Advertisement

A Review on Regression-based Monte Carlo Methods for Pricing American Options

  • Michael Kohler

Abstract

In this article we give a review of regression-based Monte Carlo methods for pricing American options. The methods require in a first step that the generally in continuous time formulated pricing problem is approximated by a problem in discrete time, i.e., the number of exercising times of the considered option is assumed to be finite. Then the problem can be formulated as an optimal stopping problem in discrete time, where the optimal stopping time can be expressed by the aid of so-called continuation values. These continuation values represent the price of the option given that the option is exercised after time t conditioned on the value of the price process at time t. The continuation values can be expressed as regression functions, and regression-based Monte Carlo methods apply regression estimates to data generated by the aid of artificial generated paths of the price process in order to approximate these conditional expectations. In this article we describe various methods and corresponding results for estimation of these regression functions.

Keywords

Option Price Payoff Function Conditional Expectation Nonparametric Regression Price Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andersen, L., Broadie, M.: Primal-dual simulation algorithm for pricing multidimensional American options. Manag. Sci. 50, 1222–1234 (2004) CrossRefGoogle Scholar
  2. Belomestny, D.: Pricing Bermudan options using regression: optimal rates of convergence for lower estimates. Preprint (2009) Google Scholar
  3. Belomestny, D., Bender, C., Schoenmakers, J.: True upper bounds for Bermudan products via non-nested Monte Carlo. Math. Finance 19, 53–71 (2009) MATHCrossRefMathSciNetGoogle Scholar
  4. Carriér, J.: Valuation of early-exercise price of options using simulations and nonparametric regression. Insur. Math. Econ. 19, 19–30 (1996) CrossRefGoogle Scholar
  5. Chen, N., Glasserman, P.: Additive and multiplicative duals for American option pricing. Finance Stoch. 11, 153–179 (2007) MATHCrossRefMathSciNetGoogle Scholar
  6. Clément, E., Lamberton, D., Protter, P.: An analysis of the Longstaff-Schwartz algorithm for American option pricing. Finance Stoch. 6, 449–471 (2002) MATHCrossRefMathSciNetGoogle Scholar
  7. Chow, Y.S., Robbins, H., Siegmund, D.: Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston (1971) MATHGoogle Scholar
  8. Egloff, D.: Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab. 15, 1–37 (2005) CrossRefMathSciNetGoogle Scholar
  9. Egloff, D., Kohler, M., Todorovic, N.: A dynamic look-ahead Monte Carlo algorithm for pricing American options. Ann. Appl. Probab. 17, 1138–1171 (2007) MATHCrossRefMathSciNetGoogle Scholar
  10. Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, Berlin (2004) MATHGoogle Scholar
  11. Györfi, L., Kohler, M., Krzyżak, A., Walk, H.: A Distribution-Free Theory of Nonparametric Regression. Springer Series in Statistics. Springer, Berlin (2002) Google Scholar
  12. Haugh, M., Kogan, L.: Pricing American Options: A Duality Approach. Oper. Res. 52, 258–270 (2004) MATHCrossRefMathSciNetGoogle Scholar
  13. Jamshidian, F.: The duality of optimal exercise and domineering claims: a Doob-Meyer decomposition approach to the Snell envelope. Stochastics 79, 27–60 (2007) MATHMathSciNetGoogle Scholar
  14. Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Applications of Math., vol. 39. Springer, Berlin (1998) MATHGoogle Scholar
  15. Kohler, M.: Nonparametric regression with additional measurement errors in the dependent variable. J. Stat. Plan. Inference 136, 3339–3361 (2006) MATHCrossRefMathSciNetGoogle Scholar
  16. Kohler, M.: A regression based smoothing spline Monte Carlo algorithm for pricing American options. AStA Adv. Stat. Anal. 92, 153–178 (2008a) CrossRefMathSciNetGoogle Scholar
  17. Kohler, M.: Universally consistent upper bounds for Bermudan options based on Monte Carlo and nonparametric regression. Preprint (2008b) Google Scholar
  18. Kohler, M., Krzyżak, A.: Pricing of American options in discrete time using least squares estimates with complexity penalties. Preprint (2009) Google Scholar
  19. Kohler, M., Krzyżak, A., Todorovic, N.: Pricing of high-dimensional American options by neural networks. Math. Finance (2010, to appear) Google Scholar
  20. Kohler, M., Krzyżak, A., Walk, H.: Upper bounds for Bermudan options on Markovian data using nonparametric regression and a reduced number of nested Monte Carlo steps. Stat. Decis. 26, 275–288 (2008) MATHCrossRefGoogle Scholar
  21. Lamberton, D., Pagès, G.: Sur l’approximation des réduites. Ann. Inst. H. Poincaré 26, 331–355 (1990) MATHGoogle Scholar
  22. Longstaff, F.A., Schwartz, E.S.: Valuing American options by simulation: a simple least-squares approach. Rev. Financ. Stud. 14, 113–147 (2001) CrossRefGoogle Scholar
  23. Rogers, L.: Monte Carlo valuation of American options. Math. Finance 12, 271–286 (2001) CrossRefGoogle Scholar
  24. Rumelhart, D.E., McClelland, J.L.: Parallel Distributed Processing: Explorations in the Microstructure of Cognition, vol. 1: Foundations. MIT Press, Cambridge (1986) Google Scholar
  25. Shiryayev, A.N.: Optimal Stopping Rules. Applications of Mathematics. Springer, Berlin (1978) MATHGoogle Scholar
  26. Stoer, J.: Numerische Mathematik, vol. 1. Springer, Berlin (1993) Google Scholar
  27. Todorovic, N.: Bewertung Amerikanischer Optionen mit Hilfe von regressionsbasierten Monte-Carlo-Verfahren. Shaker, Aachen (2007) MATHGoogle Scholar
  28. Tsitsiklis, J.N., Van Roy, B.: 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 (1999) MATHCrossRefGoogle Scholar
  29. Tsitsiklis, J.N., Van Roy, B.: Regression methods for pricing complex American-style options. IEEE Trans. Neural Netw. 12, 694–730 (2001) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany

Personalised recommendations