# Duality for pathwise superhedging in continuous time

## TL;DR

This paper establishes a duality between model-free superhedging prices and supremum expectations over martingale measures in continuous time, accommodating path-dependent options and semi-static strategies.

## Contribution

It introduces a new duality framework for pathwise superhedging in continuous time, extending to semi-static hedging and Vovk's outer measure.

## Key findings

- Superhedging price equals supremum over martingale measures.
- Applicable to path-dependent European options with semi-static strategies.
- Reduces to measures with compact support under certain stability conditions.

## Abstract

We provide a model-free pricing-hedging duality in continuous time. For a frictionless market consisting of $d$ risky assets with continuous price trajectories, we show that the purely analytic problem of finding the minimal superhedging price of a path dependent European option has the same value as the purely probabilistic problem of finding the supremum of the expectations of the option over all martingale measures. The superhedging problem is formulated with simple trading strategies, the claim is the limit inferior of continuous functions, which allows for upper and lower semi-continuous claims, and superhedging is required in the pathwise sense on a $\sigma$-compact sample space of price trajectories. If the sample space is stable under stopping, the probabilistic problem reduces to finding the supremum over all martingale measures with compact support. As an application of the general results we deduce dualities for Vovk's outer measure and semi-static superhedging with finitely many securities.