The small-maturity smile for exponential Levy models

TL;DR

This paper derives small-time expansions for out-of-the-money call options and implied volatility under exponential Lévy models, improving accuracy with second-order approximations and extending results to time-changed Lévy models and small log-moneyness regimes.

Contribution

It introduces a second-order small-time expansion for call options and implied volatility under exponential Lévy models, enhancing precision over previous first-order estimates.

Findings

Second-order expansion significantly outperforms first-order approximation.

Effect of Gaussian volatility increases call price by a specific small-time term.

Extensions to time-changed Lévy models and small log-moneyness regimes are demonstrated.

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 num\'eraire via the Esscher transform. In particular, we quantify find that the effect of a non-zero volatility $\sigma$ of the Gaussian component of the driving L\'{e}vy process is to increase the call price by $1/2\sigma^2 t^2 e^{k}\nu(k)(1+o(1))$ as $t \to 0$, where $\nu$ is the L\'evy 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 L\'evy 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.