price time series[@volatility], that quantifies the propensity of the market to fluctuate. Since volatility represents a measure of the risk associated to the fluctuating dynamics of prices, it is crucial to develop suitable models to predict its complex intermittent behavior. There is empirical evidence that it fluctuates following a stochastic dynamics subjacent to that of prices, whose dynamics, in turn, depends on the time evolving volatility. Many approaches are based on that assumption[@prices], although others propose the existence of a reciprocal feedback between both processes[@leverage].

Our approach builds on the development of a simple Langevin equation to characterize the stochastic process of volatility. The equation provides an unifying description that generalizes widely discussed models in the literature. We analyze the shape of the long-time probability density function (PDF) associated to the stochastic differential equation that characterizes each particular case of the generalized model. Most previous results focus on the tails of the PDFs. In fact, for stochastic variables, such as volatilities presenting fat-tailed PDFs[@liu; @volat1], it is specially important to reproduce extreme events in a realistic model. Now we go a step further and aim to predict the PDFs in the whole range of events.

One of the main features 