The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 718 Wednesday, February 4, 2004 1:12 PM

718 The Mathematics of Financial Modeling and Investment Management

The default swap valued under the counterparty risk requires two

default curves, one for the reference entity and one for the counterparty.

This default swap should be cheaper than the default swap with only

default risk for the reference entity. The difference is the value of the

default swap that protects the joint default. An investor who buys such

a default swap owns a default swap on the reference entity and has

implicitly sold a default swap of joint default back to the counterparty.

When the defaults of the reference entity and the counterparty are

correlated, the solution becomes much more complex. When the corre-

lation is high, it is more likely that the counterparty should default

before the reference entity, and the credit default swap should have very

little value. On the other hand, when the correlation is low (negative),

the situation where the reference entity defaults almost guarantees the

survival of the counterparty. Consequently, in such instances the coun-

terparty risk is not a concern.

VALUING BASKET DEFAULT SWAPS

In the previous section we presented a model for valuing single-name

credit default swaps. Unlike a single-name credit default swap, which

provides protection for one bond, a basket default swap provides pro-

tection against a basket of bonds. As with single-name credit default

swaps, the protection buyer of a basket default swap makes a stream of

spread payments until either maturity or default. In the event of default,

the protection buyer receives a single lump-sum payment.

Default baskets have become popular because purchasing individual

basket default swaps for a collection of bonds can be very expensive,

especially considering how unlikely it is that all the bonds in a given

basket will default simultaneously. Buying a basket default swap,

instead, provides a much cheaper solution. The most popular default

basket swap contract is the first-to-default basket. In this contract, the

seller pays (the default event occurs) when the first default is observed

among the bonds in the basket.

In this section, we describe how to extend the model to basket

default swaps. The key in the extension is estimating default correla-

tions. We begin with the valuation model and then discuss how to

model default correlations.

The Pricing Model

The number of issuers (or issues) contained in a default basket typically

varies (three to five). The payoff of a default basket contract can be a