European Option Pricing Model Based On Mixed Fractional Brownian Motion In Incomplete Market

In the past two years,with the continuous deepening of financial reform,the development of option market has entered the "fast lane".As a good risk avoidance tool,option is favored by more and more enterprises and investors due to its unique nonlinear advantages and the ability to achieve limited losses but unlimited profits.At the same time,more scholars have devoted themselves to the research of options to help investors more conveniently mine the information behind the market and make reasonable investment judgments by establishing relevant theoretical models.Since the emergence of the classic B-S-M model,option pricing has become one of the important directions for the academic circle to study financial derivatives.However,because the assumptions of this model are too idealized,and there are a lot of uncertainties in the actual financial market,such as randomness and fuzziness,the B-S-M model cannot accurately depict the real market.This paper introduces fuzzy coefficient to establish pricing model to describe market uncertainty.The theoretical part of this paper focuses on two parts:First,considering that stock price volatility has self-similarity and long-term memory,and its return series has non-stationary increment,we use mixed subfractional Brownian motion to describe the underlying asset change process;Secondly,in order to characterize the uncertainty in the market,this article introduces the concept of fuzzy coefficient and takes into account the varying degrees of investors’ preference for risk.Asymmetric selection of fuzzy coefficient values is used.Through a series of mathematical theoretical knowledge,a European option pricing model based on mixed fractional Brownian motion is established under incomplete market conditions,and an explicit solution is obtained.In order to determine whether the pricing model established in this article performs well in characterizing the true price of options,this article uses Huatai Bairui Shanghai and Shenzhen 300 ETF option data for empirical analysis.First,estimate model parameters:estimate Hurst index by R/S method,estimate volatility of subject matter by GARCH(1,1)model,estimate risk-free interest rate by Shanghai Interbank Offered Rate,and estimate expected return rate of subject matter by approximate normal distribution;Secondly,the "97.5%" method and two methods,mean square error and ambiguity,were used to select the optimal lower and upper fuzzy coefficients {k1,k2} and {k1,k2}.Two sets of fuzzy coefficients were substituted into the pricing model to calculate the fitted lower and upper prices of three out of sample option contracts.The proportion of the actual price falling within the fitted upper and lower price ranges was used as the judgment basis.It was found that the upper and lower prices determined by the "97.5%" method were The fitting performance of substituting the fuzzy coefficient into the pricing formula is better;Finally,a comparative model is introduced to determine the asymmetric fuzzy coefficient using the "97.5%" method and substitute it into the pricing formula.The fitting results are compared with the pricing model established in this paper.The results indicate that both have good fitting effects,but the model established in this paper has higher characterization accuracy,and the fitting price of the comparative model is closer to the real price.In conclusion,the European option pricing model with asymmetric fuzzy coefficients established on the basis of underlying assets subject to mixed subfractional Brownian motion has further enriched the option pricing theory,filled the research gap,and also has practical application value.