The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 713 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 713

This result is similar to the risk-free coupon bond where only risk-free

discount factors are used.

The “forward” default probability is a conditional default probabil-

ity for a forward interval conditional on surviving until the beginning of

the interval. This probability can be expressed as

QtT( , j– 1 )– QtT( , j)

pT()j = ------------------------------------------------------- (22.10)

QtT( , j– 1 )

Credit Default Swap Value

A credit default swap takes the defaulted bond as the recovery value and

pays par upon default and zero otherwise.

• μrs

V= Ee ∫e ()ds 1

μ <T[^1 – R()μ]

where μis default time.

Hence the value of the credit default swap (V) should be the loss upon

default weighted by the default probability:

n

V= ∑PtT( , j)[QtT( , j– 1 )– QtT( , j)][ 1 – RT()j] (22.11)

j= 1

where P(·) is the risk-free discount factor and R(·) is the recovery rate.

In equation (22.2) it is implicitly assumed that the discount factor is

independent of the survival probability. However, in reality, these two

may be correlated—usually higher interest rates lead to more defaults

because businesses suffer more from higher interest rates. Equation

(22.2) has no easy solution.

From the value of the credit default swap, we can derive a spread

(s), which is paid until default or maturity:

V

s= -------------------------------------------------- (22.12)

n

∑PtT( , j)QtT( , j)

j= 1

Exhibit 22.6 depicts the general default and recovery structure. The

payoff upon default of a default swap can vary. In general, the owner of