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http://ssrn.com/abstract=2031461
 
 

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Maximum Maximum of Martingales Given Marginals


Pierre Henry-Labordere


Société Générale - Paris, France

Jan K. Obloj


University of Oxford

Peter Spoida


University of Oxford

Nizar Touzi


Ecole Polytechnique, Paris

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.

Number of Pages in PDF File: 35

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

JEL Classification: C00

working papers series


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Date posted: April 1, 2012 ; Last revised: April 10, 2013

Suggested Citation

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

Contact Information

Pierre Henry-Labordere (Contact Author)
Société Générale - Paris, France ( email )
Paris-La Défense, Paris 92987
France
Jan K. Obloj
University of Oxford ( email )
Mansfield Road
Oxford, Oxfordshire OX1 4AU
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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