A Discrete Sine Transform Approach for Realized Volatility Measurement

Realized volatility afiords the ex-post empirical measurement of the latent notional volatility. However, the time-varying returns autocorrelation induced by microstructure efiects represents a challenging problem for standard volatility measures. In this study, a new nonparametric volatility measures based on the Discrete Sine Transform (DST) is proposed. We show that the DST exactly diagonalizes the covariance matrix of MA(1) process. This original result provides us an orthonomal basis decomposition of the return process which permits to optimally disentangle the underlying e‐cient price signal from the time-varying nuisance component contained in tick-by-tick return series. As a result, two nonparametric volatility estimators which fully exploit all the available information contained in high frequency data are constructed. Moreover the DST orthogonalization allow us to analytically compute the score and the Fischer information matrix of MA(1) processes. In discussing e‐cient numerical procedures for the likelihood maximizations we also suggest that DST estimator would represent the most valid starting point for the numerical maximization of the likelihood. Monte Carlo simulations based on a realistic model for microstructure efiects show the superiority of DST estimators, compared to alternative local volatility proxies for every level of the noise to signal ratio and a large class of noise contaminations. These properties make the DST approach a nonparametric method able to cope with time-varying autocorrelation, in a simple and e‐cient way, providing robust and accurate volatility estimates under a wide set of realistic conditions. Moreover, its computational e‐ciency makes it well suitable for real-time analysis of high frequency data.

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

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

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

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

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

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

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

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

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

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

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

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

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

[14]  T. Bollerslev,et al.  ANSWERING THE SKEPTICS: YES, STANDARD VOLATILITY MODELS DO PROVIDE ACCURATE FORECASTS* , 1998 .

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

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

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

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

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

[20]  F. Diebold,et al.  The Distribution of Realized Exchange Rate Volatility , 2000 .

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

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

[23]  F. Diebold,et al.  Financial Institutions Center Parametric and Nonparametric Volatility Measurement , 2002 .

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

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