Handbook of High-Frequency Trading and Modeling in Finance

Ionut Florescu, PhD, is Research Associate Professor in Financial Engineering and Director of the Hanlon Financial Systems Laboratory at Stevens Institute of Technology. His research interests include stochastic volatility, stochastic partial differential equations, Monte Carlo Methods, and numerical methods for stochastic processes. Dr. Florescu is the author of Probability and Stochastic Processes, the coauthor of Handbook of Probability, and the coeditor of Handbook of Modeling High-Frequency Data in Finance, all published by Wiley. Maria C. Mariani, PhD, is Shigeko K. Chan Distinguished Professor in Mathematical Sciences and Chair of the Department of Mathematical Sciences at The University of Texas at El Paso. Her research interests include mathematical finance, applied mathematics, geophysics, nonlinear and stochastic partial differential equations and numerical methods. Dr. Mariani is the coeditor of Handbook of Modeling High-Frequency Data in Finance, also published by Wiley. H. Eugene Stanley, PhD, is William Fairfield Warren Distinguished Professor at Boston University. Stanley is one of the key founders of the new interdisciplinary field of econophysics, and has an ISI Hirsch index H=128 based on more than 1200 papers. In 2004 he was elected to the National Academy of Sciences. Frederi G. Viens, PhD, is Professor of Statistics and Mathematics and Director of the Computational Finance Program at Purdue University. He holds more than two dozen local, regional, and national awards and he travels extensively on a world-wide basis to deliver lectures on his research interests, which range from quantitative finance to climate science and agricultural economics. A Fellow of the Institute of Mathematics Statistics, Dr. Viens is the coeditor of Handbook of Modeling High-Frequency Data in Finance, also published by Wiley.

Notes on Contributors xiii

Preface xv

1 Trends and Trades 1
Michael Carlisle, Olympia Hadjiliadis, and Ioannis Stamos

1.1 Introduction 1

1.2 A trend-based trading strategy 3

1.2.1 Signaling and trends 3

1.2.2 Gain over a subperiod 5

1.3 CUSUM timing 7

1.3.1 Cusum process and stopping time 7

1.3.2 A CUSUM timing scheme 10

1.3.3 US treasury notes, CUSUM timing 11

1.4 Example: Random walk on ticks 12

1.4.1 Random walk expected gain over a subperiod 15

1.4.2 Simple random walk, CUSUM timing 18

1.4.3 Lazy simple random walk, cusum timing 21

1.5 CUSUM strategy Monte Carlo 24

1.6 The effect of the threshold parameter 27

1.7 Conclusions and future work 39

Appendix: Tables 40

References 47

2 Gaussian Inequalities and Tranche Sensitivities 51
Claas Becker and Ambar N. Sengupta

2.1 Introduction 51

2.2 The tranche loss function 52

2.3 A sensitivity identity 54

2.4 Correlation sensitivities 55

Acknowledgment 58

References 58

3 A Nonlinear Lead Lag Dependence Analysis of Energy Futures: Oil, Coal, and Natural Gas 61
Germán G. Creamer and Bernardo Creamer

3.1 Introduction 61

3.1.1 Causality analysis 62

3.2 Data 64

3.3 Estimation techniques 64

3.4 Results 65

3.5 Discussion 67

3.6 Conclusions 69

Acknowledgments 69

References 70

4 Portfolio Optimization: Applications in Quantum Computing 73
Michael Marzec

4.1 Introduction 73

4.2 Background 75

4.2.1 Portfolios and optimization 76

4.2.2 Algorithmic complexity 77

4.2.3 Performance 78

4.2.4 Ising model 79

4.2.5 Adiabatic quantum computing 79

4.3 The models 80

4.3.1 Financial model 81

4.3.2 Graph-theoretic combinatorial optimization models 82

4.3.3 Ising and Qubo models 83

4.3.4 Mixed models 84

4.4 Methods 84

4.4.1 Model implementation 85

4.4.2 Input data 85

4.4.3 Mean-variance calculations 85

4.4.4 Implementing the risk measure 86

4.4.5 Implementation mapping 86

4.5 Results 88

4.5.1 The simple correlation model 88

4.5.2 The restricted minimum-risk model 91

4.5.3 The WMIS minimum-risk, max return model 94

4.6 Discussion 95

4.6.1 Hardware limitations 97

4.6.2 Model limitations 97

4.6.3 Implementation limitations 98

4.6.4 Future research 98

4.7 Conclusion 100

Acknowledgments 100

Appendix 4.A: WMIS Matlab Code 100

References 103

5 Estimation Procedure for Regime Switching Stochastic Volatility Model and Its Applications 107
Ionut Florescu and Forrest Levin

5.1 Introduction 107

5.1.1 The original motivation 108

5.1.2 The model and the problem 108

5.1.3 A brief historical note 109

5.2 The methodology 110

5.2.1 Obtaining filtered empirical distributions at t1,…, tT 110

5.2.2 Obtaining the parameters of the Markov chain 112

5.3 Results obtained applying the model to real data 113

5.3.1 Part i: financial applications 113

5.3.2 Part ii: physical data application. temperature data 119

5.3.3 Part iii: analysis of seismometer readings during an earthquake 121

5.3.4 Analysis of the earthquake signal: beginning 123

5.3.5 Analysis: during the earthquake 125

5.3.6 Analysis: end of the earthquake signal, aftershocks 127

5.4 Conclusion 127

5.A Theoretical results and empirical testing 128

5.A.1 How does the particle filter work? 128

5.A.2 Theoretical results about convergence and parameter estimates 129

5.A.3 Markov chain parameter estimates 131

5.A.4 Empirical testing 132

5.A.5 A list of supplementary documents 133

References 133

6 Detecting Jumps in High-Frequency Prices Under Stochastic Volatility: A Review and a Data-Driven Approach 137
Ping-Chen Tsai and Mark B. Shackleton

6.1 Introduction 137

6.2 Review on the intraday jump tests 140

6.2.1 Realized volatility measure and the BNS tests 140

6.2.2 The ABD and LM tests 142

6.3 A data-driven testing procedure 146

6.3.1 Spy data and microstructure noise 146

6.3.2 A generalized testing procedure 149

6.4 Simulation study 153

6.4.1 Model specification 153

6.4.2 Simulation results 158

6.5 Empirical results 161

6.5.1 Results on the backward-looking test 162

6.5.2 Results on the interpolated test 165

6.6 Conclusion 165

Acknowledgments 166

Appendix 6.A: Least-square estimation of HAR-MA (2) model for log(BP) of SPY 167

Appendix 6.B: Estimation of ARMA (2, 1) model for log(BP) of SPY 168

Appendix 6.C: Minimized loss function loss(𝜌1, 𝜌2) for SV2FJ_2𝜌 model, SPY 169

Appendix 6.D.1: Calibration of 𝜉 under SV2FJ_2𝜌 model at 2-min frequency, E[Nt] = 0.08 170

Appendix 6.D.2: Calibration of 𝜉 under SV2FJ_2𝜌 model at 2-min frequency, E[Nt] = 0.40 171

Appendix 6.D.3: Calibration of 𝜉 under SV2FJ_2𝜌 model at 5-min frequency, E[Nt] = 0.08 172

Appendix 6.D.4: Calibration of 𝜉 under SV2FJ_2𝜌 Model at 5-min frequency, E[Nt] = 0.40 173

Appendix 6.D.5: Calibration of 𝜉 under SV2FJ_2𝜌 model at 10-min frequency, E[Nt] = 0.08 174

Appendix 6.D.6: Calibration of 𝜉 under SV2FJ_2𝜌 model at 10-min frequency, E[Nt] = 0.40 175

References 175

7 Hawkes Processes and Their Applications to High-Frequency Data Modeling 183
Baron Law and Frederi G. Viens

7.1 Introduction 183

7.2 Point processes 184

7.3 Hawkes processes 186

7.3.1 Branching structure representation 188

7.3.2 Stationarity 188

7.3.3 Convergence 189

7.4 Statistical inference of Hawkes processes 191

7.4.1 Simulation 191

7.4.2 Estimation 194

7.4.3 Hypothesis testing 197

7.5 Applications of Hawkes processes 198

7.5.1 Modeling order arrivals 199

7.5.2 Modeling price jumps 200

7.5.3 Modeling jump-diffusion 205

7.5.4 Measuring endogeneity (Reflexivity) 205

Appendix 7.A: Point Processes 207

7.A.1 Definition 207

7.A.2 Moments 208

7.A.3 Marked point processes 209

7.A.4 Stochastic intensity 209

7.A.5 Random time change 211

Appendix 7.B: A Brief History of Hawkes processes 211

References 212

8 Multifractal Random Walk Driven by a Hermite Process 221
Alexis Fauth and Ciprian A. Tudor

8.1 Introduction 221

8.2 Preliminaries 224

8.2.1 Fractional brownian motion and hermite processes 224

8.2.2 Wiener integrals with respect to the hermite process 226

8.2.3 Infinitely divisible cascading noise 229

8.3 Multifractal random walk driven by a Hermite process 231

8.3.1 Definition and existence 231

8.3.2 Properties of the hermite multifractal random walk 233

8.4 Financial applications 234

8.4.1 Simulation of the Hmrw 235

8.4.2 Financial statistics 241

8.5 Concluding remarks 243

References 247

9 Interpolating Techniques and Nonparametric Regression Methods Applied to Geophysical and Financial Data Analysis 251
K. Basu and Maria C. Mariani

9.1 Introduction 251

9.2 Nonparametric regression models 253

9.2.1 Local polynomial regression 255

9.2.2 Lowess/loess method 257

9.2.3 Numerical applications 259

9.3 Interpolation methods 271

9.3.1 Nearest-neighbor interpolation 271

9.3.2 Bilinear interpolation 272

9.3.3 Bicubic interpolation 276

9.3.4 Biharmonic interpolation 277

9.3.5 Thin plate splines 282

9.3.6 Numerical applications 285

9.4 Conclusion 287

Acknowledgments 292

References 292

10 Study of Volatility Structures in Geophysics and Finance Using Garch Models 295
Maria C. Mariani, F. Biney, and I. SenGupta

10.1 Introduction 295

10.2 Short memory models 297

10.2.1 ARMA(p,q) model 297

10.2.2 GARCH(p,q) model 297

10.2.3 IGARCH(1,1) model 298

10.3 Long memory models 298

10.3.1 ARFIMA(p,d,q) model 299

10.3.2 ARFIMA(p,d,q)-GARCH(r,s) 299

10.3.3 Intermediate memory process 300

10.3.4 Figarch model 300

10.4 Detection and estimation of long memory 302

10.4.1 Augmented dickey–fuller test(ADF test) 302

10.4.2 KPSS test 303

10.4.3 Whittle method 304

10.5 Data collection, analysis, and result 306

10.5.1 Analysis on dow Jones index (DJIA) returns 306

10.5.2 Model selection and specification: conditional mean 306

10.5.3 Conditional mean model (returns) 309

10.5.4 Model diagnostics: ARMA(2, 2) 309

10.5.5 Test for ARCH effect 311

10.5.6 Model selection and specification: Conditional variance 313

10.5.7 Standardized residuals test 314

10.5.8 Model diagnostics 314

10.5.9 Returns and variance equation 315

10.5.10 standardized residuals test 317

10.5.11 Model diagnostic of conditional returns with conditional variance 318

10.5.12 One-step ahead prediction of last 10 observations 330

10.5.13 Analysis on high-frequency, earthquake, and explosives series 330

10.6 Discussion and conclusion 335

References 337

11 Scale Invariance and Lévy Models Applied to Earthquakes and Financial High-Frequency Data 341
M. P. Beccar-Varela, Ionut Florescu, and I. SenGupta

11.1 Introduction 341

11.2 Governing equations for the deterministic model 342

11.2.1 Application to geophysical (earthquake data) 343

11.2.2 Results 344

11.3 L´evy flights and application to geophysics 345

11.3.1 Truncated L´evy flight distribution 353

11.3.2 Results 356

11.4 Application to the high-frequency market data 360

11.4.1 Methodology 360

11.4.2 Results 361

11.5 Brief program code description 362

11.6 Conclusion 364

11.A Appendix 366

11.A.1 Stable distributions 366

11.A.2 Characterization of stable distributions 367

References 368

12 Analysis of Generic Diversity in the Fossil Record, Earthquake Series, and High-Frequency Financial Data 371
M. P. Beccar Varela, F. Biney, Maria C. Mariani, I. SenGupta, M. Shpak, and P. Bezdek

12.1 Introduction 371

12.2 Statistical preliminaries and results 373

12.2.1 Sum of exponential random variables with different parameters 374

12.3 Statistical and numerical analysis 377

12.4 Analysis with Lévy distribution 380

12.4.1 Characterization of Stable Distributions 383

12.4.2 Truncated Lévy flight (TLF) distribution 384

12.4.3 Data analysis with TLF distribution 389

12.4.4 Sum of Lévy random variables with different parameters 390

12.5 Analysis of the Stock Indices, high-frequency (tick) data, and explosive series 394

12.6 Results and discussion 409

Acknowledgments 421

12.A Appendix A—Big ‘O’ notation 421

References 422

Index 425