  The short-time asymptotic behavior of option prices for a variety of models
with jumps has received much attention in recent years. In the present work, a
novel second-order approximation for ATM option prices under the CGMY L\'evy
model is derived, and then extended to a model with an additional independent
Brownian component. Our results shed 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 case of an additional Brownian
component, the second-order term, in time-t, is of the form $ d_{2}
t^{(3-Y)/2}$, with the coefficient $d_{2}$ depending only on the overall jump
intensity parameter C and the tail-heaviness parameter Y. This extends the
known result that the leading term is $(\sigma/\sqrt{2\pi})t^{1/2}$, where
$\sigma$ is the volatility of the continuous component. In contrast, under a
pure-jump CGMY model, the dependence on the two parameters C and Y is already
reflected in the leading term, which is of the form $d_{1} t^{1/Y}$.
Information on the relative frequency of negative and positive jumps appears
only in the second-order term, which is shown to be of the form $d_{2} t$ and
whose order of decay turns out to be independent of Y. The third-order
asymptotic behavior of the option prices as well as the asymptotic behavior of
the corresponding Black-Scholes implied volatilities are also addressed. Our
numerical results show that in most cases the second-order term significantly
outperform the first-order approximation.
