Implied Filtering Densities on the Hidden State of Stochastic Volatility

Abstract

We formulate and analyse an inverse problem using derivative 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 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.

Key Words:

• Filtering
• stochastic volatility
• HMM
• volatility risk premium
• Heston model
• volatility uncertainty

Acknowledgements

We thank the anonymous referee(s) whose comments greatly helped to improve this article.

Notes

¹ The parameter r is part of a term structure of discount rates and includes an adjustment for the SPX’s dividend rate. For each maturity time T, the discount rate is computed from the put-call parity, on which we run a linear regression for parameters and over options of maturity T. The optimal parameters fit the data to the strike prices, K, to the model , where and are the market’s price of calls and puts (respectively) with maturity T, and is the indicator that we have taken the difference as opposed to . The discount rate is where is the least-squares fit; this estimator usually has very low variance if is greater than a week.