The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 728 Wednesday, February 4, 2004 1:12 PM

728 The Mathematics of Financial Modeling and Investment Management

As more assets are considered, more default combinations become

possible. With just three assets we have the following eight possibilities:

■ No assets default

■ Only asset A defaults

■ Only asset B defaults

■ Only asset C defaults

■ Asset A and asset B default

■ Asset B and asset C default

■ Asset A and asset C default

■ Asset A and asset B and asset C default

To price this basket we either need all of the joint probabilities or

the pairwise correlations ρAB, ρBC, and ρAC (see Exhibit 22.11). The

probability that the basket is triggered is given by

Ω = pA + pB + pC –pAB –pBC –pAC + pABC

Joint Poisson Process

Recent evidence (for example, Enron, WorldCom, and Quest) demon-

strated that severe economic hardship and publicity can cause chain

defaults for even very large firms. Hence, incorporating default correla-

tion is an important task in valuing credit derivatives.

As stated above, the period-end joint default probability by two ref-

erence entities is as follows:

Pr(A ∩ B) = E[ (^1) A ∩ B] = pAB
EXHIBIT 22.11 Venn Diagram for Three Issuers