European Option Pricing Under Long Memory Stochastic Volatility With Time-varying Hurst Index

In financial mathematics, the Black-Scholes option pricing model is one of the mostclassic option pricing model and is widely known. It relates option prices to current stockprices with a constant volatility risk. But Black-Scholes model is based on many assumptions,which has violated the reality of financial market conditions, so the results are not accurate.So far, there have been many researchers to improve and modify and achieve some goodmodels on the basis of Black-Scholes model, one of Which is that volatility is not a constant,but is a function of stock price or a random process. a large number of empirical analysisresults indicates that the volatility of high frequency data is not only stochastic but also haveobviously long memory. Assuming the volatility as stochastic process is conformable withrealistic financial markets, and the fractional Brownian motion has self-similarity and longmemory, the fractional brown motion instead of Brownian motion to pricing options is morecloser to the truth.According to empirical results, this article assumes that stock pricesS tis GeometricBrownian Motion,and volatility σ (t)is multfractional Brownian motion.Based on the model, this paper got the option price formula.Based on thought of Mean-Variance analysis, this article will define expectation ofoption price as option market price, and got option market priceFor the above conclusions, this paper introduces in detail the derivation process of theoption pricing formula. In the last chapter, through analysing the stock index data from1996to2012of Shanghai Composite Index, we found that stock index volatility has a strong longmemory, and Hurst index values in different periods are also different, so volatility which isdescribed by time variable Hurst index is more appropriate.