Fourier transform methods for pathwise covariance estimation in the presence of jumps

We modify (classical) jump-robust estimators of integrated realized covariance to estimate the Fourier coefficients of the instantaneous stochastic covariance. By means of a central limit theorem for those Fourier coefficient estimators we are then able to prove consistency and a (pointwise) central limit theorem for the non-parametrically reconstructed instantaneous covariance process itself. The procedure is -- by methods of Fourier analysis -- robust enough to allow for an iteration and we can therefore show theoretically and empirically how to estimate the integrated realized covariance of the instantaneous stochastic covariance process. We also explain a surprising shrinkage phenomenon for the constructed Fourier estimators, i.e., in comparison to classical (local) estimators of instantaneous variance the asymptotic estimator variance of the Fourier estimator is smaller by a factor 2/3, but an additional (pointwise small) bias appears. We apply these techniques to robust calibration problems for multivariate modeling in finance, i.e. the selection of a pricing measure by using time series and derivatives' prices information simultaneously. "Robust" here means that re-calibration is more stable over time, that the estimation procedures of, e.g., instantaneous covariance also work in the presence of jumps, and that the procedures are as robust as possible with respect to input deficiencies.