The Valuation of Currency Options with Markov-Modulated Jump Risks

Abstract

In this paper, we study the valuation of European currency options when the dynamics of the spot foreign exchange rate are driven by geometric Brownian motions with Markov-modulated Poisson processes. Under such dynamics, the jump events are described as a compound Poisson process with log-normal jump size, and the regime-switching arrival intensity is governed by a continuous-time finite-state Markov chain. Since the market is incomplete, we present a framework for option valuation using the technique of Esscher transforms. After determining the Esscher parameters, we derive the analytical pricing formulas of European currency options under Markov-modulated jump risks.