IDEAS home Printed from https://ideas.repec.org/p/hal/cesptp/hal-05265653.html
   My bibliography  Save this paper

An Entropy Regularized BSDE Approach to Bermudan Options and Games

Author

Listed:
  • Noufel Frikha

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, UP1 - Université Paris 1 Panthéon-Sorbonne)

  • Libo Li

    (School of Mathematics and Statistics - UNSW - University of New South Wales [Sydney])

  • Daniel Chee

    (School of Mathematics and Statistics - UNSW - University of New South Wales [Sydney])

Abstract

In this paper, we investigate optimal stopping problems in a continuous-time framework where only a discrete set of stopping dates is admissible, corresponding to the Bermudan option, within the so-called exploratory formulation. We introduce an associated control problem for the value function, represented as a non-cadlag reflected backward stochastic differential equation (RBSDE) with an entropy regulariser that promotes exploration, and we establish existence and uniqueness results for this entropy-regularised RBSDE. We then compare the entropy-regularised RBSDE with the theoretical value of a Bermudan option and propose a reinforcement learning algorithm based on a policy improvement scheme, for which we prove both monotone improvement and convergence. This methodology is further extended to Bermudan game options, where we obtain analogous results. Finally, drawing on the preceding analysis, we present two numerical approximation schemes - a BSDE solver based on a temporal-difference scheme and neural networks and the policy improvement algorithm - to illustrate the feasibility and effectiveness of our approach.

Suggested Citation

  • Noufel Frikha & Libo Li & Daniel Chee, 2025. "An Entropy Regularized BSDE Approach to Bermudan Options and Games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-05265653, HAL.
    • Handle: RePEc:hal:cesptp:hal-05265653
    Note: View the original document on HAL open archive server: https://hal.science/hal-05265653v1
    as

    Download full text from publisher

    File URL: https://hal.science/hal-05265653v1/document
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Leonardo Kanashiro Felizardo & Elia Matsumoto & Emilio Del-Moral-Hernandez, 2022. "Solving the optimal stopping problem with reinforcement learning: an application in financial option exercise," Papers 2208.00765, arXiv.org.
    2. Jodi Dianetti & Giorgio Ferrari & Renyuan Xu, 2024. "Exploratory Optimal Stopping: A Singular Control Formulation," Papers 2408.09335, arXiv.org, revised Oct 2024.
    3. Ivan Guo & Nicolas Langrené & Jiahao Wu, 2025. "Simultaneous upper and lower bounds of American-style option prices with hedging via neural networks," Quantitative Finance, Taylor & Francis Journals, vol. 25(4), pages 509-525, April.
    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. 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.
    6. Bernard Lapeyre & Jérôme Lelong, 2021. "Neural network regression for Bermudan option pricing," Post-Print hal-02183587, HAL.
    7. Sebastian Becker & Patrick Cheridito & Arnulf Jentzen & Timo Welti, 2019. "Solving high-dimensional optimal stopping problems using deep learning," Papers 1908.01602, arXiv.org, revised Aug 2021.
    8. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286, July.
    9. Aurélien Alfonsi & Ahmed Kebaier & Jérôme Lelong, 2025. "A Pure Dual Approach for Hedging Bermudan Options," Mathematical Finance, Wiley Blackwell, vol. 35(4), pages 745-759, October.
    10. Andres Max Reppen & Halil Mete Soner & Valentin Tissot‐Daguette, 2025. "Neural optimal stopping boundary," Mathematical Finance, Wiley Blackwell, vol. 35(2), pages 441-469, April.
    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. Junyan Ye & Hoi Ying Wong & Kyunghyun Park, 2025. "Robust Exploratory Stopping under Ambiguity in Reinforcement Learning," Papers 2510.10260, arXiv.org.

    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. Lukas Gonon, 2022. "Deep neural network expressivity for optimal stopping problems," Papers 2210.10443, arXiv.org.
    2. Zineb El Filali Ech-Chafiq & Pierre Henry-Labordere & Jérôme Lelong, 2021. "Pricing Bermudan options using regression trees/random forests," Working Papers hal-03436046, HAL.
    3. Sebastian Becker & Patrick Cheridito & Arnulf Jentzen, 2020. "Pricing and Hedging American-Style Options with Deep Learning," JRFM, MDPI, vol. 13(7), pages 1-12, July.
    4. Lukas Gonon, 2024. "Deep neural network expressivity for optimal stopping problems," Finance and Stochastics, Springer, vol. 28(3), pages 865-910, July.
    5. Jiefei Yang & Guanglian Li, 2024. "A deep primal-dual BSDE method for optimal stopping problems," Papers 2409.06937, arXiv.org.
    6. Christian Bayer & Denis Belomestny & Paul Hager & Paolo Pigato & John Schoenmakers, 2020. "Randomized optimal stopping algorithms and their convergence analysis," Papers 2002.00816, arXiv.org.
    7. Fabian Dickmann & Nikolaus Schweizer, 2014. "Faster Comparison of Stopping Times by Nested Conditional Monte Carlo," Papers 1402.0243, arXiv.org.
    8. Hainaut, Donatien & Akbaraly, Adnane, 2023. "Risk management with Local Least Squares Monte-Carlo," LIDAM Discussion Papers ISBA 2023003, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. repec:hum:wpaper:sfb649dp2006-051 is not listed on IDEAS
    10. Calypso Herrera & Florian Krach & Pierre Ruyssen & Josef Teichmann, 2021. "Optimal Stopping via Randomized Neural Networks," Papers 2104.13669, arXiv.org, revised Dec 2023.
    11. Denis Belomestny & Grigori Milstein & Vladimir Spokoiny, 2009. "Regression methods in pricing American and Bermudan options using consumption processes," Quantitative Finance, Taylor & Francis Journals, vol. 9(3), pages 315-327.
    12. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2022. "Deep Stochastic Optimization in Finance," Papers 2205.04604, arXiv.org.
    13. Mehrdoust, Farshid & Noorani, Idin & Hamdi, Abdelouahed, 2023. "Two-factor Heston model equipped with regime-switching: American option pricing and model calibration by Levenberg–Marquardt optimization algorithm," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 204(C), pages 660-678.
    14. Belomestny, Denis & Kolodko, Anastasia & Schoenmakers, John G. M., 2009. "Regression methods for stochastic control problems and their convergence analysis," SFB 649 Discussion Papers 2009-026, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    15. Xuwei Yang & Anastasis Kratsios & Florian Krach & Matheus Grasselli & Aurelien Lucchi, 2023. "Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricing," Papers 2309.04557, arXiv.org, revised Oct 2024.
    16. Cosma, Antonio & Galluccio, Stefano & Pederzoli, Paola & Scaillet, Olivier, 2020. "Early Exercise Decision in American Options with Dividends, Stochastic Volatility, and Jumps," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 55(1), pages 331-356, February.
    17. Masahiro Nishiba, 2013. "Pricing Exotic Options and American Options: A Multidimensional Asymptotic Expansion Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 20(2), pages 147-182, May.
    18. Antonio Cosma & Stefano Galluccio & Paola Pederzoli & O. Scaillet, 2012. "Valuing American Options Using Fast Recursive Projections," Swiss Finance Institute Research Paper Series 12-26, Swiss Finance Institute.
    19. Garcia, Diego, 2003. "Convergence and Biases of Monte Carlo estimates of American option prices using a parametric exercise rule," Journal of Economic Dynamics and Control, Elsevier, vol. 27(10), pages 1855-1879, August.
    20. Yi Yang & Jianan Wang & Youhua Chen & Zhiyuan Chen & Yanchu Liu, 2020. "Optimal procurement strategies for contractual assembly systems with fluctuating procurement price," Annals of Operations Research, Springer, vol. 291(1), pages 1027-1059, August.
    21. Mark Broadie & Weiwei Shen, 2016. "High-Dimensional Portfolio Optimization With Transaction Costs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(04), pages 1-49, June.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:hal:cesptp:hal-05265653. 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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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.