Models of the Short-term Rate

Abstract

The aim of this chapter is to survey the most popular models of the short-term interest rate. For convenience, we will work throughout within a continuous-time framework; a detailed presentation of a discrete-time approach to term structure modelling is done in Jarrow (1996). We start this chapter by addressing the existence and uniqueness of an arbitrage-free family of bond prices related to a given short-term rate process. To obtain more explicit results, we then assume that the short-term interest rate is modelled either as an Ito process or, even more specifically, as a one-dimensional diffusion process. The latter approach to bond price modelling has been examined by many authors during the last 20 years. In this text, we provide only a brief survey of the most widely accepted examples of diffusion processes used to model the short-term rate. The short-term rate approach to bond price modelling is not developed in subsequent chapters. This is partially explained by the abundance of literature taking this approach, and partially by the difficulty of fitting the observed term structure of interest rates and volatilities within a simple diffusion model. Instead, we develop the term structure theory for a much larger class of models which includes diffusion-type models as special cases. Nevertheless, it should be made clear that diffusion-type modelling of the short-term rate is still the most popular method for the valuing and hedging of interest rate-sensitive derivatives.