The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 717 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 717

Default Swaps with Counterparty Risk

Counterparty risk is a major concern for credit default swap investors

because major participants in the market are financial firms, which are

themselves subject to default risk.^41 Most bank/dealer counterparties

are single A or at most AA rated. If the reference entity name is a AAA

rated company, then the default probability of the bank/dealer is so

much higher than the reference entity that the bank/dealer may default

well before the reference entity. In this case, the protection buyer in a

credit default swap is more concerned with the counterparty default risk

than the default risk of the reference entity. In this section, we shall

extend the previous risk-neutral methodology to account for counter-

party risk, with the assumption that the default of the reference entity

and the default of the counterparty are uncorrelated.

We label the survival probability of the reference entity Q 1 (t,T) and

that of the counterparty Q 2 (t,T). The default probabilities of the reference

entity and counterparty in the jth period in the future are Q 1 (t,Tj) –

Q 1 (t,Tj+1) and Q 2 (t,Tj) – Q 2 (t,Tj+1), respectively. The default of either one is

Q 1 (tT, j)Q 2 (tT, j)– Q 1 (tT, j+ 1 )Q 2 (tT, j+ 1 )

The above equation represents a situation that both the reference

entity and counterparty jointly survive till Tjbut not Tj+1. Hence one of

them must have defaulted in the period (Tj,Tj+1). Subtracting the coun-

terparty default probability from the probability of either default gives

rise to the probability of the case that only the reference entity (but not

the counterparty) defaults. Hence the total probability of only the refer-

ence entity defaulting is

n

∑[Q 1 (tT, j)Q 2 (tT, j )– Q 1 (tT, j+ 1 )Q 2 (tT, j+ 1 )]– [Q 2 (tT, j)– Q 2 (tT, j+ 1 )]

j= 0

When recovery and discounting are included, we have the credit

default swap value as

n

V= ∑PtT( , j)[ 1 – RT()j][Q 1 (tT, j)Q 2 (tT, j)– Q 1 (tT, j+ 1 )Q 2 (tT, j+ 1 )

j= 0

• {Q 2 (tT, j )– Q 2 (tT, j+ 1 )} ]

(^41) See also Hull and White, “Valuing Credit Default Swaps II: Counterparty Default
Risk.”