Table of Contents

Preface xi

1 Introduction 1

1.1 A Brief History of Economics from the Physicist’s Perspective 5

1.2 Outline of the Book 10

2 The Random Walk 13

2.1 What is a Random Walk? 13

2.1.1 Definition of Random Walk 13

2.1.2 The Random Walk Formalism and Derivation of the Gaussian Distribution 17

2.1.3 The Gaussian or Normal Distribution 21

2.1.4 Wiener Process 23

2.1.5 Langevin Equation and Brownian Motion 24

2.2 Do Markets Follow a Random Walk? 27

2.2.1 What if the Time-Series Were Similar to a Random Walk? 28

2.2.2 What are the “Stylized” Facts? 31

2.2.3 Short Note on Multiplicative Stochastic Processes ARCH/GARCH 33

2.2.4 Is the Market Efficient? 34

2.3 Are there any Long-Time Correlations? 36

2.3.1 Detrended Fluctuation Analysis (DFA) 36

2.3.2 Power Spectral Density Analysis 37

2.3.3 DFA and PSD Analyses Of the Autocorrelation Function Of Absolute Returns 38

3 Beyond the Simple Random Walk 41

3.1 Deviations from Brownian Motion 43

3.2 Multifractal Random Walk 46

3.3 Rescaled Range (R/S) Analysis and the Hurst Exponent 47

3.4 Is there Long-Range Memory in the Market? 48

3.4.1 Mandelbrot and the Joseph Effect 49

3.4.2 Cycles in Economics 49

3.4.3 Log-Normal Oscillations 50
4 Understanding Interactions through Cross-Correlations 53

4.1 The Return Cross-Correlation Matrix 54

4.1.1 Eigenvalue Spectrum of Correlation Matrix 55

4.1.2 Properties of the “Deviating” Eigenvalues 58

4.1.3 Filtering the Correlation Matrix 60

4.2 Time-Evolution of the Correlation Structure 62

4.3 Relating Correlation with Market Evolution 64

4.4 Eigenvalue Spacing Distributions 67

4.4.1 Unfolding of Eigenvalues for the Market Correlation Matrix 69

4.4.2 Distribution of Eigenvalue Spacings 69

4.4.3 Distribution of Next Nearest Spacings between Eigenvalues 70

4.4.4 The Number Variance Statistic 70

4.5 Visualizing the Network Obtained from Cross-Correlations 72

4.6 Application to Portfolio Optimization 76

4.7 Model of Market Dynamics 77

4.8 So what did we Learn? 79

5 Why Care about a Power Law? 83

5.1 Power Laws in Finance 83

5.1.1 The Return Distribution 84

5.1.2 Stock Price Return Distribution 86

5.1.3 Market Index Return Distribution 92

5.1.3.1 TP Statistic 94

5.1.3.2 TE Statistic 95

5.1.3.3 Hill Estimation of Tail Exponent 97

5.1.3.4 Temporal Variations in the Return Distribution 98

5.2 Distribution of Trading Volume and Number of Trades 103

5.3 A Model for Reproducing the Power Law Tails of Returns and Activity 104

5.3.1 Reproducing the Inverse Cubic Law 110

6 The Log-Normal and Extreme-Value Distributions 115

6.1 The Log-Normal Distribution 115

6.2 The Law of Proportionate Effect 115

6.3 Extreme Value Distributions 119

6.3.1 Value at Risk 121

7 When a Single Distribution is not Enough? 125

7.1 Empirical Data on Income and Wealth Distribution 125

8 Explaining Complex Distributions with Simple Models 131

8.1 Kinetic Theory of Gases 131

8.1.1 Derivation of Maxwell–Boltzmann Distribution 131

8.1.2 Maxwell–Boltzmann Distribution in D Dimensions 135

8.2 The Asset Exchange Model 136

8.3 Gas-Like Models 137

8.3.1 Model with Uniform Savings 140

8.3.2 Model with Distributed Savings 142

9 But Individuals are not Gas Molecules 147

9.1 Agent-Based Models: Going beyond the Simple Statistical Mechanics of Colliding Particles 147

9.2 Explaining the Hidden Hand of Economy: Self-Organization in a Collection of Interacting “Selfish” Agents 149

9.2.1 Hidden Hand of Economy 149

9.2.2 A Minimal Model 150

9.2.2.1 Unlimited Money Supply and Limited Supply of Commodity 151

9.2.2.2 Limited Money Supply and Limited Supply Of Commodity 153

9.3 Game Theory Models 154

9.3.1 Minority Game and its Variants (Evolutionary, Adaptive and so on) 159

9.3.1.1 El Farol Bar Problem 159

9.3.1.2 Basic Minority Game 161

9.3.1.3 Evolutionary Minority Games 161

9.3.1.4 Adaptive Minority Games 164

9.4 The Kolkata Paise Restaurant Problem 168

9.4.1 One-Shot KPR Game 169

9.4.2 Simple Stochastic Strategies and Utilization Statistics 172

9.4.2.1 No Learning (NL) Strategy 173

9.4.2.2 Limited Learning (LL) Strategy 173

9.4.2.3 One Period Repetition (OPR) Strategy 175

9.4.2.4 Follow the Crowd (FC) Strategy 176

9.4.3 Limited Queue Length and Modified KPR Problem 176

9.4.4 Some Uniform Learning Strategy Limits 178

9.4.4.1 Numerical Analysis 179

9.4.4.2 Analytical Results 180

9.4.5 Statistics of the KPR Problem: A Summary 181

9.5 Agent-Based Models for Explaining the Power Law for Price Fluctuations, and so on 184

9.5.1 Herding Model: Cont–Bouchaud 184

9.5.2 Strategy Groups Model: Lux–Marchesi 187

9.6 Spin-Based Model of Agent Interaction 190

9.6.1 Random Network of Agents and the Mean Field Model 194

9.6.2 Agents on a Spatial Lattice 195

10 . . . and Individuals don’t Interact Randomly: Complex Networks 203

10.1 What are Networks? 204

10.2 Fundamental Network Concepts 206

10.2.1 Measures for Complex Networks 207

10.3 Models of Complex Networks 210

10.3.1 Erdős–Rényi Random Network 210

10.3.2 Watts–Strogatz Small-World Network 212

10.3.3 Modular Small-World Network 213

10.3.4 Barabasi–Albert Scale-Free Network 216

10.4 The World Trade Web 220

10.5 The Product Space of World Economy 230

10.6 Hierarchical Network within an Organization: Connection to Power-Law Income Distribution 234

10.6.1 Income as Flow along Hierarchical Structure: The Tribute Model 236

10.7 The Dynamical Stability of Economic Networks 237

11 Outlook and Concluding Thoughts 245

11.1 The Promise and Perils of Economic Growth 246

11.2 Jay Forrester’s World Model 247

Appendix A Thermodynamics and Free Particle Statistics 251

A.1 A Brief Introduction to Thermodynamics and Statistical Mechanics 251

A.1.1 Preliminary Concepts of Thermodynamics 251

A.1.2 Laws of Thermodynamics 253

A.2 Free Particle Statistics 256

A.2.1 Classical Ideal Gas: Maxwell–Boltzmann Distribution and Equation of State 257

A.2.1.1 Ideal Gas: Equation of State 258

A.2.2 Quantum Ideal Gas 260

A.2.2.1 Bose Gas: Bose–Einstein (BE) Distribution 261

A.2.2.2 Fermi Gas: Fermi–Dirac Distribution 263

Appendix B Interacting Systems: Mean Field Models, Fluctuations and Scaling Theories 265

B. 1 Interacting Systems: Magnetism 265

B.1.1 Heisenberg and Ising Models 265

B.1.2 Mean Field Approximation (MFA) 266

B.1.2.1 Critical Exponents in MFA 269

B.1.2.2 Free Energy in MFA 272

B.1.3 Landau Theory of Phase Transition 273

B.1.4 When is MFA Exact? 275

B.1.5 Transverse Ising Model (TIM) 276

B.1.5.1 MFA for TIM 278

B.1.5.2 Dynamical Mode-Softening Picture 280

B.2 Quantum Systems with Interactions 281

B.2.1 Superfluidity and Superconductivity 281

B.2.2 MFA: BCS Theory of Superconductivity 282

B.3 Effect of Fluctuations: Peierls’ Argument 286

B.3.1 For Discrete Excitations 286

B.3.2 For Continuous Excitations 289

B.4 Effect of Disorder 290

B.4.1 Annealed Disorder: Fisher Renormalization 290

B.4.2 Quenched Disorder: Harris Criterion 291

B.5 Flory Theory for Self-Avoiding Walk (SAW) Statistics 292

B.5.1 Random Walk Statistics 292

B.5.2 SAW Statistics 292

B.6 Percolation Theory 293

B.6.1 Critical Exponents 295

B.6.2 Scaling Theory 296

B.7 Fractals 297

Appendix C Renormalization Group Technique 301

C.1 Renormalization Group Technique 301

C.1.1 Widom Scaling 301

C.1.2 Formalism 303

C.1.3 RG for One-Dimension Ising Model 305

C.1.4 Momentum Space RG for 4 Dimensional Ising Model 307

C.1.5 Real Space RG for Transverse Field Ising Chain 316

C.1.6 RG Method for Percolation 319

C.1.6.1 Site Percolation in One Dimension 319

C.1.6.2 Site Percolation in Two Dimension Triangular Lattice 321

C.1.6.3 Bond Percolation in Two Dimension Square Lattice 322

Appendix D Spin Glasses and Optimization Problems: Annealing 325

D.1 Spin Glasses 325

D.1.1 Models 325

D.1.2 Critical Behavior 326

D.1.3 Replica Symmetric Solution of the S–K Model 327

D.2 Optimization and Simulated Annealing 329

D.2.1 Some Combinatorial Optimization Problems 330

D.2.1.1 The Traveling Salesman Problem (TSP) 330

D.2.2 Details of a few Optimization Techniques 333

D.3 Modeling Neural Networks 336

D.3.1 Hopfield Model of Associative Memory [20] 337

Appendix E Nonequilibrium Phenomena 339

E.1 Nonequilibrium Phenomena 339

E.1.1 Fluctuation Dissipation Theorem 339

E.1.2 Fokker–Planck Equation and Condition of Detailed Balance 340

E.1.3 Self-Organized Criticality (SOC) 340

E.1.3.1 The BTW Model and Manna Model 341

E.1.3.2 Subcritical Response: Precursors 342

E.1.4 Dynamical Hysteresis 345

E.1.5 Dynamical Transition in Fiber Bundle Models 346

Some Extensively Used Notations in Appendices 351

Index 353

About the Author

Sitabhra Sinha is professor of theoretical physics at the Institute of Mathematical Sciences (IMSc), Chennai, India. He received his doctorate from the Indian Statistical Institute, Kolkata, for research on nonlinear dynamics of neural network models in 1998. Following postdoctoral positions at the Indian Institute of Science, Bangalore, and Cornell University, New York City, he joined IMSc in 2002. His research interests include complex networks, nonlinear dynamics of biological pattern formation, theoretical/computational biophysics, and the application of statistical physics for analyzing socio-economic phenomena. He was an International Fellow of the Santa Fe Institute (2000-2002). Arnab Chatterjee is a postdoctoral researcher at the Centre de Physique Theorique at Marseille. He was formerly working as a postdoc at The Abdus Salam International Centre for Theoretical Physics at Trieste, Italy. After working at Saha Institute of Nuclear Physics, Kolkata, India, he was awarded his Ph.D from Jadavpur University. He worked on dynamic transitions in Ising models, and also on the application of statistical physics to varied interdisciplinary fields such as complex networks and econophysics. Dr. Chatterjee has studied structural properties of the transport networks and has developed the kinetic models of markets. In recent years he has worked on percolation models and even on problems related to stock market crashes, resource utilization, queuing, dynamical networks and models of social opinion dynamics. Anirban Chakraborti has been an assistant professor at the Quantitative Finance Group, Ecole Centrale Paris, France, since 2009. He received his doctorate in physics from Jadavpur University in 2003. Following postdoctoral positions at the Helsinki University of Technology, Brookhaven National Laboratory, and Saha Institute of Nuclear Physics, he joined the Banaras Hindu University as a lecturer in theoretical physics in 2005. Statistical physics of the traveling salesman problem, models of trading markets, stock market correlations, adaptive minority games and quantum entanglement are his major research interests. He is a recipient of the Young Scientist Medal of the Indian National Science Academy (2009). Bikas K. Chakrabarti is a senior professor of theoretical condensed matter physics at the Saha Institute of Nuclear Physics (SINP), Kolkata, and visiting professor of economics at the Indian Statistical Institute, Kolkata, India. He received his doctorate in physics from Calcutta University in 1979. Following postdoctoral positions at Oxford University and Cologne University, he joined SINP in 1983. His main research interests include physics of fracture, quantum glasses, etc., and the interdisciplinary sciences of optimisation, brain modelling, and econophysics. He has written several books and reviews on these topics. Professor Chakrabarti is a recipient of the S. S. Bhatnagar Award (1997), a Fellow of the Indian Academy of Sciences (Bangalore) and of the Indian National Science Academy (New Delhi). He has also received the Outstanding Referee Award of the American Physical Society (2010).