Multifractal Volatility (eBook)

Front Cover 1
Multifractal Volatility 4
Copyright Page 5
Table of Contents 6
Acknowledgments 10
Foreword 12
Credits and Copyright Exceptions 15
Chapter 1. Introduction 16
1.1 Empirical Properties of Financial Returns 16
1.2 Modeling Multifrequency Volatility 19
1.3 Pricing Multifrequency Risk 21
1.4 Contributions to Multifractal Literature 22
1.5 Organization of the Book 23
Part 1: Discrete Time 26
Chapter 2. Background: Discrete-Time Volatility Modeling 28
2.1 Autoregressive Volatility Modeling 28
2.2 Markov-Switching Models 31
Chapter 3. The Markov-Switching Multifractal (MSM) in Discrete Time 34
3.1 The MSM Model of Stochastic Volatility 35
3.1.1 Definition 35
3.1.2 Basic Properties 37
3.1.3 Low-Frequency Components and Long Memory 37
3.2 Maximum Likelihood Estimation 40
3.2.1 Updating the State Vector 40
3.2.2 Closed-Form Likelihood 41
3.3 Empirical Results 41
3.3.1 Currency Data 42
3.3.2 ML Estimation Results 42
3.3.3 Model Selection 47
3.4 Comparison with Alternative Models 49
3.4.1 In-Sample Comparison 50
3.4.2 Out-of-Sample Forecasts 50
3.4.3 Comparison with FIGARCH 57
3.5 Discussion 61
Chapter 4. Multivariate MSM 64
4.1 Comovement of Univariate Volatility Components 65
4.1.1 Comovement of Exchange Rate Volatility 65
4.1.2 Currency Volatility and Macroeconomic Indicators 70
4.2 A Bivariate Multifrequency Model 75
4.2.1 The Stochastic Volatility Specification 75
4.2.2 Properties 77
4.3 Inference 78
4.3.1 Closed-Form Likelihood 78
4.3.2 Particle Filter 78
4.3.3 Simulated Likelihood 79
4.3.4 Two-Step Estimation 81
4.4 Empirical Results 82
4.4.1 Bivariate MSM Estimates 82
4.4.2 Specification Tests 86
4.4.3 Out-of-Sample Diagnostics 88
4.4.4 Value-at-Risk 90
4.5 Discussion 92
Part 2: Continuous Time 94
Chapter 5. Background: Continuous-Time Volatility Modeling, Fractal Processes, and Multifractal Measures 96
5.1 Continuous-Time Models of Asset Prices 97
5.1.1 Brownian Motion, Time Deformation, and Jump-Diffusions 97
5.1.2 Self-Similar (Fractal) Processes 98
5.2 Multifractal Measures 99
5.2.1 The Binomial Measure 100
5.2.2 Random Multiplicative Cascades 101
5.2.3 Local Scales and the Multifractal Spectrum 104
5.2.4 The Spectrum of Multiplicative Measures 106
Chapter 6. Multifractal Diffusions Through Time Deformation and the MMAR 110
6.1 Multifractal Processes 110
6.2 Multifractal Time Deformation 111
6.3 The Multifractal Model of Asset Returns 113
6.3.1 Unconditional Distribution of Returns 113
6.3.2 Long Memory in Volatility 114
6.3.3 Sample Paths 115
6.4 An Extension with Autocorrelated Returns 116
6.5 Connection with Related Work 117
6.6 Discussion 118
Chapter 7. Continuous-Time MSM 120
7.1 MSM with Finitely Many Components 121
7.2 MSM with Countably Many Components 122
7.2.1 Limiting Time Deformation 122
7.2.2 Multifractal Price Diffusion 125
7.2.3 Connection between Discrete-Time and Continuous-Time Versions of MSM 126
7.3 MSM with Dependent Arrivals 129
7.4 Connection with Related Work 130
7.5 Discussion 134
Chapter 8. Power Variation 136
8.1 Power Variation in Currency Markets 136
8.1.1 Data 136
8.1.2 Methodology 138
8.1.3 Main Empirical Results 138
8.1.4 Comparison of MSM vs. Alternative Specifications 144
8.1.5 Global Tests of Fit 151
8.2 Power Variation in Equity Markets 152
8.3 Additional Moments 154
8.4 Discussion 156
Part III: Equilibrium Pricing 158
Chapter 9. Multifrequency News and Stock Returns 160
9.1 An Asset Pricing Model with Regime-Switching Dividends 162
9.1.1 Preferences, Consumption, and Dividends 163
9.1.2 Asset Pricing under Complete Information 164
9.2 Volatility Feedback with Multifrequency Shocks 166
9.2.1 Multifrequency Dividend News 166
9.2.2 Equilibrium Stock Returns 167
9.3 Empirical Results with Fully Informed Investors 168
9.3.1 Excess Return Data 168
9.3.2 Maximum Likelihood Estimation and Volatility Feedback 169
9.3.3 Comparison with Campbell and Hentschel (1992) 174
9.3.4 Conditional Inference 175
9.3.5 Return Decomposition 177
9.3.6 Alternative Calibrations 179
9.4 Learning about Volatility and Endogenous Skewness 180
9.4.1 Investor Information and Stock Returns 183
9.4.2 Learning Model Results 184
9.5 Preference Implications and Extension to Multifrequency Consumption Risk 187
9.6 Discussion 191
Chapter 10. Multifrequency Jump-Diffusions 192
10.1 An Equilibrium Model with Endogenous Price Jumps 193
10.1.1 Preferences, Information, and Income 193
10.1.2 Financial Markets and Equilibrium 194
10.1.3 Equilibrium Dynamics under Isoelastic Utility 196
10.2 A Multifrequency Jump-Diffusion for Equilibrium Stock Prices 198
10.2.1 Dividends with Multifrequency Volatility 198
10.2.2 Multifrequency Economies 198
10.2.3 The Equilibrium Stock Price 199
10.3 Price Dynamics with an Infinity of Frequencies 200
10.4 Recursive Utility and Priced Jumps 204
10.5 Discussion 206
Chapter 11. Conclusion 208
A. Appendices 212
A.1 Appendix to Chapter 3 212
A.1.1 Proof of Proposition 1 212
A.1.2 HAC-Adjusted Vuong Test 215
A.2 Appendix to Chapter 4 216
A.2.1 Distribution of the Arrival Vector 216
A.2.2 Ergodic Distribution of Volatility Components 216
A.2.3 Particle Filter 217
A.2.4 Two-Step Estimation 218
A.2.5 Value-at-Risk Forecasts 219
A.2.6 Extension to Many Assets 219
A.3 Appendix to Chapter 5 222
A.3.1 Properties of D 222
A.3.2 Interpretation of f(a) as a Fractal Dimension 222
A.3.3 Heuristic Proof of Proposition 3 223
A.4 Appendix to Chapter 6 224
A.4.1 Concavity of the Scaling Function t (q) 224
A.4.2 Proof of Proposition 5 224
A.4.3 Proof of Proposition 7 225
A.4.4 Proof of Proposition 8 225
A.5 Appendix to Chapter 7 226
A.5.1 Multivariate Version of Continuous-Time MSM 226
A.5.2 Proof of Proposition 9 227
A.5.3 Proof of Proposition 10 229
A.5.4 Proof of Corollary 1 231
A.5.5 Proof of Proposition 11 231
A.5.6 MSM with Dependent Arrivals 233
A.5.7 Autocovariogram of Log Volatility in MSM 234
A.5.8 Limiting MRW Process 234
A.6 Appendix to Chapter 9 235
A.6.1 Full-Information Economies 235
A.6.2 Learning Economies 238
A.6.3 Multifrequency Consumption Risk 239
A.7 Appendix to Chapter 10 239
A.7.1 Proof of Proposition 13 239
A.7.2 Multivariate Extensions 240
A.7.3 Proof of Proposition 14 241
A.7.4 Proof of Proposition 15 242
References 244
Index 266