General Error Estimates for the Longstaff–Schwartz Least-Squares Monte Carlo Algorithm

• Published in: Mathematics of operations research Vol. 45; no. 3; pp. 923 - 946
• Main Author: Zanger, Daniel Z.
• Format: Journal Article
• Language: English
• Published: Linthicum INFORMS 01.08.2020; Institute for Operations Research and the Management Sciences
• Subjects: Algorithms; American options; Approximation; Estimates; Finite element analysis; Least squares method; least-squares regression; Mathematical analysis; Monte Carlo algorithms; Monte Carlo simulation; Operations research; optimal stopping; Optimization; Polynomials; Primary: 65C05; Primary: dynamic programming/optimal control: Markov; Secondary: 62L15, 60J20, 91B70, 93E24; Secondary: Finance: asset pricing/statistics: regression; statistical learning theory
• ISSN: 0364-765X; 1526-5471
• DOI: 10.1287/moor.2019.1017

We establish error estimates for the Longstaff–Schwartz algorithm, employing just a single set of independent Monte Carlo sample paths that is reused for all exercise time steps. We obtain, within the context of financial derivative payoff functions bounded according to the uniform norm, new bounds on the stochastic part of the error of this algorithm for an approximation architecture that may be any arbitrary set of L 2 functions of finite Vapnik–Chervonenkis (VC) dimension whenever the algorithm’s least-squares regression optimization step is solved either exactly or approximately. Moreover, we show how to extend these estimates to the case of payoff functions bounded only in L p , p a real number greater than 2 < p < ∞ . We also establish new overall error bounds for the Longstaff–Schwartz algorithm, including estimates on the approximation error also for unconstrained linear, finite-dimensional polynomial approximation. Our results here extend those in the literature by not imposing any uniform boundedness condition on the approximation architectures, allowing each of them to be any set of L 2 functions of finite VC dimension and by establishing error estimates as well in the case of ɛ-additive approximate least-squares optimization, ɛ greater than or equal to 0.