The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 702 Wednesday, February 4, 2004 1:12 PM

702 The Mathematics of Financial Modeling and Investment Management

Solving for the second-period default probability one obtains p(1,2) =

14.01%.

The total survival probability till two years is surviving through the

first year (98.21%) and the second year (1 – 14.01% = 85.99%):

Q( 02 , ) = Q( 01 , )( 1 – p( 12 , )) = 98.21% × ( 1 – 14.01% ) = 84.45%

λ 1 + λ 2 = –ln 0.8445 = 16.9011%

λ 2 = 16.9011% – λ 1 = 16.9011% – 1.8062% = 15.0949%

The total default probability is either defaulting in the first period

(1.79%) or surviving through the first year (98.21%) and defaulting in

the second (14.01%).

1.79% + 98.21% × 14.01% = 15.55%

This probability can be calculated alternatively by 1 minus the two-

period survival probability:

1 – Q(0,2) = 1 – 84.45% = 15.55%

It should be noted that any forward default probability is the differ-

ence of two survivals weighted by the previous survival as shown below:

(

Q( 0 , j – 1 ) – Q( 0 , j)

pj – 1 , j) = ---------------------------------------------------- (22.8)

Q( 0 , j – 1 )

For example, the second period default probability is

p(0,2) = 1 – Q(0,2)/Q(0,1)

To express this more clearly, let us examine a two-period binomial

tree shown in Exhibit 22.3. It should be clear how the recovery amount

can change the default probabilities. Take the one-year bond as an

example. If the recovery were higher, the default probability would be

higher. This is because for a higher recovery bond to be priced at the

same price (par in our example), the default probability would need to

be higher to compensate for it. If the default probability remains the

same, then the bond should be priced above par.

So far we have not discussed any model. We simply adopt the spirit

of the reduced form models and use the market bond prices to recover