The Research Of Volatility Based On Stock Index Options

The study of option pricing,to a great extent,is the study of its volatility.Volatility pricing model can be divided into three categories: the first is the time series model based on historical data;the second is the B-S model which the volatility parameters has improved;the third is the implied volatility model.In order to give a comprehensive analysis of the volatility and use volatility to quantify people' s attitude in the face of risk,the paper starting from the above three points,applies the Tsallis entropy,Ito formula and dupire formula to combine Tsallis entropy distribution with update process and local volatility model with implied volatility respectively.Reached the purpose of expanding the existing volatility model.Firstly,this article derives B-S model formula using the diffusion coefficient methods and analyzes the factors affecting the option price.It finds that there is a stable long-term equilibrium relationship between the option price volatility and the stock price volatility and the volatility has the clustering time-varying feature.Secondly,the traditional B-S model's fluctuation parameters are optimized,Considering that the stock price has the mean reversion,long memory and the characteristics of fat-tailed,the exponential O-U process and the distribution of Tsallis entropy are used to improve the drift and random fluctuations respectively in this paper.And supposing information coming is a renewal process which is more common than Possion process,this paper deduces the partial differential equation when stock price obeys a kind of renewal jump-diffusion process using the APT theory and generalized Ito formula,at last obtains the European pricing formula by Feynman-Kac formula as well as the method of equivalent martingale.The last we use the implied volatility for modeling the important parameters in Local fluctuation model——local volatility.The implied volatility is calculated through the Newton-Rapthson iterative method when option price,expire time,interest rate and other conditions are already known,then smoothes its surface by using the cubic spline interpolation method,and applies the dupire formula to construction of the Local Volatility Surface.