A note on the Fundamental Theorem of Asset Pricing under model uncertainty

TL;DR

This paper extends the Fundamental Theorem of Asset Pricing and super-hedging results to markets with bid-ask spreads on hedging options, introducing robust no-arbitrage and proving set closedness under these conditions.

Contribution

It introduces the concept of robust no-arbitrage for markets with bid-ask spreads and proves the closedness of attainable claims in this setting.

Findings

Robust no-arbitrage is equivalent to no-arbitrage with non-redundant spread options.

The set of attainable claims is closed under the extended framework.

The results generalize classical theorems to markets with bid-ask spreads.

Abstract

We show that the results of ArXiv:1305.6008 on the Fundamental Theorem of Asset Pricing and the super-hedging theorem can be extended to the case in which the options available for static hedging (\emph{hedging options}) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of \emph{robust no-arbitrage} which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are \emph{non-redundant}. A key result is the closedness of the set of attainable claims, which requires a new proof in our setting.