The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 733 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 733

are then converted into uniform random variables by cumulative proba-

bility functions.

Once we have the vector of correlated random uniforms u we can

calculate the corresponding default times knowing that asset i defaults

in trial n at time T given by

lnuin

Tin = – -------------

λi

Comparing Default Correlation and Default Time Correlation

In addition to correlating default times, we could correlate default

events. There is no simple way to do this directly. It is better to correlate

the assets using some other mechanism and then measure the default

correlation a posteriori. The question is: If we implement a model which

correlates default times, how does the correlation relate to default cor-

relation as defined above.

In common with the case of default correlation, it is only possible

to have a 100% pairwise correlation in default times between two

assets if both assets have the same default probabilities. Otherwise, the

distributions are centered around different average default times and

having equal default times and different average default times is not

compatible.

If we assume that in both cases all assets have the same default

probability, what is the difference between correlating default times and

correlating default events? In the limit of zero correlation there is no dif-

ference as the assets default independently. In the limit of 100% correla-

tion there is a fundamental difference: If default times have a 100%

correlation, then assets must default either simultaneously or with a

fixed time difference.^50 However, if there is 100% default correlation,

then this means that the default of one asset within a certain horizon

always coincides with the default of the other within the same horizon.

In general, we would expect a 100% default correlation to imply that

both assets default together, but this is not a strict requirement. In prac-

tice, the default of one asset may occur at any time and be followed by

default of the other asset at the end of the horizon. Default correlation

is 100%, but default times have a lower correlation.

Consider also the effect of the default horizon. Given that default

times are exponentially distributed, extending the default horizon

(^50) Since the default time correlation of 100% is preserved under translations of the
form Tj = Ti + θ.