  Volatility measures the amplitude of price fluctuations. Despite it is one of
the most important quantities in finance, volatility is not directly
observable. Here we apply a maximum likelihood method which assumes that price
and volatility follow a two-dimensional diffusion process where volatility is
the stochastic diffusion coefficient of the log-price dynamics. We apply this
method to the simplest versions of the expOU, the OU and the Heston stochastic
volatility models and we study their performance in terms of the log-price
probability, the volatility probability, and its Mean First-Passage Time. The
approach has some predictive power on the future returns amplitude by only
knowing current volatility. The assumed models do not consider long-range
volatility auto-correlation and the asymmetric return-volatility
cross-correlation but the method still arises very naturally these two
important stylized facts. We apply the method to different market indexes and
with a good performance in all cases.
