Multi-fractal Volatility Forecasting And Its Application In Black-Scholes Model

With the development of Chinese financial market,the market demand for the pricing of financial derivatives has become more and more urgent.However,the classical Black-Scholes pricing model assumes that the volatility is constant,which obviously does not conform to the actual situation of the financial market.In previous studies,many scholars have used GARCH models to estimate the stock volatility;however,in recent years the research about Chinese financial market found that the financial market has multi fractal properties,namely the long-term memory.In order to reflect the characteristics in the B-S model,we will introduce a new method for measuring the volatility of multi fractal(MVM),and establish the ARFIMA model which can describe the long-term memory of time series,analyze the influence of different multi fractal volatility measurement methods and its dynamic models on the pricing of financial derivatives.In this paper,we will take the 5-minute high frequency data of Baogang stock as an example.Firstly,we study the statistical characteristics of the stock price,the daily return rate and the realized volatility series in the sample interval.The research results show that the daily return rate sequence and the realized volatility sequence are not subject to normal distribution,but have more complex multi fractal characteristics.In order to establish an accurate prediction model of volatility,and combined with the multi fractal characteristics of the sequence,we will introduces the new method based on multifractal volatility measure(MVM)which build on the realized volatility and fractal spectrum standard deviation.Then established the short memory model ARMA(ARMA-LnRV?ARMA-LnMVM etc.)and the long memory model ARFIMA(ARFIMA-LnRV?ARFIMA-LnMVM etc.)to forecast the volatility of Baogang stocks in the next 10 days.Finally,substituting the different volatility into the B-S model which forecasted by each model to estimate the price of Baogang warrants in the sample interval.And compare the difference between the theoretical price and market price of warrants in the next 10 days,according to the relative error between theoretical price and the market price to determine the impact of different volatility on the B-S model pricing accuracy.The research result shows that the new method based on multifractal volatility measure is superior to other forecasting models.the theory price of the warrants calculated by the Black-Scholes based on the ARFIMA-MVM model is closer to the market price than other volatility models.