 # 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.

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