The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 691 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 691

an option on another option. The main point is that defaults are a series

of contingent events. Later defaults are contingent upon prior no-

default. Hence, layers of contingent defaults build up a series of sequen-

tial compound options, one linking to the other.

For example, suppose there are two zero-coupon bonds expiring in

one year and two years, respectively. Both bonds have a $100 face

value. The asset value is $200 today and follows the diffusion process

given by equation (22.3). If the asset value falls below the face value in

year 1, the company is technically under default. The company may seek

additional capital to keep it alive or the company may simply declare

default and let the holders of the two debts liquidate the company. In

this case we have

A(t) = $200 million r =5%

K 1

 = $100 million T 1 – t = 1 year

K 2 = $100 million T 2 – t = 2 years

σ = 20%

The default point of a two-year model is the key to the problem.

The recovery further complicates the problem. For example, the com-

pany may default when it fails to pay the first debt ($100); or the com-

pany may default if its asset value falls below the market value of the

total debt, which is the face value of the first debt ($100) and the market

value of the second debt. This happens at a situation where the second

debt owner can audit the asset value of the firm. Furthermore, a fixed

recovery of these debts simplifies the problem. But oftentimes recoveries

of debts depend on claims on the assets at different priority levels.

Take a simple example where the company defaults when it fails to

pay its first debt. In this case the default probability is

ln 200 ln – 100 + (5% – 0.2^2 ⁄ 2 ) × 1

d 2 = --------------------------------------------------------------------------------------------= 3.6157

0.2 1

p= 1 – N(3.6157) = 0.015%

If we further assume that the first debt has a recovery rate of 0, then the

debt value is

DtT( , 1 ) = ( 1 – 0.015% )e –5% ×^1 × 100 = 95.11