The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 708 Wednesday, February 4, 2004 1:12 PM

708 The Mathematics of Financial Modeling and Investment Management

and the credit spread is small. When the recovery is high (i.e., 1 – δ is

small), the product is small and the credit spread is small.

Consider a two-year zero coupon bond. Assume that the probability

of defaulting each year is 4%, conditional on surviving to the beginning

of the year. If the bond defaults we assume that it loses 60% of its mar-

ket value. We also assume that risk-free interest rates evolve as shown in

Exhibit 22.5 where an up move and a down move have an equal proba-

bility of 50%. At any node on the tree the price is the risk-free dis-

counted expectation of the payoff at the next time step. Therefore at the

node where the risk-free rate has climbed to 7%, the value of the secu-

rity is given by

1

-----------[( 1 – 0.04) × $100 + 0.04 × ($100 – $60)] = $91.25

1.07

Using the relationship

EXHIBIT 22.5 Valuation of a Two-Year Defaultable Zero-Coupon Bond Using

Duffie-Singleton