American option pricing with randomized quasi-Monte Carlo simulations

Abstract

We study the pricing of American options using least-squares Monte Carlo combined with randomized quasi-Monte Carlo (RQMC), viewed as a variance reduction method. We find that RQMC reduces both the variance and the bias of the option price obtained in an out-of-sample evaluation of the retained policy, and improves the quality of the returned policy on average. Various sampling methods of the underlying stochastic processes are compared and the variance reduction is analyzed in terms of a functional ANOVA decomposition.

Randomized Quasi-Monte Carlo: An Introduction for Practitioners

“…Applications in finance and computer graphics can be found in [31,66]. There are combinations with splitting techniques (multilevel and without levels), with importance sampling, and with weight windows (related to particle filters) [8,30], combination with "coupling from the past" for exact sampling [39], and combination with approximate dynamic programming and for optimal stopping problems [11].…”

Lattice point sets for state sampling in approximate dynamic programming

“…This obviously introduces a further error with respect to the true optimal cost‐to‐go that, in general, may be non‐negligible, and decreases as S grows. Suitable Monte Carlo techniques, such as importance sampling (possibly combined with LDSs), can be employed to generate the S vectors efficiently depending on the distribution of θ t . However, since we are focusing on efficient sampling of the state space, we do not consider this error in the definition of .…”

“…Suitable Monte Carlo techniques, such as importance sampling (possibly combined with LDSs), can be employed to generate the S vectors efficiently depending on the distribution of t . 23,35 However, since we are focusing on efficient sampling of the state space, we do not consider this error in the definition of e J t . The termẽ J t is sometimes called "approximation error" in the literature, 40 and depends on how closely the best element in the class Ξ t approximates the unknown function.…”

“…Another kind of sequences that have been successfully employed in the ADP context are low‐discrepancy sequences (LDSs), a family of methods commonly adopted for numerical integration and number‐theoretic algorithms . Most of the works available in the literature rely on LDSs to efficiently simulate state transitions . However, LDSs have been employed in the literature also to directly sample the state space, which is the framework that we address in this paper.…”

“…20,21 Most of the works available in the literature rely on LDSs to efficiently simulate state transitions. 22,23 However, LDSs have been employed in the literature also to directly sample the state space, [10][11][12]24,25 which is the framework that we address in this paper.…”