The Annals of Applied Probability

Number of paths versus number of basis functions in American option pricing

Paul Glasserman and Bin Yu

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An American option grants the holder the right to select the time at which to exercise the option, so pricing an American option entails solving an optimal stopping problem. Difficulties in applying standard numerical methods to complex pricing problems have motivated the development of techniques that combine Monte Carlo simulation with dynamic programming. One class of methods approximates the option value at each time using a linear combination of basis functions, and combines Monte Carlo with backward induction to estimate optimal coefficients in each approximation. We analyze the convergence of such a method as both the number of basis functions and the number of simulated paths increase. We get explicit results when the basis functions are polynomials and the underlying process is either Brownian motion or geometric Brownian motion. We show that the number of paths required for worst-case convergence grows exponentially in the degree of the approximating polynomials in the case of Brownian motion and faster in the case of geometric Brownian motion.

Article information

Ann. Appl. Probab., Volume 14, Number 4 (2004), 2090-2119.

First available in Project Euclid: 5 November 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 65C05: Monte Carlo methods 65C50: Other computational problems in probability 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

Optimal stopping Monte Carlo methods dynamic programming orthogonal polynomials finance


Glasserman, Paul; Yu, Bin. Number of paths versus number of basis functions in American option pricing. Ann. Appl. Probab. 14 (2004), no. 4, 2090--2119. doi:10.1214/105051604000000846.

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