Recently, various authors proposed Monte-Carlo methods for the computation of American option prices, based on least squares regression. The purpose of this paper is to analyze an algorithm due to Longstaff and Schwartz. This algorithm involves two types of approximation. Approximation one: replace the conditional expectations in the dynamic programming principle by projections on a finite set of functions. Approximation two: use Monte-Carlo simulations and least squares regression to compute the value function of approximation one. Under fairly general conditions, we prove the almost sure convergence of the complete algorithm. We also determine the rate of convergence of approximation two and prove that its normalized error is asymptotically Gaussian.

This article presents how to use the least-squares (LS) regression method to price the American options on basis of the algorithm in a paper by Clement, Lamberton & Protter[1]. The key to LS is the approximation of the conditional expectation functions which determine the optimal exercise strategy. In this paper, through the detailed description of the algorithm and presentation of convergence, it shows how to estimate the conditional expectation by using the LS to value the American options. Moreover, we also compare the simulation results with the historical data and experiment with alternative polynomials to analyze testing results.