%0 Generic %1 henrylabordere2012maximum %A Henry-Labordere, Pierre %A Obloj, Jan %A Spoida, Peter %A Touzi, Nizar %D 2012 %K marginals martingales maximum %T Maximum Maximum of Martingales given Marginals %U http://arxiv.org/abs/1203.6877 %X 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 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 hobson-klimmek (under slightly different conditions). The two marginal case corresponds to the work of Brown, Hobson and Rogers 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 Obłój and Spoida OblSp.

@misc{henrylabordere2012maximum, 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 \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 Az\'ema-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\l\'oj and Spoida \cite{OblSp}.}, added-at = {2014-09-18T20:10:48.000+0200}, author = {Henry-Labordere, Pierre and Obloj, Jan and Spoida, Peter and Touzi, Nizar}, biburl = {https://www.bibsonomy.org/bibtex/2204f015d8c18d16b9f2e9ff68910d985/pitman}, interhash = {9e684f7606f769157834a5dde8f35811}, intrahash = {204f015d8c18d16b9f2e9ff68910d985}, keywords = {marginals martingales maximum}, note = {cite arxiv:1203.6877}, timestamp = {2014-09-18T20:10:48.000+0200}, title = {Maximum Maximum of Martingales given Marginals}, url = {http://arxiv.org/abs/1203.6877}, year = 2012 }