Copulas: a Review and Recent Developments (2007)

Sklar's Theorem (for conditional pseudo copula, Fermanian and Wegkamp(2004)). Let H be a joint distribution function on (¡1; 1) n . Assume that for everyx =(x 1 ;::: ;x n ), y =(y 1 ;::: ;y n ) 2 (¡1; 1) n ,F Xi (x i )=F Xi (y i ) for all 1 · j · n implies H(x) =H(y):Then there exist a conditional pseudo-copula C pseudo such thatH(x) =C pseudo (F X1 (x 1 );::: ;F Xn (x n ));for every x =(x 1 ;::: ;x n ) 2 (¡1; 1) n . The function C pseudo is uniquely de¯nedon RanF X1 £ :::£ RanF Xn , the product of the values taken by the F Xi . Conversely,if C pseudo is a conditional pseudo-copula and if F X1 ;::: ;F Xn are some univariatedistribution functions, then the function H is an n-dimensional distribution function.¥Note that the conditional pseudo-copula C pseudo in the last theorem is a copulaif and only if H(+1;::: ;x i ;::: ;1) = F Xi (x i ) for every i = 1;::: ;n and x =(x 1 ;::: ;x n ) 2 (¡1; 1) n .Additional statements and estimation of conditional pseudo-copulas as well asapplications to goodness of ¯t test are discussed by Fermanian and Wegkamp (2004).Fermanian and Scaillet (2005) consider statistical pitfalls arising when using copulas,in particular they discuss issues in copula estimation and the design of time-dependentcopulas. They provide a simulation study where it is shown the potential impact ofmisspeci¯ed margins on the estimation of the copula parameter.2.5 An application: using copula to bound quantile measuresConsider a decision maker faced with a number of risks, i.e. random future losses. Arisk measure à is de¯ned as a mapping from the set of random variables representingthe risks at hand to the real line, i.e., à :(¡1; 1) n ! (¡1; 1). In this section wewill always consider random variables as losses, i.e. random payments that have tobe made.As a ¯rst example, consider the p-quantile risk measure, often called VaR (Valueat-Risk)at level p 2 (0; 1), de¯ned byVaR p (X) =inffx 2 (¡1; 1) :F X (x) ¸ pg = F ¡1X(p):For a given risks (X 1 ;::: ;X n )andariskmeasureà :(¡1; 1) n ! (¡1; 1) oneisofteninterestedincomputing certain quantities of Ã(X 1 ;::: ;X n ) like some momentsor a quantile. In the actuarial and ¯nance literature can be found variety of formsof such a risk measure à corresponding to exotic options, basket derivatives, creditderivatives, operational risk, insurance covers, see Embrechts et al. (2003a), Georgeset al. (2001), Patton (2006). Typical examples of à include:16