The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 712 Wednesday, February 4, 2004 1:12 PM

712 The Mathematics of Financial Modeling and Investment Management

Since in reality, a default can occur any time, to accurately value a

default swap, we need a consistent methodology that describes the fol-

lowing: (1) how defaults occur; (2) how recovery is paid; and (3) how

discounting is handled.

Survival Probability and Forward Default Probability:

A Recap

Earlier in this chapter we introduced two important analytical con-

structs: survival probability and forward default probability. We recap

both below since we will need them in pricing credit default swaps.

Assume the risk-neutral probabilities exist. Then we can identify a

series of risk-neutral default probabilities so that the weighted average

of default and no-default payoffs can be discounted at the risk-free rate.

Let Q(t,T) to be the survival probability from now ttill some future

time T. Then Q(t,T) – Q(t,T+ τ) is the default probability between Tand

T+ τ(i.e., survive till Tbut default at T+ τ). Assume defaults can only

occur at discrete points in time, T 1 , T 2 , ..., Tn. Then the total probability

of default over the life of the credit default swap is the sum of all the per

period default probabilities:

n

∑QtT( , j)– QtT( , j+ 1 )=^1 – QT( n)=^1 – QT()

j= 0

where t = T 0 < T 1 < ... < Tn = Tand Tis the maturity time of the credit

default swap. Note that the sum of the all the per-period default proba-

bilities should equal one minus the total survival probability.

The survival probabilities have a useful application. A $1 “risky”

cash flow received at time Thas a risk-neutral expected value of Q(t,T)

and a present value of P(t,T)Q(t,T) where Pis the risk-free discount fac-

tor. A “risky” annuity of $1 can therefore be written as

∑

n

PtT( , j)QtT( , j)

j= 1

A “risky” bond with no recovery upon default and a maturity of n

can thus be written as

n

Bt()= ∑PtT( , j)QtT( , j)cj+ PtT( , n)QtT( , n )

j= 1