  We formulate and analyze an inverse problem using derivatives prices to
obtain an implied filtering density on volatility's hidden state. Stochastic
volatility is the unobserved state in a hidden Markov model (HMM) and can be
tracked using Bayesian filtering. However, derivative data can be considered as
conditional expectations that are already observed in the market, and which can
be used as input to an inverse problem whose solution is an implied conditional
density on volatility. Our analysis relies on a specification of the martingale
change of measure, which we refer to as \textit{separability}. This
specification has a multiplicative component that behaves like a risk premium
on volatility uncertainty in the market. When applied to SPX options data, the
estimated model and implied densities produce variance-swap rates that are
consistent with the VIX volatility index. The implied densities are relatively
stable over time and pick up some of the monthly effects that occur due to the
options' expiration, indicating that the volatility-uncertainty premium could
experience cyclic effects due to the maturity date of the options.
