The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 714 Wednesday, February 4, 2004 1:12 PM

714 The Mathematics of Financial Modeling and Investment Management

EXHIBIT 22.6 Payoff and Payment Structure of a Credit Default Swap

the default swap delivers the defaulted bond and in return receives prin-

cipal. Many default swaps are cash settled and an estimated recovery is

used. In either case, the amount of recovery is randomly dependent

upon the value of the reference obligation at the time of default. Models

differ in how this recovery is modeled.^38

To illustrate how to use the above formulation of credit default

swap pricing, assume (1) two “risky” zero-coupon bonds exist with one

and two years to maturity and (2) no recovery upon default. From equa-

tion (22.10) we know the credit spreads of these two “risky” zeros are

approximately their default probabilities. For example, assume the one-

year zero has a spread of 100 basis points and the two-year has a spread

of 120. The survival probabilities can be computed from equation

(22.10). For the one-year bond whose yield spread is 100 basis points,

the (one year) survival probability is

1% = –ln Q( 01 , )

Q( 01 ) = e

• 1%
  , = 0.9900

For the two-year zero-coupon bond whose yield spread is 120 basis

points, the (two year) survival probability is:

1.2% × 2 = –lnQ( 02 , )

Q( 02 , ) = e

• 1.2% × 2
  = 0.9763

(^38) We provide an example where the two variables are independent and the defaults
follow a Poisson process. The simple solution exists under the continuous time as-
sumption. The analysis is provided in the appendix to Chapter 10 in Anson, Fabozzi,
Choudhry, and Chen, Credit Derivatives: Instruments, Applications, and Pricing.