Asymptotics for exponential Lévy processes and their volatility smile: survey and new results

Abstract. Exponential Lévy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes, and the corresponding implied volatility surfaces have been analyzed in some detail. In the non-asymptotic regimes, option prices are de-scribed by the Lewis-Lipton formula which allows one to represent them as Fourier integrals; the prices can be trivially expressed in terms of their implied volatility. Recently, attempts at calculating the asymptotic limits of the im-plied volatility have yielded several expressions for the short-time, long-time, and wing asymptotics. In order to study the volatility surface in required de-tail, in this paper we use the FX conventions and describe the implied volatility as a function of the Black-Scholes delta. Surprisingly, this convention is closely related to the resolution of singularities frequently used in algebraic geometry. In this framework, we survey the literature, reformulate some known facts re-garding the asymptotic behavior of the implied volatility, and present severa