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Stability of the Weak Martingale Optimal Transport Problem

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  • Mathias Beiglbock
  • Benjamin Jourdain
  • William Margheriti
  • Gudmund Pammer

Abstract

While many questions in (robust) finance can be posed in the martingale optimal transport (MOT) framework, others require to consider also non-linear cost functionals. Following the terminology of Gozlan, Roberto, Samson and Tetali this corresponds to weak martingale optimal transport (WMOT). In this article we establish stability of WMOT which is important since financial data can give only imprecise information on the underlying marginals. As application, we deduce the stability of the superreplication bound for VIX futures as well as the stability of stretched Brownian motion and we derive a monotonicity principle for WMOT.

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  • Mathias Beiglbock & Benjamin Jourdain & William Margheriti & Gudmund Pammer, 2021. "Stability of the Weak Martingale Optimal Transport Problem," Papers 2109.06322, arXiv.org, revised Apr 2022.
    • Handle: RePEc:arx:papers:2109.06322
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    References listed on IDEAS

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    1. Veraguas, Julio Backhoff & Beiglböck, Mathias & Eder, Manu & Pichler, Alois, 2020. "Fundamental properties of process distances," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5575-5591.
    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. Julien Guyon & Romain Menegaux & Marcel Nutz, 2016. "Bounds for VIX Futures given S&P 500 Smiles," Papers 1609.05832, arXiv.org, revised Jun 2017.
    4. Julien Guyon & Romain Menegaux & Marcel Nutz, 2017. "Bounds for VIX futures given S&P 500 smiles," Finance and Stochastics, Springer, vol. 21(3), pages 593-630, July.
    5. 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.
    6. Gaoyue Guo & Jan Obloj, 2017. "Computational Methods for Martingale Optimal Transport problems," Papers 1710.07911, arXiv.org, revised Apr 2019.
    7. Alfred Galichon & Pierre Henri-Labordère & Nizar Touzi, 2013. "A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Lookback options," Sciences Po publications info:hdl:2441/5rkqqmvrn4t, Sciences Po.
    8. Julio Backhoff-Veraguas & Gudmund Pammer, 2019. "Stability of martingale optimal transport and weak optimal transport," Papers 1904.04171, arXiv.org, revised Dec 2020.
    9. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, September.
    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.
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    Citations

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    Cited by:

    1. Benjamin Jourdain & Kexin Shao, 2023. "Maximal Martingale Wasserstein Inequality," Papers 2310.08492, arXiv.org.
    2. 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.
    3. 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..
    4. Julian Sester, 2023. "On intermediate Marginals in Martingale Optimal Transportation," Papers 2307.09710, arXiv.org, revised Nov 2023.
    5. Benjamin Jourdain & Gudmund Pammer, 2023. "An extension of martingale transport and stability in robust finance," Papers 2304.09551, arXiv.org.
    6. Beatrice Acciaio & Mathias Beiglböck & Gudmund Pammer, 2021. "Weak transport for non‐convex costs and model‐independence in a fixed‐income market," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1423-1453, October.

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