Research On European Option Pricing In Sub-fractional Brownian Motion Environment

With the rapid development of financial market, the option as a kind of financial derivatives came into being, because of hedging effect, it is favored by the majority of investors. Among the research results of option pricing,the classic B-S model is the most widely used ,which assumes that the financial asset price is subject to the geometric Brownian motion,and the yield is subject to the normal distribution. But the financial empirical study shows that the change of the price of the financial asset are self-similarity and long dependency, and presents a "higher peak and fat tail" distribution. So some scholars try to use the fractional Brownian motion to improve the classic B-S pricing model, although the new model can reflect the long-dependent nature of financial assets, it will produce arbitrage. Therefore, in the process of option pricing, it is of great practical significance to find a suitable model to describe the change of the underlying asset price.Sub-fractional Brownian motion not only has the nature of self-similarity and long dependency but also has a non-stationary second-order increment, which can describe the change of the price of financial assets better than the fractional Brownian motion. In this paper,we study the pricing of European options in the sub-fractional Brownian motion environment. Firstly, the assumptions are relaxed on the basis of the classical B-S model.Then, the sub-fractional Ito formula and stochastic differential equation theory and the hedging principle of the portfolio are used to derive the stochastic differential equation which is satisfying the option, and finally the two pricing formulas of the European option are obtained by the substitution of the variable into the classical heat conduction equation, that is, European option pricing formula under sub-factional Brownian motion with dividend payments and European option pricing formula under sub-fractional Vasicek stochastic interest rate model.1European option pricing under sub-fractional Brownian motion with paying dividendsThe differential equation for underlying assets is as follows?European option pricing under sub--fractional Vasicek stochastic interest rateUnder the risk neutral measures rates satisfy stochastic differential equation is as followsdr_t=?(u-r_t)dt+?_rd?_H~1(t)Risk assets stock prices satisfy stochastic differential equations is as followsdS_t=r_tS_tdt+?_rd?_H~2(t)In the empirical simulation section, the 50ETF option is used for simulation analysis.Firstly,testing the applicability of the data. Secondly,using the moment estimation and maximum likelihood estimation method,and simulating the value of the parameter to be estimated by Monte Carlo.Again, using the classical B-S model and the sub-fractional Brownian motion model proposed in this paper to simulate the price of the asset and comparing with the real price change path. Finally, the prices of the underlying asset simulated by the different models are brought into the corresponding option pricing formulas to obtain the prices of the option.The simulation result shows that the pricing model proposed in this paper are closer to the real value of the option than the classical B-S model, which explains the validity of the proposed model. The study not only puts forward a new direction for the pricing of the option, but also takes a tentative step in practice, at the same time provides the theoretical basis for the application of option pricing in risk management.