Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results

Exponential L\'evy 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 described 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 implied volatility have yielded several expressions for the short-time, long-time, and wing asymptotics. In order to study the volatility surface in required detail, 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 regarding the asymptotic behavior of the implied volatility, and present several new results. We emphasize the role of fractional differentiation in studying the tempered stable exponential Levy processes and derive novel numerical methods based on judicial finite-difference approximations for fractional derivatives. We also briefly demonstrate how to extend our results in order to study important cases of local and stochastic volatility models, whose close relation to the L\'evy process based models is particularly clear when the Lewis-Lipton formula is used. Our main conclusion is that studying asymptotic properties of the implied volatility, while theoretically exciting, is not always practically useful because the domain of validity of many asymptotic expressions is small.