Foreign Currency Power Option Pricing Based on Esscher Transform

Abstract

In this paper, we introduce a dynamic model for the spot foreign exchange rate which is driven by a standard Brownian motion and a stationary compound Poisson process under the domestic real measure. In order to price the derivatives on the foreign exchange rate, we need to find an equivalent probability measure under which the discounted process of the foreign exchange rate by the domestic free interest rate minus the foreign free interest rate is a martingale. The Esscher transform is an efficient technique to find an equivalent martingale measure. Applying the tool of the characteristic function, we derive some Esscher transform parameters with respect to the spot foreign exchange rate. At the same time, we get the corresponding Esscher martingale measure which is the domestic risk-neutral measure Q equivalent to the domestic real measure. Moreover, we reconsider the dynamic process of the spot foreign exchange rate under the measure Q. Furthermore, we hope that the exchange rate fluctuates within a certain range, since too large fluctuation will bring a series of serious problems. In fact, the foreign exchange rate is usually stable in a certain range. Thus, studying the pricing of foreign exchange rate derivatives, we often assume that the foreign exchange rate fluctuates within a certain range. Based on the above work, we combine European option with the power option to propose a new type of the foreign exchange power option whose payoff function is controlled by multiplying an indicative function on the interval of the foreign exchange rate and further obtain the pricing formulas under this model. At last, we utilize the actual market data of the foreign exchange rate of USD/CNY to obtain the value of the foreign exchange power option and investigate the implied volatility.