Chapter
3 Introduction to Asymptotic Theory: Volatility Estimation for a Continuous Process
3.1 Estimating Integrated Volatility in Simple Cases
3.1.1 Constant Volatility
3.1.2 Deterministic Time-Varying Volatility
3.1.3 Stochastic Volatility Independent of the Driving Brownian Motion W
3.1.4 From Independence to Dependence for the Stochastic Volatility
3.2 Stable Convergence in Law
3.3 Convergence for Stochastic Processes
3.4 General Stochastic Volatility
3.5 What If the Process Jumps?
4 With Jumps: An Introduction to Power Variations
4.1.1 The Purely Discontinuous Case
4.1.2 The Continuous Case
4.2 Estimation in a Simple Parametric Example: Merton’s Model
4.2.1 Some Intuition for the Identification or Lack Thereof: The Impact of High Frequency
4.2.2 Asymptotic Efficiency in the Absence of Jumps .
4.2.3 Asymptotic Efficiency in the Presence of Jumps .
4.2.5 GMM Estimation of Volatility with Power Variations
5 High-Frequency Observations: Identifiability and Asymptotic Efficiency
5.1 Classical Parametric Models
5.1.2 Efficiency for Fully Identifiable Parametric Models
5.1.3 Efficiency for Partly Identifiable Parametric Models
5.2 Identifiability for Lévy Processes and the Blumenthal-Getoor Indices
5.2.1 About Mutual Singularity of Laws of Lévy Processes
5.2.2 The Blumenthal-Getoor Indices and Related Quantities for Lévy Processes
5.3 Discretely Observed Semimartingales: Identifiable Parameters
5.3.1 Identifiable Parameters: A Definition
5.3.2 Identifiable Parameters: Examples
5.4 Tests: Asymptotic Properties
5.5 Back to the Lévy Case: Disentangling the Diffusion Part from Jumps
5.5.1 The Parametric Case
5.5.2 The Semi-Parametric Case
5.6 Blumenthal-Getoor Indices for Lévy Processes: Efficiency via Fisher’s Information
6 Estimating Integrated Volatility: The Base Case with No Noise and Equidistant Observations
6.1 When the Process Is Continuous
6.1.1 Feasible Estimation and Confidence Bounds
6.1.2 The Multivariate Case
6.1.3 About Estimation of the Quarticity
6.2 When the Process Is Discontinuous
6.2.1 Truncated Realized Volatility
6.2.2 Choosing the Truncation Level : The One- Dimensional Case
6.2.3 Multipower Variations
6.2.4 Truncated Bipower Variations
6.2.5 Comparing Truncated Realized Volatility and Multipower Variations
6.3.1 Range-Based Volatility Estimators
6.3.2 Range-Based Estimators in a Genuine High- Frequency Setting
6.3.3 Nearest Neighbor Truncation
6.3.4 Fourier-Based Estimators
6.4 Finite Sample Refinements for Volatility Estimators
7 Volatility and Microstructure Noise
7.1 Models of Microstructure Noise
7.1.1 Additive White Noise
7.1.2 Additive Colored Noise
7.1.3 Pure Rounding Noise
7.1.4 A Mixed Case: Rounded White Noise
7.1.5 Realized Volatility in the Presence of Noise
7.2 Assumptions on the Noise
7.3 Maximum-Likelihood and Quasi Maximum-Likelihood Estimation
7.3.1 A Toy Model: Gaussian Additive White Noise and Brownian Motion
7.3.2 Robustness of the MLE to Stochastic Volatility .
7.5 Subsampling and Averaging: Two-Scales Realized Volatility
7.6 The Pre-averaging Method
7.6.1 Pre-averaging and Optimality
7.6.2 Adaptive Pre-averaging
7.7 Flat Top Realized Kernels
7.8 Multi-scales Estimators
7.9 Estimation of the Quadratic Covariation
8 Estimating Spot Volatility
8.1 Local Estimation of the Spot Volatility
8.1.1 Some Heuristic Considerations
8.1.2 Consistent Estimation
8.1.3 Central Limit Theorem
8.2 Global Methods for the Spot Volatility
8.3 Volatility of Volatility
8.4 Leverage: The Covariation between X and c
8.5 Optimal Estimation of a Function of Volatility
8.6 State-Dependent Volatility
8.7 Spot Volatility and Microstructure Noise
9 Volatility and Irregularly Spaced Observations
9.1 Irregular Observation Times: The One-Dimensional Case
9.1.1 About Irregular Sampling Schemes
9.1.2 Estimation of the Integrated Volatility and Other Integrated Volatility Powers
9.1.3 Irregular Observation Schemes: Time Changes
9.2 The Multivariate Case: Non-synchronous Observations
9.2.2 The Hayashi-Yoshida Method
9.2.3 Other Methods and Extensions
10.2 Relative Sizes of the Jump and Continuous Parts and Testing for Jumps
10.2.1 The Mathematical Tools
10.2.2 A “Linear” Test for Jumps
10.2.3 A “Ratio” Test for Jumps
10.2.4 Relative Sizes of the Jump and Brownian Parts
10.2.5 Testing the Null Ω^(c)T instead of Ω^(cW)T
10.3 A Symmetrical Test for Jumps
10.3.1 The Test Statistics Based on Power Variations
10.3.2 Some Central Limit Theorems
10.3.3 Testing the Null Hypothesis of No Jump
10.3.4 Testing the Null Hypothesis of Presence of Jumps
10.3.5 Comparison of the Tests
10.4.1 Mathematical Background
10.4.3 Finding the Jumps: The Finite Activity Case
10.5 Detection of Volatility Jumps
10.6 Microstructure Noise and Jumps
10.6.1 A Noise-Robust Jump Test Statistic
10.6.2 The Central Limit Theorems for the Noise-Robust Jump Test
10.6.3 Testing the Null Hypothesis of No Jump in the Presence of Noise
10.6.4 Testing the Null Hypothesis of Presence of Jumps in the Presence of Noise
11 Finer Analysis of Jumps: The Degree of Jump Activity
11.1 The Model Assumptions
11.2 Estimation of the First BG Index and of the Related Intensity
11.2.1 Construction of the Estimators
11.2.2 Asymptotic Properties
11.2.3 How Far from Asymptotic Optimality ?
11.2.4 The Truly Non-symmetric Case
11.3 Successive BG Indices
11.3.3 Improved Estimators
12 Finite or Infinite Activity for Jumps?
12.1 When the Null Hypothesis Is Finite Jump Activity
12.2 When the Null Hypothesis Is Infinite Jump Activity
13 Is Brownian Motion Really Necessary?
13.1 Tests for the Null Hypothesis That the Brownian Is Present
13.2 Tests for the Null Hypothesis That the Brownian Is Absent
13.2.1 Adding a Fictitious Brownian
13.2.2 Tests Based on Power Variations
14.1 Co-jumps for the Underlying Process
14.1.2 Testing for Common Jumps
14.1.3 Testing for Disjoint Jumps
14.1.4 Some Open Problems
14.2 Co-jumps between the Process and Its Volatility
14.2.1 Limit Theorems for Functionals of Jumps and Volatility
14.2.2 Testing the Null Hypothesis of No Co-jump
14.2.3 Testing the Null Hypothesis of the Presence of Co-jumps
A Asymptotic Results for Power Variations
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,Δn) and B(f, Δn)
A.3.3 CLTs for the Processes B′(f,Δn) and B′(f,Δn)
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
A.5 Localization and Strengthened Assumptions
B.1.1 Proofs for Sections 5.2 and 5.3
B.1.2 Proofs for Section 5.5
B.1.3 Proof of Theorem 5.25
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 8.14
B.2.5 Proof of Theorems 8.11 and 8.14 and Remark 8.15
B.2.6 Proof of Theorems 8.12 and 8.17
B.2.7 Proof of Theorem 8.20
B.3 Proofs for Chapter 10
B.3.1 Proof of Theorem 10.12
B.3.2 Proofs for Section 10.3
B.3.3 Proofs for Section 10.4
B.3.4 Proofs for Section 10.5
B.4 Limit Theorems for the Jumps of an Itô Semimartingale
B.5 A Comparison Between Jumps and Increments
B.6 Proofs for Chapter 11
B.6.1 Proof of Theorems 11.11, 11.12, 11.18, 11.19, and Remark 11.14
B.6.2 Proof of Theorem 11.21
B.6.3 Proof of Theorem 11.23
B.7 Proofs for Chapter 12
B.8 Proofs for Chapter 13
B.9 Proofs for Chapter 14
B.9.1 Proofs for Section 14.1
B.9.2 Proofs for Section 14.2
High-Frequency Financial Econometrics
：High-Frequency Financial Econometrics
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