Partial Observability of Implied Volatility Matrices: Identification and Covolatilities Filtering

Whereas data on implied volatilities are available for a large number of assets, this is less frequently the case of implied covolatilities. We introduce a new approach based on static and dynamic Wishart models to solve this problem of missing data. We first discuss the identification of the parameter of the (nonlinear state-space) Wishart models from observed implied volatilities. It is shown that the parameter of the Wishart models is identified, possibly up to some signs. Then we derive the filtering approach for implied covolatilities and apply it to different financial applications. The identification issues in other dynamic models based on spectral decomposition, matrix logarithm, and volatility–correlation decomposition are also discussed. We also discuss the implication of this result for the modeling of realized covariance matrices, when this latter is fully observable, by proposing new specification tests for Wishart type models.