  In Figueroa-L\'opez et al. (2013), a second order approximation for
at-the-money (ATM) option prices is derived for a large class of exponential
L\'evy models, with or without a Brownian component. The purpose of this
article is twofold. First, we relax the regularity conditions imposed in
Figueroa-L\'opez et al. (2013) on the L\'evy density to the weakest possible
conditions for such an expansion to be well defined. Second, we show that the
formulas extend both to the case of "close-to-the-money" strikes and to the
case where the continuous Brownian component is replaced by an independent
stochastic volatility process with leverage.
