A Tale of Two Time Scales

It is a common practice in finance to estimate volatility from the sum of frequently sampled squared returns. However, market microstructure poses challenges to this estimation approach, as evidenced by recent empirical studies in finance. The present work attempts to lay out theoretical grounds that reconcile continuous-time modeling and discrete-time samples. We propose an estimation approach that takes advantage of the rich sources in tick-by-tick data while preserving the continuous-time assumption on the underlying returns. Under our framework, it becomes clear why and where the “usual” volatility estimator fails when the returns are sampled at the highest frequencies. If the noise is asymptotically small, our work provides a way of finding the optimal sampling frequency. A better approach, the “two-scales estimator,” works for any size of the noise.

[1]  David Aldous,et al.  On Mixing and Stability of Limit Theorems , 1978 .

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

[3]  A. Lo,et al.  THE ECONOMETRICS OF FINANCIAL MARKETS , 1996, Macroeconomic Dynamics.

[4]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[5]  Roel C. A. Oomen,et al.  Properties of realized variance for a pure jump process: calendar time sampling versus business time sampling , 2004 .

[6]  P. Billingsley,et al.  Probability and Measure , 1980 .

[7]  P. Protter,et al.  Asymptotic error distributions for the Euler method for stochastic differential equations , 1998 .

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

[9]  R. Gencay,et al.  An Introduc-tion to High-Frequency Finance , 2001 .

[10]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

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

[12]  William L. Silber,et al.  Financial Options: From Theory to Practice , 1990 .

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

[14]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[15]  Arnaud Gloter.  Estimation des paramètres d'une diffusion cachée : intégrales de processus de diffusion et modèles à volatilité stochastique , 2000 .

[16]  Peter Reinhard Hansen,et al.  Realized Variance and IID Market Microstructure Noise , 2004 .

[17]  Maureen O'Hara,et al.  Market Microstructure Theory , 1995 .

[18]  A. Gallant,et al.  Using Daily Range Data to Calibrate Volatility Diffusions and Extract the Forward Integrated Variance , 1999, Review of Economics and Statistics.

[19]  Per A. Mykland,et al.  ANOVA for diffusions , 2003 .

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

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

[22]  G. Mason,et al.  Beyond Merton’s Utopia: Effects of Non-normality and Dependence on the Precision of Variance Estimates Using High-frequency Financial Data , 2000 .

[23]  H. Rootzén.  Limit Distributions for the Error in Approximations of Stochastic Integrals , 1980 .

[24]  E. Ghysels,et al.  A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation , 2000 .

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

[26]  P. Hall,et al.  Martingale Limit Theory and its Application. , 1984 .

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