Optimum thresholding using mean and conditional mean squared error

We consider a univariate semimartingale model for (the logarithm of) an asset price, containing jumps having possibly infinite activity. The nonparametric threshold estimator hat{IV} of the integrated variance IV := int_0^T sigma^2_s ds proposed in [21] is constructed using observations on a discrete time grid, and precisely it sums up the squared increments of the process when they are below a threshold, which depends on the observation time step and, sometimes, model parameters or latent variables, that need to be estimated. All the threshold functions satisfying given conditions allow asymptotically consistent estimates of IV, however the finite sample properties of hat{IV} can depend on the specific choice of the threshold. We aim here at optimally selecting the threshold by minimizing either the estimation mean squared error (MSE) or the conditional mean squared error (cMSE). The last criterion allows to reach a threshold which is optimal not in mean but for the specific volatility and jumps paths at hand. A parsimonious characterization of the optimum is established, which turns out to be asymptotically proportional to the Lévy’s modulus of continuity of the underlying Brownian motion. Moreover, minimizing the cMSE enables us to propose a novel implementation scheme for approximating the optimal threshold. Monte Carlo simulations illustrate the superior performance of the proposed method.