Asymptotic equivalence in Lee's moment formulas for the implied volatility and Piterbarg's conjecture

TL;DR

This paper investigates the asymptotic behavior of implied volatility through Lee's moment formulas and Piterbarg's conjecture, providing new conditions for their validity and applying results to specific stochastic models.

Contribution

It establishes conditions for the existence of limits in Lee's formulas, proves a modified Piterbarg's conjecture, and applies findings to the CEV and Heston models with jumps.

Findings

Conditions guaranteeing the existence of limits in Lee's formulas

A proof of a modified Piterbarg's conjecture

Asymptotic formulas applied to CEV and Heston models with jumps

Abstract

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.