Chapter 2: FAQs 213
The copula approach in effect allows us to readily go
from a single-default world to a multiple-default world
almost seamlessly. And by choosing the nature of the
dependence, the copula function, we can explore models
with rich ‘correlation’ structure. For example, having a
higher degree of dependence during big market moves
is quite straightforward.
TakeNuniformly distributed random variablesU 1 ,U 2 ,
...,UN, each defined on [0, 1]. The copula function is
defined as
C(u 1 ,u 2 ,...,uN)=Prob(U 1 ≤u 1 ,U 2 ≤u 2 ,...,UN≤uN).Clearly we have
C(u 1 ,u 2 ,...,0,...,uN)=0,and
C(1, 1,...,ui,...,1)=ui.That is the copula function. The way it links many
univariate distributions with a single multivariate dis-
tribution is as follows.
Letx 1 ,x 2 ,...,xNbe random variables with cumulative
distribution functions (so-calledmarginaldistributions)
ofF 1 (x 1 ),F 2 (x 2 ),...,FN(xN). Combine theFswiththe
copula function,
C(F 1 (x 1 ),F 2 (x 2 ),...,FN(xN))=F(x 1 ,x 2 ,...,xN)and it’s easy to show that this functionF(x 1 ,x 2 ,...,xN)
isthesameas
Prob(X 1 ≤x 1 ,X 2 ≤x 2 ,...,XN≤xN).In pricing basket credit derivatives we would use the
copula approach by simulating default times of each of
the constituent names in the basket. And then perform
many such simulations in order to be able to analyze