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

By Pierre Henry-Labordère, Jan Obłój, Peter Spoida, Nizar Touzi. 2012

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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 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].

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/c881ef544191fc57712bc932cb50065afbcc5121

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Citations in the corpus

Optimal Skorokhod embedding given full marginals and Azema-Yor peacocks (2015, background): “Then, for every m > x 0 and As argued in [13], the above inequality implies that also the following inequality holds: with T m ( x ) := inf { t ≥ 0 : x t ≥ m } and Indeed, if x is continuous at T m ( x ), then the two inequalities coincide.”
Martingale Benamou–Brenier: A probabilistic perspective (2017, background): “nts. Such problems arise naturally in robust ﬁnance, but are also of independent mathematical interest, for example they have important consequences for the study of martingale inequalities (see e.g. [18, 31, 51]) and the Skorokhod embedding problem [8, 36]. Early papers to investigate such problems”
Optimal transport and Skorokhod embedding (2013, background): “l context, but are also of independent mathematical interest, for example — mirroring classical optimal transport — they have important consequences for the study of martingale inequalities (see e.g. [9, 27, 42]). The ﬁrst papers to study such problems include [31, 7, 23, 19], and this ﬁeld is commo”