A Valuation Formula for Foreign Currency Options

Abstract

Traditional models used to value foreign currency options assume either that interest rates are constant (e.g. Black and Scholes, 1973; Biger and Hull, 1983; and Garman and Kohlhagen, 1983) or that bond price dynamics follow a geometric Brownian motion process (Grabbe, 1983; and Hilliard et al, 1991). The difficulty with the constant interest rate assumption is that it is violated in practice, and the problem with assuming geometric Brownian motion for the bond price dynamics is that it does not preclude the possibility of a non-zero variance of the bond price at maturity. This paper presents an alternative approach for the valuation of foreign currency options which overcomes these problems by assuming stochastic interest rates and a Brownian bridge process for the bond price dynamics. A closed-form solution is obtained for the valuation of European foreign currency call options under this formulation, and formulas for estimation of parameters of the model are also provided.