Short-time expansions for close-to-the-money options under a L\'evy jump model with stochastic volatility

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.