# Robustness of the Longstaff-Schwartz LSM Method of Pricing American Derivatives

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This Demonstration shows the Longstaff–Schwartz least squares Monte Carlo method of computing the value of an American put option. We approximate the American option by a Bermudan option with 50 exercise times. The key point of the method is the estimation of conditional expectations by least squares regression on data obtained from Monte Carlo simulation using a basis for the space of -functions. In this Demonstration we use three different kinds of bases. We show the graphs of the functions that estimate conditional expectations at the exercise points on a graph, below the value of the option computed by means of the Longstaff–Schwartz method, and the value computed by means of the binomial tree method with 1000 time steps.

Contributed by: Andrzej Kozlowski (March 2011)

Open content licensed under CC BY-NC-SA

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The Longstaff–Schwartz least-squares Monte Carlo method of valuing American type options is one of the most popular ones due to its flexibility. It can be used with many models of stock movements, but here we use the classical Black–Scholes model.

This Demonstration was inspired by the paper of Moreno and Navas, in which the robustness of the method with respect to a change of basis and the maximum degree of basis polynomials was investigated. Here we only consider three types of bases: monomial , Laguerre (used in the original Longstaff–Schwartz paper), and Chebyshev. For each choice of basis we allow the degree to be changed as well as the number of paths in a Monte Carlo sample. In addition you can choose whether the computations should be performed with scaled data or not. Scaling produces more accurate results with a Laguerre basis because the presence of exponential terms can lead to significant numerical errors.

For responsiveness we are forced to use much smaller sample sizes than the ones used in the Moreno–Navas article. Nevertheless comparison with option values computed with the Cox, Ross, Rubinstein binomial method with 1000 steps (the same as used by Moreno and Navas) shows that even with small sample sizes the accuracy of the Longstaff–Schwartz method is quite impressive.

F. A. Longstaff and E. S. Schwartz, "Valuing American Options by Simulation: A Simple Least-Squares Approach," *Review of Financial Studies*, 14(1), 2001 pp. 113–147.

M. Moreno and J. F. Navas, "On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives," *Review of Derivatives Research*, 6(2), 2003 pp. 107–127.

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