Multifractal Volatility - Theory, Forecasting, and Pricing

Front Cover

Multifractal Volatility

Copyright Page

Table of Contents

Acknowledgments

Foreword

Credits and Copyright Exceptions

Chapter 1. Introduction

1.1 Empirical Properties of Financial Returns

1.2 Modeling Multifrequency Volatility

1.3 Pricing Multifrequency Risk

1.4 Contributions to Multifractal Literature

1.5 Organization of the Book

Part 1: Discrete Time

Chapter 2. Background: Discrete-Time Volatility Modeling

2.1 Autoregressive Volatility Modeling

2.2 Markov-Switching Models

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

3.1 The MSM Model of Stochastic Volatility

3.1.1 Definition

3.1.2 Basic Properties

3.1.3 Low-Frequency Components and Long Memory

3.2 Maximum Likelihood Estimation

3.2.1 Updating the State Vector

3.2.2 Closed-Form Likelihood

3.3 Empirical Results

3.3.1 Currency Data

3.3.2 ML Estimation Results

3.3.3 Model Selection

3.4 Comparison with Alternative Models

3.4.1 In-Sample Comparison

3.4.2 Out-of-Sample Forecasts

3.4.3 Comparison with FIGARCH

3.5 Discussion

Chapter 4. Multivariate MSM

4.1 Comovement of Univariate Volatility Components

4.1.1 Comovement of Exchange Rate Volatility

4.1.2 Currency Volatility and Macroeconomic Indicators

4.2 A Bivariate Multifrequency Model

4.2.1 The Stochastic Volatility Specification

4.2.2 Properties

4.3 Inference

4.3.1 Closed-Form Likelihood

4.3.2 Particle Filter

4.3.3 Simulated Likelihood

4.3.4 Two-Step Estimation

4.4 Empirical Results

4.4.1 Bivariate MSM Estimates

4.4.2 Specification Tests

4.4.3 Out-of-Sample Diagnostics

4.4.4 Value-at-Risk

4.5 Discussion

Part 2: Continuous Time

Chapter 5. Background: Continuous-Time Volatility Modeling, Fractal Processes, and Multifractal Measures

5.1 Continuous-Time Models of Asset Prices

5.1.1 Brownian Motion, Time Deformation, and Jump-Diffusions

5.1.2 Self-Similar (Fractal) Processes

5.2 Multifractal Measures

5.2.1 The Binomial Measure

5.2.2 Random Multiplicative Cascades

5.2.3 Local Scales and the Multifractal Spectrum

5.2.4 The Spectrum of Multiplicative Measures

Chapter 6. Multifractal Diffusions Through Time Deformation and the MMAR

6.1 Multifractal Processes

6.2 Multifractal Time Deformation

6.3 The Multifractal Model of Asset Returns

6.3.1 Unconditional Distribution of Returns

6.3.2 Long Memory in Volatility

6.3.3 Sample Paths

6.4 An Extension with Autocorrelated Returns

6.5 Connection with Related Work

6.6 Discussion

Chapter 7. Continuous-Time MSM

7.1 MSM with Finitely Many Components

7.2 MSM with Countably Many Components

7.2.1 Limiting Time Deformation

7.2.2 Multifractal Price Diffusion

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

7.3 MSM with Dependent Arrivals

7.4 Connection with Related Work

7.5 Discussion

Chapter 8. Power Variation

8.1 Power Variation in Currency Markets

8.1.1 Data

8.1.2 Methodology

8.1.3 Main Empirical Results

8.1.4 Comparison of MSM vs. Alternative Specifications

8.1.5 Global Tests of Fit

8.2 Power Variation in Equity Markets

8.3 Additional Moments

8.4 Discussion

Part III: Equilibrium Pricing

Chapter 9. Multifrequency News and Stock Returns

9.1 An Asset Pricing Model with Regime-Switching Dividends

9.1.1 Preferences, Consumption, and Dividends

9.1.2 Asset Pricing under Complete Information

9.2 Volatility Feedback with Multifrequency Shocks

9.2.1 Multifrequency Dividend News

9.2.2 Equilibrium Stock Returns

9.3 Empirical Results with Fully Informed Investors

9.3.1 Excess Return Data

9.3.2 Maximum Likelihood Estimation and Volatility Feedback

9.3.3 Comparison with Campbell and Hentschel (1992)

9.3.4 Conditional Inference

9.3.5 Return Decomposition

9.3.6 Alternative Calibrations

9.4 Learning about Volatility and Endogenous Skewness

9.4.1 Investor Information and Stock Returns

9.4.2 Learning Model Results

9.5 Preference Implications and Extension to Multifrequency Consumption Risk

9.6 Discussion

Chapter 10. Multifrequency Jump-Diffusions

10.1 An Equilibrium Model with Endogenous Price Jumps

10.1.1 Preferences, Information, and Income

10.1.2 Financial Markets and Equilibrium

10.1.3 Equilibrium Dynamics under Isoelastic Utility

10.2 A Multifrequency Jump-Diffusion for Equilibrium Stock Prices

10.2.1 Dividends with Multifrequency Volatility

10.2.2 Multifrequency Economies

10.2.3 The Equilibrium Stock Price

10.3 Price Dynamics with an Infinity of Frequencies

10.4 Recursive Utility and Priced Jumps

10.5 Discussion

Chapter 11. Conclusion

A. Appendices

A.1 Appendix to Chapter 3

A.1.1 Proof of Proposition 1

A.1.2 HAC-Adjusted Vuong Test

A.2 Appendix to Chapter 4

A.2.1 Distribution of the Arrival Vector

A.2.2 Ergodic Distribution of Volatility Components

A.2.3 Particle Filter

A.2.4 Two-Step Estimation

A.2.5 Value-at-Risk Forecasts

A.2.6 Extension to Many Assets

A.3 Appendix to Chapter 5

A.3.1 Properties of D

A.3.2 Interpretation of f(a) as a Fractal Dimension

A.3.3 Heuristic Proof of Proposition 3

A.4 Appendix to Chapter 6

A.4.1 Concavity of the Scaling Function t (q)

A.4.2 Proof of Proposition 5

A.4.3 Proof of Proposition 7

A.4.4 Proof of Proposition 8

A.5 Appendix to Chapter 7

A.5.1 Multivariate Version of Continuous-Time MSM

A.5.2 Proof of Proposition 9

A.5.3 Proof of Proposition 10

A.5.4 Proof of Corollary 1

A.5.5 Proof of Proposition 11

A.5.6 MSM with Dependent Arrivals

A.5.7 Autocovariogram of Log Volatility in MSM

A.5.8 Limiting MRW Process

A.6 Appendix to Chapter 9

A.6.1 Full-Information Economies

A.6.2 Learning Economies

A.6.3 Multifrequency Consumption Risk

A.7 Appendix to Chapter 10

A.7.1 Proof of Proposition 13

A.7.2 Multivariate Extensions

A.7.3 Proof of Proposition 14

A.7.4 Proof of Proposition 15

References

Index