Options Pricing Based On Several Special Hurst Index

With the gradual development of financial markets, many scholars have found that the financial market is not fully subject to the standard Brownian motion. In most cases it follow the "biased random walk" process. The efficient market hypothesis theory can not fully explain the financial market. In order to solve more financial problems, to make a more realistic theory, fractal mathematics theory was introduced to the financial field. The standard Brownian motion is being replaced by fractional Brownian motion gradually. The finance theory has been further developed.This article mainly divided into four parts. The first part gave the definition and nature of fractional Brownian motion. Since all of the financial processes should be based on the no-arbitrage assumption, we have taken semi-martingale process method which is based on original one to close the whole process. In the second part, first we proved that there is an equivalent martingale measure, under which martingale the fractional Brownian motion market is no arbitrage, and for H?(1/n ,1/( n1)), we also gave the similar provf. Following we studied fractional Brownian motion to close the financial market which Hurst index is belong to (1/4,1/3) and (1/n ,1/( n- 1)),and got the partial differential equations based on the fractional Brownian motion in the no-arbitrage market. The third part discussed the mixture of fractional Brownian motion, and studied the situation which H₁(?) (1/4,1/3), H₂(?)(1/2,1)and H₁/+H₂> 1/to get the partial differential equations.The forth chapter use R / S analysis to check Chinese stock market, and get the situation when Hurst index belong to the interval(1/4,1/3) , and illustrated it , and also proved the applicability of this thesis.