We explore the inversion of derivatives prices to obtain an implied probability measure on volatility's hidden state. Stochastic volatility is  a hidden Markov model (HMM) , and HMMs ordinarily warrant filtering. However, derivative data is a set of conditional expectations that are already observed in the market, so rather than use filtering techniques we compute animplied distribution by inverting the market's option prices. Robustness is an issue when model parameters are probably unknown, but isn't crippling in practical settings because the data is sufficiently imprecise and prevents us from reducing the fitting error down to levels where parameter uncertainty will show . When applied to SPX  data, the estimated model and implied distributions produce variance swap rates that are consistent with the VIX , and also pick up some of the monthly effects that occur from option expiration. We find that parsimony of the Heston model is beneficial because we are able to decipher behavior in estimated parameters and implied measures, whereas the richer Heston model with jumps produces a better fit but also has implied behavior that is less revealing .
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 asseparability. 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 .
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explore the inversion of
formulate and analyze an inverse problem using
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probability measure
filtering density
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the unobserved state in
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, and HMMs ordinarily warrant
and can be tracked using Bayesian
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is a set of
can be considered as
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so rather than use filtering techniques we compute an
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implied distribution
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by inverting the market's option prices. Robustness is an issue when model parameters are probably unknown, but isn't crippling in practical settings because the data is sufficiently imprecise and prevents us from reducing the fitting error down to levels where parameter uncertainty will show
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
A
separability
A
. This specification has a multiplicative component that behaves like a risk premium on volatility uncertainty in the market
A
options
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distributions produce variance swap
densities produce variance-swap
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, and also
volatility index. The implied densities are relatively stable over time and
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from option expiration. We find that parsimony of the Heston model is beneficial because we are able to decipher behavior in estimated parameters and implied measures, whereas the richer Heston model with jumps produces a better fit but also has implied behavior that is less revealing
due to the options' expiration, indicating that the volatility-uncertainty premium could experience cyclic effects due to the maturity date of the options