Currency Trading Markets and Pricing Their Derivatives

Using a Levy process we generalize formulas in Bo et. al. (2010) to the Esscher transform parameters for the log-normal distribution which ensures the martingale condition holds for the discounted foreign exchange rate. We also derive similar results, but in the case when the dynamics of the FX rate is driven by a general Merton jump-diffusion process. Using these values of the parameters we find 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 this measure. The formulas for a European call foreign exchange option are also derived. We apply these formulas to the case of the log-double exponential and exponential distribution of jumps. We provide numerical simulations for the European call foreign exchange option prices with different parameters.