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2 3.3.8 Moving Norm, Variance, and Standard Deviation 3.3.9 Differential 3.3.10 Derivative and Derivative 3.3.11 Volatility 3.3.12 Standardized Time Series, Moving Skewness, and Kurtosis 3.3.13 Moving Correlation 3.3.14 Windowed Fourier Transform 3.4 Microscopic Operators 3.4.1 Backward Shift and Time Translation Operators 3.4.2 Regular Time Series Operator 3.4.3 Microscopic Return, Difference, and Derivative 3.4.4 Microscopic Volatility 3.4.5 Tick Frequency and Activity 4.1 Introduction: Using a Filter to Clean the Data 4.2 Data and Data Errors 4.2.1 Time Series of Ticks 4.2.2 Data Error Types 4.3 General Overview of the Filter 4.3.1 The Functionality of the Filter 4.3.2 Overview of the Filtering Algorithm and Tts Stmcture 4.4 Basic Filtering Elements and Operations 4.4.1 Credibility and Trust Capital 4.4.2 Filtering of Single Scalar Quotes: The Level Filter 4.4.3 Pair Filtering: The Credibility of Returns 4.4.4 Computing the Expected Volatility 4.4.5 Pair Filtering: Comparing Quote Origins 4.4.6 A Time Scale for Filtering 4.5 The Scalar Filtering Window 4.5.1 Entering a New Quote in the Scalar Filtering Window 4.5.2 The Trust Capital of a New Scalar Quote 4.5.3 Updating the Scalar Window 4.5.4 Dismissing Quotes from the Scalar Window 4.5.5 Updating the Statistics with Credible Scalar Quotes 4.5.6 A Second Scalar Window for Old Valid Quotes 4.6 The FullQuote Filtering Window 4.6.1 Quote Splitting Depending on the Instrument Type 4.6.2 The Basic Validity Test 4.6.3 Transforming the Filtered Variable 4.7 Univariate Filtering ADAPTIVE DATA CLEANING
4.7.1 The Results of Univariate Filtering 114 4.7.2 Filtering in Historical and RealTime Modes 115 4.7.3 Choosing the Filter Parameters 116 4.8 Special Filter Elements 116 4.8.1 Multivariate Filtering: Filtering Sparse Data 116 4.9 Behavior and Effects of the Data Filter 118 BASIC STYLIZED FACTS 5.1 Introduction 121 5.2 Price Formation Process 123 5.2.1 Negative FirstOrder Autocorrelation of Returns 123 5.2.2 Discreteness of Quoted Spreads 125 5.2.3 ShortTerm Triangular Arbitrage 127 5.3 Institutional Structure and Exogeneous Impacts 127 5.3.1 Institutional Framework 127 5.3.2 Positive Impact of Official Interventions 129 5.3.3 Mixed Effect of News 129 5.4 Distributional Properties of Returns 132 , 5.4.1 Finite Variance, Symmetry and Decreasing FatTailedness 132 5.4.2 The Tail Index of Return Distributions 135 5.4.3 Extreme Risks in Financial Markets 144 5.5 Scaling Laws 147 5.5.1 Empirical Evidence 147 5.5.2 Distributions and Scaling Laws 151 5.5.3 A Simple Model of the Market Maker Bias 154 5.5.4 Limitations of the Scaling Laws 158 5.6 Autocorrelation and Seasonality 160 5.6.1 Autocorrelations of Returns and Volatility 161 5.6.2 Seasonal Volatility: Across Markets for Instruments 163 5.6.3 Seasonal Volatility: UShaped for Exchange Traded Instruments 167 5.6.4 Deterministic Volatility in Eurofutures Contracts 169 5.6.5 BidAsk Spreads 170 MODELING SEASONAL VOLATILITY 6.1 Introduction 174 6.2 A Model of Market Activity 175 6.2.1 Seasonal Patterns of the Volatility and Presence of Markets 175
6.2.2 Modeling the Volatility Patterns with an Alternative Time Scale and an Activity Variable 176 6.2.3 Market Activity and Scaling Law 177 6.2.4 Geographical Components of Market Activity 178 6.2.5 A Model of Intraweek Market Activity 179 6.2.6 Interpretation of the Activity Modeling Results 183 6.3 A New Business Time Scale (#Scale) 188 6.3.1 Definition of the &Scale 188 6.3.2 Adjustments of the #Scale Definition 189 6.3.3 A Ratio Test for the #Scale Quality 192 6.4 Filtering Intraday Seasonalities with Wavelets 193 REALIZED VOLATILITY DYNAMICS 7.1 Introduction 197 7.2 The Bias of Realized Volatility and Its Correction 198 7.3 Conditional Heteroskedasticity 204 7.3.1 Autocorrelation of Volatility in ?Time 204 7.3.2 Short and Long Memory 207 7.4 The Heterogeneous Market Hypothesis 209 7.4.1 Volatilities of Different Time Resolutions 210 7.4.2 Asymmetric LeadLag Correlation of Volatilities 211 7.4.3 Conditional Predictability 215 VOLATILITY PROCESSES 8.1 Introduction 219 8.2 Intraday Volatility and GARCH Models 221 8.2.1 Parameter Estimation of GARCH Models 222 8.2.2 Temporal Aggregation of GARCH Models 224 8.2.3 Estimates of GARCH( 1,1) for Various Frequencies 226 8.3 Modeling Heterogeneous Volatilities 231 8.3.1 The HARCH Model 231 8.3.2 HARCH and Market Components 234 8.3.3 Generalization of the Process Equation 237 8.3.4 EMAHARCH Model 237 8.3.5 Estimating HARCH and EMAHARCH Models 239 8.3.6 HARCH in Interest Rate Modeling 242 8.4 Forecasting ShortTerm Volatility 243 8.4.1 A Framework to Measure the Forecasting Performance 243 8.4.2 Performance of ARCHType Models 246
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