# Credit

Credit risk models come in two main varieties, the structural and the reduced form.

## Structural models

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.

## Reduced form

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