The FFT Method Of Pricing European Foreign Exchange Option Under Double Exponential Jump-Diffusion Process

The G-K model of FX option pricing first derived by German and Kohlhagen relying on a B-S model indeed promoted the development of the FX option pricing. However, the limitation of some assumptions about the market is non-real, for example, the assumptions of constant volatility can’t reflect the nature of fat tail and excess kurtosis presented by the returns distribution. The assumption of constant interest rates ignored the effects of the changes of interest rates on option price in floating exchange rate system. The assumption that the exchange rate moves smoothly doesn’t consider the jumps linked to outside news. Accordingly, to avoid the drawback of the G-K model in pricing currency options, scholars began to seek more effective pricing models. Based on the analysis above, the paper studied FX option pricing as follows:(1) The pricing of FX option under double-exponential jump-diffusion process was studied. Firstly, the pricing model with double jumps was derived by ?Ito’ s formula. And then we obtained the pricing formula with unknown characteristic function according to risk-neutral pricing principle and the relationship between probability distribution and characteristic function. Next, we got the pricing formula by deriving the characteristic function of logarithmic spot exchange rate. Finally, we analyzed the influence of 1 2,,Q? ? ? on option value though numerical analysis.(2) The pricing of FX option pricing under double-exponential jump-diffusion process with stochastic volatility was studied. The dynamic process of spot exchange rate under stochastic volatility and with double jumps was derived by ?Ito’ s formula. Afterwards, by applying measure transformation and solving the partial differential equations, the closed-form characteristic function of logarithmic spot exchange rate was derived. And then, we obtained the pricing formula by FFT method. In the end, the figures about influence of,,,,,,v d f? ? ?K T r r on option price were drawn.(3) The pricing of FX option pricing under double-exponential jump-diffusion process with stochastic volatility and stochastic interest rates was studied. Under stochastic volatility and stochastic interest rates, the dynamics of logarithmic forward exchange rate with double jumps was derived. And then the closed-form characteristic function of logarithmic forward exchange rate was derived. By FFT method, we obtained the pricing formula of FX option. At last, the influence of pricing factors,,,,,d f d f d fa a ? ? ? ? on option price was analyzed by figures. Besides, we compared the outcomes of the paper by tables.