European Option Pricing Under Mixed Fractional Brownian Motion

Option pricing is an important content of financial mathematics. Black,Scholes and Merton created the beginning of option pricing, but they set interest rates and volatility as constant power, which did not satisfy the law of the financial markets.This paper is to study the problem of European option pricing by adopting the method of fast Fourier transform(FFT).(1) This paper is to study the stock price which is driven by a geometric Brownian motion and obeys the double exponential jump diffusion process, and European option pricing based on the stochastic interest rate, stochastic volatility, and stochastic jump intensity. On the basis of previous studies, this paper gives the numerical solution of the option pricing using the fast Fourier transform method, joined by the random jump intensity; And option value along with the change of strike price are analyzed by using the matlab software.(2) This paper is to research the stock prices which is driven by mixed fractional Brownian motion and obeys the normal jump diffusion process, and European option pricing based on the stochastic interest rate. Compared with the general pricing method, this paper is to verify the correctness of the fast Fourier transform method.Because Geometric Brownian motion and fractional Brownian motion are two special forms of mixed fractional Brownian motion, we can get three different corresponding option value numerical solutions based on the Brownian motion.(3) This paper is to study the stock prices which is driven by mixed fractional Brownian motion and obeys the double exponential jump diffusion process; and European option pricing problems based on the stochastic interest rate and stochastic jump intensity, obeying with Vasicek model and CIR model. Then we can obtain the numerical solution of option value in two different kinds of models.