Under The Fractional Brownian Motion Environment Supposes The Reset Option Pricing Formula

The option is a kind of financial derivation tool which appeared in the mid 20th century of America, since more than 20 years as an effective means of prevention and speculation which obtained rapid development. Regarding the traditional Black-Scholes option pricing model, do-mestic and foreign scholars have done a lot of research work, obtained a lot of financial sense of the results. However, this traditional formula is based on the efficient market hypothesis, in recent years a large number of empirical studies suggest that stock price changes do not meet the normal distribution, they exhibit a "spike, fat tail" distribution and the stock is not between the random walk, and at different times related to the existence of long-term, self-similarity and other features, which have a certain gap between the geometric Brownian motion. But the time fractional Brow-nian motion happen to have the long-term, self-similarity and other features, its characteristic exponent as well as the scale parameter and so on can well describe the financial market volatil-ity, stock price process and the general conduct of "spike, fat tail" distribution. Therefore, as the underlying assets in the stock option pricing study, the stock price follows Brownian motion of op-tion pricing will have the feasibility compared to the traditional standard Brownian motion driven and more suitable for solving financial problems in the capital market.This article focuses on option pricing research under fractional Brownian motion environment, applied stochastic processes, martingale theory, stochastic analysis and mathematical tools, trying to promote some conclusions of the general pricing formula. Specifically, the main work is as follows:①Suppose the underlying asset price follows the geometric Brownian motion, obtained Euro-pean call and put option pricing formula and parity under the model;②In one hypothesis, discussed reset option pricing when interest rate is non-random variable , obtained its pricing formula;③In one hypothesis, discussed reset option pricing when interest rate is random variable, obtained its pricing formula.