  Stochastic volatility models describe asset prices $S_t$ as driven by an
unobserved process capturing the random dynamics of volatility $\sigma_t$.
Here, we quantify how much information about $\sigma_t$ can be inferred from
asset prices $S_t$ in terms of Shannon's mutual information $I(S_t :
\sigma_t)$. This motivates a careful numerical and analytical study of
information theoretic properties of the Heston model. In addition, we study a
general class of discrete time models motivated from a machine learning
perspective. In all cases, we find a large uncertainty in volatility estimates
for quite fundamental information theoretic reasons.
