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.”