Implied Risk-Neutral Filtering Densities on Volatility's Hidden State

Abstract

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, so we can input derivative prices into an inverse problem, and the solution obtained will be an implied conditional density on volatility. Our analysis relies on a specification of the martingale change of measure, which we will refer to as 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, which indicates that the volatility-uncertainty premium could experience cyclic effects due to the maturity date of the options.