Discrete Sine Transform for Multi-Scales Realized Volatility Measures

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.

[1]  M. Dacorogna,et al.  Consistent High-Precision Volatility from High-Frequency Data , 2001 .

[2]  P. Hansen,et al.  Realized Variance and Market Microstructure Noise , 2005 .

[3]  H. Geman,et al.  Order Flow, Transaction Clock, and Normality of Asset Returns , 2000 .

[4]  Yacine Ait-Sahalia,et al.  How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise , 2003 .

[5]  L. Harris.  Estimation of Stock Price Variances and Serial Covariances from Discrete Observations , 1990 .

[6]  Neil Shephard,et al.  Regular and Modified Kernel-Based Estimators of Integrated Variance: The Case with Independent Noise , 2004 .

[7]  Joel Hasbrouck,et al.  Assessing the Quality of a Security Market: A New Approach to Transaction-Cost Measurement , 1993 .

[8]  B. Bollen,et al.  Estimating Daily Volatility in Financial Markets Utilizing Intraday Data , 2002 .

[9]  R. C. Merton,et al.  On Estimating the Expected Return on the Market: An Exploratory Investigation , 1980 .

[10]  F. Diebold,et al.  The Distribution of Exchange Rate Volatility , 1999 .

[11]  Joseph Frederick Elliott.  The Characteristic Roots of Certain Real Symmetric Matrices , 1953 .

[12]  N. Shephard,et al.  Estimating quadratic variation using realized variance , 2002 .

[13]  R. Oomen.  Properties of Bias-Corrected Realized Variance Under Alternative Sampling Schemes , 2005 .

[14]  Kalman J. Cohen,et al.  Transaction Costs, Order Placement Strategy, and Existence of the Bid-Ask Spread , 1981, Journal of Political Economy.

[15]  P. Hansen,et al.  An Unbiased Measure of Realized Variance , 2004 .

[16]  L. Glosten.  Is the Electronic Open Limit Order Book Inevitable , 1994 .

[17]  Oscar H. Bustos,et al.  Robust Estimates for ARMA Models , 1986 .

[18]  Tim Bollerslev,et al.  Some Like it Smooth, and Some Like it Rough: Untangling Continuous and Jump Components in Measuring, Modeling, and Forecasting Asset Return Volatility , 2003 .

[19]  Lan Zhang.  Efficient Estimation of Stochastic Volatility Using Noisy Observations: A Multi-Scale Approach , 2004, math/0411397.

[20]  K. French,et al.  Stock return variances: The arrival of information and the reaction of traders , 1986 .

[21]  Fulvio Corsi,et al.  Efficient Estimation of Volatility Using High Frequency Data , 2002 .

[22]  Joel Hasbrouck,et al.  The Dynamics of Discrete Bid and Ask Quotes , 1996 .

[23]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[24]  Francis X. Diebold,et al.  Modeling and Forecasting Realized Volatility , 2001 .

[25]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[26]  F. Diebold,et al.  Roughing It Up: Including Jump Components in the Measurement, Modeling, and Forecasting of Return Volatility , 2005, The Review of Economics and Statistics.

[27]  Joel Hasbrouck,et al.  Modeling Market Microstructure Time Series , 1996 .

[28]  Lan Zhang,et al.  A Tale of Two Time Scales , 2003 .

[29]  T. Ho,et al.  Order Arrival, Quote Behavior, and the Return-Generating Process , 1987 .

[30]  Michael W. Brandt,et al.  Range-Based Estimation of Stochastic Volatility Models , 2001 .

[31]  N. Shephard,et al.  Econometric analysis of realized volatility and its use in estimating stochastic volatility models , 2002 .

[32]  Jeffrey R. Russell,et al.  Microstructure noise, realized volatility, and optimal sampling , 2004 .

[33]  M. Blume,et al.  BIASES IN COMPUTED RETURNS An Application to the Size Effect , 1983 .

[34]  Kim Christensen,et al.  Realized Range-Based Estimation of Integrated Variance , 2006 .

[35]  R. Roll,et al.  A Simple Implicit Measure of the Effective Bid-Ask Spread in an Efficient Market , 2008 .

[36]  D. Dijk,et al.  Measuring volatility with the realized range , 2006 .

[37]  R. Oomen.  Properties of Realized Variance Under Alternative Sampling Schemes , 2006 .

[38]  Gary Gottlieb,et al.  Implications of the Discreteness of Observed Stock Prices , 1985 .

[39]  P. Clark.  A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices , 1973 .

[40]  Robert Todd Gregory,et al.  A collection of matrices for testing computational algorithms , 1969 .

[41]  N. Shephard,et al.  Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise , 2006 .

[42]  M. Parkinson.  The Extreme Value Method for Estimating the Variance of the Rate of Return , 1980 .

[43]  F. Comte,et al.  Long memory in continuous‐time stochastic volatility models , 1998 .

[44]  Benoit B. Mandelbrot,et al.  A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices: Comment , 1973 .