Interest Rate Theory

Abstract

In this chapter we apply the techniques developed in the previous chapters to the fast-growing fixed-income securities market. We mainly focus on the continuous-time model (since the available tools from stochastic calculus allow an elegant presentation) and comment of the discrete-time analogue ([132] gives a splendid account of discrete-time models). As we want to develop a relative pricing theory, based on the no-arbitrage assumption, we will assume prices of some underlying objects as given. In the present context we take zero-coupon bonds as the building blocks of our theory. In doing so we face the additional modelling restriction that the value of a zero-coupon bond at time of maturity is predetermined (= 1). Furthermore, since the entirety of fixed-income securities gives rise to the term-structure of interest rates (sometimes called the yield curve), which describes the relationship between the yield-to-maturity and the maturity of a given fixed-income security, we face the further task of calibrating our model to a whole continuum of initial values (and not just to a vector of prices). A first attempt at explaining the behaviour of the yield curve is in terms of a continuum of spot rates of maturities between τ and T, where τ is the shortest (instantaneous) lending/borrowing period, and T the longest maturity of interest. We model these rates as correlated stochastic variables with the degree of correlation decreasing in terms of the difference in maturity.