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This paper proposes two approaches that quantify the exact relationship amongthe viability, the absence of arbitrage, and/or the existence of thenum\'eraire portfolio under minimal assumptions and for general continuous-timemarket models. Precisely, our first and principal contribution proves theequivalence among the No-Unbounded-Profit-with-Bounded-Risk condition (NUPBRhereafter), the existence of the num\'eraire portfolio, and the existence ofthe optimal portfolio under an equivalent probability measure for any "nice"utility and positive initial capital. Herein, a 'nice" utility is any smoothvon Neumann-Morgenstern utility satisfying Inada's conditions and theelasticity assumptions of Kramkov and Schachermayer. Furthermore, theequivalent probability measure ---under which the utility maximization problemshave solutions--- can be chosen as close to the real-world probability measureas we want (but might not be equal). Without changing the underlyingprobability measure and under mild assumptions, our second contribution provesthat the NUPBR is equivalent to the "{\it local}" existence of the optimalportfolio. This constitutes an alternative to the first contribution, if oneinsists on working under the real-world probability. These two contributionslead naturally to new types of viability that we call weak and localviabilities.

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