Credit risk models come in two main varieties, the structural and the reduced form.
Structural models try to model the behaviour of the firm so as to represent the default or bankruptcy of a company in as realistic a way as possible. The classical work in this area was by Robert Merton who showed how to think of a company's value as being a call option on its assets. The strike of the option being the outstanding debt.
Merton assumes that the assets of the company A follow a random walk
If V is the current value of the outstanding debt, allowing for risk of default, then the value of the equity equals assets less liabilities:
Here S is the value of the equity. At maturity of this debt
where D is the amount of the debt to be paid back at time T.
If we can hedge the debt with a dynamically changing quantity of equity, then the Black--Scholes hedging argument applies and we find that the current value of the debt, V, satisfies and exactly the same partial differential equation for the equity of the firm S but with
S(A, T) = max(A - D,0).
The problem for S is exactly that for a call option, but now we have S instead of the option value, the underlying variable is the asset value A and the strike is D, the debt. The formula for the equity value is the Black--Scholes value for a call option.
The more popular approach to the modelling of credit risk is to use an instantaneous risk of default or hazard rate, p. This means that if at time t the company has not defaulted then the probability of default between times t and t + dt is pdt. This is just the same Poisson process seen in jump-diffusion models. If p is constant then this results in the probability of
a company still being in existence at time T, assuming that it wasn't bankrupt at time t, being simply
If the yield on a risk-free, i.e. government bond, with maturity T is r, then its value is
If we say that an equivalent bond on the risky company will pay off 1 if the company is not bankrupt and zero otherwise, then the present value of the expected payoff comes from multiplying the value of a risk-free bond by the probability that the company is not in default to get
So to represent the value of a risky bond just add a credit spread of p to the yield on the equivalent risk-free bond. Or, conversely, knowing the yields on equivalent risk-free and risky bonds one can estimate p, the implied risk of default.
This is a popular way of modelling credit risk because it is so simple and the mathematics is identical to that for interest rate models.
References and Further Reading
Black F 1976 The pricing of commodity contracts. Journal of Financial Economics 3 167-179
Black, F & Scholes, M 1973 The pricing of options and corporate liabilities. Journal of Political Economy 81 637-659
Brace, A, Gatarek, D & Musiela, M 1997 The market model of interest rate dynamics. Mathematical Finance 7 127-154
Cox, J, Ingersoll, J & Ross, S 1985 A theory of the term structure of interest rates. Econometrica 53 385-467
Hagan, P, Kumar, D, Lesniewski, A & Woodward, D 2002 Managing smile risk. Wilmott magazine, September
Haug, EG 1997 The Complete Guide to Option Pricing Formulas. McGraw-Hill
Heath, D, Jarrow, R & Morton, A 1992 Bond pricing and the term structure of interest rates: a new methodology. Econometrica 60 77-105
Heston, S 1993 A closed-form solution for options with stochastic volatility with application to bond and currency options. Review of Financial Studies 6 327-343
Ho, T & Lee, S 1986 Term structure movements and pricing interest rate contingent claims. Journal of Finance 42 1129-1142
Hull, JC & White, A 1987 The pricing of options on assets with stochastic volatilities. Journal of Finance 42 281-300
Hull, JC & White, A 1990 Pricing interest rate derivative securities. Review of Financial Studies 3 573-592
Lewis, A 2000 Option Valuation under Stochastic Volatility. Finance Press
Merton, RC 1973 Theory of rational option pricing. Bell Journal of Economics and Management Science 4 141-183
Merton, RC 1974 On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance 29 449-470
Merton, RC 1976 Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3 125-144
Rasmussen, H & Wilmott, P 2002 Asymptotic analysis of stochastic volatility models. In New Directions in Mathematical Finance (eds Wilmott, P & Rasmussen, H). John Wiley & Sons Ltd
Schonbucher, PJ 1999 A market model for stochastic implied volatility. Philosophical Transactions A 357 2071-2092
Schonbucher, PJ 2003 Credit Derivatives Pricing Models. John Wiley & Sons Ltd
Vasicek, OA 1977 An equilibrium characterization of the term structure. Journal of Financial Economics 5 177-188
Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd