The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 685 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 685

Duffie-Singleton^6 models, do not look inside the firm. Instead, they

model directly the likelihood of default or downgrade. Not only is the

current probability of default modeled, some researchers attempt to

model a “forward curve” of default probabilities which can be used to

price instruments of varying maturities. Modeling a probability has the

effect of making default a surprise—the default event is a random event

which can suddenly occur at any time. All we know is its probability.

There is no standard model for credit. Part of the reason why this is

so is that each of the models has its own set of advantages and disad-

vantages, making the choice of which to use depend heavily on what the

model is to be used for.

The Black-Scholes-Merton Model

The earliest credit model that employed the option pricing theory can be

credited to BSM. Black-Scholes, explicitly articulated that corporate lia-

bilities can be viewed as a covered call: own the asset but short a call

option. In the simplest setting, where the company has only one zero-

coupon debt, at the maturity of the debt the debt holder either gets paid

the face value of the debt—in such a case, the ownership of the com-

pany is transferred to the equity holder—or takes control of the com-

pany—in such a case, the equity holder receives nothing. The debt

holder of the company therefore is subject to default risk for he or she

may not be able to receive the face value of his or her investment. BSM

effectively turned a risky debt evaluation into a covered call evaluation

whereby the option pricing formulas can readily apply.

In BSM, the company balance sheet consists of issued equity with a

market value at time tequal to E(t). On the liability side is debt with a

face value of Kissued in the form of a zero-coupon bond that matures

at time T. The market value of this debt at time tis denoted by D(t,T).

The value of the assets of the firm at time tis given by A(t).

At time T(the maturity of the debt), the market value of the issued

equity of the company is the amount remaining after the debts have

been paid out of the firm’s assets; that is,

ET()= max{AT()– K, 0 }

This payoff is identical to that of a call option on the value of the firm’s

assets struck at the face value of the debt. The payoff is graphed as a

function of the asset value in Exhibit 22.1. The holders of the risky cor-

(^6) Darrell Duffie and Kenneth Singleton, “Modeling the Term Structure of Default-
able Bonds,” working paper, Stanford University, 1997.