  The volatility characterizes the amplitude of price return fluctuations. It
is a central magnitude in finance closely related to the risk of holding a
certain asset. Despite its popularity on trading floors, the volatility is
unobservable and only the price is known. Diffusion theory has many common
points with the research on volatility, the key of the analogy being that
volatility is the time-dependent diffusion coefficient of the random walk for
the price return. We present a formal procedure to extract volatility from
price data, by assuming that it is described by a hidden Markov process which
together with the price form a two-dimensional diffusion process. We derive a
maximum likelihood estimate valid for a wide class of two-dimensional diffusion
processes. The choice of the exponential Ornstein-Uhlenbeck (expOU) stochastic
volatility model performs remarkably well in inferring the hidden state of
volatility. The formalism is applied to the Dow Jones index. The main results
are: (i) the distribution of estimated volatility is lognormal, which is
consistent with the expOU model; (ii) the estimated volatility is related to
trading volume by a power law of the form $\sigma \propto V^{0.55}$; and (iii)
future returns are proportional to the current volatility which suggests some
degree of predictability for the size of future returns.
