The exact Taylor formula of the implied volatility

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