Econophysics

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).

• Preface xi1 Introduction 11.1 A Brief History of Economics from the Physicist’s Perspective 51.2 Outline of the Book 102 The Random Walk 132.1 What is a Random Walk? 132.1.1 Definition of Random Walk 132.1.2 The Random Walk Formalism and Derivation of the Gaussian Distribution 172.1.3 The Gaussian or Normal Distribution 212.1.4 Wiener Process 232.1.5 Langevin Equation and Brownian Motion 242.2 Do Markets Follow a Random Walk? 272.2.1 What if the Time-Series Were Similar to a Random Walk? 282.2.2 What are the “Stylized” Facts? 312.2.3 Short Note on Multiplicative Stochastic Processes ARCH/GARCH 332.2.4 Is the Market Efficient? 342.3 Are there any Long-Time Correlations? 362.3.1 Detrended Fluctuation Analysis (DFA) 362.3.2 Power Spectral Density Analysis 372.3.3 DFA and PSD Analyses Of the Autocorrelation Function Of Absolute Returns 383 Beyond the Simple Random Walk 413.1 Deviations from Brownian Motion 433.2 Multifractal Random Walk 463.3 Rescaled Range (R/S) Analysis and the Hurst Exponent 473.4 Is there Long-Range Memory in the Market? 483.4.1 Mandelbrot and the Joseph Effect 493.4.2 Cycles in Economics 493.4.3 Log-Normal Oscillations 504 Understanding Interactions through Cross-Correlations 534.1 The Return Cross-Correlation Matrix 544.1.1 Eigenvalue Spectrum of Correlation Matrix 554.1.2 Properties of the “Deviating” Eigenvalues 584.1.3 Filtering the Correlation Matrix 604.2 Time-Evolution of the Correlation Structure 624.3 Relating Correlation with Market Evolution 644.4 Eigenvalue Spacing Distributions 674.4.1 Unfolding of Eigenvalues for the Market Correlation Matrix 694.4.2 Distribution of Eigenvalue Spacings 694.4.3 Distribution of Next Nearest Spacings between Eigenvalues 704.4.4 The Number Variance Statistic 704.5 Visualizing the Network Obtained from Cross-Correlations 724.6 Application to Portfolio Optimization 764.7 Model of Market Dynamics 774.8 So what did we Learn? 795 Why Care about a Power Law? 835.1 Power Laws in Finance 835.1.1 The Return Distribution 845.1.2 Stock Price Return Distribution 865.1.3 Market Index Return Distribution 925.1.3.1 TP Statistic 945.1.3.2 TE Statistic 955.1.3.3 Hill Estimation of Tail Exponent 975.1.3.4 Temporal Variations in the Return Distribution 985.2 Distribution of Trading Volume and Number of Trades 1035.3 A Model for Reproducing the Power Law Tails of Returns and Activity 1045.3.1 Reproducing the Inverse Cubic Law 1106 The Log-Normal and Extreme-Value Distributions 1156.1 The Log-Normal Distribution 1156.2 The Law of Proportionate Effect 1156.3 Extreme Value Distributions 1196.3.1 Value at Risk 1217 When a Single Distribution is not Enough? 1257.1 Empirical Data on Income and Wealth Distribution 1258 Explaining Complex Distributions with Simple Models 1318.1 Kinetic Theory of Gases 1318.1.1 Derivation of Maxwell–Boltzmann Distribution 1318.1.2 Maxwell–Boltzmann Distribution in D Dimensions 1358.2 The Asset Exchange Model 1368.3 Gas-Like Models 1378.3.1 Model with Uniform Savings 1408.3.2 Model with Distributed Savings 1429 But Individuals are not Gas Molecules 1479.1 Agent-Based Models: Going beyond the Simple Statistical Mechanics of Colliding Particles 1479.2 Explaining the Hidden Hand of Economy: Self-Organization in a Collection of Interacting “Selfish” Agents 1499.2.1 Hidden Hand of Economy 1499.2.2 A Minimal Model 1509.2.2.1 Unlimited Money Supply and Limited Supply of Commodity 1519.2.2.2 Limited Money Supply and Limited Supply Of Commodity 1539.3 Game Theory Models 1549.3.1 Minority Game and its Variants (Evolutionary, Adaptive and so on) 1599.3.1.1 El Farol Bar Problem 1599.3.1.2 Basic Minority Game 1619.3.1.3 Evolutionary Minority Games 1619.3.1.4 Adaptive Minority Games 1649.4 The Kolkata Paise Restaurant Problem 1689.4.1 One-Shot KPR Game 1699.4.2 Simple Stochastic Strategies and Utilization Statistics 1729.4.2.1 No Learning (NL) Strategy 1739.4.2.2 Limited Learning (LL) Strategy 1739.4.2.3 One Period Repetition (OPR) Strategy 1759.4.2.4 Follow the Crowd (FC) Strategy 1769.4.3 Limited Queue Length and Modified KPR Problem 1769.4.4 Some Uniform Learning Strategy Limits 1789.4.4.1 Numerical Analysis 1799.4.4.2 Analytical Results 1809.4.5 Statistics of the KPR Problem: A Summary 1819.5 Agent-Based Models for Explaining the Power Law for Price Fluctuations, and so on 1849.5.1 Herding Model: Cont–Bouchaud 1849.5.2 Strategy Groups Model: Lux–Marchesi 1879.6 Spin-Based Model of Agent Interaction 1909.6.1 Random Network of Agents and the Mean Field Model 1949.6.2 Agents on a Spatial Lattice 19510 . . . and Individuals don’t Interact Randomly: Complex Networks 20310.1 What are Networks? 20410.2 Fundamental Network Concepts 20610.2.1 Measures for Complex Networks 20710.3 Models of Complex Networks 21010.3.1 Erdős–Rényi Random Network 21010.3.2 Watts–Strogatz Small-World Network 21210.3.3 Modular Small-World Network 21310.3.4 Barabasi–Albert Scale-Free Network 21610.4 The World Trade Web 22010.5 The Product Space of World Economy 23010.6 Hierarchical Network within an Organization: Connection to Power-Law Income Distribution 23410.6.1 Income as Flow along Hierarchical Structure: The Tribute Model 23610.7 The Dynamical Stability of Economic Networks 23711 Outlook and Concluding Thoughts 24511.1 The Promise and Perils of Economic Growth 24611.2 Jay Forrester’s World Model 247Appendix A Thermodynamics and Free Particle Statistics 251A.1 A Brief Introduction to Thermodynamics and Statistical Mechanics 251A.1.1 Preliminary Concepts of Thermodynamics 251A.1.2 Laws of Thermodynamics 253A.2 Free Particle Statistics 256A.2.1 Classical Ideal Gas: Maxwell–Boltzmann Distribution and Equation of State 257A.2.1.1 Ideal Gas: Equation of State 258A.2.2 Quantum Ideal Gas 260A.2.2.1 Bose Gas: Bose–Einstein (BE) Distribution 261A.2.2.2 Fermi Gas: Fermi–Dirac Distribution 263Appendix B Interacting Systems: Mean Field Models, Fluctuations and Scaling Theories 265B. 1 Interacting Systems: Magnetism 265B.1.1 Heisenberg and Ising Models 265B.1.2 Mean Field Approximation (MFA) 266B.1.2.1 Critical Exponents in MFA 269B.1.2.2 Free Energy in MFA 272B.1.3 Landau Theory of Phase Transition 273B.1.4 When is MFA Exact? 275B.1.5 Transverse Ising Model (TIM) 276B.1.5.1 MFA for TIM 278B.1.5.2 Dynamical Mode-Softening Picture 280B.2 Quantum Systems with Interactions 281B.2.1 Superfluidity and Superconductivity 281B.2.2 MFA: BCS Theory of Superconductivity 282B.3 Effect of Fluctuations: Peierls’ Argument 286B.3.1 For Discrete Excitations 286B.3.2 For Continuous Excitations 289B.4 Effect of Disorder 290B.4.1 Annealed Disorder: Fisher Renormalization 290B.4.2 Quenched Disorder: Harris Criterion 291B.5 Flory Theory for Self-Avoiding Walk (SAW) Statistics 292B.5.1 Random Walk Statistics 292B.5.2 SAW Statistics 292B.6 Percolation Theory 293B.6.1 Critical Exponents 295B.6.2 Scaling Theory 296B.7 Fractals 297Appendix C Renormalization Group Technique 301C.1 Renormalization Group Technique 301C.1.1 Widom Scaling 301C.1.2 Formalism 303C.1.3 RG for One-Dimension Ising Model 305C.1.4 Momentum Space RG for 4 Dimensional Ising Model 307C.1.5 Real Space RG for Transverse Field Ising Chain 316C.1.6 RG Method for Percolation 319C.1.6.1 Site Percolation in One Dimension 319C.1.6.2 Site Percolation in Two Dimension Triangular Lattice 321C.1.6.3 Bond Percolation in Two Dimension Square Lattice 322Appendix D Spin Glasses and Optimization Problems: Annealing 325D.1 Spin Glasses 325D.1.1 Models 325D.1.2 Critical Behavior 326D.1.3 Replica Symmetric Solution of the S–K Model 327D.2 Optimization and Simulated Annealing 329D.2.1 Some Combinatorial Optimization Problems 330D.2.1.1 The Traveling Salesman Problem (TSP) 330D.2.2 Details of a few Optimization Techniques 333D.3 Modeling Neural Networks 336D.3.1 Hopfield Model of Associative Memory [20] 337Appendix E Nonequilibrium Phenomena 339E.1 Nonequilibrium Phenomena 339E.1.1 Fluctuation Dissipation Theorem 339E.1.2 Fokker–Planck Equation and Condition of Detailed Balance 340E.1.3 Self-Organized Criticality (SOC) 340E.1.3.1 The BTW Model and Manna Model 341E.1.3.2 Subcritical Response: Precursors 342E.1.4 Dynamical Hysteresis 345E.1.5 Dynamical Transition in Fiber Bundle Models 346Some Extensively Used Notations in Appendices 351Index 353