Copula and its application

Sklar’s theorem:

Let F_{xy} be a joint distribution with margins F_x and F_y. Then there exists a function C:[0,1]^2->[0,1] as:

F_{xy}(x,y)= C(F_x(x), F_y(y))

If X and Y are continuous, then C is unique; otherwise, C is uniquely determined on the range of X and Y. Conversely if C is a copula and Fx and Fy are distribution functions, then the function Fxy defined above is a joint distribution with margins Fx and Fy.

If C is a Copula, the following properties are followed:

(i) C(0,u) = C(v,0) = 0

(ii) C(1,u) = C(u,1) = u

(iii) C(u2,v2) - C(u1,v2) - C(u2,v2) + C(u1,v1) ≥ 0 for all v1 ≤ v2, u1 ≤ u2

The most commonly used copulae are the Gumbel copula for extreme

distributions, the Gaussian copula for linear correlation, and the Archimedean copula

and the t-copula for dependence in the tail.

In order to price the CDS (ie to find the value of s which solves the

above equations) we need to derive or observe the default probability function Pu. But

If we think of the prices of individual bonds and CDS’s as reflecting the marginal

risks of default, and the price of a CDO tranche as reflecting the joint risk of defaults.

Given that the marginal prices (and hence probabilities of default) are observable, we

could assume a copula and then either: by observing the price of the CDO, infer the

relevant correlation structure, or, by estimating the correlation structure exogenously,

calculate the fair price of the CDO.

The Gaussian copula is used to generate Monte Carlo simulations of the defaults of

the underlying instruments, which are then used to price the CDO