Chapter 5: Models and Equations 297
If the yield on a risk-free, i.e. government bond, with
maturityTisrthen its value is
e−r(T−t).
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
e−r(T−t)×e−p(T−t)=e−(r+p)(T−t).
So to represent the value of a risky bond just add a
credit spread ofpto the yield on the equivalent risk-free
bond. Or, conversely, knowing the yields on equivalent
risk-free and risky bonds one can estimatepthe 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 3 167–79
Black, F & Scholes, M 1973 The pricing of options and corpo-
rate liabilities.Journal of Political Economy 81 637–59
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 struc-
ture of interest rates.Econometrica 53 385–467
Hagan, P, Kumar, D, Lesniewski, A & Woodward, D 2002 Man-
aging smile risk.Wilmottmagazine, September
Haug, EG 1997The Complete Guide to Option Pricing Formulas.
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