Optimum thresholding using mean and conditional mean square error

We consider a univariate semimartingale model for (the logarithm of) an asset price, containing jumps having possibly infinite activity (IA). The nonparametric threshold estimator ˆ IV n of the integrated variance IV := ∫ T 0 σ sds proposed in [6] is constructed using observations on a discrete time grid, and precisely it sums up the squared increments of the process when they are under a threshold, a deterministic function of the observation step and possibly of the coefficients of X. All the threshold functions satisfying given conditions allow asymptotically consistent estimates of IV , however the finite sample properties of ˆ IV n can depend on the specific choice of the threshold. We aim here at optimally selecting the threshold by minimizing either the estimation mean square error (MSE) or the conditional mean square error (cMSE). The last criterion allows to reach a threshold which is optimal not in mean but for the specific path 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 the optimal threshold sequence. Monte Carlo simulations illustrate the superior performance of the proposed method.