Pricing American Options by Simulation: Overview and New Results

An American option allows the holder to choose the time of exercise, so valuing such an option entails solving an optimal stopping problem, which presents a challenge for Monte Carlo methods. The first part of this talk will be an overview of methods developed in recent years to address this problem. These methods apply weighted backward induction to simulated paths, with weights defined through likelihood ratios, through calibration, or implicitly through regression. The second part of this talk analyzes conditions for convergence as both the number of paths and number of basis functions for regression grow. Using polynomials in the regressions, the number of paths must grow exponentially with the number of basis functions to assure convergence when applied to Brownian motion, faster when applied to geometric Brownian motion. This analysis is based on joint work with Bin Yu.