Multifractal volatility : theory, forecasting, and pricing / Laurent E. Calvet, Adlai J. Fisher.

Format:

Book

Author/Creator:

Calvet, Laurent E.

Contributor:

Fisher, Adlai.

Series:

Academic Press advanced finance series

Language:

English

Subjects (All):

Finance--Econometric models.

Finance.

Economic forecasting--Econometric models.

Economic forecasting.

Multifractals.

Physical Description:

xiii, 258 pages : illustrations ; 24 cm.

Place of Publication:

Burlington, MA ; London : Academic Press, [2008]

Summary:

Laurent Calvet and Adlai Fisher show in this book that a simple class of models efficiently captures seemingly disparate aspects of financial market returns Inspired by earlier uses of multifractais in the natural sciences, the authors construct multifrequency regime-switching models that are convenient to estimate, provide excellent volatilify forecasts, and easily integrate into asset pricing applications.

Contents:

Credits and Copyright Exceptions xiv

1.1 Empirical Properties of Financial Returns 1

1.2 Modeling Multifrequency Volatility 4

1.3 Pricing Multifrequency Risk 6

1.4 Contributions to Multifractal Literature 7

I Discrete Time 11

2 Background: Discrete-Time Volatility Modeling 13

2.1 Autoregressive Volatility Modeling 13

2.2 Markov-Switching Models 16

3 The Markov-Switching Multifractal (MSM) in Discrete Time 19

3.1 The MSM Model of Stochastic Volatility 20

3.1.2 Basic Properties 22

3.1.3 Low-Frequency Components and Long Memory 22

3.2 Maximum Likelihood Estimation 25

3.2.1 Updating the State Vector 25

3.2.2 Closed-Form Likelihood 26

3.3 Empirical Results 26

3.3.1 Currency Data 27

3.3.2 ML Estimation Results 27

3.3.3 Model Selection 32

3.4 Comparison with Alternative Models 34

3.4.1 In-Sample Comparison 35

3.4.2 Out-of-Sample Forecasts 35

3.4.3 Comparison with FIGARCH 42

4 Multivariate MSM 49

4.1 Comovement of Univariate Volatility Components 50

4.1.1 Comovement of Exchange Rate Volatility 50

4.1.2 Currency Volatility and Macroeconomic Indicators 55

4.2 A Bivariate Multifrequency Model 60

4.2.1 The Stochastic Volatility Specification 60

4.2.2 Properties 62

4.3 Inference 63

4.3.1 Closed-Form Likelihood 63

4.3.2 Particle Filter 63

4.3.3 Simulated Likelihood 64

4.3.4 Two-Step Estimation 66

4.4 Empirical Results 67

4.4.1 Bivariate MSM Estimates 67

4.4.2 Specification Tests 71

4.4.3 Out-of-Sample Diagnostics 73

4.4.4 Value-at-Risk 75

II Continuous Time 79

5 Background: Continuous-Time Volatility Modeling 81

5.1 Continuous-Time Models of Asset Prices 82

5.1.1 Brownian Motion, Time Deformation, and Jump-Diffusions 82

5.1.2 Self-Similar (Fractal) Processes 83

5.2 Multifractal Measures 84

5.2.1 The Binomial Measure 85

5.2.2 Random Multiplicative Cascades 86

5.2.3 Local Scales and the Multifractal Spectrum 89

5.2.4 The Spectrum of Multiplicative Measures 91

6 Multifractal Diffusions Through Time Deformation and the MMAR 95

6.1 Multifractal Processes 95

6.2 Multifractal Time Deformation 96

6.3 The Multifractal Model of Asset Returns 98

6.3.1 Unconditional Distribution of Returns 98

6.3.2 Long Memory in Volatility 99

6.3.3 Sample Paths 100

6.4 An Extension with Autocorrelated Returns 101

6.5 Connection with Related Work 102

7 Continuous-Time MSM 105

7.1 MSM with Finitely Many Components 106

7.2 MSM with Countably Many Components 107

7.2.1 Limiting Time Deformation 107

7.2.2 Multifractal Price Diffusion 110

7.2.3 Connection between Discrete-Time and Continuous-Time Versions of MSM 111

7.3 MSM with Dependent Arrivals 114

7.4 Connection with Related Work 115

8 Power Variation 121

8.1 Power Variation in Currency Markets 121

8.1.1 Data 121

8.1.2 Methodology 123

8.1.3 Main Empirical Results 123

8.1.4 Comparison of MSM vs. Alternative Specifications 129

8.1.5 Global Tests of Fit 136

8.2 Power Variation in Equity Markets 137

8.3 Additional Moments 139

III Equilibrium Pricing 143

9 Multifrequency News and Stock Returns 145

9.1 An Asset Pricing Model with Regime-Switching Dividends 147

9.1.1 Preferences, Consumption, and Dividends 148

9.1.2 Asset Pricing under Complete Information 149

9.2 Volatility Feedback with Multifrequency Shocks 151

9.2.1 Multifrequency Dividend News 151

9.2.2 Equilibrium Stock Returns 152

9.3 Empirical Results with Fully Informed Investors 153

9.3.1 Excess Return Data 153

9.3.2 Maximum Likelihood Estimation and Volatility Feedback 154

9.3.3 Comparison with Campbell and Hentschel (1992) 159

9.3.4 Conditional Inference 160

9.3.5 Return Decomposition 162

9.3.6 Alternative Calibrations 164

9.4 Learning about Volatility and Endogenous Skewness 165

9.4.1 Investor Information and Stock Returns 168

9.4.2 Learning Model Results 169

9.5 Preference Implications and Extension to Multifrequency Consumption Risk 172

10 Multifrequency Jump-Diffusions 177

10.1 An Equilibrium Model with Endogenous Price Jumps 178

10.1.1 Preferences, Information, and Income 178

10.1.2 Financial Markets and Equilibrium 179

10.1.3 Equilibrium Dynamics under Isoelastic Utility 181

10.2 A Multifrequency Jump-Diffusion for Equilibrium Stock Prices 183

10.2.1 Dividends with Multifrequency Volatility 183

10.2.2 Multifrequency Economies 183

10.2.3 The Equilibrium Stock Price 184

10.3 Price Dynamics with an Infinity of Frequencies 185

10.4 Recursive Utility and Priced Jumps 189

A.1 Appendix to Chapter 3 197

A.1.1 Proof of Proposition 1 197

A.1.2 HAC-Adjusted Vuong Test 200

A.2 Appendix to Chapter 4 201

A.2.1 Distribution of the Arrival Vector 201

A.2.2 Ergodic Distribution of Volatility Components 201

A.2.3 Particle Filter 202

A.2.4 Two-Step Estimation 203

A.2.5 Value-at-Risk Forecasts 204

A.2.6 Extension to Many Assets 204

A.3 Appendix to Chapter 5 207

A.3.1 Properties of D 207

A.3.2 Interpretation of f([alpha]) as a Fractal Dimension 207

A.3.3 Heuristic Proof of Proposition 3 208

A.4 Appendix to Chapter 6 209

A.4.1 Concavity of the Scaling Function [tau](q) 209

A.4.2 Proof of Proposition 5 209

A.4.3 Proof of Proposition 7 210

A.4.4 Proof of Proposition 8 210

A.5 Appendix to Chapter 7 211

A.5.1 Multivariate Version of Continuous-Time MSM 211

A.5.2 Proof of Proposition 9 212

A.5.3 Proof of Proposition 10 214

A.5.4 Proof of Corollary 1 216

A.5.5 Proof of Proposition 11 216

A.5.6 MSM with Dependent Arrivals 218

A.5.7 Autocovariogram of Log Volatility in MSM 219

A.5.8 Limiting MRW Process 219

A.6 Appendix to Chapter 9 220

A.6.1 Full-Information Economies 220

A.6.2 Learning Economies 223

A.6.3 Multifrequency Consumption Risk 224

A.7 Appendix to Chapter 10 224

A.7.1 Proof of Proposition 13 224

A.7.2 Multivariate Extensions 225

A.7.3 Proof of Proposition 14 226

A.7.4 Proof of Proposition 15 227.

Notes:

Includes bibliographical references (pages [229]-250) and index.

ISBN:

9780121500139

0121500136

OCLC:

213839313