# The robust superreplication problem: a dynamic approach

## TL;DR

This paper develops a dynamic framework for robust superhedging and utility maximization in discrete-time financial markets, accounting for model uncertainty and trader-specific subjective views, with proven existence and uniqueness of optimal strategies.

## Contribution

It introduces a dynamic programming approach to robust superhedging and formulates a new utility maximization problem under subjective model uncertainty.

## Key findings

- Superhedging prices evolve via a concave envelope process.
- Optimal strategies exist under certain utility conditions.
- The framework accommodates trader-specific subjective market views.

## Abstract

In the frictionless discrete time financial market of Bouchard et al.(2015) we consider a trader who, due to regulatory requirements or internal risk management reasons, is required to hedge a claim $\xi$ in a risk-conservative way relative to a family of probability measures $\mathcal{P}$. 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 and show that $\pi_t(\xi)$ can be seen as a concave envelope of $\pi_{t+1}(\xi)$ evaluated at today's prices. Then we consider an optimal investment problem for a trader who is rolling over her robust superhedge and phrase this as a robust maximisation problem, where the expected utility of inter-temporal consumption is optimised subject to a robust superhedging constraint. This utility maximisation is carrried out under a new family of measures $\mathcal{P}^u$, which no longer have to capture regulatory or institutional risk views but rather represent 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.