On the Stability the Least Squares Monte Carlo

TL;DR

This paper analyzes the stability of the Least Squares Monte Carlo algorithm for pricing American options, showing that it becomes ill-conditioned as the number of exercise dates increases, especially for continuous underlying processes.

Contribution

It provides a theoretical analysis of the stability issues in LSM, highlighting conditions under which the regression problem becomes ill-conditioned.

Findings

Regression problem is ill-conditioned for small time steps.

Stability deteriorates as the number of exercise dates increases.

Ill-conditioning is linked to the continuity of the underlying process.

Abstract

Consider Least Squares Monte Carlo (LSM) algorithm, which is proposed by Longstaff and Schwartz (2001) for pricing American style securities. This algorithm is based on the projection of the value of continuation onto a certain set of basis functions via the least squares problem. We analyze the stability of the algorithm when the number of exercise dates increases and prove that, if the underlying process for the stock price is continuous, then the regression problem is ill-conditioned for small values of the time parameter.