Short-Time Behavior of the At-the-Money Implied Volatility for the Jump-Diffusion Stochastic Volatility Bachelier Model

Abstract

In this paper we obtain expressions for the short-time behaviour of the at-the-money implied volatility (ATM-IV) level and skew for a jump-diffusion asset price. The diffusion part is assumed to be the stochastic volatility Bachelier model and the jumps are modelled by a pure-jump Lévy process with drift so that the asset price is a martingale. Regarding the level, we show that the short-time behaviour of the ATM-IV level is the same for all pure-jump Lévy processes and, regarding the skew, we give conditions on the law of the jumps for the skew to exist. To do so, we combine Malliavin Calculus techniques to obtain the formulas for the compound Poisson case and, using an adequate approximation scheme, we extend the formulas for Lévy processes with infinite activity, including some cases of Lévy processes with infinite variation paths. We also provide numerical evidence that confirm the theoretical results found in the paper.