Convergence of the Least Squares Monte Carlo Approach to American Option Valuation
从半非参数级数估计理论出发，证明了最小二乘蒙特卡洛方法在一般多期多维设定下条件期望逼近的收敛性，并给出了两期多维情形下的收敛速度，为该方法在衍生品研究中的应用提供了数学基础。
In a recent paper, Longstaff and 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.