Monte Carlo Simulation for American Options

This paper reviews the basic properties of American options and the difficulties of applying Monte Carlo valuation to American options. Asymptotic results by Keller and co-workers are described for the singularity in the early exercise boundary for time t near the final time T. Recent progress on application of Monte Carlo to American options is described including the following: Branching processes have been constructed to obtain upper and lower bounds on the American option price. A Martingale optimization formulation for the American option price can be used to obtain an upper bound on the price, which is complementary to the trivial lower bound. The Least Squares Monte Carlo (LSM) provides a direct method for pricing American options. Quasi-random sequences have been used to improve performance of LSM; a brief introduction to quasi-random sequences is presented. Conclusions and prospects for future research are discussed. In particular, we expect that the asymptotic results of Keller and co-workers could be useful for improving Monte Carlo methods.