Open Access
February 2016 The maximum maximum of a martingale with given $\mathbf{n}$ marginals
Pierre Henry-Labordère, Jan Obłój, Peter Spoida, Nizar Touzi
Ann. Appl. Probab. 26(1): 1-44 (February 2016). DOI: 10.1214/14-AAP1084


We obtain bounds on the distribution of the maximum of a martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to $n$-marginal Skorokhod embedding problem in Obłój and Spoida [An iterated Azéma-Yor type embedding for finitely many marginals (2013) Preprint]. It follows that their embedding maximizes the maximum among all other embeddings. Our motivating problem is superhedging lookback options 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 derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558–578]. This inequality, proved by elementary arguments, is derived by following the stochastic control approach of Galichon, Henry-Labordère and Touzi [Ann. Appl. Probab. 24 (2014) 312–336].


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Pierre Henry-Labordère. Jan Obłój. Peter Spoida. Nizar Touzi. "The maximum maximum of a martingale with given $\mathbf{n}$ marginals." Ann. Appl. Probab. 26 (1) 1 - 44, February 2016.


Received: 1 November 2013; Revised: 1 September 2014; Published: February 2016
First available in Project Euclid: 5 January 2016

zbMATH: 1337.60078
MathSciNet: MR3449312
Digital Object Identifier: 10.1214/14-AAP1084

Primary: 60G44 , 91G20 , 91G80
Secondary: 60J60

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

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.26 • No. 1 • February 2016
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