The Mathematics of Financial Modelingand Investment Management

22-Credit Risk Model Derivs Page 687 Wednesday, February 4, 2004 1:12 PM

Credit Risk Modeling and Credit Default Swaps 687

ET()+ DT T ( , ) = max{AT()– K, 0 }+ min{AT(),K}

= AT()

as required.

Since any corporate debt is a contingent claim on the firm’s future

asset value at the time the debt matures, this is what we must model in

order to capture the default. BSM assumed that the dynamics of the

asset value follow a lognormal stochastic process of the form

dA t

--------------- = rdt + σdW t

()

() (22.3)

At()

where r is the instantaneous risk-free rate which is assumed constant, σ

is the percentage volatility, and W(t) is the Wiener process under the

risk neutral measure (see Chapter 15).^8 This is the same process as is

generally assumed within equity markets for the evolution of stock

prices and has the property that the asset value of the firm can never go

negative and that the random changes in the asset value increase pro-

portionally with the asset value itself. As it is the same assumption used

by Black-Scholes for pricing equity options, it is possible to use the

option pricing equations developed by BSM to price risky corporate lia-

bilities.

The company can default only at the maturity time of the debt when

the payment of the debt (face value) is made. At maturity, if the asset

value lies above the face value, there is no default, else the company is in

bankruptcy and the recovery value of the debt is the asset value of the

firm. While we shall discuss more complex cases later, for this simple

one-period case, the probability of default at maturity is

K

p = ∫ φ[AT()]dAT()= 1 – Nd() 2 (22.4)

• ∞

where φ(⋅) represents the log normal density function, N(⋅) represents

the cumulative normal probability, and

(^8) The discussions of the risk neutral measure and the change of measure using the
Girsanov theorem can be found in standard finance texts. See, for example, Darrell
Duffie, Dynamic Asset Pricing (New Jersey: Princeton Press, 2000), and John Hull,
Options, Futures, and Other Derivatives (New York: Prentice Hall, 2002).