A convergence result for the Emery topology and a variant of the proof of the fundamental theorem of asset pricing

TL;DR

This paper establishes that NUPBR implies P-UT in the Emery topology, and uses this to provide a concise proof of the fundamental theorem of asset pricing within a general admissible portfolio framework.

Contribution

It introduces a new implication from NUPBR to P-UT in the Emery topology and offers a simplified proof of the fundamental theorem of asset pricing.

Findings

NUPBR implies P-UT in the Emery topology.

A short variant of the proof of the fundamental theorem of asset pricing.

Results are applicable in a general admissible portfolio setting.

Abstract

We show that \emph{No unbounded profit with bounded risk} (NUPBR) implies \emph{predictable uniform tightness} (P-UT), a boundedness property in the Emery topology which has been introduced by C. Stricker \cite{S:85}. Combining this insight with well known results from J. M\'emin and L. S{\l}ominski \cite{MS:91} leads to a short variant of the proof of the fundamental theorem of asset pricing initially proved by F. Delbaen and W. Schachermayer \cite{DS:94}. The results are formulated in the general setting of admissible portfolio wealth processes as laid down by Y. Kabanov in \cite{kab:97}.