A bias-reduction technique for Monte Carlo pricing of early-exercise options

Abstract

T. Whitehead thanks UWO, OGS, and NSERC for financial support. M. Davison and M. Reesor thank NSERC for discovery grants and MITACS for financial support. All of us thank SHARCNET for computational resources and Allan B. MacIssac for helpful comments on an earlier version of this manuscript. AbstractWe present a new method for reducing the bias present in Monte-Carlo estimators of the price of American-style contingent claims. At each exercise opportunity (in a time discretization), we assume there is an unbiased estimator of the claim value at the next exercise opportunity. We approximate the distribution of this statistic using the central limit theorem, and use this to derive an asymptotic expression for the bias. This expression is easily estimated in the context of a simulation, which allows for the straightforward computation of bias-reduced estimators of the claim value. We conclude by presenting a well-studied multivariate pricing example to show that this method offers significant improvements over the vanilla stochastic mesh technique, and that it is much more computationally efficient approach to reducing bias than nonparametric bootstrapping.

American Option Pricing with Importance Sampling and Shifted Regressions

“…where τ(s) n = k(s) n ∆t denotes the path-n LSM-s stopping time. To see why LSM-s is better adapted to the task of estimating stopping times, we consider the work by Whitehead et al (2012) and Kan and Reesor (2012) which derive approximations to the bias of Monte Carlo estimators of American option prices. Whitehead et al (2012) do this in the context of the easy to analyze stochastic tree, while Kan and Reesor (2012) provide analogous derivations for LSM estimators.…”

“…To see why LSM-s is better adapted to the task of estimating stopping times, we consider the work by Whitehead et al (2012) and Kan and Reesor (2012) which derive approximations to the bias of Monte Carlo estimators of American option prices. Whitehead et al (2012) do this in the context of the easy to analyze stochastic tree, while Kan and Reesor (2012) provide analogous derivations for LSM estimators. In both cases, the approximation to estimator bias depends on the ratio Var/N, where N is the sample size used to construct the continuation value estimator and Var is the estimated variance.…”