22-Credit Risk Model Derivs Page 729 Wednesday, February 4, 2004 1:12 PM
Credit Risk Modeling and Credit Default Swaps 729where 1 is the indicator function.^45
The BSM model is particularly useful in modeling correlated
defaults. If two firms do business together, it is likely that the two firms
may have a certain relationship between their defaults. The BSM model
provides an easy explanation as to how that may be modeled:Pr(AA ()T <KA ∩AB ()T <KB )A bivariate diffusion of firm A and firm B can easily provide what we
need. Under the BSM model, logarithm of asset price is normally distrib-
uted. Hence, the previous equation is the tail probability of a bivariate
normal distribution. The correlation between the two normally distrib-
uted log asset prices characterizes the default correlation. When the corre-
lation in the bivariate normal is 100%, the distribution becomes a
univariate normal distribution and the two firms default together. When
the correlation is –100%, one firm defaulting implies the survival of the
other firm; so there is always one that is live and one that is dead.
While the BSM model cleverly explains how default risk is priced in
the corporate debt conceptually, it remains a practical problem in that it
cannot price today’s complex credit derivatives. Hence, researchers
recently have developed a series of reduced form models that simplify
the computations of the prices.Using Common Factors to Model Joint Defaults
There are two ways to model joint defaults in a reduced form model.
One way, proposed by Duffie and Singleton, is to specify a “common
factor.”^46 When this common factor jumps, all firms default. Firms also
can do so on their own. The model can be extended to multiple common
factors: market factor, industry factor, sector factor, and so on to cap-
ture more sophisticated joint defaults.
Formally, let a firm’s jump process be^47(^45) Recall from Chapter 6 that for any random variable X the following relationship
holds: EX[]= ∫XPd. If X is the indicator function of the event A, X = 1A we can
write Ω
E[ (^1) A ]= ∫ (^1) A Pd = ∫ Pd = PA()
Ω A
(^46) Darrell Duffie and Kenneth Singleton, “Econometric Modeling of Term Structure
of Defaultable Bonds,” Review of Financial Studies (December 1999), pp. 687–720.
(^47) Darrell Duffie and Kenneth Singleton, unpublished lecture notes on credit deriva-
tives; and Darrell Duffie and Kenneth Singleton, “Simulating Correlated Defaults,”
working paper, Stanford University (September 1998).