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 [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 σ of the Gaussian component of the driving Lévy process is to increase the call price by 1/2σ ²t²e^kv(k)(1+o(1)) as t → 0, where v is the Lévy density. Using the small-time expansion for call options, we then derive a small-time expansion for the implied volatility Equation Presented at log-moneyness k, which sharpens the first order estimate Equation Presented 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 ∈ (1, 2), lim_t-0 t^-1/Y E{double}(S_t-S₀)₊ = S₀E{double} ^*(Z₊) and the corresponding at-the-money implied volatility σ̂_t(0) satisfies lim_t-0 Equation Presented, 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 Equation Presented of the process.