The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 707 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 707

immediately prior to default. In this way we avoid the contradictory sce-

nario which can arise in the Jarrow-Turnbull model in which the recovery

rate, being an exogenously specified percentage of the default-free payoff,

may actually exceed the price of the bond at the moment of default.

The debt value at time tis^32

1

DtT ( , )= ------------------{pδEDt[ ( + ∆t,T)]+ ( 1 – p)EDt[ ( + ∆t,T)]}

1 + rt∆

By recursive substitutions, we can write the current value of the

bond as its terminal payoff if no default occurs:

( ,

1 – pt∆ ( 1 – δ) n

DtT ) = ------------------------------------ XT()

1 + rt∆

Note that the instantaneous default probability being p∆tis consis-

tent with the Poisson distribution,

• dQ
  ------------= pt∆
  Q

Hence, recognizing ∆t = T/n,

DtT )= -------------------------------------------XT + () (22.9)

exp(rT)

( ,

exp(–p( 1 – δ)T)

()= exp(–(rs)T)XT

When rand sare not constants, we can write the Duffie-Singleton

model as



T



DtT ( , ) = Et exp–∫[ru()+ su()]du XT()

 

t

where s(u) = pu(1 – δ). Not only does the Duffie-Singleton model have a

closed-form solution, it is possible to have a simple intuitive interpretation

of their result. The product p(1 – δ) serves as a spread over the risk-free

discount rate. When the default probability is small, the product is small

(^32) The probability, p, can be time dependent in a more general case.