Implied Filtering Densities on Volatility's Hidden State

31 Pages Posted: 23 Dec 2012 Last revised: 21 Sep 2014

See all articles by Carlos Fuertes

Carlos Fuertes

Princeton University

Andrew Papanicolaou

NYU Tandon School of Engineering, Department of Finance and Risk Engineering

Date Written: March 25, 2014


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.

Keywords: Heston model, filtering, stochastic volatility, hidden states

JEL Classification: G12, G13, G17

Suggested Citation

Fuertes, Carlos and Papanicolaou, Andrew, Implied Filtering Densities on Volatility's Hidden State (March 25, 2014). Applied Mathematical Finance, Volume 21, Issue 6, (2014) pp. 483-522.. Available at SSRN: or

Carlos Fuertes

Princeton University ( email )

22 Chambers Street
Princeton, NJ 08544-0708
United States

Andrew Papanicolaou (Contact Author)

NYU Tandon School of Engineering, Department of Finance and Risk Engineering ( email )

6 Metrotech Center
Brooklyn, NY 11201
United States

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