The Numerical Method Of Option Pricing And Nonlinear Dynamical Model Of Volatility Estimate

The pricing problem of the American Put Option and volatility estimate are currently studied as two of the important items in the option pricing theory. This paper is devoted to a new computational algorithm of numerical solution of Black-Scholes equation of American Put Option based on wavelet and a new evaluation method on important factor- volatility, which has an important influence on the pricing of option, based on nonlinear dynamic analysis. After a simple reviewing of the Black-Scholes Option Pricing Model, the article discretely segments the solution space of partial differential equation for American put option price with multiresolution analysis in chapter 3, applying the finite difference scheme for the time variable and the orthonormal compactly supported wavelet transform approximation for the space variable. Due to adaptive and vanishing moment property of wavelet basis, the difference operator and solution vector and wavelet series have sparseness in wavelet domain. Numerical results show the effectiveness and potential of the new method to price American Put Options in decreasing computational complexity. In chapter 4, the article cites the CBOE Volatility Index, which was introduced by the Chicago Board Options Exchange in 1993 and measures market expectation of near term volatility conveyed by stock index option prices. A new evaluation method of volatility is given applying the computational algorithm of VIX,the nonlinear dynamic system and chaos theories ,which is a new forecasting algorithm base on the method of phase space reconstruction and Takagi-Sugeno fuzzy model .As an example, the forecasting algorithm was applied to analyze VIX. The results show that the option market is a high-dimensions nonlinear system and the forecasting algorithm is effective.