IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v90y2019i3d10.1007_s00186-019-00684-8.html
   My bibliography  Save this article

Martingale optimal transport in the discrete case via simple linear programming techniques

Author

Listed:
  • Nicole Bäuerle

    (Karlsruhe Institute of Technology (KIT))

  • Daniel Schmithals

    (Karlsruhe Institute of Technology (KIT))

Abstract

We consider the problem of finding consistent upper price bounds and super replication strategies for exotic options, given the observation of call prices in the market. This field of research is called model-independent finance and has been introduced by Hobson (Finance Stoch 2(4):329–347, 1998). Here we use the link to mass transport problems. In contrast to existing literature we assume that the marginal distributions at the two time points we consider are discrete probability distributions. This has the advantage that the optimization problems reduce to linear programs and can be solved rather easily when assuming a general martingale Spence Mirrlees condition. We will prove the optimality of left-monotone transport plans under this assumption and provide an algorithm for its construction. Our proofs are simple and do not require much knowledge of probability theory. At the end we present an example to illustrate our approach.

Suggested Citation

  • Nicole Bäuerle & Daniel Schmithals, 2019. "Martingale optimal transport in the discrete case via simple linear programming techniques," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 90(3), pages 453-476, December.
  • Handle: RePEc:spr:mathme:v:90:y:2019:i:3:d:10.1007_s00186-019-00684-8
    DOI: 10.1007/s00186-019-00684-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00186-019-00684-8
    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/s00186-019-00684-8?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. Beiglböck, Mathias & Henry-Labordère, Pierre & Touzi, Nizar, 2017. "Monotone martingale transport plans and Skorokhod embedding," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 3005-3013.
    2. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    3. Huesmann, Martin & Stebegg, Florian, 2018. "Monotonicity preserving transformations of MOT and SEP," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1114-1134.
    4. Mathias Beiglbock & Pierre Henry-Labord`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929, arXiv.org, revised Feb 2013.
    5. Mathias Beiglboeck & Pierre Henry-Labordere & Nizar Touzi, 2017. "Monotone Martingale Transport Plans and Skorohod Embedding," Papers 1701.06779, arXiv.org.
    6. Pierre Henry-Labordère & Nizar Touzi, 2016. "An explicit martingale version of the one-dimensional Brenier theorem," Finance and Stochastics, Springer, vol. 20(3), pages 635-668, July.
    7. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    8. Marcel Nutz & Florian Stebegg & Xiaowei Tan, 2017. "Multiperiod Martingale Transport," Papers 1703.10588, arXiv.org, revised May 2019.
    9. Henry-Labordère, Pierre & Tan, Xiaolu & Touzi, Nizar, 2016. "An explicit martingale version of the one-dimensional Brenier’s Theorem with full marginals constraint," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2800-2834.
    10. Mathias Beiglböck & Pierre Henry-Labordère & Friedrich Penkner, 2013. "Model-independent bounds for option prices—a mass transport approach," Finance and Stochastics, Springer, vol. 17(3), pages 477-501, July.
    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. David Hobson & Dominykas Norgilas, 2019. "Robust bounds for the American put," Finance and Stochastics, Springer, vol. 23(2), pages 359-395, April.
    2. Erhan Bayraktar & Shuoqing Deng & Dominykas Norgilas, 2023. "Supermartingale Brenier’s Theorem with Full-Marginal Constraint," World Scientific Book Chapters, in: Robert A Jarrow & Dilip B Madan (ed.), Peter Carr Gedenkschrift Research Advances in Mathematical Finance, chapter 17, pages 569-636, World Scientific Publishing Co. Pte. Ltd..
    3. Nutz, Marcel & Stebegg, Florian & Tan, Xiaowei, 2020. "Multiperiod martingale transport," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1568-1615.
    4. Sergey Badikov & Mark H. A. Davis & Antoine Jacquier, 2018. "Perturbation analysis of sub/super hedging problems," Papers 1806.03543, arXiv.org, revised May 2021.
    5. Mathias Beiglbock & Marcel Nutz & Florian Stebegg, 2019. "Fine Properties of the Optimal Skorokhod Embedding Problem," Papers 1903.03887, arXiv.org, revised Apr 2020.
    6. Mathias Beiglboeck & Alexander Cox & Martin Huesmann, 2017. "The geometry of multi-marginal Skorokhod Embedding," Papers 1705.09505, arXiv.org.
    7. Nicole Bauerle & Daniel Schmithals, 2019. "Consistent upper price bounds for exotic options given a finite number of call prices and their convergence," Papers 1907.09144, arXiv.org.
    8. Sebastian Herrmann & Florian Stebegg, 2017. "Robust Pricing and Hedging around the Globe," Papers 1707.08545, arXiv.org, revised Apr 2019.
    9. Lim, Tongseok, 2020. "Optimal martingale transport between radially symmetric marginals in general dimensions," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1897-1912.
    10. Linn Engstrom & Sigrid Kallblad & Johan Karlsson, 2024. "Computation of Robust Option Prices via Structured Multi-Marginal Martingale Optimal Transport," Papers 2406.09959, arXiv.org.
    11. Neufeld, Ariel & Sester, Julian, 2021. "On the stability of the martingale optimal transport problem: A set-valued map approach," Statistics & Probability Letters, Elsevier, vol. 176(C).
    12. Benjamin Jourdain & Gilles Pagès, 2022. "Convex Order, Quantization and Monotone Approximations of ARCH Models," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2480-2517, December.
    13. Henry-Labordère, Pierre & Tan, Xiaolu & Touzi, Nizar, 2016. "An explicit martingale version of the one-dimensional Brenier’s Theorem with full marginals constraint," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2800-2834.
    14. Alessandro Doldi & Marco Frittelli, 2023. "Entropy martingale optimal transport and nonlinear pricing–hedging duality," Finance and Stochastics, Springer, vol. 27(2), pages 255-304, April.
    15. David Hobson & Anthony Neuberger, 2016. "On the value of being American," Papers 1604.02269, arXiv.org.
    16. Sergey Nadtochiy & Jan Obłój, 2017. "Robust Trading Of Implied Skew," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(02), pages 1-41, March.
    17. Florian Stebegg, 2014. "Model-Independent Pricing of Asian Options via Optimal Martingale Transport," Papers 1412.1429, arXiv.org.
    18. Cox, Alexander M.G. & Kinsley, Sam M., 2019. "Discretisation and duality of optimal Skorokhod embedding problems," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2376-2405.
    19. Marcel Nutz & Florian Stebegg, 2016. "Canonical Supermartingale Couplings," Papers 1609.02867, arXiv.org, revised Nov 2017.
    20. David Hobson & Martin Klimmek, 2015. "Robust price bounds for the forward starting straddle," Finance and Stochastics, Springer, vol. 19(1), pages 189-214, 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:spr:mathme:v:90:y:2019:i:3:d:10.1007_s00186-019-00684-8. 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.