Penalised alternatives to ordinary least Squares in the Longstaff-Schwartz algorithm for pricing American options

Abstract

One of the most popular techniques for evaluating the American put option is the Longstaff-Schwartz algorithm. In this algorithm, orthogonal polynomials are typically used to estimate the maximum expected future payoff given the current value of the American option. An optimal exercise strategy then ensues for each of these paths, where the average payoff over all paths becomes equivalent to the fair price of the American option. Convergence results have been proven over the years which show that, under certain regularity conditions and using a least squares estimation approach, this average payoff converges in probability to the true price as the sample size of the paths and the order of the orthogonal polynomial go simultaneously to infinity. A number of alternative modelling and estimation approaches have been attempted to make the Longstaff-Schwartz algorithm more accurate and computationally efficient; however a detailed insight at penalised regression methods and how they fare within this context is not found in literature. In this thesis we conduct an empirical assessment of OLS, Ridge, LASSO and Elastic Net estimation to see which of these methods are the best compromise in terms of accuracy and computational efficiency. We compare these methods on three staple processes in finance, namely the Geometric Brownian Motion, Heston Stochastic Volatility and Meixner processes. Furthermore, we use OLS results for large samples and a high number of basis functions as a benchmark for accuracy.