Research On The Option Pricing Volatility Based On The Inverse Problem Of Black-scholes Equation

Recently, with the continuous development of financial mathematics, option pricing theory is more widely used to various fields. The inverse problem of option pricing is a gradual improvement of mature research which occupies an important position in the field of modern finance theory, which has the value of a strong academic research and practical economic significance.For the Black-Scholes option pricing model, this article studies with the Black-Scholes equation which has stochastic volatility. The main purpose is to solve the unknown parameters of the equation which is called the stochastic volatility. Firstly, this article introduces the basic theory of option pricing and then gives the direct problem of option pricing, and it leads to the inverse problem. In chapter three and four, the first step of the solution process is to construct a finite difference scheme of the Black-Scholes equation, then using two different mathematical methods to inverse volatility. This article overcomes the ill-posedness by using Tikhonov Regularization method and using regular-Gauss-Newton method to solve the problem. The second method to solve the volatility s of the Black-Scholes equation is the homology perturbation method and the modified homotopy perturbation method. Finally this article gives the numerical simulation results and the analysis.The results show that the volatility changes sensitive to the option price. The implied volatility maps out the real price of the option market, and it can reflect some information on future market trends. Most investors consider getting the most valuable and accurate information from the stock or options markets by using the implied volatility. In the two methods of solving the implied volatility, it takes better result when* *S,t electing all values. When more volatility was reconstructed, the result is not very well because of the nonlinear and ill-posedness. Using homotopy perturbation method can get more precise reconstruction and it takes less computing time. It is easier than using the traditional Landweber method.