The small-maturity smile for exponential Levy models

Abstract

We derive a small-time expansion for out-of-the-money call options under an exponential Levy model, using the small-time expansion for the distribution function given in Figueroa-Lopez & Houdre (2009), combined with a change of numéraire via the Esscher transform. In particular, we quantify find that the effect of a non-zero volatility $σ$ of the Gaussian component of the driving Lévy process is to increase the call price by $1/2σ^2 t^2 e^{k}ν(k)(1+o(1))$ as $t \to 0$, where $ν$ is the Lévy density. Using the small-time expansion for call options, we then derive a small-time expansion for the implied volatility, which sharpens the first order estimate given in Tankov (2010). Our numerical results show that the second order approximation can significantly outperform the first order approximation. Our results are also extended to a class of time-changed Lévy models. We also consider a small-time, small log-moneyness regime for the CGMY model, and apply this approach to the small-time pricing of at-the-money call options.