IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2211.11691.html
   My bibliography  Save this paper

Deep Signature Algorithm for Multi-dimensional Path-Dependent Options

Author

Listed:
  • Erhan Bayraktar
  • Qi Feng
  • Zhaoyu Zhang

Abstract

In this work, we study the deep signature algorithms for path-dependent options. We extend the backward scheme in [Hur\'e-Pham-Warin. Mathematics of Computation 89, no. 324 (2020)] for state-dependent FBSDEs with reflections to path-dependent FBSDEs with reflections, by adding the signature layer to the backward scheme. Our algorithm applies to both European and American type option pricing problems while the payoff function depends on the whole paths of the underlying forward stock process. We prove the convergence analysis of our numerical algorithm with explicit dependence on the truncation order of the signature and the neural network approximation errors. Numerical examples for the algorithm are provided including: Amerasian option under the Black-Scholes model, American option with a path-dependent geometric mean payoff function, and the Shiryaev's optimal stopping problem.

Suggested Citation

  • Erhan Bayraktar & Qi Feng & Zhaoyu Zhang, 2022. "Deep Signature Algorithm for Multi-dimensional Path-Dependent Options," Papers 2211.11691, arXiv.org, revised Jan 2024.
    • Handle: RePEc:arx:papers:2211.11691
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2211.11691
    File Function: Latest version
    Download Restriction: no
    ---><---

    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. Chassagneux, Jean-François & Richou, Adrien, 2019. "Rate of convergence for the discrete-time approximation of reflected BSDEs arising in switching problems," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4597-4637.
    3. 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.
    4. Rene Carmona & Michael Ludkovski, 2008. "Pricing Asset Scheduling Flexibility using Optimal Switching," Applied Mathematical Finance, Taylor & Francis Journals, vol. 15(5-6), pages 405-447.
    5. Lukas Gonon, 2022. "Deep neural network expressivity for optimal stopping problems," Papers 2210.10443, arXiv.org.
    6. Hamadène, Said & Zhang, Jianfeng, 2010. "Switching problem and related system of reflected backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 403-426, April.
    7. Imanol Perez Arribas, 2018. "Derivatives pricing using signature payoffs," Papers 1809.09466, arXiv.org.
    8. Bénézet, Cyril & Chassagneux, Jean-François & Richou, Adrien, 2022. "Switching problems with controlled randomisation and associated obliquely reflected BSDEs," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 23-71.
    9. Haojie Wang & Han Chen & Agus Sudjianto & Richard Liu & Qi Shen, 2018. "Deep Learning-Based BSDE Solver for Libor Market Model with Application to Bermudan Swaption Pricing and Hedging," Papers 1807.06622, arXiv.org, revised Sep 2018.
    10. Ming Min & Ruimeng Hu, 2021. "Signatured Deep Fictitious Play for Mean Field Games with Common Noise," Papers 2106.03272, arXiv.org.
    11. Jérôme Barraquand & Thierry Pudet, 1996. "Pricing Of American Path‐Dependent Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 6(1), pages 17-51, January.
    12. Asbjørn T. Hansen & Peter Løchte Jørgensen, 2000. "Analytical Valuation of American-Style Asian Options," Management Science, INFORMS, vol. 46(8), pages 1116-1136, August.
    13. Johan Auster & Ludovic Mathys & Fabio Maeder, 2022. "JDOI variance reduction method and the pricing of American-style options," Quantitative Finance, Taylor & Francis Journals, vol. 22(4), pages 639-656, April.
    14. Ma, Jin & Zhang, Jianfeng, 2005. "Representations and regularities for solutions to BSDEs with reflections," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 539-569, April.
    15. 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.
    Full references (including those not matched with items on IDEAS)

    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. Lin Zhao & Sweder van Wijnbergen, 2015. "Asset Pricing in Incomplete Markets: Valuing Gas Storage Capacity," Tinbergen Institute Discussion Papers 15-104/VI/DSF95, Tinbergen Institute.
    2. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2022. "Deep Stochastic Optimization in Finance," Papers 2205.04604, arXiv.org.
    3. 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.
    4. 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.
    5. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2023. "Deep stochastic optimization in finance," Digital Finance, Springer, vol. 5(1), pages 91-111, March.
    6. Yuchao Dong, 2022. "Randomized Optimal Stopping Problem in Continuous time and Reinforcement Learning Algorithm," Papers 2208.02409, arXiv.org, revised Sep 2023.
    7. Lukas Gonon, 2022. "Deep neural network expressivity for optimal stopping problems," Papers 2210.10443, arXiv.org.
    8. Magnus Perninge, 2018. "A limited-feedback approximation scheme for optimal switching problems with execution delays," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 87(3), pages 347-382, June.
    9. Beatriz Salvador & Cornelis W. Oosterlee & Remco van der Meer, 2020. "Financial Option Valuation by Unsupervised Learning with Artificial Neural Networks," Mathematics, MDPI, vol. 9(1), pages 1-20, December.
    10. Magnus Perninge & Lennart Söder, 2014. "Irreversible investments with delayed reaction: an application to generation re-dispatch in power system operation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 79(2), pages 195-224, April.
    11. Hatem Ben-Ameur & Michèle Breton & Pierre L'Ecuyer, 2002. "A Dynamic Programming Procedure for Pricing American-Style Asian Options," Management Science, INFORMS, vol. 48(5), pages 625-643, May.
    12. Aïd, René & Campi, Luciano & Langrené, Nicolas & Pham, Huyên, 2014. "A probabilistic numerical method for optimal multiple switching problems in high dimension," LSE Research Online Documents on Economics 63011, London School of Economics and Political Science, LSE Library.
    13. Jiawei Huo, 2023. "Finite Difference Solution Ansatz approach in Least-Squares Monte Carlo," Papers 2305.09166, arXiv.org, revised Nov 2023.
    14. A. Max Reppen & H. Mete Soner & Valentin Tissot-Daguette, 2022. "Neural Optimal Stopping Boundary," Papers 2205.04595, arXiv.org, revised May 2023.
    15. Mike Ludkovski, 2020. "mlOSP: Towards a Unified Implementation of Regression Monte Carlo Algorithms," Papers 2012.00729, arXiv.org, revised Oct 2022.
    16. Giovanni Mottola, 2014. "A stochastic switching control model arising in general OTC contracts with contingent CSA in presence of CVA, collateral and funding," Papers 1412.1469, arXiv.org.
    17. Kailin Ding & Zhenyu Cui & Xiaoguang Yang, 2023. "Pricing arithmetic Asian and Amerasian options: A diffusion operator integral expansion approach," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 43(2), pages 217-241, February.
    18. Christian Bayer & Denis Belomestny & Paul Hager & Paolo Pigato & John Schoenmakers, 2020. "Randomized optimal stopping algorithms and their convergence analysis," Papers 2002.00816, arXiv.org.
    19. Chassagneux, Jean-François & Richou, Adrien, 2019. "Rate of convergence for the discrete-time approximation of reflected BSDEs arising in switching problems," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4597-4637.
    20. Mark Broadie & Jerome B. Detemple, 2004. "ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications," Management Science, INFORMS, vol. 50(9), pages 1145-1177, September.

    More about this item

    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:arx:papers:2211.11691. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.