An Explicit Martingale Version of Brenier's Theorem

Pierre Henry-Labordere () and Nizar Touzi ()
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Pierre Henry-Labordere: SOCIETE GENERALE - Equity Derivatives Research Societe Generale - Société Générale
Nizar Touzi: CMAP - Centre de Mathématiques Appliquées de l'Ecole polytechnique - Inria - Institut National de Recherche en Informatique et en Automatique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique

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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.

Date: 2013-04-09
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