An Explicit Martingale Version of Brenier's Theorem

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.
Type de document :
Pré-publication, Document de travail
2013
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Contributeur : Nizar Touzi <>
Soumis le : mercredi 24 avril 2013 - 06:49:22
Dernière modification le : jeudi 10 mai 2018 - 02:05:50
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  • HAL Id : hal-00817179, version 1

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Pierre Henry-Labordere, Nizar Touzi. An Explicit Martingale Version of Brenier's Theorem. 2013. 〈hal-00817179〉

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