The exact Taylor formula of the implied volatility

TL;DR

This paper derives the exact Taylor expansion of implied volatility near expiry and at the money for multi-dimensional local diffusion models, including popular models like Heston, CEV, and SABR.

Contribution

It provides explicit limits and the exact Taylor formula for implied volatility derivatives in a broad class of local diffusion models, including locally elliptic cases.

Findings

Explicit limits of implied volatility derivatives near expiry and at the money.

Exact Taylor formula for implied volatility within a parabolic region.

Applicability to models like Heston, CEV, and SABR.

Abstract

In a model driven by a multi-dimensional local diffusion, we study the behavior of implied volatility {\sigma} and its derivatives with respect to log-strike k and maturity T near expiry and at the money. We recover explicit limits of these derivatives for (T,k) approaching the origin within the parabolic region |x-k|^2 < {\lambda} T, with x denoting the spot log-price of the underlying asset and where {\lambda} is a positive and arbitrarily large constant. Such limits yield the exact Taylor formula for implied volatility within the parabola |x-k|^2 < {\lambda} T. In order to include important models of interest in mathematical finance, e.g. Heston, CEV, SABR, the analysis is carried out under the assumption that the infinitesimal generator of the diffusion is only locally elliptic.