Variation, Jumps, Market Frictions and High Frequency Data in Financial Econometrics

We will review the econometrics of non-parametric estimation of the components of the variation of asset prices. This very active literature has been stimulated by the recent advent of complete records of transaction prices, quote data and order books. In our view the interaction of the new data sources with new econometric methodology is leading to a paradigm shift in one of the most important areas in econometrics: volatility measurement, modelling and forecasting. We will describe this new paradigm which draws together econometrics with arbitrage free financial economics theory. Perhaps the two most influential papers in this area have been Andersen, Bollerslev, Diebold and Labys(2001) and Barndorff-Nielsen and Shephard(2002), but many other papers have made important contributions. This work is likely to have deep impacts on the econometrics of asset allocation and risk management. One of our observations will be that inferences based on these methods, computed from observed market prices and so under the physical measure, are also valid as inferences under all equivalent measures. This puts this subject also at the heart of the econometrics of derivative pricing. One of the most challenging problems in this context is dealing with various forms of market frictions, which obscure the efficient price from the econometrician. Here we will characterise four types of statistical models of frictions and discuss how econometricians have been attempting to overcome them.

[1]  Norbert Wiener,et al.  The Quadratic Variation of a Function and its Fourier Coefficients , 1924 .

[2]  M. Bartlett On the Theoretical Specification and Sampling Properties of Autocorrelated Time‐Series , 1946 .

[3]  M. Bartlett Periodogram analysis and continuous spectra. , 1950, Biometrika.

[4]  F. Eicker Limit Theorems for Regressions with Unequal and Dependent Errors , 1967 .

[5]  Barr Rosenberg. The Behavior of Random Variables with Nonstationary Variance and the Distribution of Security Prices , 1972 .

[6]  R. Officer The Variability of the Market Factor of the New York Stock Exchange. , 1973 .

[7]  J. Hausman Specification tests in econometrics , 1978 .

[8]  T. W. Epps Comovements in Stock Prices in the Very Short Run , 1979 .

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

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

[11]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

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

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

[14]  K. French,et al.  Expected stock returns and volatility , 1987 .

[15]  G. Schwert Why Does Stock Market Volatility Change Over Time? , 1988 .

[16]  A. Lo,et al.  An Econometric Analysis of Nonsynchronous Trading , 1989 .

[17]  G. Schwert Indexes of U.S. Stock Prices from 1802 to 1987 , 1990 .

[18]  P. Protter Stochastic integration and differential equations , 1990 .

[19]  Kerry Back,et al.  Asset pricing for general processes , 1991 .

[20]  Christopher A. Sims,et al.  Empirical Implications of Arbitrage-Free Asset Markets , 1992 .

[21]  Dominique Picard,et al.  Non-parametric estimation of the diffusion coefficient by wavelets methods , 1992 .

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

[23]  Jeffrey R. Russell,et al.  Forecasting Transaction Rates: The Autoregressive Conditional Duration Model , 1994 .

[24]  Dean P. Foster,et al.  Continuous Record Asymptotics for Rolling Sample Variance Estimators , 1994 .

[25]  N. Shephard,et al.  Stochastic Volatility: Likelihood Inference And Comparison With Arch Models , 1996 .

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

[27]  Yue Fang,et al.  Volatility modeling and estimation of high-frequency data with Gaussian noise , 1996 .

[28]  Bent E. Sørensen,et al.  GMM Estimation of a Stochastic Volatility Model: A Monte Carlo Study , 1996 .

[29]  Robert F. Engle,et al.  The Econometrics of Ultra-High Frequency Data , 1996 .

[30]  A. Harvey,et al.  5 Stochastic volatility , 1996 .

[31]  J. Jacod,et al.  La Variation Quadratique du Brownien en Pre'sence d''Erreurs d''Arrondi , 1996 .

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

[33]  T. Bollerslev,et al.  Intraday periodicity and volatility persistence in financial markets , 1997 .

[34]  Inder Rana,et al.  An introduction to measure and integration , 1997 .

[35]  Ulrich A. Müller STATISTICS OF VARIABLES OBSERVED OVER OVERLAPPING INTERVALS , 1997 .

[36]  G. Schwert Stock Market Volatility: Ten Years after the Crash , 1997 .

[37]  S. Delattre,et al.  A central limit theorem for normalized functions of the increments of a diffusion process, in the presence of round-off errors , 1997 .

[38]  T. Bollerslev,et al.  Deutsche Mark–Dollar Volatility: Intraday Activity Patterns, Macroeconomic Announcements, and Longer Run Dependencies , 1998 .

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

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

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

[42]  K. Demeterfi,et al.  A Guide to Volatility and Variance Swaps , 1999 .

[43]  A. Shiryaev Essentials of stochastic finance , 1999 .

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

[45]  N. Shephard,et al.  Econometric analysis of realised volatility and its use in estimating stochastic volatility models , 2000 .

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

[47]  Jean Jacod,et al.  Diffusions with measurement errors. I. Local Asymptotic Normality , 2001 .

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

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

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

[51]  P. Carr,et al.  Option Pricing, Interest Rates and Risk Management: Towards a Theory of Volatility Trading , 2001 .

[52]  F. Diebold,et al.  The distribution of realized stock return volatility , 2001 .

[53]  N. Shephard,et al.  Realised power variation and stochastic volatility models , 2003 .

[54]  Arnold J Stromberg,et al.  Subsampling , 2001, Technometrics.

[55]  N. Shephard,et al.  How accurate is the asymptotic approximation to the distribution of realised variance , 2001 .

[56]  M. Yor,et al.  Stochastic Volatility for Levy Processes , 2001 .

[57]  S. Howison,et al.  On the pricing and hedging of volatility derivatives , 2004 .

[58]  N. Shephard,et al.  Econometric Analysis of Realised Covariation: High Frequency Covariance, Regression and Correlation in Financial Economics , 2002 .

[59]  N. Meddahi,et al.  A theoretical comparison between integrated and realized volatility , 2002 .

[60]  Clive G. Bowsher Modelling Security Market Events in Continuous Time: Intensity Based, Multivariate Point Process Models , 2003 .

[61]  T. Bollerslev,et al.  Correcting the Errors: A Note on Volatility Forecast Evaluation Based on High-Frequency Data and Realized Volatilities , 2002 .

[62]  Michael W. Brandt,et al.  A No-Arbitrage Approach to Range-Based Estimation of Return Covariances and Correlations , 2002 .

[63]  Corridor Variance Swaps , 2002 .

[64]  Maria Elvira Mancino,et al.  Fourier series method for measurement of multivariate volatilities , 2002, Finance Stochastics.

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

[66]  T. Bollerslev,et al.  Estimating Stochastic Volatility Diffusion Using Conditional Moments of Integrated Volatility , 2001 .

[67]  E. Barucci,et al.  On measuring volatility and the GARCH forecasting performance , 2002 .

[68]  E. Barucci,et al.  On measuring volatility of diffusion processes with high frequency data , 2002 .

[69]  N. Shephard,et al.  Power Variation and Time Change , 2006 .

[70]  T. Bollerslev,et al.  Analytical Evaluation of Volatility Forecasts , 2002 .

[71]  P. Carr,et al.  Time-Changed Levy Processes and Option Pricing ⁄ , 2002 .

[72]  N. Shephard,et al.  Econometric Analysis of Realized Covariation: High Frequency Based Covariance, Regression, and Correlation in Financial Economics , 2004 .

[73]  Laurent E. Calvet,et al.  Multifractality in Asset Returns: Theory and Evidence , 2002, Review of Economics and Statistics.

[74]  N. Shephard,et al.  Econometrics of Testing for Jumps in Financial Economics Using Bipower Variation , 2005 .

[75]  A. Lunde,et al.  Wavelet Estimation of Integrated Volatility , 2003 .

[76]  E. Ghysels,et al.  Série Scientifique Scientific Series Predicting Volatility: Getting the Most out of Return Data Sampled at Different Frequencies , 2022 .

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

[78]  David S. Bates Empirical option pricing: a retrospection , 2003 .

[79]  R. Engle,et al.  A Multiple Indicators Model for Volatility Using Intra-Daily Data , 2003 .

[80]  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 .

[81]  R. Engle,et al.  A Multiple Indicators Model for Volatility Using Intra-Daily Data , 2003 .

[82]  Piazza S. Francesco,et al.  A CLOSER LOOK AT THE EPPS EFFECT , 2003 .

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

[84]  N. Shephard,et al.  Impact of jumps on returns and realised variances: econometric analysis of time-deformed Lévy processes , 2006 .

[85]  P. Hansen,et al.  Consistent Ranking of Volatility Models , 2006 .

[86]  A. Gallant,et al.  Alternative models for stock price dynamics , 2003 .

[87]  S. Koopman,et al.  Forecasting Daily Variability of the S&P 100 Stock Index Using Historical, Realised and Implied Volatility Measurements , 2004 .

[88]  M. Martens Estimating Unbiased and Precise Realized Covariances , 2004 .

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

[90]  Jean Jacod,et al.  Fisher's Information for Discretely Sampled Lévy Processes , 2004 .

[91]  Jean Jacod,et al.  A Central Limit Theorem for Realised Power and Bipower Variations of Continuous Semimartingales , 2004 .

[92]  Fulvio Corsi,et al.  A Simple Long Memory Model of Realized Volatility , 2004 .

[93]  P. Hansen,et al.  A Forecast Comparison of Volatility Models: Does Anything Beat a Garch(1,1)? , 2004 .

[94]  R. Oomen Properties of Bias Corrected Realized Variance in Calendar Time and Business Time , 2004 .

[95]  Paul Wilmott,et al.  GARCH and volatility swaps , 2004 .

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

[97]  Integrated volatility measuring from unevenly sampled observations , 2004 .

[98]  Yacine Aït-Sahalia,et al.  Disentangling diffusion from jumps , 2004 .

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

[100]  P. Mykland,et al.  How Often to Sample a Continuous-Time Process in the Presence of Market Microstructure Noise , 2003 .

[101]  Donald W. K. Andrews,et al.  Identification and Inference for Econometric Models , 2005 .

[102]  P. Hansen,et al.  A Realized Variance for the Whole Day Based on Intermittent High-Frequency Data , 2005 .

[103]  N. Yoshida,et al.  On covariance estimation of non-synchronously observed diffusion processes , 2005 .

[104]  R. Renò,et al.  Dynamic Principal Component Analysis of Multivariate Volatility via Fourier Analysis , 2005 .

[105]  G. Iori,et al.  Cross-correlation Measures in the High-frequency Domain , 2005 .

[106]  Christian Schlag,et al.  An Economic Motivation for Variance Contracts , 2005 .

[107]  Neil Shephard,et al.  Limit theorems for multipower variation in the presence of jumps , 2006 .

[108]  P. Phillips,et al.  A Two-Stage Realized Volatility Approach to the Estimation for Diffusion Processes from Discrete Observations , 2005 .

[109]  Andrew J. Patton,et al.  Volatility Forecast Evaluation and Comparison Using Imperfect Volatility Proxies , 2005 .

[110]  George Tauchen,et al.  Cross-Stock Comparisons of the Relative Contribution of Jumps to Total Price Variance , 2012 .

[111]  Kim Christensen,et al.  Asymptotic theory for range-based estimation of integrated variance of a continuous semi-martingale , 2005 .

[112]  Jeremy H. Large Estimating quadratic variation when quoted prices jump by a constant increment , 2005 .

[113]  Zhou Zhou,et al.  “A Tale of Two Time Scales: Determining Integrated Volatility with Noisy High-Frequency Data” , 2005 .

[114]  Marc Yor,et al.  Pricing options on realized variance , 2005, Finance Stochastics.

[115]  Jeffrey R. Russell,et al.  Realized covariation , realized beta , and microstructure noise , 2005 .

[116]  M. Nielsen,et al.  Finite Sample Accuracy of Integrated Volatility Estimators , 2005 .

[117]  E. Ghysels,et al.  Why Do Absolute Returns Predict Volatility So Well , 2006 .

[118]  Jeannette H. C. Woerner Power and Multipower Variation: inference for high frequency data , 2006 .

[119]  Giuseppe Cavaliere Stochastic Volatility: Selected Readings , 2006 .

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

[121]  Asger Lunde,et al.  Realized Variance and Market Microstructure Noise , 2006 .

[122]  Michael S. Gibson,et al.  Dynamic Estimation of Volatility Risk Premia and Investor Risk Aversion from Option-Implied and Realized Volatilities , 2007 .

[123]  Jean Jacod,et al.  Volatility estimators for discretely sampled Lévy processes , 2007 .

[124]  Jeremy H. Large Estimating Quadratic Variation When Quoted Prices Change by a Constant Increment , 2007 .