  This paper investigates the problem of maximizing expected terminal utility
in a (generically incomplete) discrete-time financial market model with finite
time horizon. In contrast to the standard setting, a possibly non-concave
utility function $U$ is considered, with domain of definition $\mathbb{R}$.
Simple conditions are presented which guarantee the existence of an optimal
strategy for the problem. In particular, the asymptotic elasticity of $U$ plays
a decisive role: existence can be shown when it is strictly greater at
$-\infty$ than at $+\infty$.
