THE MAXIMUM MAXIMUM OF A MARTINGALE WITH GIVEN n MARGINALS

• 成果类型: Article
• 署名作者: Henry-Labordere, Pierre; Obloj, Jan; Spoida, Peter; Touzi, Nizar
• 署名单位: University of Oxford; Institut Polytechnique de Paris; ENSTA Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1084
• 发表日期: 2016
• 页码: 1-44
• 关键词: PROBABILITY-MEASURES; Contingent claims; arbitrage

摘要

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 Obloj and Spoida [An iterated Azema-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-Labordere and Touzi [Ann. Appl. Probab. 24 (2014) 312-336].