How Non-Arbitrage, Viability and Num\'eraire Portfolio are Related

TL;DR

This paper establishes fundamental links between market viability, absence of arbitrage, and the existence of a numéraire portfolio in continuous-time models, providing new equivalences and concepts under minimal assumptions.

Contribution

It proves the equivalence of NUPBR, the existence of the numéraire portfolio, and optimal portfolios under an equivalent measure, introducing new viability concepts.

Findings

NUPBR is equivalent to the existence of the numéraire portfolio.

Optimal portfolios exist under a measure close to the real-world probability.

New viability concepts called weak and local viabilities are introduced.

Abstract

This paper proposes two approaches that quantify the exact relationship among the viability, the absence of arbitrage, and/or the existence of the num\'eraire portfolio under minimal assumptions and for general continuous-time market models. Precisely, our first and principal contribution proves the equivalence among the No-Unbounded-Profit-with-Bounded-Risk condition (NUPBR hereafter), the existence of the num\'eraire portfolio, and the existence of the optimal portfolio under an equivalent probability measure for any "nice" utility and positive initial capital. Herein, a 'nice" utility is any smooth von Neumann-Morgenstern utility satisfying Inada's conditions and the elasticity assumptions of Kramkov and Schachermayer. Furthermore, the equivalent probability measure ---under which the utility maximization problems have solutions--- can be chosen as close to the real-world probability measure as we want (but might not be equal). Without changing the underlying probability measure and under mild assumptions, our second contribution proves that the NUPBR is equivalent to the "{\it local}" existence of the optimal portfolio. This constitutes an alternative to the first contribution, if one insists on working under the real-world probability. These two contributions lead naturally to new types of viability that we call weak and local viabilities.