  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.
