IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v22y2020i3d10.1007_s11009-019-09764-y.html
   My bibliography  Save this article

Discrete-type Approximations for Non-Markovian Optimal Stopping Problems: Part II

Author

Listed:
  • Sérgio C. Bezerra

    (Universidade Federal da Paraíba, Rua dos Escoteiros)

  • Alberto Ohashi

    (Universidade de Brasília)

  • Francesco Russo

    (Unité de Mathématiques Appliquées)

  • Francys Souza

    (Universidade de Campinas)

Abstract

In this paper, we present a Longstaff-Schwartz-type algorithm for optimal stopping time problems based on the Brownian motion filtration. The algorithm is based on Leão et al. (??2019) and, in contrast to previous works, our methodology applies to optimal stopping problems for fully non-Markovian and non-semimartingale state processes such as functionals of path-dependent stochastic differential equations and fractional Brownian motions. Based on statistical learning theory techniques, we provide overall error estimates in terms of concrete approximation architecture spaces with finite Vapnik-Chervonenkis dimension. Analytical properties of continuation values for path-dependent SDEs and concrete linear architecture approximating spaces are also discussed.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:metcap:v:22:y:2020:i:3:d:10.1007_s11009-019-09764-y
    DOI: 10.1007/s11009-019-09764-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-019-09764-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s11009-019-09764-y?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. Dorival Le~ao & Alberto Ohashi & Francesco Russo, 2017. "Discrete-type approximations for non-Markovian optimal stopping problems: Part I," Papers 1707.05234, arXiv.org, revised Jun 2019.
    2. 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.
    3. Alexandra Chronopoulou & Frederi G. Viens, 2012. "Stochastic volatility and option pricing with long-memory in discrete and continuous time," Quantitative Finance, Taylor & Francis Journals, vol. 12(4), pages 635-649, December.
    4. Carriere, Jacques F., 1996. "Valuation of the early-exercise price for options using simulations and nonparametric regression," Insurance: Mathematics and Economics, Elsevier, vol. 19(1), pages 19-30, December.
    5. repec:dau:papers:123456789/4273 is not listed on IDEAS
    6. Daniel Z. Zanger, 2018. "Convergence Of A Least†Squares Monte Carlo Algorithm For American Option Pricing With Dependent Sample Data," Mathematical Finance, Wiley Blackwell, vol. 28(1), pages 447-479, January.
    7. Burq, Zaeem A. & Jones, Owen D., 2008. "Simulation of Brownian motion at first-passage times," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 77(1), pages 64-71.
    8. 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.
    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. Ludkovski, Michael, 2009. "A simulation approach to optimal stopping under partial information," Stochastic Processes and their Applications, Elsevier, vol. 119(12), pages 4061-4087, December.
    11. 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.
    12. 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.
    13. Denis Belomestny & John Schoenmakers & Fabian Dickmann, 2013. "Multilevel dual approach for pricing American style derivatives," Finance and Stochastics, Springer, vol. 17(4), pages 717-742, October.
    14. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    15. L. C. G. Rogers, 2002. "Monte Carlo valuation of American options," Mathematical Finance, Wiley Blackwell, vol. 12(3), pages 271-286, July.
    16. 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)

    Citations

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


    Cited by:

    1. Bradley Sturt, 2021. "A nonparametric algorithm for optimal stopping based on robust optimization," Papers 2103.03300, arXiv.org, revised Mar 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. 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.
    2. Bradley Sturt, 2021. "A nonparametric algorithm for optimal stopping based on robust optimization," Papers 2103.03300, arXiv.org, revised Mar 2023.
    3. 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.
    4. Jérôme Lelong, 2019. "Pricing path-dependent Bermudan options using Wiener chaos expansion: an embarrassingly parallel approach," Working Papers hal-01983115, HAL.
    5. Zhiyi Shen & Chengguo Weng, 2019. "A Backward Simulation Method for Stochastic Optimal Control Problems," Papers 1901.06715, arXiv.org.
    6. J'er^ome Lelong, 2019. "Pricing path-dependent Bermudan options using Wiener chaos expansion: an embarrassingly parallel approach," Papers 1901.05672, arXiv.org, revised Jul 2020.
    7. 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.
    8. Calypso Herrera & Florian Krach & Pierre Ruyssen & Josef Teichmann, 2021. "Optimal Stopping via Randomized Neural Networks," Papers 2104.13669, arXiv.org, revised Dec 2023.
    9. 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.
    10. 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.
    11. Ammann, Manuel & Kind, Axel & Wilde, Christian, 2008. "Simulation-based pricing of convertible bonds," Journal of Empirical Finance, Elsevier, vol. 15(2), pages 310-331, March.
    12. 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.
    13. Burcu Aydoğan & Ümit Aksoy & Ömür Uğur, 2018. "On the methods of pricing American options: case study," Annals of Operations Research, Springer, vol. 260(1), pages 79-94, January.
    14. Zbigniew Palmowski & Tomasz Serafin, 2020. "A Note on Simulation Pricing of π -Options," Risks, MDPI, vol. 8(3), pages 1-19, August.
    15. Hampus Engsner, 2021. "Least Squares Monte Carlo applied to Dynamic Monetary Utility Functions," Papers 2101.10947, arXiv.org, revised Apr 2021.
    16. Mark S. Joshi, 2016. "Analysing the bias in the primal-dual upper bound method for early exercisable derivatives: bounds, estimation and removal," Quantitative Finance, Taylor & Francis Journals, vol. 16(4), pages 519-533, April.
    17. Axel Kind, 2005. "Pricing American-Style Options By Simulation," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 19(1), pages 109-116, June.
    18. Christian Bayer & Ra'ul Tempone & Soren Wolfers, 2018. "Pricing American Options by Exercise Rate Optimization," Papers 1809.07300, arXiv.org, revised Aug 2019.
    19. Zineb El Filali Ech-Chafiq & Pierre Henry Labordère & Jérôme Lelong, 2023. "Pricing Bermudan options using regression trees/random forests," Post-Print hal-03436046, HAL.
    20. Maximilian Mair & Jan Maruhn, 2013. "On the primal-dual algorithm for callable Bermudan options," Review of Derivatives Research, Springer, vol. 16(1), pages 79-110, April.

    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:spr:metcap:v:22:y:2020:i:3:d:10.1007_s11009-019-09764-y. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.