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Maximum Maximum of Martingales Given Marginals

35 Pages Posted: 1 Apr 2012 Last revised: 10 Apr 2013

Pierre Henry-Labordere

Société Générale - Paris, France

Jan Obłój

University of Oxford - Mathematical Institute; University of Oxford - Oxford-Man Institute of Quantitative Finance; University of Oxford - Saint John's College

Peter Spoida

University of Oxford

Nizar Touzi

Ecole Polytechnique, Paris

Date Written: April 2013

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-Labord\`ere and Touzi, 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 Az\'ema-Yor solution of the Skorokhod embedding problem obtained by Hobson and Klimmek (under slightly different conditions). The two marginal case corresponds to the work of Brown, Hobson and Rogers.

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\l\'oj and Spoida.

Keywords: Optimal control, robust pricing and hedging, volatility uncertainty, optimal transportation, pathwise inequalities, lookback option

JEL Classification: C00

Suggested Citation

Henry-Labordere, Pierre and Obłój, Jan and Spoida, Peter and Touzi, Nizar, Maximum Maximum of Martingales Given Marginals (April 2013). Available at SSRN: https://ssrn.com/abstract=2031461 or http://dx.doi.org/10.2139/ssrn.2031461

Pierre Henry-Labordere (Contact Author)

Société Générale - Paris, France ( email )

Paris-La Défense, Paris 92987
France

Jan K. Obloj

University of Oxford - Mathematical Institute ( email )

AWB, ROQ, Woodstock Rd
Oxford, OX2 6GG
United Kingdom

University of Oxford - Oxford-Man Institute of Quantitative Finance ( email )

Eagle House
Walton Well Road
Oxford, Oxfordshire OX2 6ED
United Kingdom

University of Oxford - Saint John's College ( email )

St Giles
Oxford, Oxon OX1 3JP
United Kingdom

Peter Spoida

University of Oxford ( email )

Mansfield Road
Oxford, Oxfordshire OX1 4AU
United Kingdom

Nizar Touzi

Ecole Polytechnique, Paris ( email )

1 rue Descartes
Paris, 75005
France

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