We consider a univariate semimartingale model for (the logarithm of) an asset price, containing jumps havingpossibly infinite activity. The nonparametric threshold estimator hatIV of the integrated variance IV := int_0^T sigma^2_s dsproposed in [21] is constructed using observations on a discrete time grid, and precisely it sums up the squaredincrements 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 satisfyinggiven conditions allow asymptotically consistent estimates of IV, however the finite sample properties of hatIV candepend on the specific choice of the threshold. We aim here at optimally selecting the threshold by minimizingeither the estimation mean squared error (MSE) or the conditional mean squared error (cMSE). The last criterionallows 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 proportionalto the Lévy’s modulus of continuity of the underlying Brownian motion. Moreover, minimizing thecMSE enables us to propose a novel implementation scheme for approximating the optimal threshold. MonteCarlo simulations illustrate the superior performance of the proposed method.
Optimum thresholding using mean and conditional mean squared error
MANCINI, CECILIA;
2019-01-01
Abstract
We consider a univariate semimartingale model for (the logarithm of) an asset price, containing jumps havingpossibly infinite activity. The nonparametric threshold estimator hatIV of the integrated variance IV := int_0^T sigma^2_s dsproposed in [21] is constructed using observations on a discrete time grid, and precisely it sums up the squaredincrements 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 satisfyinggiven conditions allow asymptotically consistent estimates of IV, however the finite sample properties of hatIV candepend on the specific choice of the threshold. We aim here at optimally selecting the threshold by minimizingeither the estimation mean squared error (MSE) or the conditional mean squared error (cMSE). The last criterionallows 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 proportionalto the Lévy’s modulus of continuity of the underlying Brownian motion. Moreover, minimizing thecMSE enables us to propose a novel implementation scheme for approximating the optimal threshold. MonteCarlo simulations illustrate the superior performance of the proposed method.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.