Bounds on Stock Price probability distributions in Local-Stochastic Volatility models

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.