Analysis on the convergence of the Least Squares Monte Carlo method and the Finite Differences Method on the valuation of American options

Abstract

American options, which permit early exercise, pose considerable challenges in financial valuation due to the intricacies involved in identifying optimal exercise strategies. Conventional analytical models are often inadequate for these options, particularly in high-dimensional contexts, thereby necessitating the use of robust numerical techniques such as the Least Squares Monte Carlo (LSM) and Finite Difference (FD) methods. The LSM method is distinguished by its adaptability in addressing high-dimensional issues, employing a combination of Monte Carlo simulations and regression analysis to estimate continuation values, which renders it particularly effective for complex, path-dependent options. Conversely, the FD method utilizes a grid-based framework to solve the partial differential equations (PDEs) that dictate option pricing, providing stable and dependable solutions, especially in one-dimensional scenarios. This study compares two numerical methods—Least Squares Monte Carlo and Finite Difference Methods —for pricing American options under a SABR-based volatility calibration. LSMC offers flexibility and moderate runtime in higher-dimensional contexts, while FDM provides a deterministic, systematically refined solution. Both approaches are benchmarked against binomial-lumps and Bjerksund Stensland references for a short-dated American call, with emphasis on computational efficiency, accuracy, and the practical implications of adopting a lognormal SABR model to ensure market-consistent volatility. We analyze convergence rates, computational runtimes, and pricing accuracy across varying grid resolutions, simulation sizes, and basis selections. Our results delineate the trade-offs between flexibility, dimensional scalability, numerical stability and model run time, offering investors clear guidelines on choosing the optimal technique, LSMC for high-dimensional, path-dependent payoffs and FDM for one-dimensional problems demanding tight error control under SABR-driven market dynamics.

Keywords: American options, Implied Volatility (IV), Least Squares Monte Carlo (LSMC), Finite Differences Methods (FDM), Stochastic Alpha, Beta, Rho (SABR) Model, Control Variate (CV), Antithetic Variate (AV)