Some estimates in extended stochastic volatility models of Heston type

We show that in lognormal-like stochastic volatility models with additional local volatility functions, the tails of the cumulative distribution of log-returns behave as exp(-c|y|), where c is a positive constant depending on time and on model parameters. This estimate stems from the proof of a stronger result: using some estimates for the probability that an Itô process remains in a tube around a deterministic curve, 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 set an optimization procedure on the set of admissible curves, leading to the desired lower bound on the terminal c.d.f.. Even though the involved constants are less sharp than the ones derived for stochastic volatility models with a particular structure such as Heston [1,6,12], these lower bounds entail moment explosion.