The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 697 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 697

default is endogenous in the BSM model while it is exogenous in the Jar-

row-Turnbull and Duffie-Singleton models. As we will see, specifying

defaults exogenously greatly simplifies the problem because it ignores

the constraint of defining what causes default and simply looks at the

default event itself. The computations of debt values of different maturi-

ties are independent, unlike in the BSM model that defaults of the later-

maturity debts are contingent on defaults of earlier-maturity debts.

The Poisson Process

The theoretical framework for reduced form models is the Poisson pro-

cess.^23 To see what it is, let us begin by defining a Poisson process that

at time t has a value Nt. The values taken by Nt are an increasing set of

integers 0, 1, 2, ... and the probability of a jump from one integer to the

next occurring over a small time interval dt is given by

Pr [ Ntdt + – Nt = 1 ] = λdt

where λ is known as the intensity parameter in the Poisson process.

Equally, the probability of no event occurring in the same time

interval is simply given by

Pr [ Ntdt + – Nt = 0 ] = 1 – λdt

For the time being we shall assume the intensity parameter to be a fixed

constant. In later discussions and especially when pricing is covered in the

next chapter, we will let it be a function of time or even a stochastic vari-

able (known as a Cox process^24 ). These more complex situations are

beyond the scope of this chapter. It will be seen shortly that the intensity

parameter represents the annualized instantaneous forward default prob-

ability at time t. As dt is small, there is a negligible probability of two

jumps occurring in the same time interval.

The Poisson process can be seen as a counting process (0 or 1) for

some as yet undefined sequence of events. In our case, the relationship

between Poisson processes and reduced form models is that the event

which causes the Poisson process to jump from zero to 1 can be viewed

as being a default.

(^23) A Poisson process is a point process. Point processes were briefly introduced in
Chapter 13.
(^24) David Lando, “On Cox Processes and Credit Risky Securities,” Review of Deriv-
atives Research 2 (1998), pp. 99–120. Cox processes were briefly covered in Chapter
13 of this book.