Currency Option Valuation using Esscher and Fourier Transforms

Abstract

The size and volatility of the exchange rate market warrants the use of currency options in order to hedge the exchange rate risk. This study sought to build a robust pricing model that takes into account some of the stylized facts of exchange rates reported in literature. The normal inverse Gaussian distribution is chosen to model the exchange rate returns as: one, its higher order moments exist, thus can capture skewness and excess kurtosis unlike the two-parameter normal distribution; and two, since it is a normal mixture, the aggregational Gaussianity property holds. To price options written on these exchange rates under the risk-neutral measure, a change of measure is required as this proposed model renders the market incomplete, which implies the existence of a range of equivalent martingale measures (by the fundamental theorem of asset pricing). To choose a unique martingale measure, we apply the Esscher measure. Pricing equations are, thus, derived as the expected discounted value of the payo s with respect to this risk-neutral measure. The fast Fourier transform is then applied to compute option prices due to its computational e ciency. From the results presented herein, the NIG distribution results in a better t of the empirical distribution of the exchange rate returns and the corresponding option pricing model, the Esscher-NIG model, outperforms the Black-Scholes model in pricing performance.