The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 730 Wednesday, February 4, 2004 1:12 PM

730 The Mathematics of Financial Modeling and Investment Management

Ji = aiqM + qi

where qM is the market jump process and qi is the idiosyncratic jump

process. The coefficient ai is to capture different correlation levels. The

joint event is then

corr(Ji,Jj)= aiajvar[qM]

Correlating Default Times

Before we discuss how the default correlation is introduced, we need to

discuss how single issuer default is modeled. The approach used is

equivalent to the Jarrow-Turnbull model.^48 A hazard rate, λ(t), is intro-

duced where λ(t)dt is the probability of defaulting in a small time inter-

val dt. This leads to the definition of the survival probability

Q( 0 ,T)= exp–∫

T

 λs

0

()sd 

The probability of surviving to a time T and then defaulting in the

next instant is therefore given by the density function:

()exp–∫

T

• dQ = λT  λs
  0

()sd dT

In the simple case when the hazard rate is constant over time so that

λ(t) = λwe have

• dQ = λexp(–λT)dT

From this we see that the probability of defaulting at time T as

given by –dQ shows that default times are exponentially distributed. By

extension, the average time to default is given by computing

∞

1

〈〉T = λ ∫Texp(–λT)dT = ---

0 λ

(^48) Robert Jarrow and Stuart Turnbull, “Pricing Derivatives on Financial Securities
Subject to Default Risk,” Journal of Finance 20, no. 1 (1995), pp. 53–86.