The Small-Maturity Smile for Exponential Lévy Models

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 Figueroa-López&Houdré[FLH09], combined with a change of numéraire via the Esscher transform. In particular, we 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 2 σteν(k)(1+ o(1)) as t → 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 σ̂ t (k) at log-moneyness k, which sharpens the first order estimate σ̂ 2 t (k) ∼ 1 2 k t log(1/t) given in [Tnkv10]. 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 ∈ (1, 2), limt→0 t E(St−S0)+ = S0E(Z+) and the corresponding at-the-money implied volatility σ̂t(0) satisfies limt→0 σ̂t(0)/t 1/Y −1/2 = √ 2π E(Z+), where Z is a symmetric Y stable random variable under P and Y is the usual parameter for the CGMY model appearing in the Lévy density ν(x) = Cx e1{x>0} +C|x| e1{x<0} of the process.