Finite Difference Methods For Pricing The American Put Options

Options is one of the core tools of financial derivatives. It plays an important role in the effective management of risk and speculation. Risk mangement oppupies the right evaluation of options . The critical thing that the options securities exist reasonably and develop properly is how to value its fair price. A wide variety of the options traded in exchanges are American option. It is thus important to find appropriate ways to price American options. Thus, unlike European options, no explicit closed-form formulas have been found for American options, approximation methods have to be used in practice. At present, the most commonly used methods include: Monte Carlo methods, binomial tree, finite element and finite difference methods, etc. We recall that the classic Black-Scholes model for an American option leads to a free boundary problem with a degenerate partial differential operator. In this work, we use finite difference methods for pricing the American put stock options. This paper presented two difference schemes for pricing the American put stock options: one is implicit scheme, the other is modified Crank-Nicolson scheme. The analysis of stability, convergence and error estimation of solutions is presented by utilizing energy methods. Using Gauss-Seidel iteration method to solve difference inequality equations is also presented. Numerical examples show the convergence and efficiency of our algorithm.