  In this paper, we study stochastic volatility models in regimes where the
maturity is small, but large compared to the mean-reversion time of the
stochastic volatility factor. The problem falls in the class of
averaging/homogenization problems for nonlinear HJB-type equations where the
"fast variable" lives in a noncompact space. We develop a general argument
based on viscosity solutions which we apply to the two regimes studied in the
paper. We derive a large deviation principle, and we deduce asymptotic prices
for out-of-the-money call and put options, and their corresponding implied
volatilities. The results of this paper generalize the ones obtained in Feng,
Forde and Fouque [SIAM J. Financial Math. 1 (2010) 126-141] by a moment
generating function computation in the particular case of the Heston model.
