Implied Filtering Densities on Volatility's Hidden State

TL;DR

This paper develops a method to infer the hidden state of volatility from market data using an inverse problem approach, incorporating a risk premium, and demonstrates its effectiveness with SPX options data.

Contribution

It introduces a novel inverse problem framework with a separability assumption to derive implied filtering densities on volatility's hidden state from derivatives prices.

Findings

Estimated densities align with VIX index

Densities are stable over time

Capture cyclic effects related to options expiration

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, 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.