Foreword
Preface
Symbols
Acknowledgements
1 A Brief Introduction: Stochastic Modelling of Big Data in Finance
1.1 Introduction
1.2 Big Data in Finance: Limit Order Books
1.2.1 Description of Limit Order Books Mechanism
1.2.2 Big Data in Finance: Lobster Data
1.2.3 More Big Data in Finance: Xetra and Frankfurt Markets (Deutsche Boerse Group), on September 23, 2013 and CISCO Data on November 3, 2014
1.3 Stochastic Modelling of Big Data in Finance: Limit Order Books (LOB)
1.3.1 Semi-Markov Modelling of LOB
1.3.2 General Semi-Markov Modelling of LOB
1.3.3 Modelling of LOB with a Compound Hawkes Processes
1.3.4 Modelling of LOB with a General Compound Hawkes Processes
1.3.5 Modelling of LOB with a Non-linear General Compound Hawkes Processes
1.3.6 Modelling of LOB with a Multivariable General Compound Hawkes Processes
1.4 Illustration and Justification of Our Method to Study Big Data in Finance
1.4.1 Numerical Results: Lobster Data (Apple, Google and Microsoft Stocks)
1.4.2 Numerical Results: Xetra and Frankfurt Markets stocks (Deutsche Boerse Group), on September 23, 2013
1.4.3 Numerical Results: CISCO Data, November 3, 2014
1.5 Methodological Aspects of Using the Models
1.6 Conclusion
Bibliography
I Semi-Markovian Modelling of Big Data in Finance
2 A Semi-Markovian Modelling of Big Data in Finance
2.1 Introduction
2.2 A Semi-Markovian Modelling of Limit Order Markets
2.2.1 Markov Renewal and Semi-Markov Processes
2.2.2 Semi-Markovian Modelling of Limit Order Books
2.3 Main Probabilistic Results
2.3.1 Duration until the next price change
2.3.2 Probability of Price Increase
2.3.3 The stock price seen as a functional of a Markov renewal process
2.4 Diffusion Limit of the Price Process
2.4.1 Balanced Order Flow case: Pa(1,1)=Pa(−1,−1) and Pb(1,1)=Pb(−1,−1)
2.4.2 Other cases: either Pa(1,1)
2.5 Numerical Results
2.6 More Big Data
2.6.1 More Data
2.6.2 Estimated Probabilities
2.6.3 Assumption on Distributions f and f˜
2.6.4 Diffusion Limit (Not-Fixed Spread)
2.6.5 The Optimal Liquidation/Acquisition Problems
2.6.6 Market Making
2.7 Conclusion
Bibliography
3 General Semi-Markovian Modelling of Big Data in Finance
3.1 Introduction
3.1.1 Motivation for Generalizing the Model
3.1.2 Data
3.2 Reviewing the Assumptions with Our New Data Sets
3.2.1 Liquidity of Our Data
3.2.2 Empirical Distributions of Initial Queue Sizes and Calculated Conditional Probabilities
3.2.3 Inter-arrival Times of Book Events
3.2.4 Asymptotic Analysis
3.3 General Semi-Markov Model for the Limit Order Book with Two States
3.3.1 Diffusion Limits
3.3.2 Implementation
3.3.3 Numerical Results
3.3.4 Application of the Model
3.3.4.1 Examination of the Data
3.3.4.2 Model Implementation
3.3.4.3 Results for Constructed Sample Day
3.4 General Semi-Markov Model for the Limit Order Book with arbitrary number of states
3.4.1 Justification
3.4.2 Diffusion Limits
3.4.3 Implementation
3.4.4 Numerical Results
3.5 Discussion on Price Spreads
3.6 Conclusion
Bibliography
II Modelling of Big Data in Finance with Hawkes Processes
4 A Brief Introduction to Hawkes Processes
4.1 Introduction
4.2 Definition of Hawkes Processes (HPs)
4.3 Compound Hawkes Processes
4.3.1 Special Cases of Compound Hawkes Processes in Limit Order Books
4.4 Limit Theorems for Hawkes Processes: LLN and FCLT
4.4.1 Law of Large Numbers (LLN) for Hawkes Processes
4.4.2 Functional Central Limit Theorems (FCLT) for Hawkes Processes
4.5 Limit Theorems for Poisson Processes: LLN and FCLT
4.5.1 Law of Large Numbers (LLN) for Poisson Processes
4.5.2 Functional Central Limit Theorems (FCLT) for Hawkes Processes
4.6 Stylized Properties of Hawkes Process
4.6.1 Non-exponential Inter-arrival Times
4.6.2 Clustering Effect of Trades
4.6.3 Non-independency of Mid-price Changes
4.7 Conclusion
Bibliography
5 Stochastic Modelling of Big Data in Finance with CHP
5.1 Introduction
5.2 Definitions of HP, CHP and RSCHP
5.2.1 One-dimensional Hawkes Process
5.2.2 Compound Hawkes Process (CHP)
5.2.3 Regime-switching Compound Hawkes Process
5.3 Diffusion Limits and LLNs for CHP and RSCHP in Limit Order Books
5.3.1 Diffusion Limits for CHP in Limit Order Books
5.3.2 LLN for CHP
5.3.3 Corollary: Extension to a Point Process
5.3.4 Diffusion Limits for RSCHP in Limit Order Books
5.3.5 LLN for RSCHP
5.4 Numerical Examples and Parameters Estimations
5.4.1 Parameters Estimation for CISCO Data
5.4.2 Error of Estimation
5.4.3 Graphs based on Parameters Estimation for CISCO Data (5 Days, 3-7 November 2014 ([Cartea et al., 2015])) from Section 4.1
5.4.4 Remark on Regime-switching Case (Section 3.4)
5.5 Conclusion
Bibliography
6 Stochastic Modelling of Big Data in Finance with GCHP
6.1 A Brief Introduction and Literature Review
6.2 Diffusion Limits and LLNs
6.2.1 Diffusion Limit and LLN for NLCHPnSDO
6.2.2 Diffusion Limit and LLN for GCHPnSDO
6.2.3 Diffusion Limits and LLNs for Special Cases of GCHPnSDO
6.3 Empirical Results
6.3.1 CHPDO
6.3.2 GCHP2SDO
6.3.3 GCHPNSDO
6.3.4 Quantitative Analysis
6.3.5 Remarks
6.3.6 Figures to Chapter 6
6.4 Conclusion
Bibliography
7 Quantitative and Comparative Analyses of Big Data with GCHP
7.1 Introduction
7.2 Theoretical Analysis
7.2.1 One-dimensional Hawkes Process
7.2.1.1 Definition
7.2.1.2 Calibration
7.2.2 General Compound Hawkes Process
7.2.2.1 Definition
7.2.2.2 Diffusive Limit
7.3 Application
7.3.1 Limit Order Book
7.3.2 Data
7.3.3 Descriptive Data Analysis
7.3.3.1 QQ-plot
7.3.3.2 Autocorrelation
7.3.3.3 Clustering Feature
7.4 Hawkes Process and Models Calibrations
7.4.1 Hawkes Process’ Parameters Calibration
7.4.2 Mid Price Modelling and Calibration
7.4.2.1 GCHPDO
7.4.2.2 GCHP2SDO
7.4.2.3 GCHPnSDO
7.5 Error Measurement
7.6 Conclusion
Bibliography
III Multivariate Modelling of Big Data in Finance
8 Multivariate General Compound Hawkes Processes in BDF
8.1 Introduction
8.2 Hawkes Processes and Limit Theorems
8.2.1 One-dimensional Hawkes Process
8.2.2 Multivariate Hawkes Process (MHP) and Limit Theorems
8.3 Multivariate General Compound Hawkes Processes (MGCHP) and Limit Theorems
8.4 FCLT II for MGCHP: Deterministic Centralization
8.5 Numerical Example
8.5.1 Data Description
8.5.2 Maximum Likelihood Estimation (MLE)
8.5.3 Calibration and Empirical Analysis by FCLT I for MGCHP with Stochastic Centralization
8.5.4 Empirical Analysis by FCLT II for MGCHP with Deterministic Centralization
8.6 Conclusion
Bibliography
9 Multivariate General Compound Point Processes in BDF
9.1 Introduction
9.2 Definition of Multivariate General Compound Point Process (MGCPP)
9.2.1 Assumptions for Multivariate Point Processes
9.2.2 Definition for MGCPP
9.3 LLNs and Diffusion Limits for MGCPP
9.3.1 LLN for MGCPP
9.3.2 Diffusion Limits for MGCPP: Stochastic Centralization
9.3.3 Numerical Examples for FCLT: Stochastic Centralization
9.3.3.1 Data Description and Parameter Estimations
9.3.3.2 Comparison with MGCHP with Two Dependent Orders
9.3.3.3 MGCPP with N-State Dependent Orders
9.4 Diffusion Limit for the MGCPP: Deterministic Centralization
9.4.1 FCLT for MGCPP: Deterministic Centralization
9.4.2 Numerical Examples for FCLT: Deterministic Centralization
9.4.3 Rolling Cross-Validation
9.5 Conclusion
Bibliography
IV Appendix: Basics in Stochastic Processes
A Basics in Stochastic Processes
A.1 Discrete-time Markov Chains
A.1.1 Continuous-time Markov Chains.
A.1.2 Ergodicity and Reducibility of Markov Chains
A.2 Markov Renewal Processes
A.3 Semi-Markov Processes
A.4 Jump Markov Processes
A.5 Wiener Processes and Diffusion Processes
A.6 Counting and Poisson Process
A.6.1 Counting Process (CP)
A.6.2 Poisson Process (PP)
A.6.3 Compound Poisson Process (CPP)
A.7 Hawkes Processes
A.8 Martingales
A.9 Martingale Characterization of Markov and Semi-Markov Processes
A.9.1 Martingale Characterization of Markov Chains
A.9.2 Martingale Characterization of Markov Processes
A.9.3 Martingale Characterization of Semi-Markov Processes
A.10 Conclusion
Bibliography
Index
