The minimum maximum of a continuous martingale with given initial and terminal laws

Abstract

Let $(M_t)_{0 \leq t \leq 1} be a continuous martingale with initial law $M_0 \sim \mu_0$, and terminal law $M_1 \sim \mu_1$, and let $S = \sup_{0 \leq t \leq 1} M_t$. In this paper we prove that there exists a greatest lower bound with respect to stochastic ordering of probability measures, on the law of $S$. We give an explicit construction of this bound. Furthermore a martingale is constructed which attains this minimum by solving a Skorokhod embedding problem. The form of this martingale is motivated by a simple picture. The result is applied to the robust hedging of a forward start digital option.