  We show that in a large class of stochastic volatility models with additional
skew-functions (local-stochastic volatility models) the tails of the cumulative
distribution of the log-returns behave as exp(-c|y|), where c is a positive
constant depending on time and on model parameters. We obtain this estimate
proving a stronger result: using some estimates for the probability that Ito
processes remain around a deterministic curve from Bally et al. '09, we lower
bound the probability that the couple (X,V) remains around a two-dimensional
curve up to a given maturity, X being the log-return process and V its
instantaneous variance. Then we find the optimal curve leading to the bounds on
the terminal cdf. The method we rely on does not require inversion of
characteristic functions but works for general coefficients of the underlying
SDE (in particular, no affine structure is needed). Even though the involved
constants are less sharp than the ones derived for stochastic volatility models
with a particular structure, our lower bounds entail moment explosion, thus
implying that Black-Scholes implied volatility always displays wings in the
considered class of models. In a second part of this paper, using Malliavin
calculus techniques, we show that an analogous estimate holds for the density
of the log-returns as well.
