European Option Pricing Under The Fractional Black-scholes Model

The classic Black-Scholes option pricing formula is built on a large number of rigorous and unrealistic assumptions, which making the practical application of Black-Scholes option pricing are not very good. In the past few years, financial market has been considered a complex and nonlinear dynamic system. A large number of studies have found that many financial time series have shown scaling law and long-term dependence. According to behavioral finance point of view and long-range dependence in stock returns in the empirical results, we studied the underlying stocks payment of dividends and option pricing with transaction costs and volatility estimation.According to the view of behavioral finance, we replace Brownian motion in the classic Black-Scholes model by fractional Brownian motion, replace Bayes theorem by anchoring- adjusting strategy, and replace Ito formula in continue-time trade setting by Taylor formula under limited rational investors, to resolve European option pricing with transaction costs and payment of dividends under the discrete fractional Black-Scholes model. By discussing a mean-self-financing Delta-hedging argument, we obtain European option pricing equation and European option pricing formula by approximate hedging. Furthermore, we get the minimum value of European call option while considering transaction costs and the payment of dividends, and the minimum value can be regarded as its real value. Particularly, we obtain a new estimation method of volatility. Finally, we solve volatility on different scales and the Hurst index, and we know that scaling and long-term dependence have a significant influence on the volatility.