On The Pricing Of European Options Based On Mixed Fractional Brownian Motion

European option is one of the most important options in the financial derivatives market. The classical B-S model on European option pricing is carried by Fisher Black, Myron Scholes and Robert Merton in 1970s. The model made a series of assumptions on the market. For example, asset logarithmic returns follow the normal distribution. The risk-free interest rate is constant. Besides, there is no risk-free arbitrage and the market is without friction.But in China’s financial markets, some of these assumptions are not consistent with reality. For example, asset logarithmic returns do not follow the normal distribution, but have the shape of "spikes" and "fat tail". Thus, we can consider using the characteristics of fractional market to improve the B-S model, in order to get a better model for the European option pricing.This article which is based on fractal market theory and the existing classical European option pricing model (B-S model) gives the mixed fractional Brownian motion model on the European option pricing and gets a formula on pricing European option. Through the empirical analysis, we can confirm the logarithmic returns of the stock market do not follow the normal distribution, but have a peak, fat tail and the characteristics of fractal.At last, this paper does some empirical analysis on the formulas’ effectiveness, by the formula of European option pricing, we can calculate some options’ price which have different execution price. Next, we compared these prices with the closing prices in simulation market and the prices which are calculated from the classical B-S model. The empirical results show that the price in mixed fractional Brownian motion model is closer to the true value than which in the B-S model about the CSI 300 Index. Therefore we can conclude that the mixed fractional Brownian motion for European option pricing is feasible.