The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 715 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 715

These survival probabilities can then be used to compute forward

default probabilities defined in equation (22.8):

p   1

Q( 00 , )– Q( 01 , ) 1 – 99.00%

()= ---------------------------------------------- = ------------------------------ = 1.00%

Q( 00 , ) 1

and

p   2

Q( 01 , )– Q( 02 , ) 99.00% – 97.63%

()= ---------------------------------------------- = ------------------------------------------------ = 1.39%

Q( 01 , ) 99.00%

Since we assume a 5% flat risk-free rate for two years, the risk-free dis-

count factors are

P( 01 ) = e

• 5%
  ,
  P( 02 , ) = e


• 5% × 2

for one and two years, respectively. Assuming a 20% recovery ratio, we

can then calculate, using equation (22.11), what the total protection

value (V) of the default swap contract is providing

V = e –5%( 1 – 0.99)( 1 – 0.2)+ e –5% ×^2 (0.99 – 0.9763)( 1 – 0.2)

= 0.00761 + 0.010134

= 0.017744 = 177.44 basis points

As mentioned, the default swap premium is not paid in full at the

inception of the swap but paid in a form of spread until either default or

maturity, whichever is earlier. From equation (22.12), we can compute

the spread of the default swap as follows:

0.017744

s = ---------------------------------------------------------------------------------------------------------------------

0.99 ×exp(–0.05)+ 0.9763 ×exp(–0.05 × 2 )

0.017744

= -------------------------= 0.009724

1.824838

which is 9.724 basis points for each period, provided that default does not

occur. This is a payment in arrears. That is, if default occurs in the first

period, no payment is necessary. If default occurs in the second period,

there is one payment; if default never occurs, there are two payments.