Free Boundary And Numerical Methods For American Options Pricing

The option pricing and volatility estimate is financial project, financial mathematics problem of leading edge as well as a hot one at present. For a kind of American options which didn't have pricing formula, the article studies the free boundary problem of its pricing. Combining with the free boundary, the article gives faster and more exact numerical method for pricing of American options.The part of this text introduction has done the reviewing of generality to the financial derivative and pricing theory. At the second chapter, the article expatiates the instauration of the Black-Scholes Differential Equation in detail. Then the article deduces the Black-Scholes pricing formula of European options by Fourier transform. At the third chapter of the article, combining with the free boundary, finite element method used for American put options pricing is improved. First, the option pricing problem is transformed to variational inequality equations by variable substitution, then both semi discrete and fully discretized finite element approximation schemes are established. It is proved that the numerical methods are stable and convergent under and norms. Numerical example shows the convergence and efficiency of the algorithm.A fast numerical method for computing American option pricing problems governed by the Black–Scholes equation is presented in the fourth chapter. An accurate artificial boundary condition on the far boundary is found. It makes the computational domain smaller. Then this boundary condition is discretized and combined with a simple numerical method to determine the location of the free boundary. The finite difference method is used to solve the resulting problem. Computational results of some American call option problems show that the new treatment is very efficient and gives better accuracy than the normal finite difference method.