The Small-Maturity Smile for Exponential Lévy Models

Abstract

We derive a small-time expansion for out-of-the-money call options under an exponential Lévy model, using the small-time expansion for the distribution function given in [J. Figueroa-López and C. Houdré, Stochastic Process. Appl., 119 (2009), pp. 3862–3889], combined with a change of numéraire via the Esscher transform. In particular, we find that the effect of a nonzero volatility $\sigma$ of the Gaussian component of the driving Lévy process is to increase the call price by $\frac{1}{2}\sigma^2 t^2 e^{k}\nu(k)(1+o(1))$ as $t \to 0$, where $\nu$ is the Lévy density. Using the small-time expansion for call options, we then derive a small-time expansion for the implied volatility $\hat{\sigma}_{t}^{2}(k)$ at log-moneyness k, which sharpens the first order estimate $\hat{\sigma}_{t}^{2}(k)\sim \frac{\frac{1}{2}k^2}{t\log (1/t)}$ given in [P. Tankov, Pricing and hedging in exponential Lévy models: Review of recent results, in Paris-Princeton Lectures on Mathematical Finance, Springer, Berlin, 2011, pp. 319–359]. 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; we show that for $Y\in(1,2)$, $\lim_{t\to{}0}t^{-1/Y}\mathbb{E}(S_t-S_0)_{+}=S_{0}\mathbb{E}^{*}(Z_{+})$ and the corresponding at-the-money implied volatility $\hat{\sigma}_t(0)$ satisfies $\lim_{t \to 0}\hat{\sigma}_t(0)/t^{1/Y-1/2}=\sqrt{2\pi}\,\mathbb{E}^{*}(Z_{+})$, where Z is a symmetric Y-stable random variable under $\mathbb{P}^*$ and Y is the usual parameter for the CGMY model appearing in the Lévy density $\nu(x)=C x^{-1-Y}e^{-M x}{\bf 1}_{\{x>0\}}+C |x|^{-1-Y}e^{-G|x|}{\bf 1}_{\{x<0\}}$ of the process.

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