The Application Research Of Fractional Brownian Motion In Option Pricing

Since the Black-Scholes option pricing model generated in 1973, there are more and more researches on the Black-Scholes option pricing models, all the researches promote the development of the Black-Scholes model, the domestic financial markets generally use this model to analyze. However, it is found in the study and practice that the price changes of stocks in the actual financial markets do not meet random walk process, i.e. it means changes of stock price is not subject to the log-normal distribution, but to obey a distribution of Fat tail. So it’s not realistic to use the traditional Black-Scholes option pricing model to solve theoretical price of the options and the deviation will also be bigger. In order to solve the self-similarity and long-term dependence issues of share prices in real markets, the fractional Brownian motion is presented, it has the characteristics of self-similarity and long-term dependence, which can describe the stock prices changes in actual financial markets more effectively. In recent years, researches and applications on fractional Brownian motion in option pricing has become increasingly popular.This thesis introduces the definition and the basic property of Brownian motion and fractional Brownian motion in detail. In the second chapter, some important options pricing formulas of fractional Brown motion are introduced. It introduces the binomial option pricing model and the classic Black-Scholes option pricing model in the third chapter. In the fourth chapter, under fractional Brown motion, maximum option pricing formula and European call option pricing formula with a critical condition are presented. Observing these option pricing formulas, if the value of the Hurst exponent H is certain, we can get the theoretical price at any time. In the fifth chapter, by using European call option pricing formula with a critical condition that is under fractional Brown motion, we carry on an empirical analysis of 50 ETF of Shanghai Stock Exchange in China.This paper mainly introduces the traditional Black-Scholes option pricing model and the fractional Black-Scholes option pricing model. And then, the respective advantages and the disadvantages are compared with each other. At the same time, it makes further efforts to describe the application of the two models in option pricing by using empirical analysis. The Black-Scholes option pricing model has been used in almost all of the empirical analysis in the existing empirical analysis of 50 ETF. This paper uses the option pricing model of the traditional Black-Scholes and the one in the fractional Brownian motion to get empirical analysis of 50 ETF of Shanghai Stock Exchange in China. First of all, utilizing the Eviews to test normality of the closing price sequence of the 50 ETF of Shanghai Stock Exchange in China, the result shows that the 50 ETF of the Shanghai stock market is not satisfied with the log-normal distribution. Secondly, using the method of rescaled range analysis to analyze the sequence, it suggests that there is a fractal structure in the 50 ETF fund market of Shanghai Stock Exchange in China. Therefore, we have enough reason to use option pricing model in the fractional Brownian motion to make empirical analysis of 50 ETF. Comparing the deviation between the theoretical price and the actual price which is calculated by the two models, the conclusion can be made that the option pricing model in the fractional Brownian motion is better than the Black-Scholes option pricing models in the analysis of China’s Shanghai Stock 50 ETF.