Approximating the exact value of an American option

Abstract

An American option is a derivative security that can be exercised at any time before expiration. Under standard hypotheses it can be shown that its arbitrage-free price is the solution of an optimal stopping problem. Usually, if the underlying asset follows a diffusion, the stopping time problem does not have a closed form solution. Therefore, discrete time models have been proposed to determine an approximated solution. I formulate some conditions on the discrete process to insure convergence of the approximations to the exact value. I also show how to apply such conditions to check the correctness of some of the most popular discretization schemes.