## 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 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].

Original language | English (US) |
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Pages (from-to) | 1-44 |

Number of pages | 44 |

Journal | Annals of Applied Probability |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2016 |

## Keywords

- Lookback option
- Martingale
- Maximum process
- Optimal control
- Optimal transportation
- Pathwise inequalities
- Robust pricing and hedging
- Volatility uncertainty

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty