us introduce the Lagrangian
N
N
L = Cn − λ1 wi µ(i) − µ − λ2 wi − 1 , (92)
i=0
where λ1 and λ2 are two Lagrange multipliers. Differentiating with respect to w0 yields
i=0
λ2 = µ0 λ1, (93)
which by substitution in equation (92) gives
N
L = Cn − λ1 wi (µ(i) − µ0) − (µ − µ0) . (94)
i=1
Let us now differentiate L with respect to wi, i ≥ 1, we obtain
so that
n w ∗ i n−1 Cn(i) − λ1(µ(i) − µ0) = 0, (95)
w ∗ 1
i = λ1
n−1
Applying the normalization constraint yields
1
w0 + λ1
n−1
N
i=1
1
µ(i) − µ0
n−1
. (96)
n Cn(i)
1
µ(i) − µ0
n−1
n Cn(i)
30
= 1, (97)

thus
and finally
1
λ1
n−1 =
w ∗ i = (1 − w0)
N
i=1
N
i=1
1 − w0
µ(i)−µ0
n Cn(i)
1
µ(i)−µ0 n−1
Cn(i)
µ(i)−µ0
Cn(i)
Let us now define the portfolio Π exclusively made of risky assets with weights
˜wi =
N
i=1
1
µ(i)−µ0 n−1
Cn(i)
µ(i)−µ0
Cn(i)
455
1
n−1
, (98)
1
n−1
. (99)
1
n−1
, i ≥ 1. (100)
The optimal portfolio P can be split in two funds : the risk-free asset whose weight is w0 and a risky fund
Π with weight (1 − w0). The expected return of the portfolio P is thus
where µΠ denotes the expected return of portofolio Π:
µΠ =
µ = w0 µ0 + (1 − w0)µΠ, (101)
N
i=1 µ(i)
N
i=1
1
µ(i)−µ0 n−1
Cn(i)
µ(i)−µ0
Cn(i)
1
n−1
The risk associated with P and measured by the cumulant Cn of order n is
Cn = (1 − w0) n
N i=1 Cn(i)
N
i=1
n
µ(i)−µ0 n−1
Cn(i)
µ(i)−µ0
Cn(i)
1
n−1
. (102)
n . (103)
Putting together the three last equations allows us to obtain the equation of the efficient frontier:
µ = µ0 +
which is a straight line in the plane (Cn 1/n , µ).
B.2 General case
(µ(i) − µ0) n
n−1
Cn(i) 1
n−1
n−1
n
1
· Cn n , (104)
Let us now consider the more realistic case when the risky assets are dependent and/or when the risk is
measured by any risk measure ρα obeying the axioms I-IV presented in section 2, where α denotes the
degres of homogeneity of ρα. Equation (94) always holds (with Cn replaced by ρα), and the differentiation
with respect to wi, i ≥ 1 yields the set of equations:
∂ρα
∂wi
(w ∗ 1, · · · , w ∗ N) = λ1 (µ(i) − µ0), i ∈ {1, · · · , N}. (105)
31

456 14. Gestion de Portefeuilles multimoments et équilibre de marché
Since ρα(w1, · · · , wN) is a homogeneous function of order α, its first-order derivative with respect to wi is
also a homogeneous function of order α − 1. Using this homogeneity property allows us to write
∂ρα
∂wi
−1 ∂ρα
λ1 (w
∂wi
∗ 1, · · · , w ∗ N) = (µ(i) − µ0), i ∈ {1, · · · , N}, (106)
1
−
λ1
α−1 w ∗ 1
−
1, · · · , λ1
α−1 w ∗
N = (µ(i) − µ0), i ∈ {1, · · · , N} . (107)
Denoting by { ˆw1, · · · , ˆwN} the solution of
this shows that the optimal weights are
∂ρα
( ˆw1, · · · , ˆwN) = (µ(i) − µ0), i ∈ {1, · · · , N}, (108)
∂wi
w ∗ i = λ1
1
α−1 ˆwi. (109)
Now, performing the same calculation as in the case of independent risky assets, the efficient portfolio P
can be realized by investing a weight w0 of the initial wealth in the risk-free asset and the weight (1 − w0)
in the risky fund Π, whose weights are given by
˜wi =
Therefore, the expected return of every efficient portfolio is
ˆwi
N i=1 ˆwi
. (110)
µ = w0 · µ0 + (1 − w0) · µΠ, (111)
where µΠ denotes the expected return of the market portfolio Π, while the risk, measured by ρα is
so that
ρα = (1 − w0) α ρα(Π) , (112)
µ = µ0 + µΠ − µ0
ρα(Π) 1/α ρα 1/α . (113)
This expression is the natural generalization of the relation obtained by (Markovitz 1959) for mean-variance
efficient portfolios.
32

C Composition of the market portfolio
In this appendix, we derive the relationship between the composition of the market portfolio and the composition
of the optimal portfolio Π obtained by the minimization of the risks measured by ρα(n).
C.1 Homogeneous case
We first consider a homogeneous market, peopled with agents choosing their optimal portfolio with respect
to the same risk measure ρα. A given agent p invests a fraction w0(p) of his wealth W (p) in the risk-free
asset and a fraction 1 − w0(p) in the optimal portfolio Π. Therefore, the total demand Di of asset i is the
sum of the demand Di(p) over all agents p in asset i:
Di =
Di(p) , (114)
p
457
=
W (p) · (1 − w0(p)) · ˜wi , (115)
p
= ˜wi ·
W (p) · (1 − w0(p)) , (116)
p
where the ˜wi’s are given by (110). The aggregated demand D over all assets is
D =
Di, (117)
i
=
˜wi ·
W (p) · (1 − w0(p)), (118)
i
p
=
W (p) · (1 − w0(p)). (119)
p
By definition, the weight of asset i, denoted by wm i , in the market portfolio equals the ratio of its capitalization
(the supply Si of asset i) over the total capitalization of the market S = Si. At the equilibrium,
demand equals supply, so that
w m i = Si Di
=
S D = ˜wi. (121)
Thus, at the equilibrium, the optimal portfolio Π is the market portfolio.
C.2 Heterogeneous case
(120)
We now consider a heterogenous market, defined such that the agents choose their optimal portfolio with
respect to different risk measures. Some of them choose the usual mean-variance optimal portfolios, others
prefer any mean-ρα efficient portfolio, and so on. Let us denote by Πn the mean-ρα(n) optimal portfolio
made only of risky assets. Let φn be the fraction of agents who choose the mean-ρα(n) efficient portfolios.
By normalization,
n φn = 1. The demand Di(n) of asset i from the agents optimizing with respect to
ρα(n) is
Di(n) =
W (p) · (1 − w0(p)) · ˜wi(n), (122)
p∈Sn
= ˜wi(n)
W (p) · (1 − w0(p)), (123)
p∈Sn
33

458 14. Gestion de Portefeuilles multimoments et équilibre de marché
where Sn denotes the set of agents, among all the agents, who follow the optimization stragtegy with respect
to ρα(n). Thus, the total demand of asset i is
Di =
N φn · Di(n), (124)
n
= N
φn · ˜wi(n)
W (p) · (1 − w0(p)), (125)
n
where N is the total number of agents. This finally yields the total demand D for all assets and for all agents
D =
Di, (126)
i
p∈Sn
= N
φn · ˜wi(n)
W (p) · (1 − w0(p)), (127)
i
= N
n
n
φn
p∈Sn
W (p) · (1 − w0(p)), (128)
p∈Sn
since
i ˜wi(n) = 1, for every n. Thus, setting
γn = φn p∈Sn
n φn
p∈Sn
W (p) · (1 − w0(p))
, (129)
W (p) · (1 − w0(p))
the market portfolio is the weighted sum of the mean-ρα(n) optimal portfolios Πn:
w m i = Si
S
= Di
D
= γn · ˜wi(n) . (130)
34
n

D Generalized capital asset princing model
Our proof of the generalized capital asset princing model is similar to the usual demontration of the CAPM.
Let us consider an efficient portfolio P. It necessarily satisfies equation (105) in appendix B :
459
∂ρα
(w
∂wi
∗ 1, · · · , w ∗ N) = λ1 (µ(i) − µ0), i ∈ {1, · · · , N}. (131)
Let us now choose any portfolio R made only of risky assets and let us denote by wi(R) its weights. We
can thus write
N
i=1
wi(R) · ∂ρα
(w
∂wi
∗ 1, · · · , w ∗ N) = λ1
N
wi(R) · (µ(i) − µ0), (132)
i=1
= λ1 (µR − µ0). (133)
We can apply this last relation to the market portfolio Π, because it is only composed of risky assets (as
proved in appendix B). This leads to wi(R) = w ∗ i and µR = µΠ, so that
N
i=1
which, by the homogeneity of the risk measures ρα, yields
Substituting equation (131) into (135) allows us to obtain
where
w ∗ i · ∂ρα
(w
∂wi
∗ 1, · · · , w ∗ N) = λ1 (µΠ − µ0), (134)
α · ρα(w ∗ 1, · · · , w ∗ N) = λ1 (µΠ − µ0) . (135)
µj − µ0 = β j α · (µΠ − µ0), (136)
β j α =
∂
1
ln ρα α
∂wj
, (137)
calculated at the point {w∗ 1 , · · · , w∗ N }. Expression (135) with (137) provides our CAPM, generalized with
respect to the risk measures ρα.
In the case where ρα denotes the variance, the second-order centered moment is equal to the second-order
cumulant and reads
Since
we find
C2 = w ∗ 1 · Var[X1] + 2w ∗ 1w ∗ 2 · Cov(X1, X2) + w ∗ 2 · Var[X2], (138)
= Var[Π] . (139)
1 ∂C2
·
2 ∂w1
= w ∗ 1 · Var[X1] + w ∗ 2 · Cov(X1, X2) , (140)
= Cov(X1, Π), (141)
β = Cov(X1, XΠ)
Var[XΠ]
which is the standard result of the CAPM derived from the mean-variance theory.
35
, (142)

460 14. Gestion de Portefeuilles multimoments et équilibre de marché
E Calculation of the moments of the distribution of portfolio returns
Let us start with equation (72) in the 2-asset case :
ˆPS(k)
1
=
2π 1 − ρ2
dy1dy2 exp − 1
2 ytV −1
y1
y + ik χ1w1sgn(y1)
√
2
y2
+χ2w2sgn(y2)
√
2
Expanding the exponential and using the definition (67) of moments, we get
Posing
1
Mn =
2π 1 − ρ2
dy1 dy2
γq1q2 (n, p) = χ1 p χ2 n−p
this leads to
2π 1 − ρ 2
n
p=0
n
χ
p
p
dy1dy2 sgn(y1) p
y1
√
2
Mn =
1χn−p 2 wp 1wn−p 2
×sgn(y2) n−p
q1p
y2
√
2
sgn(y1) p
y1
√
2
q2(n−p)
sgn(y2) n−p
y2
√
2
Let us defined the auxiliary variables α and β such that
α = (V −1 )11 = (V −1 )22 = 1
1−ρ 2 ,
β = −(V −1 )12 = −(V −1 )21 = ρ
1−ρ 2 .
n
p=0
q1
+
q2
q1p
×
. (143)
1
−
e 2 ytV −1y . (144)
q2(n−p)
1
−
e 2 ytV −1y , (145)
n
w
p
p
1wn−pγq1q2 2 (n, p) . (146)
(147)
Performing a simple change of variable in (145), we can transform the integration such that it is defined
solely within the first quadrant (y1 ≥ 0, y2 ≥ 0), namely
γq1q2 (n, p) = χ1 p n−p 1 + (−1)n
χ2
2π 1 − ρ2 +∞ +∞
dy1 dy2
0
0
q1p q2(n−p) y1 y2
α
− √2 √2 e 2 (y2 1 +y2 2 ) ×
× e βy1y2
p −βy1y2
+ (−1) e
. (148)
This equation imposes that the coefficients γ vanish if n is odd. This leads to the vanishing of the moments
of odd orders, as expected for a symmetric distribution. Then, we expand e βy1y2 + (−1) p e −βy1y2 in series.
Permuting the sum sign and the integral allows us to decouple the integrations over the two variables y1 and
y2:
γq1q2 (n, p) = χ1 p n−p 1 + (−1)n
χ2
2π 1 − ρ2 +∞
[1 + (−1)
s=0
p+s ] βs
s!
36
+∞
dy1
0
+∞
× dy2
0
y q1p+s
1
2 q1p 2
y q2(n−p)+s
2
2 q2 (n−p)
2
α
−
e 2 y2 1
α
−
e 2 y2 1
×
. (149)

This brings us back to the problem of calculating the same type of integrals as in the uncorrelated case.
Using the expressions of α and β, and taking into account the parity of n and p, we obtain:
γq1q2 (2n, 2p) = χ1 2p χ2 2n−2p (1 − ρ2 1
q1p+q2(n−p)+ ) 2
π
×Γ q2(n − p) + s + 1
2
γq1q2 (2n, 2p + 1) = χ1 2p+1 χ2 2n−2p−1 (1 − ρ2 ) q1p+q2(n−p)+ q1−q2 +1
2
×Γ
q1p + s + 1 + q1
2
Γ
+∞
s=0
(2ρ) 2s
(2s)! Γ
q1p + s + 1
×
2
461
, (150)
+∞
(2ρ)
π
s=0
2s+1
(2s + 1)! ×
q2(n − p) + s + 1 − q2
. (151)
2
Using the definition of the hypergeometric functions 2F1 (Abramovitz and Stegun 1972), and the relation
(9.131) of (Gradshteyn and Ryzhik 1965), we finally obtain
γq1q2 (2n, 2p) = χ1 2p χ2 2n−2p Γ q1p + 1
2 Γ q2(n − p) + 1
2
2F1 −q1p, −q2(n − p);
π
1
; ρ2(152)
,
2
γq1q2 (2n, 2p + 1) = χ1 2p+1 χ2 2n−2p−1 2Γ q1p + 1 + q1
2 Γ q2(n − p) + 1 − q2
2 ρ ×
π
× 2F1 −q1p − q1 − 1
, −q2(n − p) +
2
q2 + 1
;
2
3
; ρ2 . (153)
2
In the asymmetric case, a similar calculation follows, with the sole difference that the results involves four
terms in the integral (148) instead of two.
37

462 14. Gestion de Portefeuilles multimoments et équilibre de marché
F Conditions under which it is possible to increase the return and decrease
large risks simultaneously
We consider N independent assets {1 · · · N}, whose returns are denoted by µ(1) · · · µ(N). We aggregate
these assets in a portfolio. Let w1 · · · wN be their weights. We consider that short positions are forbidden
and that
i wi = 1. The return µ of the portfolio is
µ =
The risk of the portfolio is quantified by the cumulants of the distribution of µ.
N
wiµ(i). (154)
i=1
Let us denote µ ∗ n the return of the portfolio evaluated for the asset weights which minimize the cumulant of
order n.
F.1 Case of two assets
Let Cn be the cumulant of order n for the portfolio. The assets being independent, we have
Cn = Cn(1)w1 n + Cn(2)w2 n , (155)
= Cn(1)w1 n + Cn(2)(1 − w1) n . (156)
In the following, we will drop the subscript 1 in w1, and only write w. Let us evaluate the value w = w ∗ at
the minimum of Cn, n > 2 :
dCn
dw = 0 ⇐⇒ Cn(1)w n−1 − Cn(2)(1 − w) n−1 = 0, (157)
⇐⇒ Cn(1)
Cn(2) =
1 − w ∗
w ∗
n−1
, (158)
and assuming that Cn(1)/Cn(2) > 0, which is satisfied according to our positivity axiom 1, we obtain
w ∗ =
This leads to the following expression for µ ∗ n :
µ ∗ n =
Cn(2) 1
n−1
Cn(1) 1
n−1 + Cn(2) 1
n−1
µ(1) · Cn(2) 1
n−1 + µ(2) · Cn(1) 1
n−1
Cn(1) 1
n−1 + Cn(2) 1
n−1
Thus, after simple algebraic manipulations, we find
⇐⇒ (µ(1) − µ(2)) Cn(1) 1
n−1 Cn+k(2)
µ ∗ n < µ ∗ n+k
1
n+k−1 − Cn(2) 1
. (159)
. (160)
n−1 Cn+k(1)
which concludes the proof of the result announced in the main body of the text.
38
1
n+k−1
> 0, (161)

F.2 General case
We now consider a portfolio with N independent assets. Assuming that the cumulants Cn(i) have the same
sign for all i (according to axiom 1), we are going to show that the minimum of Cn is obtained for a portfolio
whose weights are given by
and we have
µ ∗ n =
wi =
N
i=1
Indeed, the cumulant of the portfolio is given by
subject to the constraint
Cn =
1
N
j=i Cn(j) n−1
1
N
j=1 Cn(j) n−1
µ(i) N
j=i
1
N
j=1 Cn(j) n−1
N
i=1
Cn(i) w n i
463
, (162)
1
Cn(j) n−1
Introducing a Lagrange multiplier λ, the first order conditions yields
so that
Cn(i) w n−1
i
. (163)
(164)
N
wi = 1. (165)
i=1
− λ = 0, ∀i ∈ {1, · · · , N}, (166)
w n−1
i
λ
= . (167)
Cn(i)
Since all the Cn(i) are positive, we can find a λ such that all the wi are real and positive, which yields the
announced result (162). From here, there is no simple condition that ensures µ ∗ n < µ ∗ n+k . The simplest way
is to calculate diretly these quantities using the formula (163).
to compare µ ∗ n and µ ∗ n+k
39

464 14. Gestion de Portefeuilles multimoments et équilibre de marché
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42

µ µ2 1/2 µ4 1/4 µ6 1/6 µ8 1/8
0.10% 0.92% 1.36% 1.79% 2.15%
0.12% 0.96% 1.43% 1.89% 2.28%
0.14% 1.05% 1.56% 2.06% 2.47%
0.16% 1.22% 1.83% 2.42% 2.91%
0.18% 1.47% 2.21% 2.92% 3.55%
0.20% 1.77% 2.65% 3.51% 4.22%
Table 1: This table presents the risk measured by µ 1/n
n
(daily) return µ.
µ
µ 1/2
2
µ
µ 1/4
4
467
for n=2,4,6,8, for a given value of the expectedd
µ
µ 1/6
6
µ
C 1/4
4
µ
C 1/6
6
Wall Mart 0.0821 0.0555 0.0424 0.0710 0.0557
EMC 0.0801 0.0552 0.0430 0.0730 0.0612
Intel 0.0737 0.0512 0.0397 0.0694 0.0532
Hewlett Packard 0.0724 0.0472 0.0354 0.0575 0.0439
IBM 0.0705 0.0465 0.0346 0.0574 0.0421
Merck 0.0628 0.0415 0.0292 0.0513 0.0331
Procter & Gamble 0.0590 0.0399 0.0314 0.0510 0.0521
General Motors 0.0586 0.0362 0.0247 0.0418 0.0269
SBC Communication 0.0584 0.0386 0.0270 0.0477 0.0302
General Electric 0.0569 0.0334 0.0233 0.0373 0.0258
Applied Material 0.0525 0.0357 0.0269 0.0462 0.0338
MCI WorldCom 0.0441 0.0173 0.0096 0.0176 0.0098
Medtronic 0.0432 0.0278 0.0202 0.0333 0.0237
Coca-Cola 0.0430 0.0278 0.0207 0.0335 0.0252
Exxon-Mobil 0.0410 0.0256 0.0178 0.0299 0.0197
Texas Instrument 0.0324 0.0224 0.0171 0.0301 0.0218
Pfizer 0.0298 0.0184 0.0131 0.0213 0.0148
Table 2: This table presents the values of the generalized Sharpe ratios for the set of seventeen assets listed
in the first column. The assets are ranked with respect to their Sharpe ratio, given in the second column. The
third and fourth columns give the generalized Sharpe ratio calculated with respect to the fourth and sixth
centered moments µ4 and µ6 while the fifth and sixth columns give the generalized Sharpe ratio calculated
with respect to the fourth and sixth cumulants C4 and C6.
43

468 14. Gestion de Portefeuilles multimoments et équilibre de marché
Positive Tail Negative Tail
< χ+ > < c+ > χ+ c+ < χ− > < c− > χ− c−
CHF 2.45 1.61 2.33 1.26 2.34 1.53 1.72 0.93
DEM 2.09 1.65 1.74 1.03 2.01 1.58 1.45 0.91
JPY 2.10 1.28 1.30 0.76 1.89 1.47 0.99 0.76
MAL 1.00 1.22 1.25 0.41 1.01 1.25 0.44 0.48
POL 1.55 1.02 1.30 0.73 1.60 2.13 1.25 0.62
THA 0.78 0.75 0.75 0.54 0.82 0.73 0.30 0.38
UKP 1.89 1.52 1.38 0.92 2.00 1.41 1.82 1.09
Table 3: Table of the exponents c and the scale parameters χ for different currencies. The subscript ”+”
or ”-” denotes the positive or negative part of the distribution of returns and the terms between brackets
refer to parameters estimated in the bulk of the distribution while naked parameters refer to the tails of the
distribution.
Positive Tail Negative Tail
< χ+ > < c+ > χ+ c+ < χ− > < c− > χ− c−
Applied Material 12.47 1.82 8.75 0.99 11.94 1.66 8.11 0.98
Coca-Cola 5.38 1.88 4.46 1.04 5.06 1.74 2.98 0.78
EMC 13.53 1.63 13.18 1.55 11.44 1.61 3.05 0.57
General Electric 5.21 1.89 1.81 1.28 4.80 1.81 4.31 1.16
General Motors 5.78 1.71 0.63 0.48 5.32 1.89 2.80 0.79
Hewlett Packart 7.51 1.93 4.20 0.84 7.26 1.76 1.66 0.52
IBM 5.46 1.71 3.85 0.87 5.07 1.90 0.18 0.33
Intel 8.93 2.31 2.79 0.64 9.14 1.60 3.56 0.62
MCI WorldCom 9.80 1.74 11.01 1.56 9.09 1.56 2.86 0.58
Medtronic 6.82 1.95 6.09 1.11 6.49 1.54 2.55 0.67
Merck 5.36 1.91 4.56 1.16 5.00 1.73 1.32 0.59
Pfizer 6.41 2.01 5.84 1.27 6.04 1.70 0.26 0.35
Procter & Gambel 4.86 1.83 3.53 0.96 4.55 1.74 2.96 0.82
SBC Communication 5.21 1.97 1.26 0.59 4.89 1.59 1.56 0.60
Texas Instrument 9.06 1.78 4.07 0.72 8.24 1.84 2.18 0.54
Wall Mart 7.41 1.83 5.81 1.01 6.80 1.64 3.75 0.78
Table 4: Table of the exponents c and the scale parameters χ for different stocks. The subscript ”+” or ”-”
denotes the positive or negative part of the distribution and the terms between brackets refer to parameters
estimated in the bulk of the distribution while naked parameters refer to the tails of the distribution.
44

Mean (10 −3 ) Variance (10 −3 ) Skewness Kurtosis min max
Applied Material 2.11 1.62 0.41 4.68 -14% 21%
Coca-Cola 0.81 0.36 0.13 5.71 -11% 10%
EMC 2.76 1.13 0.23 4.79 -18% 15%
Exxon-Mobil 0.92 0.25 0.30 5.26 -7% 11%
General Electric 1.38 0.30 0.08 4.46 -7% 8%
General Motors 0.64 0.39 0.12 4.35 -11% 8%
Hewlett Packard 1.17 0.81 0.16 6.58 -14% 21%
IBM 1.32 0.54 0.08 8.43 -16% 13%
Intel 1.71 0.85 -0.31 6.88 -22% 14%
MCI WorldCom 0.87 0.85 -0.18 6.88 -20% 13%
Medtronic 1.70 0.55 0.23 5.52 -12% 12%
Merck 1.32 0.35 0.18 5.29 -9% 10%
Pfizer 1.57 0.46 0.01 4.28 -10% 10%
Procter&Gambel 0.90 0.41 -2.57 42.75 -31% 10%
SBC Communication 0.86 0.39 0.06 5.86 -13% 9%
Texas Instrument 2.20 1.23 0.50 5.26 -12% 24%
Wall Mart 1.35 0.52 0.16 4.79 -10% 9%
Table 5: This table presents the main statistical features of the daily returns of the set of seventeen assets
studied here over the time interval from the end of January 1995 to the end of December 2000.
45
469

470 14. Gestion de Portefeuilles multimoments et équilibre de marché
x n P(x)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
x P(x)
x 2 P(x)
x 4 P(x)
0
0 1 2 3 4 5
x
6 7 8 9 10
Figure 1: This figure represents the function x n · e −x for n = 1, 2 and 4. It shows the typycal size of the
fluctuations involved in the moment of order n.
46

μ (daily return)
x 10−3
2.5
2
1.5
1
0.5
Mean−μ 2 Efficient Frontier
Mean−μ 4 Efficient Frontier
Mean−μ 6 Efficient Frontier
Mean−μ 8 Efficient Frontier
0
0 0.01 0.02 0.03
1/n
μ
n
0.04 0.05 0.06
Figure 2: This figure represents the generalized efficient frontier for a portfolio made of seventeen risky
assets. The optimization problem is solved numerically, using a genetic algorithm, with risk measures given
respectively by the centered moments µ2, µ4, µ6 and µ8. The straight lines are the efficient frontiers when
we add to these assets a risk-free asset whose interest rate is set to 5% a year.
47
471

472 14. Gestion de Portefeuilles multimoments et équilibre de marché
μ (daily return)
x 10−3
1.5
1
0.5
Mean−μ 2 Efficient Frontier
Mean−μ 4 Efficient Frontier
Mean−μ 6 Efficient Frontier
Mean−μ 8 Efficient Frontier
0
0 0.005 0.01 0.015 0.02 0.025
1/n
μ
n
Figure 3: This figure represents the generalized efficient frontier for a portfolio made of seventeen risky
assets and a risk-free asset whose interest rate is set to 5% a year. The optimization problem is solved
numerically, using a genetic algorithm, with risk measures given by the centered moments µ2, µ4, µ6 and
µ8.
48

w i
w i
0.2
0.15
0.1
0.05
Mean−μ 2
0
0 0.2 0.4 0.6 0.8 1
w
0
0.3
0.25
0.2
0.15
0.1
0.05
Mean−μ 6
0
0 0.2 0.4 0.6 0.8 1
w
0
w i
w i
0.25
0.2
0.15
0.1
0.05
Mean−μ 4
0
0 0.2 0.4 0.6 0.8 1
w
0
0.3
0.25
0.2
0.15
0.1
0.05
Mean−μ 8
0
0 0.2 0.4 0.6 0.8 1
w
0
Figure 4: Dependence of the five largest weights of risky assets in the efficient portfolios found in figure 3
as a function of the weight w0 invested in the risk-free asset, for the four risk measures given by the centered
moments µ2, µ4, µ6 and µ8. The same symbols always represent the same asset.
49
473

474 14. Gestion de Portefeuilles multimoments et équilibre de marché
μ
Cn 1/n
Figure 5: The dark and grey thick curves represent two efficient frontiers for a portfolio without risk-free
interest rate obtained with two measures of risks. The dark and grey thin straight lines represent the efficient
frontiers in the presence of a risk-free asset, whose value is given by the intercept of the straight lines with
the ordinate axis. This illustrates the existence of an inversion of the dependence of the slope of the efficient
frontier with risk-free asset as a function of the order n of the measures of risks, which can occur only when
the efficient frontiers without risk-free asset cross each other.
50

Figure 6: Schematic representation of the nonlinear mapping Y = u(X) that allows one to transform a variable
X with an arbitrary distribution into a variable Y with a Gaussian distribution. The probability densities
for X and Y are plotted outside their respective axes. Consistent with the conservation of probability, the
shaded regions have equal area. This conservation of probability determines the nonlinear mapping.
51
475

476 14. Gestion de Portefeuilles multimoments et équilibre de marché
Empirical Cumulative Distribution
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
CHF−UKP
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cumulative Normal Ditribution
Figure 7: Quantile of the normalized sum of the Gaussianized returns of the Swiss Franc and The British
Pound versus the quantile of the Normal distribution, for the time interval from Jan. 1971 to Oct. 1998.
Different weights in the sum give similar results.
52

Empirical Cumulative Distribution
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
KO−PG
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cumulative Normale Ditribution
Figure 8: Quantile of the normalized sum of the Gaussianized returns of Coca-Cola and Procter&Gamble
versus the quantile of the Normal distribution, for the time interval from Jan. 1970 to Dec. 2000. Different
weights in the sum give similar results.
53
477

478 14. Gestion de Portefeuilles multimoments et équilibre de marché
Empirical Cumulative Distribution
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MRK−GE
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cumulative Normal Ditribution
Figure 9: Quantile of the normalized sum of the Gaussianized returns of Merk and General Electric versus
the quantile of the Normal distribution, for the time interval from Jan. 1970 to Dec. 2000. Different weights
in the sum give similar results.
54

Z 2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
CHF−UKP
0
0 0.1 0.2 0.3 0.4 0.5
χ
0.6 0.7 0.8 0.9 1
2
Figure 10: Cumulative distribution of z 2 = y t V −1 y versus the cumulative distribution of chi-square (denoted
χ 2 ) with two degrees of freedom for the couple Swiss Franc / British Pound, for the time interval from
Jan. 1971 to Oct. 1998. This χ 2 should not be confused with the characteristic scale used in the definition
of the modified Weibull distributions.
55
479

480 14. Gestion de Portefeuilles multimoments et équilibre de marché
Z 2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
KO−PG
0
0 0.1 0.2 0.3 0.4 0.5
χ
0.6 0.7 0.8 0.9 1
2
Figure 11: Cumulative distribution of z 2 = y t V −1 y versus the cumulative distribution of the chi-square
χ 2 with two degrees of freedom for the couple Coca-Cola / Procter&Gamble, for the time interval from Jan.
1970 to Dec. 2000. This χ 2 should not be confused with the characteristic scale used in the definition of the
modified Weibull distributions.
56

Z 2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
MRK−GE
0
0 0.1 0.2 0.3 0.4 0.5
χ
0.6 0.7 0.8 0.9 1
2
Figure 12: Cumulative distribution of z 2 = y t V −1 y versus the cumulative distribution of the chi-square
χ 2 with two degrees of freedom for the couple Merk / General Electric, for the time interval from Jan. 1970
to Dec. 2000. This χ 2 should not be confused with the characteristic scale used in the definition of the
modified Weibull distributions.
57
481

482 14. Gestion de Portefeuilles multimoments et équilibre de marché
10 0
10 0
10 −1
10 −1
10 −1
10 0
10 0
MAL +
MAL −
Figure 13: Graph of Gaussianized Malaysian Ringgit returns versus Malaysian Ringgit returns, for the time
interval from Jan. 1971 to Oct. 1998. The upper graph gives the positive tail and the lower one the negative
tail. The two straight lines represent the curves y = √ 〈c±〉
x 2 〈χ±〉 and y = √ c± x 2 χ±
58
10 1
10 1
10 2
10 2

10 0
10 −1
10 0
10 −1
10 −1
10 −1
10 0
10 0
UKP +
UKP −
Figure 14: Graph of Gaussianized British Pound returns versus British Pound returns, for the time interval
from Jan. 1971 to Oct. 1998. The upper graph gives the positive tail and the lower one the negative tail. The
two straight lines represent the curves y = √ 〈c±〉
x 2 〈χ±〉 and y = √ c± x 2 χ±
59
10 1
10 1
10 2
10 2
483

484 14. Gestion de Portefeuilles multimoments et équilibre de marché
10 0
10 −1
10 0
10 −1
10 −1
10 −1
10 0
10 0
GE +
GE −
Figure 15: Graph of Gaussianized General Electric returns versus General Electric returns, for the time
interval from Jan. 1970 to Dec. 2000. The upper graph gives the positive tail and the lower one the negative
tail. The two straight lines represent the curves y = √ 〈c±〉
x 2 〈χ±〉 and y = √ c± x 2 χ±
60
10 1
10 1
10 2
10 2

10 0
10 −1
10 0
10 −1
10 −1
10 −1
10 0
10 0
IBM +
IBM −
Figure 16: Graph of Gaussianized IBM returns versus IBM returns, for the time interval from Jan. 1970 to
Dec. 2000. The upper graph gives the positive tail and the lower one the negative tail. The two straight lines
represent the curves y = √ 〈c±〉
x 2 〈χ±〉 and y = √ c± x 2 χ±
61
10 1
10 1
10 2
10 2
485

486 14. Gestion de Portefeuilles multimoments et équilibre de marché
10 0
10 −1
10 0
10 −1
10 −1
10 −1
10 0
10 0
WMT +
WMT −
Figure 17: Graph of Gaussianized Wall Mart returns versus Wall Mart returns, for the time interval from
Sep. 1972 to Dec. 2000. The upper graph gives the positive tail and the lower one the negative tail. The two
straight lines represent the curves y = √ 〈c±〉
x 2 〈χ±〉 and y = √ c± x 2 χ±
62
10 1
10 1
10 2
10 2

κ
22
20
18
16
14
12
10
8
6
4
2
Excess Kurtosis for CHF / JPY
0
0 0.1 0.2 0.3 0.4 0.5
w
CHF
0.6 0.7 0.8 0.9 1
Figure 18: Excess kurtosis of the distribution of the price variation wCHF xCHF + wJP Y xJP Y of the
portfolio made of a fraction wCHF of Swiss franc and a fraction wJP Y = 1 − wCHF of the Japanese Yen
against the US dollar, as a function of wCHF . Thick solid line : empirical curve, thin solid line : theoretical
curve, dashed line : theoretical curve with ρ = 0 (instead of ρ = 0.43), dotted line: theoretical curve with
qCHF = 2 rather than 1.75 and dashed-dotted line: theoretical curve with qCHF = 1.5. The excess kurtosis
has been evaluated for the time interval from Jan. 1971 to Oct. 1998.
63
487

488 14. Gestion de Portefeuilles multimoments et équilibre de marché
C 4
65
60
55
50
45
40
35
30
25
20
Fourth Cumulant for CHF / UKP
15
0 0.1 0.2 0.3 0.4 0.5
w
CHF
0.6 0.7 0.8 0.9 1
Figure 19: Fourth cumulant for a portfolio made of a fraction wCHF of Swiss Franc and 1−wCHF of British
Pound. The thick solid line represents the empirical cumulant while the dotted line represents the theoretical
cumulant under the symmetric assumption. The dashed line shows the theoretical cumulant when the slight
asymmetry of the assets has been taken into account. This cumulant has been evaluated for the time interval
from Jan. 1971 to Oct. 1998.
64

Return μ
0.12
0.11
0.1
0.09
0.08
0.07
Efficient Frontier forIBM−HWP
1 1.2 1.4 1.6
min
C /C , n={1,2,3}
2n 2n
1.8 2 2.2
Figure 20: Efficient frontier for a portfolio composed of two stocks: IBM and Hewlett-Packard. The dashed
line represents the efficient frontier with respect to the second cumulant, i.e., the standard Markovitz efficient
frontier, the dash-dotted line represents the efficient frontier with respect to the fourth cumulant and the solid
line is the efficient frontier with respect to the sixth cumulant. The data set used covers the time interval
from Jan. 1977 to Dec 2000.
65
489

490 14. Gestion de Portefeuilles multimoments et équilibre de marché
Return μ
0.13
0.12
0.11
0.1
0.09
0.08
0.07
0.06
Efficient Frontier forIBM−KO
0.05
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
min
C /C , n={1,2,3}
2n 2n
Figure 21: Efficient frontier for a portfolio composed of two stocks: IBM and Coca-Cola. The dashed line
represents the efficient frontier with respect to the second cumulant, i.e., the standard Markovitz efficient
frontier, the dash-dotted line represents the efficient frontier with respect to the fourth cumulant and the solid
line the efficient frontier with repect to the sixth cumulant. The data set used covers the time interval from
Jan. 1970 to Dec 2000.
66

Conclusions et Perspectives
L’objet de notre étude était de contribuer à une meilleure compréhension des risques extrêmes sur les
marchés financiers afin de permettre d’esquisser les contours de nouvelles méthodes de contrôle de ce
type de risques et d’en présenter certaines conséquences pour ce qui est de la gestion de portefeuilles.
Pour cela, nous avons suivi une approche dont le but était de chercher à isoler les différentes sources de
risques selon que leur origine provienne des fluctuations individuelles des actifs ou bien de leur comportement
collectif. Cette distinction nous parait particulièrement importante pour la gestion des risques
dans la mesure où il nous semble nécessaire pour mener à bien cette tâche (i) d’être capable d’estimer les
grands risques associés à chaque actif - ce qui ne repose que sur l’étude de leur “variabilité” intrinsèque
- et (ii) de savoir si l’on peut espérer diversifier ces grands risques et donc d’en réduire l’impact par la
formation de portefeuilles - ce qui fait appel à leurs caractéristiques collectives.
C’est pourquoi nous avons tout d’abord procédé à une étude à la fois empirique et théorique des variations
de cours des actifs financiers, afin d’en déterminer les propriétés statistiques et d’essayer de mettre à
jour certains mécanismes micro-structurels et comportementaux permettant de justifier les observations
empiriques. L’étude s’est focalisée sur trois points :
– les aspects statiques (monovariés), avec l’étude des distributions marginales de rendements,
– mais aussi les aspects dynamiques de par l’étude de la modélisation de la dépendance temporelle à
l’aide de processus de marches aléatoires multifractales,
– ainsi que la prise en compte de l’interaction entre les actifs au travers de l’étude de la représentation
de la structure de dépendance en terme de copule et surtout l’estimation des dépendances extrêmes à
l’aide du coefficient de dépendance de queue.
Ceci nous a conduit, pour ce qui est du premier point, à nous pencher sur l’hypothèse d’après laquelle
les rendements sont distribués selon des lois de puissances (ou plus généralement des lois régulièrement
variables). Conformément à certains travaux récents, nous avons confirmé qu’il ne pouvait cependant être
exclu que des distributions exponentielles étirées sont à même de représenter les distributions de rendements
aussi bien (sinon mieux) que les distributions régulièrement variables généralement utilisées. La
prise en compte de cette nouvelle représentation paramétrique des distributions de rendements est importante
et nous semble devoir être employée conjointement avec la description en terme de distributions
régulièrement variables (ce que nous avons fait dans la suite de notre étude) afin de ne pas négliger le
risque de modèle inhérent à toute description paramétrique des données.
Concernant l’étude dynamique de l’évolution des cours, nous avons préféré nous tourner vers l’utilisation
de processus de marche aléatoire multifractale, plutôt que vers les traditionnels processus de la famille
ARCH, car ils semblent les seuls à pouvoir rendre compte de la manière suivant laquelle la volatilité
retourne à la moyenne (relaxe) après un grand choc. En effet, nous avons montré que le processus de
marche aléatoire multifractale prédisait une relaxation hyperbolique dont l’exposant diffère selon que
le choc est endogène ou exogène, ce qui a pu être effectivement observé sur les données. En outre, un
tel comportement ne peut pas être expliqué par les processus de type ARCH. Donc, le processus de
marche aléatoire multifractale semble bien adapté à la description de la dynamique des cours des actifs
491

492 Conclusion
financiers, et permet une meilleure compréhension de l’incorporation du flux d’information dans le prix
des actifs, ce qui présente un grand intérêt pour la prédiction de la volatilité, par exemple, mais aussi pour
la gestion de scenarii permettant de mieux appréhender l’impact de telle ou telle nouvelle sur l’évolution
du prix futur d’un actif et notamment la durée des périodes de fortes turbulences ou de calme anormal
des marchés. Notons toutefois qu’il conviendra d’améliorer cette description par la prise en compte de
l’asymétrie entre fluctuations à la hausse et à la baisse (effet de levier) qui a été totalement négligée
jusqu’ici.
Pour synthétiser ces deux premiers points, et essayer de percer certains des mécanismes microscopiques
permettant de les justifier, nous nous sommes intéressés aux modèles d’agents en interaction. Ceci avait
pour but d’intégrer les comportements non totalement rationnels des acteurs économiques afin de comprendre
comment - en dépit de cette rationnalité limitée - peut émerger une structure de marché la plupart
du temps efficiente, mais aussi pourquoi et comment les marchés s’écartent parfois violemment
de cette structure efficiente. Dans cette étude, nous avons mis en exergue l’importance des comportements
mimétiques et antagonistes entre agents afin de préciser leur rôle dans l’émergence et l’explosion
des bulles spéculatives. Là encore, cette approche offre une possibilité de simuler certaines phases de
marchés et fournit un moyen de mieux anticiper les risques à venir.
Une des limites actuelles des modèles d’agents en interaction est de se concentrer uniquement sur la
modélisation de marchés où un seul actif (plus éventuellement un actif sans risque) peut être échangé.
Nous avons, nous aussi, choisi de suivre cette voie pour des raisons de simplicité. En effet, à partir
du moment où l’on considère un marché doté de plusieurs actifs, il convient de donner aux agents les
moyens de choisir entre les divers actifs et donc d’introduire une fonction d’utilité - a priori différente
- pour chaque agent, ce qui pose quelques difficultés, au premier rang desquelles, comme souligné au
chapitre 10, le choix de cette fonction. Il nous semble clair qu’une des évolutions futures des modèles
d’agents en interaction devra apporter des réponses à ce problème qui à notre avis constitue l’une des
toutes prochaines évolutions à apporter à ce type de modélisation.
Le troisième point que nous avons abordé concernait la description de la dépendance entre les actifs. Nous
avons pour cela commencé par essayer de déterminer de manière globale la structure de dépendance
en tentant d’estimer la copule des rendements des actifs financiers. Nous avons pu conclure que pour
des rendements modérés, une copule gaussienne semblait tout à fait à même de rendre compte de leur
dépendance, mais risquait de sous-estimer les dépendances extrêmes. Pour confirmer cette hypothèse,
nous avons voulu estimer le coefficient de dépendance de queue, qui mesure la propension de deux
actifs financiers à subir ensemble de grands mouvements. La difficulté à estimer de manière directe cette
quantité nous a d’abord conduit à la calculer théoriquement en nous appuyant sur une description des
fluctuations des prix des actifs à l’aide d’un modèle à facteur. Puis, de la calibration des paramètres de ce
modèle, nous avons pu déduire la valeur du coefficient de dépendance de queue pour chaque pair d’actifs.
Ceci a alors confirmé l’inadéquation de la copule gaussienne à décrire la dépendance entre les grandes
variations des cours.
Cette étude devra être poursuivie, car nous n’avons pas pu trouver de copule réellement satisfaisante
quant à la description complète de la dépendance entre actifs. Si nous sommes désormais convaincus qu’il
existe une dépendance de queue entre les actifs, il reste néanmoins à essayer de déterminer quelle copule
est la mieux adaptée pour rendre compte de la structure de dépendance extrême entre les actifs. Ceci
est important car, comme nous l’avons déjà signalé, les paramètres intervenant dans la représentation
paramétrique de la copule permettent de résumer toute la dépendance de la même manière que le coefficient
de corrélation capture complètement la dépendance pour une distribution multivariée gaussienne.
Ces paramètres constituent, à notre avis, les “bonnes variables” qu’il convient de rechercher afin de nous
permettre de modéliser à l’échelle macroscopique le comportement des actifs et notamment la relation
entre risque et rendement. Que l’on pense au lien existant entre le coefficient de corrélation et la variance

Conclusion 493
en univers gaussien d’une part et coefficient beta mesurant le risque systématique associé à chaque actif
selon le CAPM d’autre part.
En outre, notre étude de la structure de dépendance entre actifs n’a été que descriptive et n’en a pas
sondé les causes microscopiques. Cela rejoint en fait ce que nous avons écrit plus haut concernant la
nécessité de développer des modèles d’agents considérant des marchés où peuvent être échangés plusieurs
actifs. Ce sera alors un moyen de mieux comprendre les raisons de l’existence de divers facteurs
et leur influence sur cette dépendance entre actifs : les facteurs économiques, tels que l’appartenance
à un même secteur d’activité par exemple, sont-ils des éléments primordiaux de cette dépendance, ou
bien les agents jouent-ils un rôle prépondérant ? Existe-t-il des phases où la structure de dépendance est
dominée par les fondamentaux économiques et d’autre part l’activité spéculative, comme on l’observe
pour le développement des fluctuations dans le cas marginal ? Toutes ces questions restent ouvertes et en
attente de réponses.
Après avoir considéré les deux facettes des grands risques que sont d’une part l’aspect individuel ou marginal
et d’autre part l’aspect collectif, nous avons regroupé nos résultats afin d’en déduire les conséquences
pratiques sur la gestion de portefeuilles en vue de la prévention de ces grands risques. Ceci nous a tout
d’abord amené à discuter de la manière dont il convient de les mesurer. Nous avons pour cela utilisé deux
approches : l’une basée sur la mesure du capital économique et l’autre prenant en compte la propension
du portefeuille à ne pas trop s’écarter des objectifs de rentabilité que l’on en espère. Ces deux voies ont
été explorées et nous avons discuté de leur mise en œuvre pratique ainsi que de leurs conséquences sur
les prix d’équilibre des actifs. Nous nous sommes bornés à considérer des mesures de risques et à aborder
le problème de l’optimisation de portefeuille dans un cadre mono-périodique (ou statique), alors qu’il
apparaît très clairement que ces deux problèmes doivent se poser en termes dynamiques. De nombreuses
recherches sont actuellement menées pour définir de manière axiomatique la notion de risque dans un
cadre dynamique ainsi que pour en étudier les conséquences sur la gestion de portefeuilles. Nos futures
recherches sur ce sujet devront s’inscrire dans ce courant.
Enfin, nous focalisant sur l’impact des risques extrêmes, il est notamment apparu que l’existence d’une
dépendance de queue entre les actifs en limitait les possibilités de diversification. Nous avons alors montrer
comment on parvenait à en minimiser les effets en ne sélectionnant que les actifs présentant les
plus faibles dépendances de queues. Cependant, si les risques extrêmes ne sont pas totalement diversifiables,
nous devrions être en droit de nous attendre à ce que le marché les rémunére. Nous n’avons pas
étudié cette question qui nous semble cruciale mais envisageons de traiter ce problème lors de recherches
ultérieures.
Nous voyons donc que les travaux que nous venons de présenter, loin d’être achevés et d’apporter des
réponses définitives, posent de nombreuses questions et ouvrent donc de multiples perspectives de recherches
futures.

494 Conclusion

Annexe A
Evaluation de la conduite du projet de
thèse
Ce nouveau chapitre s’inscrit dans le cadre de la poursuite d’un projet pilote développé par le ministère
de l’éducation nationale, de la recherche et de la technologie (MENRT) afin d’aider les jeunes docteurs
à s’insérer plus facilement dans la vie active, que ce soit dans le milieu de la recherche académique ou
bien au sein d’entreprises privées. L’objet de ce chapitre est de mener une réflexion sur la conduite de la
thèse, non seulement en termes de retombées scientifiques mais également d’un point de vue comptable,
avec l’évaluation du coût du projet, et de faire le bilan des expériences et compétences acquises au cours
de ces années.
A.1 Bref résumé du sujet
L’objet de cette thèse était l’étude des risques extrêmes sur les marchés financiers. Pour nous attaquer à ce
problème, nous avons tout abord cherché à modéliser les mécanismes permettant d’expliquer l’apparition
de mouvements extrêmes sur ce type de marchés afin d’en avoir une meilleure compréhension et notamment
d’en isoler certaines causes. A partir de là, nous avons, dans un premier temps, pu développer de
nouvelles mesures de risques permettant de mieux rendre compte des grandes fluctuations observées sur
les cours des actifs financiers. Cela nous a ensuite permis d’élaborer de nouvelles théories de portefeuilles
visant à se prémunir au mieux contre les risques extrêmes. Enfin, nous en avons déduit d’intéressants
résultats concernant le prix des actifs sur les marchés financiers à l’équilibre.
Notre étude s’est basée sur l’hypothèse que les marchés financiers peuvent être considérés comme des
systèmes complexes auto-organisés : de l’interaction entre agents économiques – aux objectifs divers et
souvent opposés – résultent les propriétés macroscopiques des marchés. Ceci nous a notamment conduit
à abandonner les hypothèses traditionnelles fondées sur l’existence d’agents économiques représentatifs
et rationnels, dont découle le concept de marché efficient et qui conduit à décrire la dynamique des
cours par un mouvement brownien, dont on sait pertinemment aujourd’hui, qu’il est inadapté à une telle
modélisation.
495

496 A. Evaluation de la conduite du projet de thèse
A.2 Eléments de contexte
A.2.1 Choix du sujet
Après avoir préparé le diplôme d’études approfondies en physique théorique de l’Ecole Normale Supérieure
de Lyon durant l’année scolaire 1999-2000, j’ai décidé de m’inscrire en thèse. A l’issue d’un tel DEA, il
est naturel de choisir un sujet de thèse portant sur la physique statistique ou la physique des particules.
Cependant, il était bien clair pour moi que de tels sujets ne conduisent que rarement à un poste dans la
recherche publique – leur nombre étant extrêmement faible – et ne suscitent presque jamais l’intérêt des
acteurs privés, qui jugent ces thèmes de recherche trop fondamentaux et sans applications concrètes. Ceci
risquait donc de poser quelques problèmes en vue de ma future insertion dans la vie professionnelle et
laissait présager dès le départ d’une nécessaire réorientation à l’issue d’une telle thèse. Aussi, je préférais
d’emblée me détourner de la voie habituelle.
En fait, le domaine de la finance m’attirait depuis longtemps, à tel point que j’avais un instant envisagé
de suivre un DEA de finance plutôt qu’un DEA de physique théorique. Mais, sachant que l’étude des
marchés financiers au moyen des outils de la physique statistique se développait activement en physique,
au sein d’une branche appelée “ éconophysique ”, je m’étais alors résolu à apprendre les bases
de la physique théorique dans le but de réinvestir ces connaissances dans l’étude et la modélisation des
mécanismes à l’œuvre sur les marchés financiers. C’est donc tout naturellement que m’est venu l’idée de
préparer une thèse portant sur la problématique des marchés financiers.
Durant les mois qui ont précédé le début de ma thèse j’ai rencontré et discuté avec de nombreuses
personnes, tant des enseignants et/ou chercheurs en physique et finance que des professionnels, afin
d’obtenir leur avis et conseil sur le choix de thèse que je souhaitais réaliser et sur la meilleure façon d’y
parvenir. J’avoue qu’à ce moment-là les avis que j’ai reçus étaient partagés et que de nombreuses mises
en garde sur les difficultés auxquelles je risquais d’être confronté m’ont été faites. Cependant, cela n’a
nullement entamé mon enthousiasme, et je dirais même qu’au contraire, cela m’a poussé à relever ce qui
m’apparaissait alors comme un véritable défi.
A.2.2 Choix de l’encadrement et du laboratoire d’accueil
Le thème central de ma thèse étant choisi, il me fallait encore trouver un directeur de thèse pouvant
m’encadrer sur un tel sujet ainsi qu’un laboratoire d’accueil. Le choix du directeur de thèse fut en fait
assez rapide, car si de plus en plus de chercheurs s’intéressent à ce domaine encore émergeant qu’est
l’éconophysique, très peu sont à même de diriger une thèse sur ce sujet. C’est donc fort logiquement que
je pris contact avec Didier Sornette, directeur de recherche au CNRS, affecté au Laboratoire de Physique
de la Matière Condensée (Université de Nice – Sophia Antipolis) où il dirige l’équipe de “ physique
pluridisciplinaire ” dont le but est l’étude des phénomènes critiques auto-organisés dans des domaines
aussi divers que la rupture de matériaux, la prédiction des tremblements de terre ou l’étude des marchés
financiers. Très vite nous décidâmes de nous rencontrer, ce qui me permis de découvrir plus en détails
les thèmes de recherche abordés au sein de cette équipe et de prévoir le cadre de ma thèse. Il fut aussi
décidé que j’effectuerais mon stage de DEA au sein de l’équipe, ce qui me permettrait d’une part de me
familiariser avec les thèmes et méthodes que j’aurais à employer au cours de ma thèse, et d’autre part de
vérifier que le travail qui m’attendait en thèse correspondait bien à mes aspirations.
Il apparut aussi assez rapidement qu’il serait très intéressant de s’adjoindre les services d’un professeur
de finance afin d’avoir de réels échanges avec ce milieu, de mieux cerner leurs problèmes et ainsi ne pas
s’égarer dans des recherches stériles car coupées des réalités du monde financier. De plus, cette colla-

A.2. Eléments de contexte 497
boration permettait de garantir le caractère pluridisciplinaire de la thèse dans laquelle je m’engageais.
C’est pourquoi je suis allé rencontrer Jean-Paul Laurent, professeur à l’Institut de Science Financière
et d’Assurances (Université Lyon I), afin de lui exposer mon projet et de lui demander de bien vouloir
co-diriger ma thèse, ce qu’il accepta tout de suite.
A.2.3 Financement du projet
Ce projet étant essentiellement théorique, et les retombées pratiques a priori difficilement évaluables, il
semblait délicat d’obtenir un financement autre que public. En fait, en tant qu’élève de l’Ecole Normale
Supérieure de Lyon, je pouvais postuler pour l’obtention d’une bourse spécifique, dite allocation couplée,
consistant en une allocation de recherche du MENRT et d’un monitorat (à l’université de Nice). Le
sujet de thèse peu orthodoxe que je présentais pouvait me desservir quant à l’obtention de cette bourse,
mais finalement, celui-ci fut jugé innovant et en plein accord avec la politique ministérielle visant à
promouvoir la pluridisciplinarité dans les thèmes de recherche, ce qui me permit d’obtenir l’attribution de
cette allocation couplée. Une telle bourse représente un montant annuel d’environ 24000 euros, charges
patronales comprises.
A ce montant, s’ajoute un pourcentage du coût salarial de mes directeur et co-directeur de thèse. Ce coût
est difficilement évaluable mais on peut estimer que chacun de mes encadrants a consacré entre dix et
vingt pour-cent de son temps à l’encadrement de ma thèse ce qui, globalement doit représenter prés de
15000 euros par an. On peut donc estimer le coût total en ressources humaines à approximativement
39000 euros par an.
Du point de vue investissement matériel, le coût se révèle beaucoup plus modeste. Mon travail de
thèse étant essentiellement théorique, il n’a nécessité que l’achat d’un ordinateur, quelques logiciels
et quelques livres. On peut donc chiffrer cette dépense à 3000 euros.
J’ai aussi eu à effectuer de nombreux déplacements soit pour assister à des conférences soit pour rencontrer
mon co-directeur à Lyon ou Paris. Sur toute la durée de ma thèse, les sommes engagées se chiffrent
à 4000 euros. J’ai également participé à deux écoles organisées par le département de la formation continue
du CNRS et pris en charge par lui. Le coût engagé par le CNRS à ces deux occasions avoisine les
2500 euros.
Enfin, il faut ajouter à cela la fraction qui m’est imputable des frais de fonctionnement du laboratoire qui
m’a accueilli pendant ces deux ans et demi. Ces frais sont difficilement évaluables avec précision, mais
il semble raisonnable de les estimer à 5000 euros.
En résumé, les frais engagés sont les suivants :
Ressources humaines (30 mois) 97500 euro
Frais de matériels 3000 euro
Frais de déplacement 4000 euro
Formation continue 2500 euro
Frais de fonctionnement 5000 euro
Total 112000 euro
Le total des frais engagés atteint donc 112000 euros sur trente mois, soit un coût annuel de 44800 euros.
Les frais de ressources humaines sont bien évidemment prépondérants, puisqu’ils représentent plus de
85% du coût total du projet, ce qui est en ligne avec ce qui est généralement observé pour tout sujet de
recherche théorique.

498 A. Evaluation de la conduite du projet de thèse
A.3 Evolution du projet
A.3.1 Elaboration du projet
Le projet initial visait, tout d’abord, à développer une méthode de mesure des grands risques financiers en
univers non gaussien et à les modéliser à l’aide des outils employés en physique statistique et théorie des
champs, qui semblaient particulièrement bien adaptés à cette problématique. Nous souhaitions ensuite
pouvoir appliquer ces méthodes au problème de la gestion de portefeuilles. Donc, dès le départ le projet
était conçu de manière pluridisciplinaire avec une part de physique théorique – ou du moins faisant appel
aux outils de la physique théorique – et une part de gestion financière.
Volontairement, le cadre du projet a été défini de manière assez générale afin de lui laisser le plus de
souplesse possible pour pouvoir évoluer au fur et à mesure de l’avancée des recherches. En effet, si
le cadre global des recherches à conduire apparaissait dès le départ assez clairement, il était difficile
d’évaluer très précisément les résultats auxquels nous devions nous attendre. En outre, de part la codirection
de cette thèse et son caractère pluridisciplinaire, il fallait que les termes du projet conviennent
à chacune des parties. Ce point aurait pu être sujet à quelques difficultés, mais tout s’est en fait très
bien déroulé et il n’y a eu aucun problème. Je dois dire que ceci tient en grande partie au caractère de
mes encadrants qui ont toujours fait preuve d’une grande ouverture d’esprit et m’ont laissé beaucoup de
liberté quant aux choix des directions de recherche à suivre.
A.3.2 Conduite du projet
Tout au long de ma thèse, j’ai bénéficié d’un encadrement toujours très présent et en même temps d’une
grande liberté. Cet état de fait est lié au choix de mes encadrants et je savais dès le départ à quoi m’attendre.
En effet, de par ses activités, mon directeur de thèse n’était en France que six mois par an. Ceci
m’a très rapidement poussé à devoir faire preuve d’autonomie. De plus, j’avais choisi un co-directeur occupant
un poste à l’Université Lyon I, ce qui m’a imposé une certaine mobilité et conduit à effectuer de
nombreux allers-retours à Lyon ou Paris. Ceci m’a, en contrepartie, permis de rencontrer des personnes
extérieures à mon laboratoire de rattachement, ce qui a conduit à développer des collaborations très utiles
et enrichissantes.
Ceci étant, malgré la distance qui nous séparait parfois, j’ai toujours eu quelqu’un ”à l’autre bout du fil”,
si ce n’est en face de moi pour répondre à mes questions et me guider lors de mes hésitations. Ceci a été
extrêmement important tout au long de ma thèse et plus particulièrement au cours des premiers mois où
je n’ai fait que suivre les directives de mes encadrants. Puis, passés les six premiers mois, je me suis mis
à choisir moi-même les directions de recherche que je souhaitais suivre, à en évaluer la faisabilité et à
les concrétiser. Bien sûr, tout cela se faisait en plein accord avec mes encadrants et après discussion et
concertation avec eux.
Ponctuellement, j’ai participé à des projets a priori un peu éloignés du thème central de ma thèse. Mais
au final, cela a toujours été stimulant et générateur d’idées nouvelles qui se sont parfaitement intégrées
au corps déjà établi de mes recherches, y apportant un complément ou un éclairage différent.
En résumé, s’il s’est toujours maintenu à l’intérieur du cadre originellement établi, le déroulement de ma
thèse était loin d’être planifié dès le départ et s’est plutôt produit sous l’effet des découvertes et avancées
successives.
De plus, l’avancement de la thèse s’est effectué de manière rapide puisqu’il n’a pas été nécessaire d’aller
au bout des trois ans que dure normalement une thèse. En fait, un an et demi après le début du projet, j’ai

A.4. Compétences acquises et enseignements personnels 499
émis l’idée de soutenir avant terme ce qui est apparu raisonnable à mes encadrants, qui en ont accepté le
principe. J’estimais alors avoir encore besoin de six mois pour terminer les projets en cours et quelques
mois pour rédiger le manuscrit de thèse. Sur ce point, la rédaction régulière d’articles au fur et à mesure
de l’avancée des travaux m’a grandement facilité la tâche. Au final les délais que je m’étais fixés ont
effectivement été tenus et j’ai soutenu au bout d’un peu plus de deux années de thèse.
A.3.3 Retombées scientifiques
Les retombées scientifiques ont été nombreuses et importantes. Les résultats obtenus nous ont notamment
permis d’isoler et de mieux comprendre certaines des origines des phénomènes extrêmes observés sur les
marchés financiers. Il nous a alors été possible d’établir, sur des bases théoriques solides, de nouvelles
mesures des risques extrêmes, qui ont en retour autorisé le développement concret de théories de portefeuille
prenant en compte ce type de risques, ce qui était l’objectif ultime de notre projet. Globalement,
tous les objectifs premiers ont été atteints et dans certains cas largement dépassés.
En outre, les retombées scientifiques du projet ont été rapides et régulières. En effet, très tôt les premiers
résultats sont apparus. Dès les quatre premiers mois, un article a été soumis. Puis, à partir de là, environ
un papier tous les deux ou trois mois était rédigé, si bien qu’après un peu plus de deux années de thèse,
six articles sont parus à la fois dans des journaux de physique et des revues de finance, et environ autant
sont soumis ou à paraître.
La régularité de publication tient avant tout à la diversification de mes thèmes de recherches. C’était le
principal intérêt d’avoir défini un cadre d’étude relativement large. Il est intéressant de remarquer que
sur les plus de dix papiers aujourd’hui produits, seul un quart était attendu, dans le sens où ils portent
de manière très directe sur des thèmes de recherche parfaitement identifiés dès le début de la thèse. Les
autres sont les fruits de la progression et des idées nouvelles qui sont apparues au fur et à mesure de
l’avancement des travaux.
Il est bien évident qu’une telle quantité de travail n’a pu être réalisée seul, et le nombre des publications
tient aussi aux collaborations – avec des partenaires publics mais aussi privés – qui se sont tissées tout
au long de ma thèse et qui se sont toutes révélées très fructueuses.
A.4 Compétences acquises et enseignements personnels
La thèse constitue pour moi une expérience riche d’enseignement. Sur le plan scientifique, elle m’a tout
d’abord permis d’appliquer les connaissances que j’avais acquises en physique théorique lors de ma
formation initiale (physique statistique, systèmes complexes et phénomènes critiques notamment). Ceci
a été pour moi le moyen de définitivement m’approprier ce savoir. J’ai aussi dû ponctuellement mettre à
jour et compléter ces connaissances lorsque mes recherches me conduisaient à aborder des thèmes que je
n’avais pas eu à considérer lors de mes études antérieures. Ceci fait qu’aujourd’hui je pense pouvoir dire
que je possède un niveau de connaissances élevées sur les méthodes employées en physique pour l’étude
des phénomènes critiques.
Par ailleurs, et c’est là où j’ai dû fournir l’effort le plus important, il m’a fallu acquérir la somme de
connaissances en théorie financière qui me faisait défaut (théorie de la décision et analyse du risque,
théories de portefeuille et modèles d’équilibre de marchés, pour ne citer que quelques exemples), ainsi
qu’en statistique et théorie des probabilités. Cet investissement a été long (et je ne le considère pas comme
totalement achevé), mais j’estime avoir acquis un bon niveau de compétences dans ces domaines. Pour
cela, j’ai – entre autre – participé à des écoles organisées par le CNRS. Celles-ci se sont révélées très

500 A. Evaluation de la conduite du projet de thèse
formatrices et m’ont permis tout d’abord de combler mes lacunes puis d’acquérir des connaissances de
haut niveau au contact des personnes les plus expertes dans ces domaines. En outre, les connaissances
que j’ai acquises en mathématique m’ont permis de réaliser la synthèse et d’approfondir les savoirs que
j’avais accumulés en physique au cours de mes études antérieures.
Sur le plan purement technique, j’ai développé notamment de bonnes compétences informatiques. J’ai
appris la programmation en langage C, afin de pouvoir réaliser des simulations numériques évoluées. J’ai
également appris à utiliser des logiciels tels que MatLab pour le traitement de données et Maple pour
l’aide au calcul formel.
En outre, j’ai dû apprendre à gérer mon temps de travail ainsi que le travail en groupe. Ceci me semble
être la clé du succès de tout projet (comme la thèse par exemple, mais cela va bien au-delà). En effet,
j’ai continuellement travaillé sur plusieurs sujets en même temps, généralement avec la même personne
mais plusieurs fois au sein de collaboration regroupant des personnes géographiquement dispersées sur
l’ensemble du globe. Ceci conduit à s’astreindre à une certaine rigueur dans l’organisation afin de ne pas
se laisser déborder et de maintenir une progression régulière des différents projets. En fait, un travail en
collaboration n’avançant qu’au rythme du plus lent de ces membres, le retard peut parfois s’accumuler de
manière importante. C’est pourquoi il m’a aussi fallu apprendre à limiter mes ambitions et ainsi refuser
de participer à certains projets afin de ne pas trop me disperser. Par ailleurs, mes travaux se trouvant à
l’interface de deux mondes – celui de la physique et celui de la finance – j’ai eu sans cesse à interagir
avec des interlocuteurs parlant des langages différents et ayant des sensibilités bien distinctes. Il a donc
été nécessaire que je m’adapte à cela et notamment que j’apprenne à communiquer avec des personnes
ayant à la base des formations et cultures très différentes de la mienne, ce qui s’est fait petit à petit mais
sans grandes difficultés.
Enfin, bénéficiant d’un monitorat à l’Université de Nice, j’ai eu à assurer une charge d’enseignement de
64 heures par an. Cette charge s’est trouvée repartie entre différents niveaux, allant du premier cycle à la
préparation à l’agrégation, et s’est déroulée devant des publics divers : étudiants en physique, géologie,
ou encore ponctuellement économie. Ces quelques heures d’enseignement ont été réellement profitables.
Elles ont bien sûr été l’occasion de mettre en pratique les conseils que j’avais reçus lors de ma formation
à l’enseignement au cours de mon année de préparation à l’agrégation, mais elles ont en fait surtout été
le moyen d’apprendre à transmettre des connaissances. En effet, l’effort de communication en direction
d’un public étudiant est bien plus important et totalement différent de celui auquel doit s’habituer le
chercheur qui expose ses travaux lors d’une conférence devant un public de spécialistes. En outre, les
questions – parfois naïves – des étudiants imposent une remise en cause, ou du moins une réorganisation,
permanente de ses propres connaissances. Enfin, cela m’a là encore conduit à travailler en coordination,
si ce n’est en collaboration, avec mes collègues enseignants au sein d’équipes pédagogiques, forgeant un
peu plus mon expérience du travail en groupe.
A.5 Conclusion
J’estime que ma thèse s’est extrêmement bien déroulée et j’en garderai le souvenir d’une expérience très
positive. J’attribue cela essentiellement à la très bonne entente qui a toujours régné entre mes encadrants
et moi-même. Je ne conçois pas avoir pu atteindre les mêmes objectifs si des conditions de travail aussi
agréables n’avaient pas été réunies. Cela souligne l’importance des relations humaines dans la bonne
marche de tout projet. Ceci me parait d’autant plus vrai que les groupes de travail sont petits et qu’en
conséquence les membres sont plus dépendants les uns des autres.
Cette expérience très motivante m’a conforté dans l’intention de poursuivre ma carrière dans le secteur

A.5. Conclusion 501
de la recherche – académique ou privée. En outre, l’enseignement – ou la formation au sens large –
me semble désormais indissociable de cette activité. Je pense que mes capacités d’enseignant se sont
nourries de mon expérience de chercheur et que réciproquement, les échanges lors de conférences ou
séminaires ont fourni généralement matière à de nouvelles recherches. Je dois à ces années de thèse
d’avoir découvert la formidable complémentarité entre d’une part une activité quelque peu solitaire qu’est
la recherche et d’autre part une activité tournée vers l’extérieur qu’est l’enseignement ou la communication.
Le tableau que j’ai dressé de mes années de thèse peut paraître idyllique, il n’en est pas moins sincère.
J’ai sûrement bénéficié d’une part de chance – inhérente à toute entreprise – mais cela ne saurait tout
expliquer. Je pense que l’organisation méthodique de mon projet, ce qui commence bien avant le début
officiel de la thèse et passe par le choix avisé du sujet et des encadrants, est la clé de la réussite de cette
entreprise. Pour ma part, je m’étais astreint à cette démarche plus de six mois avant le début de ma thèse,
ce qui, me semble-t-il, a porté ses fruits bien au-delà de mes espérances initiales.

502 A. Evaluation de la conduite du projet de thèse

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Risques extrêmes en finance : Statistique, théorie et gestion de portefeuille
Résumé : Cette thèse propose une étude des risques extrêmes sur les marchés financiers, considérés
comme un exemple typique de système complexe auto-organisé. Nous commençons par décrire et modéliser
les propriétés statistiques individuelles des actifs financiers afin d’en déduire une estimation précise
des grands risques et d’en comprendre les mécanismes sous-jacents en relation avec la micro-structure
des marchés et le comportement des agents économiques. A l’aide des copules et modèles à facteurs,
nous analysons ensuite les propriétés de dépendance extrêmes des actifs financiers afin de mieux cerner
les possibilités et les limites de la diversification des grands risques. Enfin, nous étudions les mesures
de risques les plus à mêmes de quantifier des risques extrêmes et appliquons l’ensemble de ces résultats
à l’obtention de portefeuilles les moins sensibles à ce type de risques. Parallèlement, nous exposons
certaines conséquences de ce type d’allocation sur les équilibres de marché.
Mots-clés : système complexe auto-organisé, risques extrêmes, marchés financiers, gestion de portefeuille,
éconophysique, bulles spéculatives, distributions à queues épaisses, copules.
Extreme risks in finance : Statistics, theory and portfolio management
Summary : This thesis proposes a study of extreme risks observed on financial markets, considered as
a typical example of self-organized complex systems. We start by describing and modeling individual
statistical properties of financial assets in order to obtain the most accurate estimation of large risks
and to provide a better understanding of the underlying mechanisms in relation with the markets microstructure
and the economic agents’ behaviors. Using copulas as well as factor models, we then analyze
the dependence properties between assets, and more specifically their extreme counterparts, in order to
understand the opportunities for diversification of large risks, but also their limits. Finally, we study the
risk measures that are the most appropriate for the assessment of extreme risks and apply all these results
to define the portofolios that are the least sensitive to extreme risks. In parallel, we develop consequences
of such assets allocations with respect to market equilibria.
Key-words : self-organized complex systems, extreme risks, financial markets, portfolio management,
econophysics, speculative bubbles, heavy tail distributions, copulas.