The maximum maximum of a martingale with given marginals
arXiv:1203.6877 · doi:10.1214/14-AAP1084
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 -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].
Published at http://dx.doi.org/10.1214/14-AAP1084 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
References in corpus (8)
- A theoretical framework for the pricing of contingent claims in the presence of model uncertainty
- The Skorokhod embedding problem and its offspring
- Arbitrage and duality in nondominated discrete-time models
- A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options
- Optimal transportation under controlled stochastic dynamics
- A trajectorial interpretation of Doob's martingale inequalities
- Pathwise Construction of Stochastic Integrals
- Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping