Pricing high-dimensional Bermudan options using the stochastic grid method

Abstract

This paper considers the problem of pricing options with early-exercise features whose pay-off depends on several sources of uncertainty. We propose a stochastic grid method for estimating the optimal exercise policy and use this policy to obtain a low-biased estimator for high-dimensional Bermudan options. The method has elements of the least-squares method (LSM) of Longstaff and Schwartz [Valuing American options by simulation: A simple least-squares approach, Rev. Finan. Stud. 3 (2001), pp. 113–147], the stochastic mesh method of Broadie and Glasserman [A stochastic mesh method for pricing high-dimensional American option, J. Comput. Finance 7 (2004), pp. 35–72], and stratified state aggregation along the pay-off method of Barraquand and Martineau [Numerical valuation of high-dimensional multivariate American securities, J. Financ. Quant. Anal. 30 (1995), pp. 383–405], with certain distinct advantages over the existing methods. We focus on the numerical results for high-dimensional problems such as max option and arithmetic basket option on several assets, with basic error analysis for a general one-dimensional problem.

Keywords:

• American options
• high dimensional
• Monte Carlo
• Gram Charlier
• stochastic grid method
• regression
• stochastic mesh method
• least squares method (LSM)
• Bermudan options

2000 AMS Subject Classifications :

• 65C05
• 65C30
• 62P05
• 91B28
• 60G40

Acknowledgements

The second author thanks CWI-Centrum Wiskunde & Informatica, Amsterdam.

Notes

A Bermudan option is an option where the buyer has the right to exercise at a set (discretely spaced) of times. This is intermediate between a European option which allows exercise at a single time, namely expiry and an American option, which allows exercise at any time. With an increasing number of exercise opportunities Bermudan option values approach the value of an American option.

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