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"An Asymptotic Expansion Approach to Currency Options with a Market Model of Interest Rates under Stochastic Volatility Processes of Spot Exchange Rates"

By Akihiko Takahashi and Kohta Takehara

Abstract

This paper proposes an asymptotic expansion scheme of currency options with a libor market model of interest rates and stochastic volatility models of spot exchange rates. In particular, we derive closed-form approximation formulas for the density functions of the underlying assets and for pricing currency options based on the third order asymptotic expansion scheme; we do not model a foreign exchange rate's variance such as in Heston[1993], but its volatility that follows a general time-inhomogeneous Markovian process, and we allow the correlations among all the factors, that is domestic and foreign interest rates, a spot foreign exchange rate and its volatility. Finally, we provide numerical examples and apply the pricing formula to the calibration of volatility surfaces in the JPY/USD option market.

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