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Maximum Maximum of Martingales given Marginals


  • Pierre Henry-Labordere

    (Société Générale - Société Générale)

  • Jan Obloj

    () (MI - Mathematical Institute [Oxford] - University of Oxford [Oxford])

  • Peter Spoida

    () (MI - Mathematical Institute [Oxford] - University of Oxford [Oxford])

  • Nizar Touzi

    () (CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique)


We consider the problem of superhedging under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We present a general duality result which converts this problem into a min-max calculus of variations problem where the Lagrange multipliers correspond to the static part of the hedge. Following Galichon, Henry-Labordére and Touzi \cite{ght}, we apply stochastic control methods to solve it explicitly for Lookback options with a non-decreasing payoff function. The first step of our solution recovers the extended optimal properties of the Azéma-Yor solution of the Skorokhod embedding problem obtained by Hobson and Klimmek \cite{hobson-klimmek} (under slightly different conditions). The two marginal case corresponds to the work of Brown, Hobson and Rogers \cite{brownhobsonrogers}. The robust superhedging cost is complemented by (simple) dynamic trading and leads to a class of semi-static trading strategies. The superhedging property then reduces to a functional inequality which we verify independently. The optimality follows from existence of a model which achieves equality which is obtained in Ob\lój and Spoida \cite{OblSp}.

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  • Pierre Henry-Labordere & Jan Obloj & Peter Spoida & Nizar Touzi, 2013. "Maximum Maximum of Martingales given Marginals," Working Papers hal-00684005, HAL.
  • Handle: RePEc:hal:wpaper:hal-00684005
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    References listed on IDEAS

    1. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    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. Alexander Cox & Jan Obłój, 2011. "Robust pricing and hedging of double no-touch options," Finance and Stochastics, Springer, vol. 15(3), pages 573-605, September.
    4. Mathias Beiglbock & Pierre Henry-Labord`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929,, revised Feb 2013.
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    Cited by:

    1. repec:wsi:ijtafx:v:16:y:2013:i:08:n:s0219024913500428 is not listed on IDEAS
    2. Marcel Nutz, 2013. "Superreplication under Model Uncertainty in Discrete Time," Papers 1301.3227,, revised Feb 2014.
    3. Dolinsky, Yan & Soner, H. Mete, 2015. "Martingale optimal transport in the Skorokhod space," Stochastic Processes and their Applications, Elsevier, vol. 125(10), pages 3893-3931.

    More about this item


    lookback option; Optimal control; robust pricing and hedging; volatility uncertainty; optimal transportation; pathwise inequalities; lookback option.;

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