The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 731 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 731

Knowing that defaults are normally distributed makes it easy to

simulate default times for independent assets. We need to generate uni-

form random numbers in the range [0,1] and then given a term structure

for the hazard rate, imply out the corresponding default time. For exam-

ple, if we denote the uniform random draw by u, the corresponding

default time T* is given by solving

u = exp(–λT*)

to give

log ()u

T* = – ----------------

λ

This is an efficient method for simulating default. Every random draw

produces a corresponding default time. In terms of its usefulness, the

only question is whether the default time is before or after the maturity

of the contract being priced.

There are many ways to introduce a default correlation between the

different reference entities in a credit default basket. One way is to cor-

relate the default times. This correlation is defined as

〈TATB〉– 〈TA〉〈 〉 TB

ρ(TA,TB)= ------------------------------------------------------------------------------

〈^2 T^2

TA〉– 〈TA〉 B TB

(^2) 〈 〉 – 〈 〉 2
It is important to stress that this is not the same as the default corre-
lation. Although correlating default times has the effect of correlating
default, there are two reasons they are not equivalent. First, there is no
need to define a default horizon when correlating default times. To mea-
sure this correlation, we would observe a sample of assets over a long
(infinite) period and compute the times at which each asset defaults.
There is no notion of a time horizon for this correlation.
Second, since the default time correlation equals 100% when Tj = Ti
and when Tj = Ti + θ, it is possible to have 100% default time correla-
tion with assets defaulting at fixed intervals.
Under a Poisson assumption,
1 1
〈TA〉= ------ and 〈 〉 TB = ------
λA λB