A diffusion model for exchange rates I: Theoretical introduction

The rates of exchange between r continuously traded currencies are modelled by a diffusion process. The motivation for this kind of model is in terms of long term trends in the exchange rates combined with random fluctuations in these rates which occur as a result of continous evaluation of information by the dealers in the currency market together with a continuous revision of expectations about the short-term course of the exchange rates of the currencies they are dealing in. The formulation as a diffusion process permits considerable theoretical development. On a qualitative level, stability of exchange rates is investigated using Hashiminsky's criterion and the components of an unstable regime quite well identified. Government intervention in exchange markets may be studied at a simple level by using local times. The properties of local times suggest a certain amount of insight into how optimal intervention performs. An important feature of the diffusion formulation is the derivation of explicit formulae for the numerical calculation of quantities of interest in terms of the model. This is illustrated by partial differential equations for local times and an equation obtained from the Feynman-Kac formula which permits interpretation in terms of forward exchange rates. Numerical techniques permit explicit calculation from such equations.