The Exact Taylor Formula of the Implied Volatility
44 Pages Posted: 14 Oct 2015 Last revised: 7 Oct 2016
Date Written: October 2016
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.
Keywords: implied volatility, local diffusions, Markov processes, asymptotic expansion, local-stochastic volatility
JEL Classification: G12, G13
Suggested Citation: Suggested Citation