Skip to main content
SpringerLink
Log in

Monte-Carlo Valuation of American Options: Facts and New Algorithms to Improve Existing Methods

  • Conference paper
  • First Online:
Numerical Methods in Finance

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 12))

Abstract

The aim of this paper is to discuss efficient algorithms for the pricing of American options by two recently proposed Monte-Carlo type methods, namely the Malliavian calculus and the regression based approaches. We explain how both techniques can be exploited with improved complexity and efficiency. We also discuss several techniques for the estimation of the corresponding hedging strategies. Numerical tests and comparisons, including the quantization approach, are performed.

MSC Code: G1G60, G1G20

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Hereafter, we say that a point x j dominates a point x k if x j i > x k i for all i ≤ d.

  2. 2.

    For all the computations, we use a core i7 2,9 GHz processor.

  3. 3.

    Here and below, the number of points corresponds to the sum of the numbers of points used at each time step. There are distributed according to [4]

References

  1. L. Andersen and M. Broadie: A Primal-Dual Simulation Algorithm for Pricing Multi-Dimensional American Options. Management Science, 50 (9), 1222–1234 (2004).

    Google Scholar 

  2. V. Bally, M.E. Caballero, B. Fernandez and N. El Karoui: Reflected BSDEÕs, PDEÕs and variational inequalities. INRIA Report 4455, (2002).

    Google Scholar 

  3. V. Bally, G. Pagès: Error analysis of the quantization algorithm for obstacle problems. Stochastic Processes and their Applications 106, 1–40, 2003.

    Google Scholar 

  4. V. Bally, G. Pagès, J. Printems : A quantization method for pricing and hedging multi-dimensional American style options. Mathematical Finance 15, 1 (2005)

    Google Scholar 

  5. O. Bardou, S. Bouthemy, G. Pagès: Optimal quantization for the pricing of swing options. Applied Mathematical Finance 16(2), 183–217 (2009)

    Google Scholar 

  6. D. Belomestny, Ch. Bender, J. Schoenmakers: True upper bounds for Bermudan products via non-nested Monte Carlo. Mathematical Finance, 19(1), 53–71 (2009).

    Google Scholar 

  7. J.-L. Bentley and M.-I. Shamos: Divide-and-Conquer in multidimensional space. Proc. Eighth ACM Annual Symp. on Theory of Comput 220–230 (1976)

    Google Scholar 

  8. M. de Berg, M. van Kreveld, M. Overmars, O. Schwarzkopf: Computational geometry, Springer (2000)

    Google Scholar 

  9. B. Bouchard, J.-F. Chassagneux : Discrete time approximation for continuously and discretely reflected BSDE’s. Stochastic Processes and their Applications, 118, 2269–2293 (2008)

    Google Scholar 

  10. B. Bouchard, I. Ekeland, N. Touzi: On the Malliavin approach to Monte Carlo approximation of conditional expectations. Finance and Stochastics, 8(1), 45–71 (2004)

    Google Scholar 

  11. B. Bouchard, E. Elie and N. Touzi: Discrete-Time Approximation of BSDEs and Probabilistic Schemes for Fully Nonlinear PDEs. Radon Series Comp. Appl. Math. 8, 133, de Gruyter ed. (2009)

    Google Scholar 

  12. B. Bouchard and N. Touzi : Discrete-time approximation and Monte Carlo simulation of backward stochastic differential equations. Stochastic Processes and their Applications, 111, 175–206 ( 2004)

    Google Scholar 

  13. M. Broadie, P. Glasserman: Estimating security price derivatives using simulation. Manag. Sci., 42, 269–285 (1996)

    Google Scholar 

  14. J.-F. Carrière: Valuation of the Early-Exercise Price for Options using Simulations and Nonparametric Regression. Insurance : mathematics and Economics, 19, 19–30 (1996)

    Google Scholar 

  15. A. R. Choudhury, A. King, S. Kumar, Y. Sabharwal: Optimizations in financial engineering: The Least-Squares Monte Carlo method of Longstaff and Schwartz. In Proc. of 2008 IEEE International Symposium on Parallel and Distributed Processing. (IPDPS 2008):pp 1–11 April 2008

    Google Scholar 

  16. E. Clément, D. Lamberton, P. Protter: An analysis of a least squares regression method for American option pricing. Finance and Stochastics, 6, 449–472, 2002.

    Google Scholar 

  17. J. Detemple, R. Garcia, M. Rindisbacher: Asymptotic Properties of Monte Carlo Estimators of Derivatives. Management science, 51(11), 1657–1675 (2005)

    Google Scholar 

  18. D. Egloff: Monte Carlo algorithms for optimal stopping and statistical learning. Ann. Appl. Probab. , 15 (2), 1396–1432 (2005).

    Google Scholar 

  19. N. El Karoui: Les aspects probabilistes du contrôle stochastique, Ecole d’Eté de Probabilités de Saint Flour, IX, Lecture Notes in Mathematics 876, Springer Verlag (1979)

    Google Scholar 

  20. E. Fournier, J.-M. Lasry, J. Lebuchoux, P.-L. Lions: Applications of Malliavin calculus to Monte Carlo methods in finance II. Finance and Stochastics , 5, 201–236 ( 2001)

    Google Scholar 

  21. E. Fournier, J.-M. Lasry, J. Lebuchoux, P.-L. Lions, N. Touzi: Applications of Malliavin calculus to Monte Carlo methods in finance. Finance and Stochastics, 3, 391–412 (1999)

    Google Scholar 

  22. P. Glasserman and B. Yu: Number of paths versus number of basis functions in American option pricing. Ann. Appl. Probab. 14(4), 2090–2119 (2004).

    Google Scholar 

  23. E. Gobet: Revisiting the Greeks for European and American options. Proceedings of the ”International Symposium on Stochastic Processes and Mathematical Finance” at Ritsumeikan University, Kusatsu, Japan (2003)

    Google Scholar 

  24. E. Gobet and J. P. Lemor: Numerical simulation of BSDEs using empirical regression methods : theory and practice. In Proceedings of the Fifth Colloquium on BSDEs (29th May - 1st June 2005, Shan- gai), Available on http ://hal.archives-ouvertes.fr/hal-00291199/fr/, (2006).

    Google Scholar 

  25. E. Gobet, J.P. Lemor, X. Warin: A regression-based Monte-Carlo method to solve backward stochastic differential equations. Annals of Applied Probability, 15(3), 2172–2002 (2005)

    Google Scholar 

  26. M. B. Haugh and L. Kogan: Pricing American Options: A Duality Approach. Operation research, 52(2), 258–270 (2004)

    Google Scholar 

  27. J. JaJa, C. Mortensen, Q. Shi: Space Efficient and Fast Algorithms for Multidimensional Dominance Reporting and Counting. Proceedings of the 2004 Annual Symposium on Algorithms and Computation, Hong Kong (2004)

    Google Scholar 

  28. J.P. Lemor: Approximation par projections et simulations de Monte-Carlo des équations différentielles stochastiques rétrogrades. PhD thesis, Ecole Polytechnique, http ://www.imprimerie.polytechnique.fr/Theses/Files/lemor.pdf, (2005).

  29. J.P. Lemor, E. Gobet, and X. Warin: Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations. Bernoulli, 12(5), 889–916 (2006).

    Google Scholar 

  30. P.-L. Lions, H. Regnier: Calcul du prix et des sensibilités d’une option américaine par une méthode de Monte Carlo, preprint (2001)

    Google Scholar 

  31. F. Longstaff and E. Schwartz: Valuing American options by simulation: A simple least-squares. Review of Financial Studies, 1(14), 113–147, 2001.

    Google Scholar 

  32. J. Ma and J. Zhang: Representations and regularities for solutions to BSDEs with reflections. Stochastic processes and their applications, 115, 539–569 (2005)

    Google Scholar 

  33. C. Makris, A.-K. Tsakalidis: Algorithms for three dimensional dominance searching in linear space. Information Processing Letters, 66, 6 (1998)

    Google Scholar 

  34. J. McNames: A Fast Nearest-Neighbor Algorithm Based on a Principal Axis Search Tree. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(9), 964–976 (2001)

    Google Scholar 

  35. G. Pagès, J. Printems: Optimal quadratic quantization for numerics: the Gaussian case. Monte Carlo Methods & Applications, 2, 9, 135–166 (2003)

    Google Scholar 

  36. G. Pagès, J. Printems: Functional quantization for numerics with an application to option pricing. Monte Carlo Methods & Applications , 4, 11, 407–446 (2005)

    Google Scholar 

  37. V.-V. Piterbarg: Risk sensitivities of Bermuda options. Technical report, Bank of America, http://ssrn.com/abstract=367920 (2002)

  38. F.-P. Preparata, M.-I.Shamos:Computational geometry (an introduction), Springer (1985)

    Google Scholar 

  39. J.-N. Tsitsiklis, B. Van Roy: Optimal Stopping of Markov Processes: Hilbert Spaces theory, Approximations Algorithms and an application to pricing high-dimensional financial derivatives. IEEE Transactions on Automatic Control, 10(44), 1840–1851 (1999)

    Google Scholar 

  40. D. Zanger: Convergence of a least-squares Monte Carlo algorithm for bounded approximating sets. Applied Mathematical Finance, 16(2), 123–150 (2009).

    Google Scholar 

Download references

Acknowledgements

We are grateful to Christos Makris, Paul Masurel and to the two anonymous referees for helpful suggestions.

Author information

Authors and Affiliations

  1. CEREMADE and Crest-ENSAE, Université Paris-Dauphine, place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Bruno Bouchard

  2. EDF R&D, Département Optimisation SImulation RIsques et Statistiques (OSIRIS), 92141, Clamart, France

    Xavier Warin

  3. Laboratoire de Finance des Marchés de l’Energie (FiME), Université Paris Dauphine, Paris, France

    Xavier Warin

Authors
  1. Bruno Bouchard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xavier Warin
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. Operations Research &, Financial Engineering, Princeton University, Charlton Street, Princeton, 08544, New Jersey, USA

    René A. Carmona

  2. Centre INRIA Bordeaux Sud-Ouest, Institut de Mathématiques, Université Bordeaux I, Cours de la Libération 351, Talence Cedex, 33405, France

    Pierre Del Moral

  3. Centre INRIA Bordeaux Sud-Ouest, Institut de Mathématiques, Université Bordeaux I, cours de la Libération 351, Talence Cedex, 33405, France

    Peng Hu

  4. EDF Recherche et Développement, Avenue Générale de Gaulle 1, Clamart, 92140, France

    Nadia Oudjane

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bouchard, B., Warin, X. (2012). Monte-Carlo Valuation of American Options: Facts and New Algorithms to Improve Existing Methods. In: Carmona, R., Del Moral, P., Hu, P., Oudjane, N. (eds) Numerical Methods in Finance. Springer Proceedings in Mathematics, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25746-9_7

Download citation

Publish with us

Policies and ethics

Search

Navigation