  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.
