Maximum Maximum of Martingales given Marginals

Pierre Henry-Labordere, Jan Obloj (), Peter Spoida () and Nizar Touzi ()
Additional contact information
Pierre Henry-Labordere: Société Générale
Jan Obloj: MI - Mathematical Institute [Oxford] - University of Oxford
Peter Spoida: MI - Mathematical Institute [Oxford] - University of Oxford
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: 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}.

Keywords: Optimal control; robust pricing and hedging; volatility uncertainty; optimal transportation; pathwise inequalities; lookback option.; lookback option (search for similar items in EconPapers)
Date: 2013-04-07
Note: View the original document on HAL open archive server: https://hal.science/hal-00684005v2
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Citations: View citations in EconPapers (3)

Published in 2013

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