## Mark Broadie

*Enhanced Monte Carlo estimates for American option prices*

Coauthor(s): Paul Glasserman, Gautam Jain.

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**Abstract:**

A methodology to price American options with finitely many exercise opportunities simulates the evolution of underlying assets via random trees that branch at each of the possible early exercise dates. From these trees, two consistent price estimates are obtained, one biased high and one biased low. These two estimates can be combined to provide a valid, though conservative confidence interval for the option price.

This article develops several enhancements to improve the efficiency of the two esimates so that the resulting confidence interval is small. We suggest "pruning" the trees by eliminating branching whenever possible, thus shortening the simulation time and allowing for faster convergence of the estimates. It is shown that branching at the penultimate exercise point is certainly not required whenver a formula for pricing the corresponding European option is available.

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 significantly. Further improvement is possible by combining pruning with this technique. Excellent results are also obtained by selecting the branches from Latin hypercube samples.

An option with infinitely many exercise opportunities can be approximated well by extrapolating a series of options with finitely many exercise points.

**Source:** *The Journal of Derivatives*

**Exact Citation:**

Broadie, Mark, Paul Glasserman, and Gautam Jain. "Enhanced Monte Carlo estimates for American option prices." *The Journal of Derivatives* 5, no. 1 (Fall 1997): 25-44.

**Volume:** 5

**Number:** 1

**Pages:** 25-44

**Date:** Fall
1997