Research And Improvement Of Option Pricing Under Fractional Brownian Motion

At present,Chinese financial market is increasingly complete and here is a growing demand for financial literacy.People expect to dispose of assets in a reasonable and planned way to obtain the maximum income.In this process,as a financial tool that can hedge risks,option has gradually come into people’s field of vision,and the study of option pricing has attracted people’s extensive attention.The most classic research on option pricing is the Black-Scholes option pricing formula proposed by American scholars Black and Scholes in 1973,which is still an important basis and reference for us to study option pricing until today.Of course,in order to get a pricing model that is closer to market price,scholars in various countries are improving in different directions.In this context,fractional Brownian motion arises in response to the assumption that the underlying asset satisfies geometric Brownian motion.This thesis studies option pricing based on fractional Brownian motion.On the one hand,a new exotic option pricing model,two-asset Asian maximum rainbow option is proposed.And taking call option as an example,different aspects of the geometric average and arithmetic average are analyzed respectively.The analytical solutions of two-asset geometric Asian maximum call rainbow option are obtained by extending the maximum and minimum parity formula under Brownian motion to fractional Brownian motion,and the accuracy of analytical solutions are further verified by the comparison between Monte Carlo simulation value and analytical solutions.For two-asset arithmetic Asian maximum call rainbow option,since it is difficult to obtain analytical solutions,this thesis obtains numerical solutions through Monte Carlo simulation,and optimizes the variance reduction technology to obtain more accurate simulation values.The results show that smaller standard error can be obtained by using dual variable method and thus more accurate simulation value is obtained under the arithmetic condition.On the basis of the above,this thesis also makes a comparative analysis of the price of such options under geometric and arithmetic average conditions.The results show that the price of two-asset arithmetic Asian maximum call rainbow option is always higher and the theoretical proof is also given.On the other hand,this thesis optimizes the pricing of European options and improves the volatility by combining Black-Scholes model with Heston and AHBS models.Then 50ETF option datas are selected for empirical analysis and the results show that the improved model can better fit the option price of the market.Thus it provides a certain reference for investors to make decisions.