The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 732 Wednesday, February 4, 2004 1:12 PM

732 The Mathematics of Financial Modeling and Investment Management

and

T^2 21

A 〈 〉

(^2) = ------ and 〈 〉– T
〈 〉– TA B
1
TB 〈 〉^2 = ------
λA λB
so we have
ρ(TA,TB)= 〈TATB〉 λAλB – 1
Copula Function
To generate correlated default times, we use the normal Copula function
methodology as proposed by Li.^49 A Copula function (see Chapter 6) is
simply a specification of how the univariate marginal distributions com-
bine to form a multivariate distribution. For example, if we have N cor-
related uniform random variables U 1 , U 2 , ..., UN then
Cu( 1 , u 2 , ,... uN)= Pr{U 1 <u 1 ,U 2 <u 2 , ,...UN <uN}
is the joint distribution function that gives the probability that all of the
uniforms are in the specified range.
In a similar manner we can define the Copula function for the default
times of N assets:
CF( 1 (T 1 ),F 2 (T 2 ), ,...FN(TN))
= Pr{U 1 <F 1 (T 1 ),U 2 <F 2 (T 2 ), ,... UN <FN(TN)}
where Fi(Ti) = Pr{ti < t}.
There are several possible choices but here we define the Copula
function Θto be the multivariate normal distribution function with cor-
relation matrix ρ. We also define Φ–1 as the inverse of a univariate nor-
mal function. The Copula function is therefore given by

• 1
  C u = u 1


• 1
  () Θ(Φ u 2 u 3 u 4 ...


• 1
  (), Φ(), Φ


• 1
  (), Φ(), Φ


• 1
  (uN),ρρρρ)

where ρρρρis the correlation matrix.

What this specification says is that in order to generate correlated

default times, we must first generate N correlated multivariate gaussians

denoted by u 1 , u 2 , u 3 , ..., uN—one for each asset in the basket. These

(^49) David X. Li, Credit Metrics Monitor, Risk Metrics Group (April 1999).