Penalised regression adaptations of 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, a Monte Carlo type algorithm where orthogonal polynomials are typically used to estimate the expected future payoff given the current value of the American option from simulated paths of the process. 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 which show that, under certain regularity conditions, and using a least squares estimation approach, this average payoff converges to the true price as the sample size of the paths and the order of the orthogonal polynomial go simultaneously to infinity. Various alternative modelling and estimation approaches have been attempted to make the Longstaff-Schwartz algorithm more accurate and computationally efficient. However, studies on the use of penalised regression approaches are scarce. Under different sample path and polynomial order settings, in this paper, an empirical assessment is conducted of the benchmark least squares method in comparison with the Ridge, LASSO and elastic net estimation to see which of these methods is the best in terms of accuracy and computational efficiency. This comparison is done on three staple processes in finance, namely geometric Brownian motion, the Heston stochastic volatility and a model based on the Meixner jump processes, while convergence properties for the standard Longstaff-Schwarz approach is used to determine a benchmark for accuracy. Across most settings in the simulation design, LASSO resulted in the best precision across the four algorithm variations, followed closely by elastic net. Ridge regression often produced results which were less accurate than LASSO and elastic net, however, these were also often more precise than the least squares approach. A discussion ensues on the computational aspect, where for each process it is determined which methods achieve the best compromise between precision and execution time.