IDEAS home Printed from https://ideas.repec.org/a/taf/apmtfi/v16y2009i2p123-150.html
   My bibliography  Save this article

Convergence of a Least-Squares Monte Carlo Algorithm for Bounded Approximating Sets

Author

Listed:
  • Daniel Zanger

Abstract

We analyse the convergence properties of the Longstaff-Schwartz algorithm for approximately solving optimal stopping problems that arise in the pricing of American (Bermudan) financial options. Based on a new approximate dynamic programming principle error propagation inequality, we prove sample complexity error estimates for this algorithm for the case in which the corresponding approximation spaces may not necessarily possess any linear structure at all and may actually be any arbitrary sets of functions, each of which is uniformly bounded and possesses finite VC-dimension, but is not required to satisfy any further material conditions. In particular, we do not require that the approximation spaces be convex or closed, and we thus significantly generalize the results of Egloff, Clement et al., and others. Using our error estimation theorems, we also prove convergence, up to any desired probability, of the algorithm for approximating sets defined using L2 orthonormal bases, within a framework depending subexponentially on the number of time steps. In addition, we prove estimates on the overall convergence rate of the algorithm for approximation spaces defined by polynomials.

Suggested Citation

  • Daniel Zanger, 2009. "Convergence of a Least-Squares Monte Carlo Algorithm for Bounded Approximating Sets," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(2), pages 123-150.
    • Handle: RePEc:taf:apmtfi:v:16:y:2009:i:2:p:123-150
    DOI: 10.1080/13504860802516881
    as

    Download full text from publisher

    File URL: http://www.tandfonline.com/doi/abs/10.1080/13504860802516881
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/13504860802516881?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    2. Lars Stentoft, 2004. "Convergence of the Least Squares Monte Carlo Approach to American Option Valuation," Management Science, INFORMS, vol. 50(9), pages 1193-1203, September.
    3. Philip Protter & Emmanuelle Clément & Damien Lamberton, 2002. "An analysis of a least squares regression method for American option pricing," Finance and Stochastics, Springer, vol. 6(4), pages 449-471.
    4. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    5. Lars Stentoft, 2004. "Assessing the Least Squares Monte-Carlo Approach to American Option Valuation," Review of Derivatives Research, Springer, vol. 7(2), pages 129-168, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Zhiyi Shen & Chengguo Weng, 2019. "A Backward Simulation Method for Stochastic Optimal Control Problems," Papers 1901.06715, arXiv.org.
    2. Maciej Klimek & Marcin Pitera, 2014. "The least squares method for option pricing revisited," Papers 1404.7438, arXiv.org, revised Nov 2015.
    3. Daniel Z. Zanger, 2020. "General Error Estimates for the Longstaff–Schwartz Least-Squares Monte Carlo Algorithm," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 923-946, August.
    4. Sérgio C. Bezerra & Alberto Ohashi & Francesco Russo & Francys Souza, 2020. "Discrete-type Approximations for Non-Markovian Optimal Stopping Problems: Part II," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1221-1255, September.
    5. Daniel Zanger, 2013. "Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing," Finance and Stochastics, Springer, vol. 17(3), pages 503-534, July.
    6. S'ergio C. Bezerra & Alberto Ohashi & Francesco Russo & Francys de Souza, 2017. "Discrete-type approximations for non-Markovian optimal stopping problems: Part II," Papers 1707.05250, arXiv.org, revised Dec 2019.
    7. Hampus Engsner, 2021. "Least Squares Monte Carlo applied to Dynamic Monetary Utility Functions," Papers 2101.10947, arXiv.org, revised Apr 2021.
    8. Jo~ao F. Doriguello & Alessandro Luongo & Jinge Bao & Patrick Rebentrost & Miklos Santha, 2021. "Quantum algorithm for stochastic optimal stopping problems with applications in finance," Papers 2111.15332, arXiv.org, revised Jul 2023.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Reesor, R. Mark & Stentoft, Lars & Zhu, Xiaotian, 2024. "A critical analysis of the Weighted Least Squares Monte Carlo method for pricing American options," Finance Research Letters, Elsevier, vol. 64(C).
    2. Nelson Areal & Artur Rodrigues & Manuel Armada, 2008. "On improving the least squares Monte Carlo option valuation method," Review of Derivatives Research, Springer, vol. 11(1), pages 119-151, March.
    3. Berridge, S.J. & Schumacher, J.M., 2002. "An Irregular Grid Approach for Pricing High Dimensional American Options," Discussion Paper 2002-99, Tilburg University, Center for Economic Research.
    4. Katarzyna Toporek, 2012. "Simple is better. Empirical comparison of American option valuation methods," Ekonomia journal, Faculty of Economic Sciences, University of Warsaw, vol. 29.
    5. Nordahl, Helge A., 2008. "Valuation of life insurance surrender and exchange options," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 909-919, June.
    6. Fabozzi, Frank J. & Paletta, Tommaso & Tunaru, Radu, 2017. "An improved least squares Monte Carlo valuation method based on heteroscedasticity," European Journal of Operational Research, Elsevier, vol. 263(2), pages 698-706.
    7. Zhiyi Shen & Chengguo Weng, 2019. "A Backward Simulation Method for Stochastic Optimal Control Problems," Papers 1901.06715, arXiv.org.
    8. Alexander Boogert & Cyriel de Jong, 2007. "Gas Storage Valuation Using a Monte Carlo Method," Birkbeck Working Papers in Economics and Finance 0704, Birkbeck, Department of Economics, Mathematics & Statistics.
    9. Daniel Zanger, 2013. "Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing," Finance and Stochastics, Springer, vol. 17(3), pages 503-534, July.
    10. Zhu, Lei & Zhang, ZhongXiang & Fan, Ying, 2015. "Overseas oil investment projects under uncertainty: How to make informed decisions?," Journal of Policy Modeling, Elsevier, vol. 37(5), pages 742-762.
    11. Nan Chen & Yanchu Liu, 2014. "American Option Sensitivities Estimation via a Generalized Infinitesimal Perturbation Analysis Approach," Operations Research, INFORMS, vol. 62(3), pages 616-632, June.
    12. Chen Liu & Henry Schellhorn & Qidi Peng, 2019. "American Option Pricing With Regression: Convergence Analysis," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(08), pages 1-31, December.
    13. Joseph Y. J. Chow & Hamid R. Sayarshad, 2016. "Reference Policies for Non-myopic Sequential Network Design and Timing Problems," Networks and Spatial Economics, Springer, vol. 16(4), pages 1183-1209, December.
    14. Hongjun Ha & Daniel Bauer, 2022. "A least-squares Monte Carlo approach to the estimation of enterprise risk," Finance and Stochastics, Springer, vol. 26(3), pages 417-459, July.
    15. Maciej Klimek & Marcin Pitera, 2014. "The least squares method for option pricing revisited," Papers 1404.7438, arXiv.org, revised Nov 2015.
    16. Daniel Z. Zanger, 2020. "General Error Estimates for the Longstaff–Schwartz Least-Squares Monte Carlo Algorithm," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 923-946, August.
    17. Jeechul Woo & Chenru Liu & Jaehyuk Choi, 2024. "Leave‐one‐out least squares Monte Carlo algorithm for pricing Bermudan options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 44(8), pages 1404-1428, August.
    18. Floryszczak, Anthony & Le Courtois, Olivier & Majri, Mohamed, 2016. "Inside the Solvency 2 Black Box: Net Asset Values and Solvency Capital Requirements with a least-squares Monte-Carlo approach," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 15-26.
    19. Jin, Xing & Yang, Cheng-Yu, 2016. "Efficient estimation of lower and upper bounds for pricing higher-dimensional American arithmetic average options by approximating their payoff functions," International Review of Financial Analysis, Elsevier, vol. 44(C), pages 65-77.
    20. Gabriel J Power & Charli D. Tandja M. & Josée Bastien & Philippe Grégoire, 2015. "Measuring infrastructure investment option value," Journal of Risk Finance, Emerald Group Publishing, vol. 16(1), pages 49-72, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:taf:apmtfi:v:16:y:2009:i:2:p:123-150. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/RAMF20 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.