The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 701 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 701

recovered. In the Duffie-Singleton model, a fraction of the market debt

value is recovered. And in the Jarrow-Turnbull and other barrier mod-

els, an arbitrary recovery value is assumed (it can be beta distributed).^28

From the observed bond prices, we can easily retrieve default proba-

bilities from bond prices. Suppose there are two bonds, a one-year bond

trading at $100 with a $6 annual coupon and a two-year bond trading

at $100 with a $7 annual coupon. Assuming a recovery of $50 per $100

par value, the first bond price is calculated as

p( 01 , ) × 50 + 106 × ( 1 – p( 01 , ))

100 = --------------------------------------------------------------------------------------

15% +

The default probability is then found by solving for p(0,1):

105 = 106 – 56 × p( 01 , )

p( 01 , ) = 1.79%

We use pt to represent the forward/conditional default probability at

time t. Hence, p 1 is the default probability of the first period. In the first

period, the survival probability is simply 1 minus the default probability:

Q( 01 , ) = 1 – p( 01 , ) = 1 – 1.79% = 98.21%

and therefore

λ = –ln 0.9821 = 1.8062%

The second bond is priced, assuming a recovery of $20 out of $100:

 p( 12 , ) × 20 + ( 1 – p( 12 , )) × 107 

p( 01 ) × 20 + Q( 01 ) × 7 + --------------------------------------------------------------------------------------

 1.05 

, ,

100 = ---------------------------------------------------------------------------------------------------------------------------------------------------------------------

1.05

 p( 12 , ) × 20 + ( 1 – p( 12 , )) × 107 

1.79% × 20 + 98.21% × 7 + --------------------------------------------------------------------------------------

 1.05 

= ----------------------------------------------------------------------------------------------------------------------------------------------------------------------

1.05

(^28) For more details, see Chen, “Credit Risk Modeling: A General Framework.”