Stochastic Mesh Method for Optimal Stopping Problems

Abstract

This paper considers the application of the stochastic mesh method in solving the multidimensional optimal stopping problem for a diffusion process with nonlinear payoff functions. A special discretization scheme of the diffusion process is presented to solve the problem in the case of geometric average Asian option payoff functions. This discretization scheme makes it possible to eliminate singularities in transition probabilities. Next, two estimates are given of the solution of the problem by the stochastic mesh method for the case of the stochastic mesh transition probabilities defined as averaged densities. The consistency of the estimates is proven. It is shown that the variance of the estimates is inversely proportional to the number of points in each mesh layer. The result extends the application area of the stochastic mesh method and methods for treating Asian options. A numerical example of the result of applying the obtained estimates to the call and put options compared to the obtained option prices using a regular mesh is presented.