  In the present work, a novel second-order approximation for ATM option prices
is derived for a large class of exponential L\'{e}vy models with or without
Brownian component. The results hereafter shed new light on the connection
between both the volatility of the continuous component and the jump parameters
and the behavior of ATM option prices near expiration. In the presence of a
Brownian component, the second-order term, in time-$t$, is of the form
$d_{2}\,t^{(3-Y)/2}$, with $d_{2}$ only depending on $Y$, the degree of jump
activity, on $\sigma$, the volatility of the continuous component, and on an
additional parameter controlling the intensity of the "small" jumps (regardless
of their signs). This extends the well known result that the leading
first-order term is $\sigma t^{1/2}/\sqrt{2\pi}$. In contrast, under a
pure-jump model, the dependence on $Y$ and on the separate intensities of
negative and positive small jumps are already reflected in the leading term,
which is of the form $d_{1}t^{1/Y}$. The second-order term is shown to be of
the form $\tilde{d}_{2} t$ and, therefore, its order of decay turns out to be
independent of $Y$. The asymptotic behavior of the corresponding Black-Scholes
implied volatilities is also addressed. Our approach is sufficiently general to
cover a wide class of L\'{e}vy processes which satisfy the latter property and
whose L\'{e}vy densitiy can be closely approximated by a stable density near
the origin. Our numerical results show that the first-order term typically
exhibits rather poor performance and that the second-order term can
significantly improve the approximation's accuracy, particularly in the absence
of a Brownian component.
