A Study Based On Fractional Brown Motion With Time Varying Hurst Index And GARCH Model In European Option Pricing

China's HUSHEN 300 index's future has traded rapidly and developped soundly since it's officially trading on April 16,2010. Judging from the development of the world's financial markets, the smooth running of stock index future market is the premise of the launch of stock index options. On November 8,2013, China financial futures exchange introduced a simulation of HUSHEN 300 index option trading, which indicated that the conditions of the development of stock index option trading market in our country are getting mature, and all parties are preparing for the launch of stock index option. In the coming of HUSHEN 300 index option, this paper establishes a new paradigm for the pricing of European option under fractional Brownian motion (the upcoming HUSHEN 300 index option is European option). Through this article, attempts to promote the application of using fractional Brownian motion in financial modelingRelative to the standard Brownian motion, fractional Brownian motion is a more generalized stochastic process, and closer in line with the reality of the financial markets. For the fractal Gaussian noise is neither a martingale nor a semi-martingale, so the standard financial stochastic analysis of the standard Brownian motion is no longer applicable to the fractional Brownian motion. By far, many scholars have given the definition of stochastic integral of the fractional Brownian motion, and the most important two of them are the path dependent integral and the Wick-Ito^ integral. On the basis of predecessors'research, T.E.Duncan, Y.Hu and B.Pasik-Duncan(2000) established the stochastic integral theory for fractional Brownian motion based on Wick product when 1/2<H<1 and the fractal Wick-Ito theory. On the other hand, this paper introduces the research results of fractal Gaussian noise in detail, and it mainly includes the chaos expansion of fractional Brownian motion,fractal Girsanov theorem and the conditional expectation,quasi martingale, etc. Like Gaussian noise superposition can generate standard Brownian motion, fractal Gaussian noise superposition can generate fractional Brownian motion. In deep understanding of the research result of fractal Gaussian noise, this paper improves the algorithm of generating fractional Brownian motion, improves the Circulant embedding method of generating fractal Gaussian noise, and generates a more accurate fractional Brownian motion.The existing option pricing model based on fractional Brownian motion assumes that volatility is a constant, and the choosing of the length of financial time series has a great deal of freedom when calculating the volatility, so how to choose the financial time series depends on many things. Empirical analysis results show that the sequence of the yield of financial asset is conditional heteroscedasticity and volatility has clustering features, this paper uses the GARCH model for volatility. Based on the financial stochastic analysis of fractional Brownian motion, the pricing formula of European call option has been given in this paper, when the volatility follows the family of GARCH model. For the empirical study data of my paper uses the intraday price, the volatility of each trading day has been considered as a constant. Of course, if using high frequency data such as tick by tick data, minutes of trading data, a more accurate result will be obtained.Based on the multifractal theory of financial markets, a model of financial market based on fractional Brownian motion with a time varying Hurst index has been given in this paper. For the high similarity design of the HUSHEN300 index option and the Kospi200 index option, the empirical research of this paper uses the Kospi200 index option daily transaction data. In chapter six, according to the proposed model, the fitting of KOSPI200 C 201409 260.0 [KR4201J92602] from April 1,2014 to July 30,2014 is very well. In order to survey the performance out of the sample data, the research object is KOSPI200 C 201409 265.0 (KR4201J92651) from March 14,2014 to July 30,2014, from March 14,2014 to June 2,2014 respectively, and the 54 option trading price as base data is used to estimate the parameters in the model, based on this results one step prediction has been made. All the parameters of the model will be updated with new information. The final result is also well.Either the theoretical analysis or the empirical results, both suggest that the fractional Brownian motion is more appropriate to the financial markets. The results of empirical studies of Kospi200 index option suggest the model proposed in this paper also pricing the HUSHEN 300 index option well. Although the early stages of HUSHEN 300 index option market may not run as perfectly as Kospi200 index option market, the development of derivatives market of China must be in a direction of getting better and better.