TY - JOUR

T1 - Implied Filtering Densities on the Hidden State of Stochastic Volatility

AU - Fuertes, Carlos

AU - Papanicolaou, Andrew

N1 - Funding Information:
This research has been partially supported by NSF grant #DMS-0739195.
Publisher Copyright:
© 2014, © 2014 Taylor & Francis.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

KW - Filtering

KW - HMM

KW - Heston model

KW - stochastic volatility

KW - volatility risk premium

KW - volatility uncertainty

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U2 - 10.1080/1350486X.2014.891357

DO - 10.1080/1350486X.2014.891357

M3 - Article

AN - SCOPUS:84926101307

SN - 1350-486X

VL - 21

SP - 483

EP - 522

JO - Applied Mathematical Finance

JF - Applied Mathematical Finance

IS - 6

ER -