  The correlated stochastic volatility models constitute a natural extension of
the Black and Scholes-Merton framework: here the volatility is not a constant,
but a stochastic process correlated with the price log-return one. At present,
several stochastic volatility models are discussed in the literature, differing
in the dynamics attached to the volatility. The aim of the present work is to
compare the most recent results about three popular models: the Vasicek, Heston
and exponential Ornstein-Uhlenbeck models. We analyzed for each of them the
theoretical results known in the literature (volatility and return
distribution, higher-order moments and different-time correlations) in order to
test their predictive effectiveness on the outcomes of original numerical
simulations, paying particular attention to their ability to reproduce
empirical statistical properties of prices. The numerical results demonstrate
that these models can be implemented maintaining all their features, especially
in view of financial applications like market risk management or option
pricing. In order to critically compare the models, we also perform an
empirical analysis of financial time series from the Italian stock market,
showing the exponential Ornstein-Uhlenbeck model's ability to capture the
stylized facts of volatility and log-return probability distributions.
