THE STATISTICAL MECHANICS OF FINANCIAL MARKETS

By KAREN JOHNSON

CONTENTS 1 INTRODUCTION MOTIVATION WHY PHYSICISTS? WHY MODELS OF PHYSICS? PHYSICS AND FINANCE – HISTORICAL AIMS OF THIS BOOK 2 BASIC INFORMATION ON CAPITAL MARKETS RISK ASSETS THREE IMPORTANT DERIVATIVES FORWARD CONTRACTS FUTURES CONTRACT OPTIONS DERIVATIVE POSITIONS MARKET ACTORS PRICE FORMATION AT ORGANIZED EXCHANGES ORDER TYPES PRICE FORMATION BY AUCTION CONTINUOUS TRADING: THE XETRA COMPUTER TRADING SYSTEM 3 RANDOM WALKS IN FINANCE AND PHYSICS IMPORTANT QUESTIONS BACHELIER’S “TH´EORIE DE LA SP´ECULATION” PRELIMINARIES PROBABILITIES IN STOCK MARKET OPERATIONS EMPIRICAL DATA ON SUCCESSFUL OPERATIONS IN STOCK MARKETS BIOGRAPHICAL INFORMATION ON LOUIS BACHELIER (1870–1946) EINSTEIN’S THEORY OF BROWNIAN MOTION OSMOTIC PRESSURE AND DI?USION IN SUSPENSIONS BROWNIAN MOTION EXPERIMENTAL SITUATION FINANCIAL DATA PERRIN’S OBSERVATIONS OF BROWNIAN MOTION ONE-DIMENSIONAL MOTION OF ELECTRONIC SPINS 4 THE BLACK–SCHOLES THEORY OF OPTION PRICES ASSUMPTIONS AND NOTATION ASSUMPTIONS NOTATION PRICES FOR DERIVATIVES FORWARD PRICE FUTURES PRICE LIMITS ON OPTION PRICES MODELING FLUCTUATIONS OF FINANCIAL ASSETS STOCHASTIC PROCESSES THE STANDARD MODEL OF STOCK PRICES THE ITˆO LEMMA LOG-NORMAL DISTRIBUTIONS FOR STOCK PRICES OPTION PRICING THE BLACK–SCHOLES DI?ERENTIAL EQUATION SOLUTION OF THE BLACK–SCHOLES EQUATION RISK-NEUTRAL VALUATION AMERICAN OPTIONS THE GREEKS SYNTHETIC REPLICATION OF OPTIONS IMPLIED VOLATILITY VOLATILITY INDICES 5 SCALING IN FINANCIAL DATA AND IN PHYSICS STATIONARITY OF FINANCIAL MARKETS GEOMETRIC BROWNIAN MOTION PRICE HISTORIES STATISTICAL INDEPENDENCE OF PRICE FLUCTUATIONS STATISTICS OF PRICE CHANGES OF FINANCIAL ASSETS PARETO LAWS AND L´EVY FLIGHTS DE?NITIONS THE GAUSSIAN DISTRIBUTION AND THE CENTRAL LIMIT THE- OREM L´EVY DISTRIBUTIONS NON-STABLE DISTRIBUTIONS WITH POWER LAWS SCALING, L´EVY DISTRIBUTIONS, AND L´EVY FLIGHTS IN NATURE CRITICALITY AND SELF-ORGANIZED CRITICALITY, DI?USION AND SUPERDI?USION MICELLES FLUID DYNAMICS THE DYNAMICS OF THE HUMAN HEART AMORPHOUS SEMICONDUCTORS AND GLASSES SUPERPOSITION OF CHAOTIC PROCESSES TSALLIS STATISTICS NEW DEVELOPMENTS: NON-STABLE SCALING, TEMPORAL AND INTERASSET CORRELATIONS IN FINANCIAL MARKETS NON-STABLE SCALING IN FINANCIAL ASSET RETURNS THE BREADTH OF THE MARKET NON-LINEAR TEMPORAL CORRELATIONS STOCHASTIC VOLATILITY MODELS CROSS-CORRELATIONS IN STOCK MARKETS 6 TURBULENCE AND FOREIGN EXCHANGE MARKETS TURBULENT FLOWS PHENOMENOLOGY STATISTICAL DESCRIPTION OF TURBULENCE RELATION TO NON-EXTENSIVE STATISTICAL MECHANICS FOREIGN EXCHANGE MARKETS WHY FOREIGN EXCHANGE MARKETS? EMPIRICAL RESULTS STOCHASTIC CASCADE MODELS THE MULTIFRACTAL INTERPRETATION 7 DERIVATIVE PRICING BEYOND BLACK–SCHOLES AN INTEGRAL FRAMEWORK FOR DERIVATIVE PRICING APPLICATION TO FORWARD CONTRACTS OPTION PRICING (EUROPEAN CALLS) MONTE CARLO SIMULATIONS OPTION PRICING IN A TSALLIS WORLD PATH INTEGRALS: INTEGRATING THE FAT TAILS INTO OPTION PRICING PATH INTEGRALS: INTEGRATING PATH DEPENDENCE INTO OPTION PRICING 8 MICROSCOPIC MARKET MODELS ARE MARKETS E?CIENT? COMPUTER SIMULATION OF MARKET MODELS TWO CLASSICAL EXAMPLES RECENT MODELS THE MINORITY GAME THE BASIC MINORITY GAME A PHASE TRANSITION IN THE MINORITY GAME RELATION TO FINANCIAL MARKETS SPIN GLASSES AND AN EXACT SOLUTION EXTENSIONS OF THE MINORITY GAME 9 THEORY OF STOCK EXCHANGE CRASHES EXAMPLES EARTHQUAKES AND MATERIAL FAILURE STOCK EXCHANGE CRASHES WHAT CAUSES CRASHES? ARE CRASHES RATIONAL? WHAT HAPPENS AFTER A CRASH? A RICHTER SCALE FOR FINANCIAL MARKETS 10 RISK MANAGEMENT 101 IMPORTANT QUESTIONS 102 WHAT IS RISK? 103 MEASURES OF RISK 1031 VOLATILITY 1032 GENERALIZATIONS OF VOLATILITY AND MOMENTS 1033 STATISTICS OF EXTREMAL EVENTS 1034 VALUE AT RISK 1035 COHERENT MEASURES OF RISK 1036 EXPECTED SHORTFALL 104 TYPES OF RISK 1041 MARKET RISK 1042 CREDIT RISK 1043 OPERATIONAL RISK 1044 LIQUIDITY RISK 105 RISK MANAGEMENT 1051 RISK MANAGEMENT REQUIRES A STRATEGY 1052 LIMIT SYSTEMS 1053 HEDGING 1054 PORTFOLIO INSURANCE 1055 DIVERSI?CATION 1056 STRATEGIC RISK MANAGEMENT 11 ECONOMIC AND REGULATORY CAPITAL FOR FINANCIAL INSTITUTIONS 111 IMPORTANT QUESTIONS 112 ECONOMIC CAPITAL 1121 WHAT DETERMINES ECONOMIC CAPITAL? 1122 HOW CALCULATE ECONOMIC CAPITAL? 1123 HOW ALLOCATE ECONOMIC CAPITAL? 1124 ECONOMIC CAPITAL AS A MANAGEMENT TOOL 113 THE REGULATORY FRAMEWORK 1131 WHY BANKING REGULATION? 1132 RISK-BASED CAPITAL REQUIREMENTS 1133 BASEL I: REGULATION OF CREDIT RISK 1134 INTERNAL MODELS 1135 BASEL II: THE NEW INTERNATIONAL CAPITAL ADEQUACY FRAMEWORK 1136 OUTLOOK: BASEL III AND BASEL IV NOTES AND REFERENCES