The Annals of Applied Probability

The maximum maximum of a martingale with given $\mathbf{n}$ marginals

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

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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].

Article information

Ann. Appl. Probab., Volume 26, Number 1 (2016), 1-44.

Received: November 2013
Revised: September 2014
First available in Project Euclid: 5 January 2016

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Zentralblatt MATH identifier

Primary: 60G44: Martingales with continuous parameter 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 91G20: Derivative securities
Secondary: 60J60: Diffusion processes [See also 58J65]

Maximum process martingale robust pricing and hedging volatility uncertainty optimal transportation optimal control pathwise inequalities lookback option


Henry-Labordère, Pierre; Obłój, Jan; Spoida, Peter; Touzi, Nizar. The maximum maximum of a martingale with given $\mathbf{n}$ marginals. Ann. Appl. Probab. 26 (2016), no. 1, 1--44. doi:10.1214/14-AAP1084.

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