Singular Currency Option Pricing Under Stochastic Interest Rate

Option pricing issue has always been the core of modern mathematical finance, in the study of the pricing of financial derivatives, option pricing models and methods are one of the most widely used in the study. Now, many countries have generally adopted the floating exchange rate system, and under the background of the financial deregulation, exchange rate on the international foreign exchange market has shown a lager volatility and fluctuations, which has caused instability in financial asset prices. However, among the most foreign exchange derivatives, foreign exchange options is one of the most effective hedging tools, so the foreign currency option pricing is crucial for research.In this paper, under the condition of Hull-White interest rate models and jump-diffusion process, details of derivation and proof was done on the European foreign exchange call and put option pricing.Further to a new option-power options by using risk neutral pricing theory and fractal theory of Ito integrals, we got the results of foreign exchange call option pricing formula and the ladder power-option pricing formula.Last, empirical analysis has been done on the fractal properties of foreign exchange markets, according to R/S analysis method, we found Hurst values of USD/RMB、 USD/JPY、EUR/USD and GBP/USD are between 0.5 and 0.6, and rate of returns sequences are not normal distribution, showed that the foreign exchange markets exist obviously fractal features, using FBM to describe fluctuations behavior of exchange rate price will be more valuable. In addition, by analysing the abnormal fluctuation of rate of return series at certain points, we find, due to some occurrence of economic and political events, the jumping behavior of currency prices are also abnormal, so based on fractional Brownian motion, the addition of jump-diffusion process is necessary. Finally, by numerical calculation, we found that the introduction of jump processes will increase the volatility of option prices, makes the option price moving in the same direction, while Hurst values increase in fractional Brown motion, which will cause the option price change in reverse, and exponent n also causes option prices change in the same direction.