Robust Fundamental Theorem for Continuous Processes

TL;DR

This paper establishes a robust version of the fundamental theorem of asset pricing for continuous-time markets under model uncertainty, linking no-arbitrage conditions to the existence of equivalent martingale measures.

Contribution

It introduces a new no-arbitrage concept under model uncertainty and proves its equivalence to the existence of martingale measures, extending classical results to more general settings.

Findings

Equivalence between robust no-arbitrage and martingale measures.

Existence of optimal superhedging strategies.

Representation of superhedging prices via martingale measures.

Abstract

We study a continuous-time financial market with continuous price processes under model uncertainty, modeled via a family $\mathcal{P}$ of possible physical measures. A robust notion ${\rm NA}_{1}(\mathcal{P})$ of no-arbitrage of the first kind is introduced; it postulates that a nonnegative, nonvanishing claim cannot be superhedged for free by using simple trading strategies. Our first main result is a version of the fundamental theorem of asset pricing: ${\rm NA}_{1}(\mathcal{P})$ holds if and only if every $P\in\mathcal{P}$ admits a martingale measure which is equivalent up to a certain lifetime. The second main result provides the existence of optimal superhedging strategies for general contingent claims and a representation of the superhedging price in terms of martingale measures.