The Annals of Applied Probability

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

Paul Glasserman and Bin Yu

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 5 November 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1099674090

Digital Object Identifier
doi:10.1214/105051604000000846

Mathematical Reviews number (MathSciNet)
MR2100385

Zentralblatt MATH identifier
1062.60041

Subjects
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]

Keywords
Optimal stopping Monte Carlo methods dynamic programming orthogonal polynomials finance

Citation

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. https://projecteuclid.org/euclid.aoap/1099674090


Export citation

References

  • Bertsekas, D. and Tsitsiklis, J. (1996). Neuro-Dynamic Programming. Athena Scientific, Belmont, MA.
  • Broadie, M. and Glasserman, P. (1997). Pricing American-style securities by simulation. J. Econom. Dynam. Control 21 1323–1352.
  • Broadie, M. and Glasserman, P. (1997). A stochastic mesh method for pricing high-dimensional American options. PaineWebber Series in Money, Economics and Finance. #PW9804, Columbia Business School, Columbia Univ.
  • Carrière, J. (1996). Valuation of early-exercise price of options using simulations and nonparametric regression. Insurance Math. Econom. 19 19–30.
  • Clément, E., Lamberton, D. and Protter, P. (2002). An analysis of a least squares regression algorithm for American option pricing. Finance Stoch. 6 449–471.
  • Duffie, D. (2001). Dynamic Asset Pricing Theory, 3rd ed. Princeton Univ. Press.
  • Egloff, D. (2003). Monte Carlo algorithms for optimal stopping and statistical learning. Working paper, Zürich Kantonalbank, Zürich, Switzerland.
  • Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer, New York.
  • Haugh, M. and Kogan, L. (2004). Pricing American options: A duality approach. Oper. Res. To appear.
  • Lamberton, D. and Lapeyre, B. (1996). Introduction to Stochastic Calculus Applied to Finance. Chapman and Hall, London.
  • Longstaff, F. A. and Schwartz, E. S. (2001). Valuing American options by simulation: A simple least-squares approach. Review of Financial Studies 14 113–147.
  • Muir, T. and Metzler, W. (1960). A Treatise on the Theory of Determinants. Dover, New York.
  • Musiela, M. and Rutkowski, M. (1997). Martingale Methods in Financial Modeling. Springer, New York.
  • Owen, A. B. (2000). Assessing linearity in high dimensions. Ann. Statist. 28 1–19.
  • Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
  • Rogers, L. C. G. (2002). Monte Carlo valuation of American options. Math. Finance 12 271–286.
  • Tsitsiklis, J. and Van Roy, B. (1999). Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives. IEEE Trans. Automat. Control 44 1840–1851.
  • Tsitsiklis, J. and Van Roy, B. (2001). Regression methods for pricing complex American-style options. IEEE Transactions on Neural Networks 12 694–703.
  • Wilmott, P., Howison, D. and Dewynne, J. (1995). The Mathematics of Financial Derivatives. Cambridge Univ. Press.