The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 725 Wednesday, February 4, 2004 1:12 PM

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Credit Risk Modeling and Credit Default Swaps 725

EXHIBIT 22.7 Venn Diagram Representation of Correlated Default for Two Assets

sponds to all scenarios in which both assets default before time T. Its

area is the probability of joint default, pAB.

The probability of either asset defaulting is

Ω= pA + pB – pAB

In the zero correlation limit, when the assets are independent, the prob-

ability of both assets defaulting is given by pAB = pA pB. Substituting

this into the above formula for the default correlation shows when the

assets are independent, ρD(T) = 0 as expected (see Exhibit 22.8).

In the limit of high default correlation, the default of the stronger

asset always results in the default of the weaker asset. In the limit the

joint default probability is given by pAB = min[pA,pB]. This is shown in

Exhibit 22.9 in the case where pA > pB. In this case we have a maximum

default correlation of

ρ

pB ( 1 – pA )

pA ( 1 – pB )

= -

Once again, the price of a first-to-default basket is the area enclosed by

the circles. In this case one circle encloses the other and the first-to-

default basket price becomes the larger of the two probabilities:

Ωρρ = = pA + pB – pAB = max[pA,pB]