The Annals of Applied Probability

A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options

Daniel Egloff, Michael Kohler, and Nebojsa Todorovic

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Abstract

Under the assumption of no-arbitrage, the pricing of American and Bermudan options can be casted into optimal stopping problems. We propose a new adaptive simulation based algorithm for the numerical solution of optimal stopping problems in discrete time. Our approach is to recursively compute the so-called continuation values. They are defined as regression functions of the cash flow, which would occur over a series of subsequent time periods, if the approximated optimal exercise strategy is applied. We use nonparametric least squares regression estimates to approximate the continuation values from a set of sample paths which we simulate from the underlying stochastic process. The parameters of the regression estimates and the regression problems are chosen in a data-dependent manner. We present results concerning the consistency and rate of convergence of the new algorithm. Finally, we illustrate its performance by pricing high-dimensional Bermudan basket options with strangle-spread payoff based on the average of the underlying assets.

Article information

Source
Ann. Appl. Probab., Volume 17, Number 4 (2007), 1138-1171.

Dates
First available in Project Euclid: 10 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1186755235

Digital Object Identifier
doi:10.1214/105051607000000249

Mathematical Reviews number (MathSciNet)
MR2344302

Zentralblatt MATH identifier
1136.91010

Subjects
Primary: 91B28 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 93E20: Optimal stochastic control
Secondary: 65C05: Monte Carlo methods 93E24: Least squares and related methods 62G05: Estimation

Keywords
Optimal stopping American options Bermudan options nonparametric regression Monte Carlo methods

Citation

Egloff, Daniel; Kohler, Michael; Todorovic, Nebojsa. A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options. Ann. Appl. Probab. 17 (2007), no. 4, 1138--1171. doi:10.1214/105051607000000249. https://projecteuclid.org/euclid.aoap/1186755235


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References

  • Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Academic Press, New York.
  • Boessarts, P. (1989). Simulation estimators of optimal early exercise. Working paper, Carnegie-Mellon Univ.
  • Broadie, M. and Glasserman, P. (1997). Pricing American-style securities using simulation. J. Economic Dynamics and Control 21 1323--1352.
  • Broadie, M. and Glasserman, P. (1998). Monte Carlo methods for pricing high-dimensional American options: An overview. In Monte Carlo Methodologies and Applications for Pricing and Risk Management 149--161. Risk Books.
  • Chow, Y. S., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston.
  • Clément, E., Lamberton, D. and Protter, P. (2002). An analysis of the Longstaff--Schwartz algorithm for American option pricing. Finance and Stochastics 6 449--471.
  • Cover, T. M. (1968). Rates of convergence of nearest neighbor procedures. In Proceedings of the Hawaii International Conference on Systems Sciences 413--415.
  • de Boor, C. (1978). A Practical Guide to Splines. Springer, New York.
  • Devroye, L. (1982). Necessary and sufficient conditions for the almost everywhere convergence of nearest neighbor regression function estimates. Z. Wahrsch. Verw. Gebiete 61 467--481.
  • Egloff, D. (2005). Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab. 15 1--37.
  • El Karoui, N. (1981). Les aspects probabilistes du contrôle stochastique. Lecture Notes in Math. 876 74--239. Springer, Berlin.
  • Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer, New York.
  • Glasserman, P. and Yu, B. (2004). Number of paths versus number of basis functions in American option pricing. Ann. Appl. Probab. 14 1--30.
  • Györfi, L., Kohler, M., Krzyżak, A. and Walk, H. (2002). A Distribution-Free Theory of Nonparametric Regression. Springer, New York.
  • Haugh, M. and Kogan, L. (2004). Pricing american options: A duality approach. Operations Research 52 258--270.
  • Karatzas, I. and Shreve, S. E. (1998). Methods of mathematical finance. Springer, New York.
  • Kohler, M. (2006). Nonparametric regression with additional measurement errors in the dependent variable. J. Statist. Plann. Inference 136 3339--3361.
  • Lamberton, D. and Pagès, G. (1990). Sur l'approximation des réduites. Ann. Inst. H. Poincaré Probab. Statist. 26 331--355.
  • Laprise, S. B., Su, Y., Wu, R., Fu, M. C. and Madan, D. B. (2001). Pricing American options: A comparision of Monte Carlo simulation approaches. J. Comput. Finance 4 39--88.
  • Lepeltier, J. P. and Marchal, B. (1976). Problème de martingale et équation différentielles stochastiques associés à un opérateur inteégrodifférentiel. Ann. Inst. H. Poincaré Sect. B 12 43--103.
  • Longstaff, F. A. and Schwartz, E. S. (2001). Valuing American options by simulation: A simple least-square approach. Review of Financial Studies 14 113--147.
  • Neveu, J. (1975). Discrete-Parameter Martingales, 2nd ed. North-Holland, Amsterdam.
  • Rogers, L. C. G. (2002). Monte Carlo valuing of American options. Math. Finance 12 271--286.
  • Schumaker, L. (1981). Spline Functions: Basic Theory. Wiley, New York.
  • Schweizer, M. (2002). On Bermudan options. In Advances in Finance and Stochastics. Essays in Honour of Dieter Sondermann (K. Sandmann and P. J. Schönbucher, eds.) 257--270. Springer, Berlin.
  • Shiryayev, A. N. (1978). Optimal Stopping Rules. Springer, New York.
  • Stone, C. J. (1982). Optimal rates of convergence for nonparametric regression. Ann. Statist. 10 1040--1053.
  • Tilley, J. A. (1993). Valuing American options in a path simulation model. Transactions of the Society of Actuaries 45 83--104.
  • Tsitsiklis, J. N. 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 Autom. Control 44 1840--1851.
  • van de Geer, S. (2000). Empirical Process in M-Estimation. Cambridge Univ. Press, New York.