Enhanced Monte Carlo Estimates for American Option Prices

Abstract

A simulation-based methodology to price American options with nite exercise opportunities has recently been introduced by Broadie and Glasserman [1995a]. This method simulates the evolution of underlying assets via random trees that branch at each of the possible earlyexercise dates. From these trees, two consistent and asymptotically unbiased price estimates, one biased high and one biased low, are obtained. These two estimates can be used to give a conservative condence interval for the option price. In this paper, we develop several enhancements to improve the eciency of the two estimates so that the resulting condence interval is small. Since branching can be computationally very expensive, we suggest \pruning the trees by eliminating branching whenever possible, thus cutting down the simulation time and allowing for faster convergence of the estimates. In particular, it is shown that branching at the penultimate exercise point is certainly not required whenever a formula for pricing the corresponding European option is available. Next, in order to further improve the estimators, we forego the idea of generating branches from independent samples. Indeed, we demonstrate that if half of the branches at a node are generated using the antithetic variates of the other half, both bias and variance are reduced signicantly. Further enhancement is possible by combining pruning with this technique. It is also shown that by selecting the branches from Latin hypercube samples, better results are obtained. We conclude by showing that an option with innite exercise opportunities can be well approximated by extrapolating a series of options with nitely many exercise points.

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