IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2204.12250.html
   My bibliography  Save this paper

Martingale Schr\"odinger Bridges and Optimal Semistatic Portfolios

Author

Listed:
  • Marcel Nutz
  • Johannes Wiesel
  • Long Zhao

Abstract

In a two-period financial market where a stock is traded dynamically and European options at maturity are traded statically, we study the so-called martingale Schr\"odinger bridge Q*; that is, the minimal-entropy martingale measure among all models calibrated to option prices. This minimization is shown to be in duality with an exponential utility maximization over semistatic portfolios. Under a technical condition on the physical measure P, we show that an optimal portfolio exists and provides an explicit solution for Q*. This result overcomes the remarkable issue of non-closedness of semistatic strategies discovered by Acciaio, Larsson and Schachermayer. Specifically, we exhibit a dense subset of calibrated martingale measures with particular properties to show that the portfolio in question has a well-defined and integrable option position.

Suggested Citation

  • Marcel Nutz & Johannes Wiesel & Long Zhao, 2022. "Martingale Schr\"odinger Bridges and Optimal Semistatic Portfolios," Papers 2204.12250, arXiv.org.
    • Handle: RePEc:arx:papers:2204.12250
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2204.12250
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Marco Avellaneda & Robert Buff & Craig Friedman & Nicolas Grandechamp & Lukasz Kruk & Joshua Newman, 2001. "Weighted Monte Carlo: A New Technique For Calibrating Asset-Pricing Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 91-119.
    3. Thaleia Zariphopoulou, 2001. "A solution approach to valuation with unhedgeable risks," Finance and Stochastics, Springer, vol. 5(1), pages 61-82.
    4. Freddy Delbaen & Peter Grandits & Thorsten Rheinländer & Dominick Samperi & Martin Schweizer & Christophe Stricker, 2002. "Exponential Hedging and Entropic Penalties," Mathematical Finance, Wiley Blackwell, vol. 12(2), pages 99-123, April.
    5. Acciaio, Beatrice & Larsson, Martin & Schachermayer, Walter, 2017. "The space of outcomes of semi-static trading strategies need not be closed," LSE Research Online Documents on Economics 69804, London School of Economics and Political Science, LSE Library.
    6. J. Jacod & A.N. Shiryaev, 1998. "Local martingales and the fundamental asset pricing theorems in the discrete-time case," Finance and Stochastics, Springer, vol. 2(3), pages 259-273.
    7. Marcel Nutz & Johannes Wiesel & Long Zhao, 2022. "Limits of Semistatic Trading Strategies," Papers 2204.12251, arXiv.org.
    8. Marco Avellaneda, 1998. "Minimum-Relative-Entropy Calibration of Asset-Pricing Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 1(04), pages 447-472.
    9. 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.
    10. Beatrice Acciaio & Martin Larsson & Walter Schachermayer, 2017. "The space of outcomes of semi-static trading strategies need not be closed," Finance and Stochastics, Springer, vol. 21(3), pages 741-751, July.
    11. Gaoyue Guo & Jan Obloj, 2017. "Computational Methods for Martingale Optimal Transport problems," Papers 1710.07911, arXiv.org, revised Apr 2019.
    12. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    13. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2014. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Post-Print hal-03460952, HAL.
    14. Marco Frittelli, 2000. "The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets," Mathematical Finance, Wiley Blackwell, vol. 10(1), pages 39-52, January.
    15. A. Galichon & P. Henry-Labord`ere & N. Touzi, 2014. "A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options," Papers 1401.3921, arXiv.org.
    16. Marco Avellaneda & Robert Buff & Craig Friedman & Nicolas Grandechamp & Lukasz Kruk & Joshua Newman, 2001. "Weighted Monte Carlo: A New Technique For Calibrating Asset-Pricing Models," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar(Volume II), chapter 9, pages 239-265, World Scientific Publishing Co. Pte. Ltd..
    17. 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. Marcel Nutz & Johannes Wiesel & Long Zhao, 2023. "Martingale Schrödinger bridges and optimal semistatic portfolios," Finance and Stochastics, Springer, vol. 27(1), pages 233-254, January.
    2. Marcel Nutz & Johannes Wiesel & Long Zhao, 2022. "Limits of Semistatic Trading Strategies," Papers 2204.12251, arXiv.org.
    3. 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.
    4. Julian Sester, 2023. "On intermediate Marginals in Martingale Optimal Transportation," Papers 2307.09710, arXiv.org, revised Nov 2023.
    5. Marcel Nutz & Johannes Wiesel & Long Zhao, 2023. "Limits of semistatic trading strategies," Mathematical Finance, Wiley Blackwell, vol. 33(1), pages 185-205, January.
    6. Joshua Zoen-Git Hiew & Tongseok Lim & Brendan Pass & Marcelo Cruz de Souza, 2023. "Geometry of vectorial martingale optimal transport and robust option pricing," Papers 2309.04947, arXiv.org, revised Sep 2023.
    7. Sergey Badikov & Mark H. A. Davis & Antoine Jacquier, 2018. "Perturbation analysis of sub/super hedging problems," Papers 1806.03543, arXiv.org, revised May 2021.
    8. Daniel Bartl, 2016. "Exponential utility maximization under model uncertainty for unbounded endowments," Papers 1610.00999, arXiv.org, revised Feb 2019.
    9. Linn Engstrom & Sigrid Kallblad & Johan Karlsson, 2024. "Computation of Robust Option Prices via Structured Multi-Marginal Martingale Optimal Transport," Papers 2406.09959, arXiv.org.
    10. Marcel Nutz & Johannes Wiesel, 2024. "On the Martingale Schr\"odinger Bridge between Two Distributions," Papers 2401.05209, arXiv.org.
    11. Tongseok Lim, 2023. "Replication of financial derivatives under extreme market models given marginals," Papers 2307.00807, arXiv.org.
    12. 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.
    13. 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.
    14. Julio Backhoff-Veraguas & Gregoire Loeper & Jan Obloj, 2024. "Geometric Martingale Benamou-Brenier transport and geometric Bass martingales," Papers 2406.04016, arXiv.org.
    15. Wiesel Johannes & Zhang Erica, 2023. "An optimal transport-based characterization of convex order," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-15, January.
    16. Alessandro Doldi & Marco Frittelli & Emanuela Rosazza Gianin, 2024. "On entropy martingale optimal transport theory," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 47(1), pages 1-42, June.
    17. Daniel Bartl & Michael Kupper & David J. Prömel & Ludovic Tangpi, 2019. "Duality for pathwise superhedging in continuous time," Finance and Stochastics, Springer, vol. 23(3), pages 697-728, July.
    18. David Hobson & Dominykas Norgilas, 2019. "Robust bounds for the American put," Finance and Stochastics, Springer, vol. 23(2), pages 359-395, April.
    19. Florian Stebegg, 2014. "Model-Independent Pricing of Asian Options via Optimal Martingale Transport," Papers 1412.1429, arXiv.org.
    20. Benjamin Jourdain & Gudmund Pammer, 2023. "An extension of martingale transport and stability in robust finance," Papers 2304.09551, arXiv.org.

    More about this item

    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:2204.12250. 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.