A non-parametric estimation for implied volatility

We provide a non-parametric method for stochastic volatility modelling. Our method allows the implied volatility to be governed by a general Levy-driven Ornstein{Uhlenbeck process, the density function of which is hidden to market participants. Using discrete-time observation we estimate the density function of the stochastic volatility process via developing a cumulant M-estimator for the Levy measure. In contrast to other non-parametric estimators (such as kernel estimators), our estimator is guaranteed to be of the correct type. We implement this method with the aid of a support-reduction algorithm, which is an ecient iterative unconstrained optimisation method. For the empirical analysis, we use discretely observed data from two implied volatility indices, VIX and VDAX. We also present an out-of-sample test to compare the performance of our method with other parametric models.