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.