A new perspective on the fundamental theorem of asset pricing for large financial markets

TL;DR

This paper extends the fundamental theorem of asset pricing to large financial markets with infinitely many assets by introducing a new no-arbitrage condition, NAFLVR, and proves its equivalence to the existence of a separating measure.

Contribution

It formulates the NAFLVR condition for large markets and proves a version of the FTAP under this condition, including uncountably infinite assets, relaxing previous assumptions.

Findings

NAFLVR is a meaningful no-arbitrage condition for large markets.

Existence of a separating measure is equivalent to NAFLVR.

Counterexample shows separating measure does not imply an equivalent σ-martingale measure.

Abstract

In the context of large financial markets we formulate the notion of \emph{no asymptotic free lunch with vanishing risk} (NAFLVR), under which we can prove a version of the fundamental theorem of asset pricing (FTAP) in markets with an (even uncountably) infinite number of assets, as it is for instance the case in bond markets. We work in the general setting of admissible portfolio wealth processes as laid down by Y. Kabanov \cite{kab:97} under a substantially relaxed concatenation property and adapt the FTAP proof variant obtained in \cite{CT:14} for the classical small market situation to large financial markets. In the case of countably many assets, our setting includes the large financial market model considered by M. De Donno et al. \cite{DGP:05} and its abstract integration theory. The notion of (NAFLVR) turns out to be an economically meaningful "no arbitrage" condition (in particular not involving weak-$*$-closures), and, (NAFLVR) is equivalent to the existence of a separating measure. Furthermore we show -- by means of a counterexample -- that the existence of an equivalent separating measure does not lead to an equivalent $\sigma$-martingale measure, even in a countable large financial market situation.