INTRODUCTION

We noted in Chapter 9 that under constancy of the interest rate, risk-neutral valuation does not provide scope for arbitrage. A risk-neutral probability measure is also known as an equivalent martingale measure. A martingale is a process whose expected future value is given by its current value. Section 13.2 provides a motivation for considering a martingale in Financial Economics using Arrow–Debreu securities. Section 13.3 explains the reasoning behind calling a risk-neutral measure an equivalent martingale measure. The models we use in this section are discrete-time models. In Section 13.4 we consider a continuous-time model and identify the corresponding equivalent martingale measure.

In our discussion of Sections 13.2–13.4 we assume that interest rates are constant. However, in reality interest rate may be floating. Because of short lifespans, the impact of interest rate changes on option prices is rather negligible. There are many other securities with longer duration that are influenced by interest rate changes. Analysis of such securities under the variability of interest rate is of high importance. In the analysis of immunization of interest rate risk presented in Chapter 7, the variation in the rate of interest was assumed to be rather minor. But in practice, variation may be more general. Variable interest rates may be realized in many ways. A coupon bond maturing at a fixed date with random coupons implies variability of the underlying interest rate.