Contents
1. Introduction .............................................. 1
1.1 Motivation ............................................ 1
1.2 WhyPhysicists?WhyModelsofPhysics? ................. 4
1.3 PhysicsandFinance–Historical ......................... 6
1.4 Aimsof thisBook ...................................... 8
2. Basic Information on Capital Markets .................... 13
2.1 Risk .................................................. 13
2.2 Assets ................................................ 13
2.3 Three ImportantDerivatives............................. 15
2.3.1 ForwardContracts................................ 16
2.3.2 FuturesContract................................. 16
2.3.3 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Derivative Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 MarketActors ......................................... 20
2.6 PriceFormationatOrganizedExchanges .................. 21
2.6.1 OrderTypes..................................... 21
2.6.2 Price Formation by Auction . . . . . . . . . . . . . . . . . . . . . . . 22
2.6.3 Continuous Trading:
TheXETRAComputerTradingSystem............. 23
3. Random Walks in Finance and Physics ................... 27
3.1 ImportantQuestions.................................... 27
3.2 Bachelier’s “Th′ eorie de la Sp′ eculation”.................... 28
3.2.1 Preliminaries .................................... 28
3.2.2 Probabilities in Stock Market Operations . . . . . . . . . . . . 32
3.2.3 Empirical Data on Successful Operations
inStockMarkets ................................. 39
3.2.4 Biographical Information
on Louis Bachelier (1870–1946) . . . . . . . . . . . . . . . . . . . . 40
3.3 Einstein’sTheoryofBrownianMotion .................... 41
3.3.1 Osmotic Pressure and Di?usion in Suspensions . . . . . . . 41
3.3.2 BrownianMotion................................. 43
3.4 Experimental Situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.1 Financial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.2 Perrin’s Observations of Brownian Motion . . . . . . . . . . . 46
3.4.3 One-Dimensional Motion of Electronic Spins . . . . . . . . . 47
4. The Black–Scholes Theory of Option Prices ............... 51
4.1 ImportantQuestions.................................... 51
4.2 AssumptionsandNotation............................... 52
4.2.1 Assumptions..................................... 52
4.2.2 Notation ........................................ 53
4.3 Prices forDerivatives ................................... 53
4.3.1 Forward Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.2 FuturesPrice ................................... 55
4.3.3 Limits on Option Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 Modeling Fluctuations of Financial Assets . . . . . . . . . . . . . . . . . 58
4.4.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.2 The Standard Model of Stock Prices . . . . . . . . . . . . . . . . 67
4.4.3 The It? oLemma .................................. 68
4.4.4 Log-normal Distributions for Stock Prices . . . . . . . . . . . 70
4.5 Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5.1 The Black–Scholes Di?erential Equation . . . . . . . . . . . . . 72
4.5.2 Solution of the Black–Scholes Equation . . . . . . . . . . . . . 75
4.5.3 Risk-NeutralValuation............................ 80
4.5.4 American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5.5 TheGreeks...................................... 83
4.5.6 Synthetic Replication of Options . . . . . . . . . . . . . . . . . . . 87
4.5.7 Implied Volatility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5.8 Volatility Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5. Scaling in Financial Data and in Physics .................. 101
5.1 ImportantQuestions.................................... 101
5.2 StationarityofFinancialMarkets......................... 102
5.3 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3.1 PriceHistories ................................... 106
5.3.2 Statistical Independence of Price Fluctuations . . . . . . . 106
5.3.3 Statistics of Price Changes of Financial Assets . . . . . . . 111
5.4 Pareto Laws and L′ evyFlights............................ 120
5.4.1 De?nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.4.2 The Gaussian Distribution and the Central Limit The-
orem............................................ 123
5.4.3 L′ evyDistributions................................ 126
5.4.4 Non-stable Distributions with Power Laws . . . . . . . . . . . 129
5.5 Scaling, L′ evy Distributions,
and L′ evy Flights in Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.5.1 Criticality and Self-Organized Criticality,
Di?usionandSuperdi?usion ....................... 131
5.5.2 Micelles ......................................... 133
5.5.3 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.5.4 TheDynamicsof theHumanHeart................. 137
5.5.5 Amorphous Semiconductors and Glasses . . . . . . . . . . . . . 138
5.5.6 Superposition of Chaotic Processes . . . . . . . . . . . . . . . . . 141
5.5.7 Tsallis Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.6 New Developments: Non-stable Scaling, Temporal
and Interasset Correlations in Financial Markets . . . . . . . . . . . 146
5.6.1 Non-stable Scaling in Financial Asset Returns . . . . . . . . 147
5.6.2 TheBreadthof theMarket ........................ 151
5.6.3 Non-linearTemporalCorrelations .................. 154
5.6.4 Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . . . . . 159
5.6.5 Cross-Correlations inStockMarkets ................ 161
6. Turbulence and Foreign Exchange Markets ............... 173
6.1 ImportantQuestions.................................... 173
6.2 TurbulentFlows........................................ 173
6.2.1 Phenomenology .................................. 174
6.2.2 StatisticalDescriptionofTurbulence................ 178
6.2.3 Relation to Non-extensive Statistical Mechanics . . . . . . 181
6.3 ForeignExchangeMarkets............................... 182
6.3.1 WhyForeignExchangeMarkets?................... 182
6.3.2 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.3.3 StochasticCascadeModels ........................ 189
6.3.4 The Multifractal Interpretation . . . . . . . . . . . . . . . . . . . . 191
7. Derivative Pricing Beyond Black–Scholes ................. 197
7.1 ImportantQuestions.................................... 197
7.2 An Integral Framework for Derivative Pricing . . . . . . . . . . . . . . 197
7.3 ApplicationtoForwardContracts ........................ 199
7.4 OptionPricing(EuropeanCalls) ......................... 200
7.5 MonteCarloSimulations ................................ 204
7.6 Option Pricing in a Tsallis World . . . . . . . . . . . . . . . . . . . . . . . . . 208
7.7 Path Integrals: Integrating the Fat Tails
into Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.8 Path Integrals: Integrating Path Dependence
into Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8. Microscopic Market Models .............................. 221
8.1 ImportantQuestions.................................... 221
8.2 AreMarketsE?cient? .................................. 222
8.3 ComputerSimulationofMarketModels ................... 226
8.3.1 TwoClassicalExamples........................... 226
8.3.2 Recent Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
8.4 TheMinorityGame .................................... 246
8.4.1 TheBasicMinorityGame ......................... 247
8.4.2 A Phase Transition in the Minority Game . . . . . . . . . . . 249
8.4.3 RelationtoFinancialMarkets...................... 250
8.4.4 Spin Glasses and an Exact Solution . . . . . . . . . . . . . . . . . 252
8.4.5 Extensionsof theMinorityGame................... 255
9. Theory of Stock Exchange Crashes ....................... 259
9.1 ImportantQuestions.................................... 259
9.2 Examples.............................................. 260
9.3 EarthquakesandMaterialFailure ........................ 264
9.4 StockExchangeCrashes................................. 270
9.5 WhatCausesCrashes?.................................. 276
9.6 AreCrashesRational? .................................. 278
9.7 WhatHappensAfteraCrash? ........................... 279
9.8 ARichterScale forFinancialMarkets..................... 285