Estimating Quadratic Variation When Quoted Prices Change by a Constant Increment

Financial assets' quoted prices normally change through frequent revisions, or jumps. For markets where quotes are almost always revised by the minimum price tick, this paper proposes a new estimator of Quadratic Variation which is robust to microstructure effects. It compares the number of alternations, where quotes are revised back to their previous price, to the number of other jumps. Many markets exhibit a lack of autocorrelation in their quotes' alternation pattern. Under quite general 'no leverage' assumptions, whenever this is so the proposed statistic is consistent as the intensity of jumps increases without bound. After an empirical implementation, some useful corollaries of this are given.

[1]  Allyn M. Munger.  See Appendix C. , 1963 .

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

[3]  I. Monroe.  Processes that can be Embedded in Brownian Motion , 1978 .

[4]  Paul R. Milgrom,et al.  Bid, ask and transaction prices in a specialist market with heterogeneously informed traders , 1985 .

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

[6]  A. Kyle.  Continuous Auctions and Insider Trading , 1985 .

[7]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[8]  W. Newey,et al.  A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelationconsistent Covariance Matrix , 1986 .

[9]  C. Ball,et al.  Estimation Bias Induced by Discrete Security Prices , 1988 .

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

[11]  Leonard M. Adleman,et al.  Proof of proposition 3 , 1992 .

[12]  L. Harris.  Minimum Price Variations, Discrete Bid-Ask Spreads, and Quotation Sizes , 1994 .

[13]  Chester Spatt,et al.  An Empirical Analysis of the Limit Order Book and the Order Flow in the Paris Bourse , 1995 .

[14]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[15]  Calyampudi Radhakrishna Rao,et al.  Statistical methods in finance , 1996 .

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

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

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

[19]  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 .

[20]  Joel Hasbrouck,et al.  Security Bid/Ask Dynamics with Discreteness and Clustering: Simple Strategies for Modeling and Estimation , 1998 .

[21]  Jeffrey R. Russell,et al.  Econometric Analysis of Discrete-Valued Irregularly-Spaced Financial Transactions Data Using a New Autoregressive Conditional Multinomial Model , 1998 .

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

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

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

[25]  R. Engle,et al.  Predicting VNET: A model of the dynamics of market depth , 2001 .

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

[27]  Neil Shephard,et al.  Estimating quadratic variation using realised volatility , 2001 .

[28]  Telling from Discrete Data Whether the Underlying Continuous-Time Model is a Diffusion , 2002 .

[29]  Eric Ghysels,et al.  Rolling-Sample Volatility Estimators , 2002 .

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

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

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

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

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

[35]  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 .

[36]  Yong Zeng.  A Partially Observed Model for Micromovement of Asset Prices with Bayes Estimation via Filtering , 2003 .

[37]  N. Shephard,et al.  Power and bipower variation with stochastic volatility and jumps , 2003 .

[38]  Fulvio Corsi,et al.  A Discrete Sine Transform Approach for Realized Volatility Measurement , 2003 .

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

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

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

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

[43]  R. Oomen.  Modelling Realized Variance when Returns are Serially Correlated , 2004 .

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

[45]  Jeffrey R. Russell,et al.  A Discrete-State Continuous-Time Model of Financial Transactions Prices and Times , 2005 .

[46]  Wang Yan,et al.  Measuring the Information Content of Stock Trades , 2004 .

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

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

[49]  Ian Domowitz,et al.  Resiliency in an Automated Auction , 2004 .

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

[51]  Jeffrey R. Russell,et al.  Separating Microstructure Noise from Volatility , 2004 .

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

[53]  Comment on “ Realized Variance and Market Microstructure Noise ” by Peter Hansen and Asger Lunde , 2005 .

[54]  N. Shephard,et al.  Variation, Jumps, Market Frictions and High Frequency Data in Financial Econometrics , 2005 .

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

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

[57]  Comment on "Realized Variance and Market , 2005 .

[58]  N. Shephard.  Stochastic Volatility: Selected Readings , 2005 .

[59]  Frank de Jong,et al.  Aggressive Orders and the Resiliency of a Limit Order Market , 2005 .

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

[61]  V. Corradi,et al.  Semi-Parametric Comparison of Stochastic Volatility Models using Realized Measures , 2006 .

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

[63]  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.

[64]  Yingying Li,et al.  Are Volatility Estimators Robust with Respect to Modeling Assumptions? , 2007, 0709.0440.

[65]  Jeremy H. Large.  Measuring the resiliency of an electronic limit order book , 2007 .

[66]  Neil Shephard,et al.  Designing Realised Kernels to Measure the Ex-Post Variation of Equity Prices in the Presence of Noise , 2008 .

[67]  Jeremy H. Large,et al.  Moving Average-Based Estimators of Integrated Variance , 2008 .