Copulas: a Review and Recent Developments (2007)

More documents

Recommendations

Info

Fisher-Tippett Theorem (version for copulas). Let H 2 MDA(W ) and W (x 1 ;x 2 )=C ¤ (W 1 (x 1 );W 2 (x 2 )). Then(i) The marginals W i are GEV and F Xi 2 MDA(W i ), i =1; 2;(ii) C ¤ (u t 1;u t 2)=C t ¤(u 1 ;u 2 ) for all t>0. ¥It is interesting to note that the limit copula C ¤ is determined only by copulaC associated to H. Thus H 2 MDA(W ) is equivalent to (C; F X1 ;F X2 ) 2MDA(C ¤ ;W 1 ;W 2 ). Then we may writeTheorem. (C; F 1 ;F 2 ) 2 MDA(C ¤ ;W 1 ;W 2 ) if and only if(i) F i 2 MDA(W i ) ;i=1; 2;(ii) C 2 MDA(C ¤ ). ¥Since the copula of the limiting distribution is unique, we need the followingde¯nition.De¯nition (copula domain of attraction). Let C ¤ be an extreme value copula.We say that C is in the domain of attraction of C ¤ , and denote C 2 CDA(C ¤ ), if andonly if C(F X1 ;F X2 ) 2 MDA(C ¤ (W 1 ;W 2 )), for F Xi continuous and F Xi 2 MDA(W i ),i =1; 2.It follows from the max-stability property, see Embrechts et al. (1997), thatC ¤ 2 CDA(C ¤ ), for all extreme value copula C ¤ .Copula modelling of extremes may also be addressed by modelling joint excessesover high thresholds. In this case there are many suitable parametric families available,see Joe (1997). Frees and Valdez (1998) worked out the expression of the copulapertaining to the bivariate Pareto distribution (Clayton copula). Juri and WÄuthrich(2002) characterize the limiting dependence structure in the upper-tails of two randomvariables assuming their dependence structure is Archimedean. Charpentier (2003)studied conditional copulas, i.e. copulas conditional to extreme events and derivedtheir properties. He studied in detail the case of Archimedean copulas and providedapplications in credit risk. Charpentier (2004) focused on the dependence structureof extreme events and compared asymptotic results by considering componentwisemaxima and joint excesses over high thresholds. He took the distributional approachand obtained copula-convergence theorems.Estimation of copulas for extreme values may follow the approaches already mentioned.Results in the literature include the investigation of the behavior of the maximumlikelihood estimators of copula parameters through simulations by Caperaµa etal. (1997) in the case of the symmetric and asymmetric logistic model, and by Genest(1987) in the case of the Frank family. Genest (1987) found that the method ofmoments estimator appears to have smaller mean squared error than the maximumlikelihood estimator at small samples. Hsing et. al. (2004) derive a nonparametricestimation procedure for estimating the limiting copula of componentwise maxima.For statistical tests on copula speci¯cation see Ane and Kharoubi (2003), Genest and30
Rivest (1993), Fermanian and Scaillet (2005).Important applications are found in the ¯elds of Insurance, Environment, Finance.Since marginal ¯ts present no di±culties using extreme value models, one can focuson modelling dependency through a variety of copula families. In Finance, the numberof publications applying copula based techniques is large. The main motivationcomes from the observed greater integration among world markets, which results instronger co-movements, increasing risk and reducing the opportunities for internationaldiversi¯cation. Li (2000) provides a Gaussian copula-based methodology toprice ¯rst-to-default credit derivatives. The issues of measuring ¯nancial risk andassessing the e®ect of copula choice on risk measures were addressed by a number ofauthors, including Embretchs et al. (2003), Ane and Kharoubi (2003), Mendes andSouza (2004), among others. The modelling of daily ¯nancial returns also motivatedMendesandArslan(2005)toobtaintheexpression for the GT-copula. This is thecopula associated to the multivariate GT-distributions which includes as a specialcase the copula pertaining to the multivariate t-distribution. For a survey of ¯nancialapplications see, for example, Bouye et al. (2000) and Embretchs et al. (2003b).E®ective risk management techniques must include extreme value techniques. Inparticular, one non-linear measure of dependence at extreme levels, the tail dependencecoe±cient has become very popular.De¯nition (upper and lower tail dependence coe±cients). Let X 1 and X 2 berandom variables with distribution functions F X1 and F X2 . The coe±cient of uppertail dependence is de¯ned by¸U = lim®!0 + ¸U(®) = lim®!0 + PrfX 1 >F ¡1X 1(1 ¡ ®)jX 2 >F ¡1X 2(1 ¡ ®)g ;provided a limit ¸U 2 [0; 1] exists. If ¸U 2 (0; 1], then X 1 and X 2 are said tobe asymptotically dependent in the upper tail. If ¸U =0,theyareasymptoticallyindependent. Similarly, the lower tail dependence coe±cient is given by¸L = lim®!1 ¡ ¸U(®) = lim®!1 ¡ PrfX 1
Page 3 and 4: can be employed in probability theo
Page 5 and 6: y Genest et al. (1993) and Shi and
Page 7 and 8: Table 1: Distribution of (C 1 jC 2
Page 9 and 10: It should be mentioned here that gr
Page 11 and 12: Nelsen et al. (2003) have used Bert
Page 13 and 14: 2.4 Time dependent copulasIn practi
Page 15 and 16: De¯nition (conditional pseudo-copu
Page 17 and 18: (i) Ã 1 (x 1 ;::: ;x n )=x 1 + :::
Page 19 and 20: (1999)) even in the cases where the
Page 21 and 22: Let R u = P nj=1 IfU j · ug and R
Page 23: usually wish to aggregate two-dimen
Page 26 and 27: 3.3 Copula representation via a loc
Page 28 and 29: such thatC(u; v) =© ru;v¡© ¡1 (
Page 32 and 33: et al. (2000). To measure contagion
Page 34 and 35: and the non-exchangeable copula (AL
Page 36 and 37: 0 2 4 6 8 10 120 1 2 3 4 5 60 2 4 6
Page 38 and 39: Bouye, E., Durrleman, V. Nikeghbali
Page 40 and 41: Embrechts, P., Lindskog, F., McNeil
Page 42 and 43: Hsing, T, KlÄuppelberg, C., Kuhn,
Page 44 and 45: Nelsen, R., Quesada-Molina, J., Rod

Copulas: a Review and Recent Developments (2007)

Create successful ePaper yourself

Delete template?

Save as template?