The maximum maximum of a martingale with given \(n\) marginals

DOI10.1214/14-AAP1084zbMATH Open1337.60078arXiv1203.6877OpenAlexW3098180202MaRDI QIDQ259564

Peter Spoida, Jan Obłój, Pierre Henry-Labordère, Nizar Touzi

Publication date: 11 March 2016

Published in: The Annals of Applied Probability (Search for Journal in Brave)

Abstract: 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 Ob{l}'oj and Spoida [An iterated Az'ema-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-Labord`ere and Touzi [Ann. Appl. Probab. 24 (2014) 312-336].

Full work available at URL: https://arxiv.org/abs/1203.6877

zbMATH Keywords

optimal controlmartingaleshedgingoptimal transportationlookback optionmaximum processpathwise inequalitiesrobust pricingvolatility uncertainty

Mathematics Subject Classification ID

Extreme value theory; extremal stochastic processes (60G70) Optimal stochastic control (93E20) Martingales with continuous parameter (60G44) Diffusion processes (60J60) Financial applications of other theories (91G80) Derivative securities (option pricing, hedging, etc.) (91G20) Existence of optimal solutions to problems involving randomness (49J55)

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