  The asymptotic behavior of the implied volatility associated with a general
call pricing function has been extensively studied in the last decade. The main
topics discussed in this paper are Lee's moment formulas for the implied
volatility, and Piterbarg's conjecture, describing how the implied volatility
behaves in the case where all the moments of the stock price are finite. We
find various conditions guaranteeing the existence of the limit in Lee's moment
formulas. We also prove a modified version of Piterbarg's conjecture and
provide a non-restrictive sufficient condition for the validity of this
conjecture in its original form. The asymptotic formulas obtained in the paper
are applied to the implied volatility in the CEV model and in the Heston model
perturbed by a compound Poisson process with double exponential law for jump
sizes.
