ASYMPTOTIC EQUIVALENCE IN LEE'S MOMENT FORMULAS FOR THE IMPLIED VOLATILITY, ASSET PRICE MODELS WITHOUT MOMENT EXPLOSIONS, AND PITERBARG'S CONJECTURE

Abstract

In this paper, we study the asymptotic behavior of the implied volatility in stochastic asset price models. We provide necessary and sufficient conditions for the validity of asymptotic equivalence in Lee's moment formulas, and obtain new asymptotic formulas for the implied volatility in asset price models without moment explosions. As an application, we prove a modified version of Piterbarg's conjecture. The asymptotic formula suggested by Piterbarg may be considered as a substitute for Lee's moment formula for the implied volatility at large strikes in the case of models without moment explosions. We also characterize the asymptotic behavior of the implied volatility in several special asset price models, e.g., the CEV model, the finite moment log-stable model of Carr and Wu, the Heston model perturbed by a compound Poisson process with double exponential law for jump sizes, and SV1 and SV2 models of Rogers and Veraart.