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An Explicit Martingale Version of Brenier's Theorem

Author

Listed:
  • Pierre Henry-Labordere

    (SOCIETE GENERALE - Equity Derivatives Research Societe Generale - Société Générale)

  • Nizar Touzi

    (CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

Abstract

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in \cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in \cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.

Suggested Citation

  • Pierre Henry-Labordere & Nizar Touzi, 2013. "An Explicit Martingale Version of Brenier's Theorem," Working Papers hal-00790001, HAL.
    • Handle: RePEc:hal:wpaper:hal-00790001
    Note: View the original document on HAL open archive server: https://hal.science/hal-00790001v3
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Mathias Beiglbock & Marcel Nutz & Nizar Touzi, 2015. "Complete Duality for Martingale Optimal Transport on the Line," Papers 1507.00671, arXiv.org, revised Jun 2016.
    2. David Hobson & Martin Klimmek, 2015. "Robust price bounds for the forward starting straddle," Finance and Stochastics, Springer, vol. 19(1), pages 189-214, January.
    3. Hadrien de March & Pierre Henry-Labordere, 2019. "Building Arbitrage-Free Implied Volatility: Sinkhorn'S Algorithm And Variants," Working Papers hal-02011533, HAL.
    4. Bruno Bouchard & Marcel Nutz, 2013. "Arbitrage and duality in nondominated discrete-time models," Papers 1305.6008, arXiv.org, revised Mar 2015.
    5. Hadrien De March & Pierre Henry-Labordere, 2019. "Building arbitrage-free implied volatility: Sinkhorn's algorithm and variants," Papers 1902.04456, arXiv.org, revised Jul 2023.
    6. Florian Stebegg, 2014. "Model-Independent Pricing of Asian Options via Optimal Martingale Transport," Papers 1412.1429, arXiv.org.
    7. Gaoyue Guo & Xiaolu Tan & Nizar Touzi, 2015. "Optimal Skorokhod embedding under finitely-many marginal constraints," Papers 1506.04063, arXiv.org, revised Aug 2016.

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