06130636

j

Figueroa-L\'opez, Jos\'e E. Forde, Martin The small-maturity smile for exponential L\'evy models. SIAM J. Financ. Math. 3, No. 1, 33-65, electronic only (2012). 2012 Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA EN 60G51 60F99 91G20 91G60 exponential L\'evy models time-changed L\'evy models option pricing short-time asymptotics implied volatility Zbl 1179.60026 Zbl 1205.91161

doi:10.1137/110820658

Summary: We derive a small-time expansion for out-of-the-money call options under an exponential L\'evy model, using the small-time expansion for the distribution function given in [{\it J. Figueroa-L\'opez} and {\it C. Houdr\'e}, Stochastic Process. Appl. 119, No. 11, 3862--3889 (2009; Zbl 1179.60026)], combined with a change of num\'eraire via the Esscher transform. In particular, we find that the effect of a nonzero volatility $\sigma$ of the Gaussian component of the driving L\'evy 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 L\'evy 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 [{\it P. Tankov}, ``Pricing and hedging in exponential L\'evy models: review of recent results'', Lecture Notes in Mathematics 2003, 319--359 (2011; Zbl 1205.91161)]. 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; 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 L\'evy density $\nu(x)=C x^{-1-Y}e^{-M x}\bold{1}_{\{x>0\}}+C |x|^{-1-Y}e^{-G|x|}\bold{1}_{\{x<0\}}$ of the process.