The robust superreplication problem: a dynamic approach

Abstract

In the frictionless discrete time financial market of Bouchard and Nutz [Ann. Appl. Probab., 25 (2015), pp. 823--859] we consider a trader who is required to hedge $\xi$ in a risk-conservative way relative to a family of probability measures ${\cal P}$. We first describe the evolution of $\pi_t(\xi)$---the superhedging price at time $t$ of the liability $\xi$ at maturity $T$---via a dynamic programming principle, show that $\pi_t(\xi)$ can be seen as a concave envelope of $\pi_{t+1}(\xi)$ evaluated at today's prices, and prove its dual characterization. Under suitable assumptions, we show that the robust superreplication price is equal to the classical $P$-superhedging price for an extreme prior $P\in {\cal P}$. Then we consider an optimal investment problem for the trader who is rolling over her robust superhedge and phrase this as a robust maximization problem, where the expected utility of intertemporal consumption is optimized subject to a robust superhedging constraint. This utility maximization is carried out under a subset ${\cal P}^u$ of ${\cal P}$ representing the trader's subjective views on market dynamics. Under suitable assumptions on the trader's utility functions, we show that optimal investment and consumption strategies exist and further specify when, and in what sense, these may be unique