Discrete Sine Transform for Multi-Scales Realized Volatility Measures

Abstract

In this study new realized volatility measures based on Multi-Scale regression and Discrete Sine Transform (DST) approaches are presented. We show that Multi-Scales estimators similar to that recently proposed by Zhang (2004) can be constructed within a simple regression based approach by exploiting the linear relation existing between the market microstructure bias and the realized volatilities computed at different frequencies. These regression based estimators can be further improved and robustified by using the DST approach to prefilter market microstructure noise. The motivation for the DST approach rests on its ability to diagonalize MA type of processes which arise naturally in discrete time models of tick-by-tick returns with market microstructure noise. Hence, the DST provides a natural orthonormal basis decomposition of observed returns which permits to optimally disentangle the volatility signal of the underlying price process from the market microstructure noise. Robustness of the DST approach with respect to more general dependent structure of the microstructure noise is also analytically shown. Then, the combination of such Multi-Scale regression approach with the DST gives us a Multi-Scales DST realized volatility estimator which is robust against a wide class of noise contaminations and model misspecifications. Thanks to the DST orthogonalization which also allows us to analytically derive closed form expressions for the Cramer-Rao bounds of MA(1)processes, an evaluation of the absolute efficiency of volatility estimators under the i.i.d. noise assumption becomes available, indicating that the Multi-Scales DST estimator possesses a finite sample variance very close to the optimal Cramer-Rao bounds. Monte Carlo simulations based on realistic models for price dynamics and market microstructure effects, show the superiority of DST estimators, compared to alternative volatility proxies for a wide range of noise to signal ratios and different types of noise contaminations. Empirical analysis based on six years of tick-by-tick data for S&P 500 index-future, FIB 30, and 30 years U.S. Treasury Bond future, confirms the accuracy and robustness of DST estimators on different types of real data.