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Monotone martingale transport plans and Skorokhod embedding

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  • Beiglböck, Mathias
  • Henry-Labordère, Pierre
  • Touzi, Nizar

Abstract

We show that the left-monotone martingale coupling is optimal for any given performance function satisfying the martingale version of the Spence–Mirrlees condition, without assuming additional structural conditions on the marginals. We also give a new interpretation of the left monotone coupling in terms of Skorokhod embedding which allows us to give a short proof of uniqueness.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:9:p:3005-3013
    DOI: 10.1016/j.spa.2017.01.004
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    References listed on IDEAS

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    1. Luciano Campi & Ismail Laachir & Claude Martini, 2017. "Change of numeraire in the two-marginals martingale transport problem," Finance and Stochastics, Springer, vol. 21(2), pages 471-486, April.
    2. Mathias Beiglbock & Marcel Nutz & Nizar Touzi, 2015. "Complete Duality for Martingale Optimal Transport on the Line," Papers 1507.00671, arXiv.org, revised Jun 2016.
    3. 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.
    4. 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.
    5. David Hobson & Martin Klimmek, 2011. "Model independent hedging strategies for variance swaps," Papers 1104.4010, arXiv.org, revised May 2011.
    6. Pierre Henry-Labord`ere & Jan Ob{l}'oj & Peter Spoida & Nizar Touzi, 2012. "The maximum maximum of a martingale with given $n$ marginals," Papers 1203.6877, arXiv.org, revised Jan 2016.
    7. 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.
    8. 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.
    9. Bruno Bouchard & Marcel Nutz, 2013. "Arbitrage and duality in nondominated discrete-time models," Papers 1305.6008, arXiv.org, revised Mar 2015.
    10. Campi, Luciano & Laachir, Ismail & Martini, Claude, 2017. "Change of numeraire in the two-marginals martingale transport problem," LSE Research Online Documents on Economics 68783, London School of Economics and Political Science, LSE Library.
    11. 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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    Cited by:

    1. 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.
    2. Nutz, Marcel & Stebegg, Florian & Tan, Xiaowei, 2020. "Multiperiod martingale transport," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1568-1615.
    3. 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.
    4. Sebastian Herrmann & Florian Stebegg, 2017. "Robust Pricing and Hedging around the Globe," Papers 1707.08545, arXiv.org, revised Apr 2019.
    5. 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..
    6. Huesmann, Martin & Stebegg, Florian, 2018. "Monotonicity preserving transformations of MOT and SEP," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1114-1134.
    7. Mathias Beiglbock & Marcel Nutz & Florian Stebegg, 2019. "Fine Properties of the Optimal Skorokhod Embedding Problem," Papers 1903.03887, arXiv.org, revised Apr 2020.
    8. 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).
    9. Aur'elien Alfonsi & Rafael Coyaud & Virginie Ehrlacher & Damiano Lombardi, 2019. "Approximation of Optimal Transport problems with marginal moments constraints," Papers 1905.05663, arXiv.org.
    10. Mathias Beiglboeck & Alexander Cox & Martin Huesmann, 2017. "The geometry of multi-marginal Skorokhod Embedding," Papers 1705.09505, arXiv.org.
    11. David Hobson & Dominykas Norgilas, 2019. "Robust bounds for the American put," Finance and Stochastics, Springer, vol. 23(2), pages 359-395, April.

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