Convergence of the Least Squares Monte Carlo Approach to American Option Valuation

In a recent paper Longstaff & Schwartz (2001) suggest a method to American option valuation based on simulation. The method is termed the Least Squares Monte-Carlo (LSM) method, and although it has become widely used not much is known about the properties of the estimator. This paper corrects this shortcoming using theory from the literature on seminonparametric series estimators. A central part of the LSM method is the approximation of a set of conditional expectation functions. We show that the approximations converge to the true expectation functions under general assumptions in a multiperiod multidimensional setting. We obtain convergence rates in the two period multidimensional case, and we discuss the relation between the optimal rate of convergence and the properties of the conditional expectation. Furthermore, we show that the actual price estimates converge to the true price. This provides the mathematical foundation for the use of the LSM method in derivatives research.