A Stochastic Model of Foreign Exchange Dynamics and an Exact Option Pricing Formula

Abstract

When applying the theory of options to foreign exchange, the stochastic specification for the spot price of the currency underlying the option must be able to accommodate a fundamental result of the theory of international trade, that the spot price of a currency would converge over time to its purchasing power parity. Fluctuations in the exchange rate are anchored, so to speak, by its long period equilibrium value. In the existing literature on options, the stochastic process is generally assumed to be geometric Brownian motion, so that the exchange rate would display the characteristics of a random walk. To incorporate the non-random walk effects of purchasing power parity, we develop a two dimensional stochastic process to model exchange rate dynamics. After obtaining a closed form expression for the transition density function, we proceed to derive an exact formula to price options on the currency, which in particular is conditional upon its purchasing power parity.