Risk-Neutral Valuation pp 327-374 | Cite as

# Interest Rate Theory

Chapter

## 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 (Jarrow (1996) 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
with .

*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. Discretizing the maturity spectrum, we are tempted to start with a generalized Black-Scholes model (as in §6.2)$$d{r_i}\left( t \right) = {a_i}\left( t \right) + \sum\limits_{j = 1}^d {{b_{ij}}} \left( t \right)d{W_j}\left( t \right),\;i = 1, \ldots ,\;d $$

*W*= (*W*1_{1}...,*W*_{d}) a standard*d*-dimensional Brownian motion. The degree of (instantaneous) correlation of the different rates can then be described in terms of their covariationt$$d\langle {r_{i,}}{r_j}\rangle (t) = \sum\limits_{k = 1}^d {{b_{ik}}(t)dt = {\rho _{ij}}} (t)dt.$$

## Keywords

Price Process Forward Rate Contingent Claim Martingale Measure Bond Price
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© Springer-Verlag London 2004