  We consider a stochastic volatility model where the moment generating
function of the logarithmic price is finite only on part of the real line.
Using a new Tauberian result obtained in [1] and [2], we show that the
knowledge of the moment generating function near its critical moment gives a
sharp asymptotic expansion (with an error of order o(1)) of the local
volatility and implied volatility for small and large strikes. We apply our
theoretical estimates to Gatheral's SVI parametrization of the implied
volatility and Heston's model.

Pricing of Securities;Probability