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 make sense . 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.
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.
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make sense
be well defined