Copulas: a Review and Recent Developments (2007)
Copulas: a Review and Recent Developments (2007) Copulas: a Review and Recent Developments (2007)
Z¾ C;à (s) =dC(F X1 (x 1 );::: ;F Xn (x n ));fÃ(X 1 ;::: ;X n )·sg½ C;à (s) =inf¡ ¡x 1 ;::: ;x n¡1 2(¡1;1) Cdual F X1 (x 1 );::: ;F Xn¡1 (x n¡1 );F Xn á1n¡1 (s)¢¢ ;where à n¡1(s) ¡1 is the left continuous (generalized) inverse function of à n¡1 (s).Theorem (best possible bounds, Embrechts et al. (2003a)). Under the abovenotations if a copula C associated to (X 1 ;:::;X n ) satis¯es C ¸ C 0 and C dual · C1dualfor some given n-copulas C 0 and C 1 , then¿ C0 ;Ã(s) · ¾ C;à (s) · ½ C1 ;Ã(s):Note that, ¾ C;à (s) =P (Ã(X 1 ;:::;X n ) · s) =¤ à (s). Also, the bounds ¿ C0 ;Ã(s)and ½ C1 ;Ã(s) are themselves distribution functions. Since there is not necessarily aunique copula satisfying the inequalities in the last theorem, it is assumed additionallythat there exists a copula C such that C ¸ C 0 and C dual · C1 dual . The copulas C 0and C 1 represent the partial information available about the dependence structure of(X 1 ;:::;X n ). In Embrechts et al. (2003b) is proposed a methodology in order to ¯ndthe bounds for VaR p (¤ à (s)), recently re¯ned by Embrechts and Puccetti (2006).In most cases, the bounds in last theorem does not have closed form. In general,one has to use numerical approximations as proposed by Williamson and Downs(1990). Their algorithm is based on the discretization of ¿ C0 ;Ã(s), ½ C1 ;Ã(s) andonaduality principle due to Frank et al. (1987).Some problems related to the control and optimization of VaRin a portfolio withnon-normal or non-log-normal returns have taken the researchers to look for anotherquantile measures. Such a measure is called Tail Value-at-Risk at level p, andisde¯ned byZ 1TVaR p (X) = 1 VaR p (X)dq; p 2 (0; 1):1 ¡ p pIt is the arithmetic average of the VaR p (X), from p on. Therefore, TVaRis a measureof risk more suitable than VaR because it takes into account the losses beyond theVaR level. Note that always TVaR p (X) > VaR p (X), TVaR 0 (X) = E(X) andTVaR p (X) is a non-decreasing function of p. The fundamental di®erences betweenVaRand TVaR canbesummarizedasfollows: VaRis the \optimistic" lower boundof the tail loses and the TVaR is the expected value of tail losses. Therefore, TVaRcan be interpreted as a \conservative" risk measure.Rockfellar and Uryasev (2000, 2002) use CV aR instead of TVaR and provide analternative de¯nition of CV aR as a coherent measure (i.e. satisfy the sub-additivity,monotonicity, homogeneity and translation invariance axioms, see Artzner et al.18¥
(1999)) even in the cases where the associated distributions are discontinuous. Besides,the TVaR is characterized as a solution of a optimizing problem. The basiccontribution in Rockfellar and Uryasev (2000, 2002) is the following. They present apractical method of optimization to evaluate TVaR and VaR simultaneously whichis convenient in evaluation derivatives, (options, futures), market, operational creditand risk, ¯nancial risk, etc. Note that TVaR uses only the upper tail distributioninformation. Nevertheless, TVaR is not adequate to extreme events, i.e. in the caseof \low frequencies and high losses". Interesting applications of the above theorem incase of TVaR and for a class of distortion risk measures suggested by Wang (2000)and discussed by Darkiewicz et al. (2005) and Goncalves et al. (2005).3 Several new methods of copula modellingIn this section we discuss three topics related to our current research and show someof results obtained.3.1 Order statistics copulaThere are very few results in literature relating the order statistics and associatedcopulas. The random variables max(X; Y )andmin(X; Y ) are the order statistics forX and Y . Then, P (max(X; Y ) · t) =C(F (t);G(t)), e.g. Nelsen (1999), p. 25, andP (min(X; Y ) · t) =F (t)+G(t) ¡ C(F (t);G(t)):The above relations are generalized by Georges et al. (2001) as follows: Let(X 1 ;::: ;X n )beasetofcontinuousrandomvariableswithF Xi (x) =P (X i · x); i=1; 2;::: ;n: Denote by C n the associated copula and let X r:n be r-th order statistic(1 · r · n). Then its distribution function F r:n (t) =P (X r:n · t) isgivenbynXF r:n (t) = (¡1) k¡l k()l X #C n (v 1 ;::: ;v n ) ;" kXk=r l=rwhere P denotes summation over the set((v 1 ;::: ;v n ) 2 [0; 1] n j v i 2fF Xi (t); 1g;)nX± f1g (v i )=n ¡ kwith ± f1g (v i )=1ifv i =1,and0otherwise. Forr = 1 the last formula givesi=1F 1:n (t) =1¡ C n (S X1 (t);::: ;S Xn (t));where C n is the survival copula and S Xi (t) =1¡ F Xi (t); i=1;::: ;n.Wealsonotethat X n:n = max(X 1 ;::: ;X n ) and its distribution function is the diagonal section ofthe multivariate distribution F n:n (t) =C n (F X1 (t);:::;F Xn (t)).19
- 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: (i) Ã 1 (x 1 ;::: ;x n )=x 1 + :::
- 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 30 and 31: Fisher-Tippett Theorem (version for
- 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
(1999)) even in the cases where the associated distributions are discontinuous. Besides,the TVaR is characterized as a solution of a optimizing problem. The basiccontribution in Rockfellar <strong>and</strong> Uryasev (2000, 2002) is the following. They present apractical method of optimization to evaluate TVaR <strong>and</strong> VaR simultaneously whichis convenient in evaluation derivatives, (options, futures), market, operational credit<strong>and</strong> risk, ¯nancial risk, etc. Note that TVaR uses only the upper tail distributioninformation. Nevertheless, TVaR is not adequate to extreme events, i.e. in the caseof \low frequencies <strong>and</strong> high losses". Interesting applications of the above theorem incase of TVaR <strong>and</strong> for a class of distortion risk measures suggested by Wang (2000)<strong>and</strong> discussed by Darkiewicz et al. (2005) <strong>and</strong> Goncalves et al. (2005).3 Several new methods of copula modellingIn this section we discuss three topics related to our current research <strong>and</strong> show someof results obtained.3.1 Order statistics copulaThere are very few results in literature relating the order statistics <strong>and</strong> associatedcopulas. The r<strong>and</strong>om variables max(X; Y )<strong>and</strong>min(X; Y ) are the order statistics forX <strong>and</strong> Y . Then, P (max(X; Y ) · t) =C(F (t);G(t)), e.g. Nelsen (1999), p. 25, <strong>and</strong>P (min(X; Y ) · t) =F (t)+G(t) ¡ C(F (t);G(t)):The above relations are generalized by Georges et al. (2001) as follows: Let(X 1 ;::: ;X n )beasetofcontinuousr<strong>and</strong>omvariableswithF Xi (x) =P (X i · x); i=1; 2;::: ;n: Denote by C n the associated copula <strong>and</strong> let X r:n be r-th order statistic(1 · r · n). Then its distribution function F r:n (t) =P (X r:n · t) isgivenbynXF r:n (t) = (¡1) k¡l k()l X #C n (v 1 ;::: ;v n ) ;" kXk=r l=rwhere P denotes summation over the set((v 1 ;::: ;v n ) 2 [0; 1] n j v i 2fF Xi (t); 1g;)nX± f1g (v i )=n ¡ kwith ± f1g (v i )=1ifv i =1,<strong>and</strong>0otherwise. Forr = 1 the last formula givesi=1F 1:n (t) =1¡ C n (S X1 (t);::: ;S Xn (t));where C n is the survival copula <strong>and</strong> S Xi (t) =1¡ F Xi (t); i=1;::: ;n.Wealsonotethat X n:n = max(X 1 ;::: ;X n ) <strong>and</strong> its distribution function is the diagonal section ofthe multivariate distribution F n:n (t) =C n (F X1 (t);:::;F Xn (t)).19
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