Abstract
We derive results similar to Bo et al. (2010), but in the case of dynamics of the FX rate driven by a general Merton jump-diffusion process. The main results of our paper are as follows: 1) formulas for the Esscher transform parameters which ensure that the martingale condition for the discounted foreign exchange rate is a martingale for a general Merton jump-diffusion process are derived; using the values of these parameters we proceed to a risk-neural measure and provide new formulas for the distribution of jumps, the mean jump size, and the Poisson Process intensity with respect to the measure; pricing formulas for European foreign exchange call options have been given as well; 2) obtained formulas are applied to the case of the exponential processes; 3) numerical simulations of European call foreign exchange option prices for different parameters are also provided.
Highlights
The existing academic literature on the pricing of foreign currency options could be divided into two categories: 1) both domestic and foreign interest rates were assumed to be constant whereas the spot exchange rate was assumed to be stochastic (see, e.g., Jarrow et al (1981, [1]); 2) models for pricing foreign currency options incor
In Garman et al (1983, [11]) and Grabbe (1983, [3]), foreign exchange option valuation formulas were derived under the assumption that the exchange rate followed a diffusion process with continuous sample paths
Ahn et al (2007, [15]) derived explicit formulas for European foreign exchange call and put options values when the exchange rate dynamics is governed by jump-diffusion processes
Summary
The existing academic literature on the pricing of foreign currency options could be divided into two categories: 1) both domestic and foreign interest rates were assumed to be constant whereas the spot exchange rate was assumed to be stochastic (see, e.g., Jarrow et al (1981, [1]); 2) models for pricing foreign currency options incor-. Takahashi et al (2006, [12]) proposed a new approximation formula for the valuation of currency options using jump-diffusion stochastic volatility processes for spot exchange rates in a stochastic interest rates environment. They applied the market models developed by Brace et al (1998), Jamshidian (1997, [13]) and Miltersen et al (1997, [14]) to model the term structure of interest rates. Main results of our research are as follows: 1) In section 2, we generalize formulas in [20] for Esscher transform parameters assuring that martingale condition for discounted foreign exchange rate is a martingale for a general Merton jump-diffusion process (see (30)). 3) In section 4, we provide numerical simulations of European call foreign exchange option prices for different parameters: S K , where S is the initial spot FX rate, and K is the strike FX rate for a maturity time T
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