The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 698 Wednesday, February 4, 2004 1:12 PM

698 The Mathematics of Financial Modeling and Investment Management

Another way to look at the Poisson process is to see how long it

takes until the first default event occurs. This is called the default time

distribution. It can be proven that the default time distribution obeys an

exponential distribution as follows:

Pr(Tt> ) = e –λ (Tt– )

This distribution function also characterizes the survival probability

before time t:

• Qt T ( , ) = Pr(Tt> ) = e


• λ (Tt)

The Jarrow-Turnbull Model

The Jarrow-Turnbull model is a simple model of default and recovery

based on the Poisson default process described above.^25 In their model,

Jarrow and Turnbull assume that no matter when default occurs, the

recovery payment is paid at maturity time T. Then the coupon bond

value can be written as

T n

Bt ( , ()∫–dQt u ) ud + ∑Pt Tj)cje

• λ (Tj – t)
  ()= Pt T )RT ( , ( ,
  t j =^1
  n
  = Pt T ( , )RT()( 1 – e –λ (Tt– ))+ ( ,

∑Pt Tj)cje

• λ (Tj – t)

j = 1

where:

P(t,T) = the risk-free discount factor

cj = the j-th coupon

Q(t,T) = the survival probability up to time t

R = the recovery ratio

It is seen that the conditional default probability is integrated out and dis-

appears from the final result. As a consequence, by assuming recovery

payment to be at maturity, Jarrow and Turnbull assume away any depen-

dency between the bond price and the conditional default probability.

It is worth noting that when the recovery rate is 0, for a zero-cou-

pon bond the value of the intensity parameter is also the bond’s forward

(^25) Jarrow and Turnbull, “Pricing Derivatives on Financial Securities Subject to De-
fault Risk.”