A technique recently developed by Longstaff and Schwartz (LS) significantly enhances the Monte Carlo technology for pricing American options, by using regression to streamline the analysis of the subsidiary future paths in modeling the early exercise decisions. But the procedure still requires a large amount of computation and the degree of difficulty explodes as the number of stochastic factors in the problem increases. In this article, the authors show several alterations to the LS approach that can increase its efficiency substantially. The most important involves setting up the problem as a kind of Monte Carlo analysis of Monte Carlo models. For each run, they construct three estimates of the option value using different numbers of simulated paths, for example 1,000, 2,000, and 4,000 paths. This process is then repeated for many runs. Averaging across these multiple Monte Carlo estimates produces a set of three average values that are then used in an extrapolation procedure to estimate the option value that would be produced with an infinite number of paths. The improvement in performance from this two-part strategy is striking, as the authors demonstrate by pricing a Bermudan put option on five underlyings.