I Preliminary Material 1

Transcription

1 Contents Preface Notation xvii xxiii I Preliminary Material 1 1 From Diffusions to Semimartingales Diffusions The Brownian Motion Stochastic Integrals A Central Example: Diffusion Processes Lévy Processes The Law of a Lévy Process Examples Poisson Random Measures Integrals with Respect to Poisson Random Measures Path Properties and Lévy-Itô Decomposition Semimartingales Definition and Stochastic Integrals Quadratic Variation Itô s Formula Characteristics of a Semimartingale and the Lévy- Itô Decomposition Itô Semimartingales The Definition Extension of the Probability Space The Grigelionis Form of an Itô Semimartingale. 47 vii

2 viii Contents A Fundamental Example: Stochastic Differential Equations Driven by a Lévy Process Processes with Conditionally Independent Increments Processes with Independent Increments A Class of Processes with F-Conditionally Independent Increments Data Considerations Mechanisms for Price Determination Limit Order and Other Market Mechanisms Market Rules and Jumps in Prices Sample Data: Transactions, Quotes and NBBO High-Frequency Data Distinctive Characteristics Random Sampling Times Market Microstructure Noise and Data Errors Non-normality Models for Market Microstructure Noise Additive Noise Rounding Errors Strategies to Mitigate the Impact of Noise Downsampling Filtering Transactions Using Quotes II Asymptotic Concepts 79 3 Introduction to Asymptotic Theory: Volatility Estimation for a Continuous Process Estimating Integrated Volatility in Simple Cases Constant Volatility Deterministic Time-Varying Volatility Stochastic Volatility Independent of the Driving Brownian Motion W From Independence to Dependence for the Stochastic Volatility Stable Convergence in Law Convergence for Stochastic Processes General Stochastic Volatility What If the Process Jumps? With Jumps: An Introduction to Power Variations 109

3 Contents ix 4.1 Power Variations The Purely Discontinuous Case The Continuous Case The Mixed Case Estimation in a Simple Parametric Example: Merton s Model Some Intuition for the Identification or Lack Thereof: The Impact of High Frequency Asymptotic Efficiency in the Absence of Jumps Asymptotic Efficiency in the Presence of Jumps GMM Estimation GMM Estimation of Volatility with Power Variations References High-Frequency Observations: Identifiability and Asymptotic Efficiency Classical Parametric Models Identifiability Efficiency for Fully Identifiable Parametric Models Efficiency for Partly Identifiable Parametric Models Identifiability for Lévy Processes and the Blumenthal- Getoor Indices About Mutual Singularity of Laws of Lévy Processes The Blumenthal-Getoor Indices and Related Quantities for Lévy Processes Discretely Observed Semimartingales: Identifiable Parameters Identifiable Parameters: A Definition Identifiable Parameters: Examples Tests: Asymptotic Properties Back to the Lévy Case: Disentangling the Diffusion Part from Jumps The Parametric Case The Semi-Parametric Case Blumenthal-Getoor Indices for Lévy Processes: Efficiency via Fisher s Information References

4 x Contents III Volatility Estimating Integrated Volatility: The Base Case with No Noise and Equidistant Observations When the Process Is Continuous Feasible Estimation and Confidence Bounds The Multivariate Case About Estimation of the Quarticity When the Process Is Discontinuous Truncated Realized Volatility Choosing the Truncation Level: The One- Dimensional Case Multipower Variations Truncated Bipower Variations Comparing Truncated Realized Volatility and Multipower Variations Other Methods Range-Based Volatility Estimators Range-Based Estimators in a Genuine High- Frequency Setting Nearest Neighbor Truncation Fourier-Based Estimators Finite Sample Refinements for Volatility Estimators References Volatility and Microstructure Noise Models of Microstructure Noise Additive White Noise Additive Colored Noise Pure Rounding Noise A Mixed Case: Rounded White Noise Realized Volatility in the Presence of Noise Assumptions on the Noise Maximum-Likelihood and Quasi Maximum-Likelihood Estimation A Toy Model: Gaussian Additive White Noise and Brownian Motion Robustness of the MLE to Stochastic Volatility Quadratic Estimators

5 Contents xi 7.5 Subsampling and Averaging: Two-Scales Realized Volatility The Pre-averaging Method Pre-averaging and Optimality Adaptive Pre-averaging Flat Top Realized Kernels Multi-scales Estimators Estimation of the Quadratic Covariation References Estimating Spot Volatility Local Estimation of the Spot Volatility Some Heuristic Considerations Consistent Estimation Central Limit Theorem Global Methods for the Spot Volatility Volatility of Volatility Leverage: The Covariation between X and c Optimal Estimation of a Function of Volatility State-Dependent Volatility Spot Volatility and Microstructure Noise References Volatility and Irregularly Spaced Observations Irregular Observation Times: The One-Dimensional Case About Irregular Sampling Schemes Estimation of the Integrated Volatility and Other Integrated Volatility Powers Irregular Observation Schemes: Time Changes The Multivariate Case: Non-synchronous Observations The Epps Effect The Hayashi-Yoshida Method Other Methods and Extensions References IV Jumps Testing for Jumps Introduction

6 xii Contents 10.2 Relative Sizes of the Jump and Continuous Parts and Testing for Jumps The Mathematical Tools A Linear Test for Jumps A Ratio Test for Jumps Relative Sizes of the Jump and Brownian Parts Testing the Null Ω (c) T instead of Ω(cW) T A Symmetrical Test for Jumps The Test Statistics Based on Power Variations Some Central Limit Theorems Testing the Null Hypothesis of No Jump Testing the Null Hypothesis of Presence of Jumps Comparison of the Tests Detection of Jumps Mathematical Background A Test for Jumps Finding the Jumps: The Finite Activity Case The General Case Detection of Volatility Jumps Microstructure Noise and Jumps A Noise-Robust Jump Test Statistic The Central Limit Theorems for the Noise-Robust Jump Test Testing the Null Hypothesis of No Jump in the Presence of Noise Testing the Null Hypothesis of Presence of Jumps in the Presence of Noise References Finer Analysis of Jumps: The Degree of Jump Activity The Model Assumptions Estimation of the First BG Index and of the Related Intensity Construction of the Estimators Asymptotic Properties How Far from Asymptotic Optimality? The Truly Non-symmetric Case Successive BG Indices Preliminaries First Estimators

7 Contents xiii Improved Estimators References Finite or Infinite Activity for Jumps? When the Null Hypothesis Is Finite Jump Activity When the Null Hypothesis Is Infinite Jump Activity References Is Brownian Motion Really Necessary? Tests for the Null Hypothesis That the Brownian Is Present Tests for the Null Hypothesis That the Brownian Is Absent Adding a Fictitious Brownian Tests Based on Power Variations References Co-jumps Co-jumps for the Underlying Process The Setting Testing for Common Jumps Testing for Disjoint Jumps Some Open Problems Co-jumps between the Process and Its Volatility Limit Theorems for Functionals of Jumps and Volatility Testing the Null Hypothesis of No Co-jump Testing the Null Hypothesis of the Presence of Co-jumps References A Asymptotic Results for Power Variations 477 A.1 Setting and Assumptions A.2 Laws of Large Numbers A.2.1 LLNs for Power Variations and Related Functionals A.2.2 LLNs for the Integrated Volatility A.2.3 LLNs for Estimating the Spot Volatility A.3 Central Limit Theorems A.3.1 CLTs for the Processes B(f, n ) and B(f, n ). 488 A.3.2 A Degenerate Case A.3.3 CLTs for the Processes B (f, n ) and B (f, n ) 492

8 xiv Contents A.3.4 CLTs for the Quadratic Variation A.4 Noise and Pre-averaging: Limit Theorems A.4.1 Assumptions on Noise and Pre-averaging Schemes 497 A.4.2 LLNs for Noise A.4.3 CLTs for Noise A.5 Localization and Strengthened Assumptions B Miscellaneous Proofs 507 B.1 Proofs for Chapter B.1.1 Proofs for Sections 5.2 and B.1.2 Proofs for Section B.1.3 Proof of Theorem B.2 Proofs for Chapter B.2.1 Preliminaries B.2.2 Estimates for the Increments of X and c B.2.3 Estimates for the Spot Volatility Estimators B.2.4 A Key Decomposition for Theorems 8.11 and B.2.5 Proof of Theorems 8.11 and 8.14 and Remark B.2.6 Proof of Theorems 8.12 and B.2.7 Proof of Theorem B.3 Proofs for Chapter B.3.1 Proof of Theorem B.3.2 Proofs for Section B.3.3 Proofs for Section B.3.4 Proofs for Section B.4 Limit Theorems for the Jumps of an Itô Semimartingale 578 B.5 A Comparison Between Jumps and Increments B.6 Proofs for Chapter B.6.1 Proof of Theorems 11.11, 11.12, 11.18, 11.19, and Remark B.6.2 Proof of Theorem B.6.3 Proof of Theorem B.7 Proofs for Chapter B.8 Proofs for Chapter B.9 Proofs for Chapter B.9.1 Proofs for Section B.9.2 Proofs for Section Bibliography 633

9 Contents xv Index 657