Stock markets as evolving complex systems [Elektronische Ressource] : simulations and statistical inferences / vorgelegt von Hans-Jürgen Holtrup

Informations

• Publié par: universitat_duisburg-essen
• Publié le: 01 janvier 2007
• Nombre de lectures: 28
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

StockMarkets as EvolvingComplexSystems.
Simulations andStatistical Inferences.
Dissertation
zur Erlangung des akademischen Grades eines Doktors der
Wirtschaftswissenschaften Dr.rer.pol
durch den Fachbereich Wirtschaftswissenschaften der Universität
Duisburg-Essen, Standort Essen
vorgelegt von
Dipl-Volkswirt Hans-Jürgen Holtrup aus Dorsten
Tag der Prüfung: 18. Januar 2006
Erstgutachter: Prof. Dr. W. Gaab
Zweitgutachter: Prof. Dr. AssenmacherContents
1 Introduction 4
I E¢ cient Markets and other Concepts 7
2 The E¢ cient Market Hypothesis and its Challenges 8
2.1 The Random Walk Hypothesis . . . . . . . . . . . . . . . . . . . 10
2.2 Theoretical and Empirical Challenges to the EMH . . . . . . . . 12
2.2.1 Bounded Rationality . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Market Anomalies . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Big Crashes . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.4 Behavioural Finance . . . . . . . . . . . . . . . . . . . . . 22
2.2.5 Herd Behaviour . . . . . . . . . . . . . . . . . . . . . . . . 27
3 The Theory of Complex Systems 30
3.1 Characteristics of Complex Systems . . . . . . . . . . . . . . . . 32
3.2 Examples of Complex Systems . . . . . . . . . . . . . . . . . . . 34
3.3 The Explanatory Range of a Theory of Complex Systems . . . . 44
II Stylised Facts of Financial Markets 46
4 Heavy tails 49
4.1 Heavy tailed distributions . . . . . . . . . . . . . . . . . . . . . . 50
4.1.1 The Class L . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.2 The Class S . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.3 Power-law distributions . . . . . . . . . . . . . . . . . . . 52
4.1.4 The Pareto distribution . . . . . . . . . . . . . . . . . . . 53
4.1.5 The LØvy Stable distribution . . . . . . . . . . . . . . . . 54
4.2 Alternatives to the stable distribution . . . . . . . . . . . . . . . 59
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.4 Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Empirical Methods for Heavy Tailed Distributions . . . . . . . . 65
4.5.1 Quantile Plots . . . . . . . . . . . . . . . . . . . . . . . . 65
14.5.2 Estimation Methods for Heavy Tailed Distributions . . . 65
4.5.3 Tail Estimators . . . . . . . . . . . . . . . . . . . . . . . . 66
4.5.4 Sample Quantiles Methods . . . . . . . . . . . . . . . . . 68
4.5.5 Maximum Likelihood Estimation . . . . . . . . . . . . . . 69
4.5.6 Estimators based on the Characteristic Function of LSD . 70
4.5.7 The performance of the estimators . . . . . . . . . . . . . 72
4.6 Empirical Results in the Literature . . . . . . . . . . . . . . . . . 75
4.7 Own Empirical Tests on the Tail parameter . . . . . . . . . . . . 78
4.7.1 The Data Sets . . . . . . . . . . . . . . . . . . . . . . . . 78
4.7.2 QQ-plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.7.3 Estimation Results for daily Price Records . . . . . . . . 80
4.7.4 Own Estimation with high-frequency Price Records . . . 85
5 FractalDimensionsandScalingLawsforFinancialTimeSeries 86
5.1 Fractal Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.1.1 The self-similar Dimension . . . . . . . . . . . . . . . . . 90
5.1.2 The Box Dimension . . . . . . . . . . . . . . . . . . . . . 91
5.1.3 The Pointwise Dimension . . . . . . . . . . . . . . . . . . 94
5.1.4 The Multifractal Spectrum . . . . . . . . . . . . . . . . . 95
5.2 The Scaling Properties of Fractional Brownian Motion . . . . . . 97
5.2.1 The scaling of Brownian Motion . . . . . . . . . . . . . . 97
5.2.2 The Scaling of fractional Brownian Motion . . . . . . 99
5.3 Multiscaling and Multifractality . . . . . . . . . . . . . . . . . . . 101
5.3.1 Estimation of the Zeta-(q)-function . . . . . . . . . . . . . 102
5.3.2 Empirical Evidence of Multiscaling (Multifractality) . . . 104
6 Autocorrelations and Volatility Clustering in the Stock Mar-
kets 117
6.1 First-order short run Correlations . . . . . . . . . . . . . . . . . . 118
6.1.1 First-order long-run Correlations . . . . . . . . . . . . . . 122
6.1.2 Empirical Evidence of long Memory in Raw Returns . . . 133
6.1.3 Own Estimations for Raw Returns . . . . . . . . . . . . . 136
6.2 Second-order Correlations . . . . . . . . . . . . . . . . . . . . . . 138
6.2.1 Empirical evidence of long memory in the volatility process138
6.2.2 Own Estimations for long Memory in the Volatility Process140
III The Simulation of Financial Markets 143
7 Stochastic Simulations 146
7.1 The Basis of Stochastic Modellings of Economic Systems . . . . . 148
7.1.1 The Multiplicity of Microstates . . . . . . . . . . . . . . . 148
7.1.2 Entropy and the Gibbs-distribution . . . . . . . . . . . . 151
7.1.3 Detailed Balance . . . . . . . . . . . . . . . . . . . . . . . 153
7.2 Ising related Models for Financial Markets . . . . . . . . . . . . . 156
7.2.1 The General Structure . . . . . . . . . . . . . . . . . . . . 156
27.2.2 The Mechanics of the System . . . . . . . . . . . . . . . . 157
7.3 Previous Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.3.1 Chowdury and Stau⁄er (1999) . . . . . . . . . . . . . . . 161
7.3.2 Kaizoji (2000) . . . . . . . . . . . . . . . . . . . . . . . . 165
7.3.3 Bornholdt (2001) . . . . . . . . . . . . . . . . . . . . . . . 167
7.3.4 Kaizoji, Bornholdt and Fujiwara (2002) . . . . . . . . . . 170
7.3.5 Iori (2002). . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.3.6 The Cont-Bouchaud Percolation Simulation (2000) . . . . 178
7.3.7 Stau⁄er and Penna (1998) . . . . . . . . . . . . . . . . . . 182
7.3.8 Stau⁄er and Sornette (1999) . . . . . . . . . . . . . . . . 183
7.3.9 Chang and Stau⁄er (1999). . . . . . . . . . . . . . . . . . 185
7.3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.4 A new Ising Model with heterogenous Traders and Information
In ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.4.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.4.2 Results Variation A . . . . . . . . . . . . . . . . . . . . . 193
7.4.3 Results Variation B . . . . . . . . . . . . . . . . . . . . . 203
7.4.4 First Conclusions . . . . . . . . . . . . . . . . . . . . . . . 207
8 Deterministic Simulation Models 208
8.1 The Levy, Levy and Solomon Model . . . . . . . . . . . . . . . . 209
8.1.1 Fundamentally Based Investors . . . . . . . . . . . . . . . 210
8.1.2 Non Fundamental Orientated Investors. . . . . . . . . . . 212
8.1.3 The Simulation . . . . . . . . . . . . . . . . . . . . . . . . 213
8.2 Other Deterministic Simulations . . . . . . . . . . . . . . . . . . 217
8.2.1 The Stigler Model (1964) . . . . . . . . . . . . . . . . . . 217
8.2.2 The Kim-Markowitz Model (1989) . . . . . . . . . . . . . 218
8.2.3 TheModelofArthur,Holland,LeBaron,PalmerandTayler
(1997) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.2.4 The Model of Lux and Marchesi (1999) . . . . . . . . . . 223
8.3 A new Deterministic Simulation with Di⁄erent Trader Types . . 226
8.3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . 226
8.3.2 The Simulation . . . . . . . . . . . . . . . . . . . . . . . . 232
9 Conclusion 240
10 Bibliographie 243
11 Appendix: The Ising Model 265
11.1 The Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
11.1.1 The Background . . . . . . . . . . . . . . . . . . . . . . . 265
11.1.2 Energy Minimisation . . . . . . . . . . . . . . . . . . . . . 265
11.1.3 Entropy Maximisation . . . . . . . . . . . . . . . . . . . . 266
11.2 The Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
12 List of Symbols 269
3Chapter 1
Introduction
Financial markets have always been playing an important part in economic
research. Many economists see …nancial markets, whether stock, foreign ex-
change or future markets, as prime examples of complete markets. Information
that might be important for the value of the traded asset is quickly propagated
through the media. There is no personal a⁄ection for speci c equities assumed,
and the focus is only to buy cheap and to sell high. Furthermore, shares can be
traded without personal contacts by placing buy and sell orders. This in turn
should reduce transaction costs by a considerable amount. In fact, …nancial
markets can claim to posses a very e¢ cient mechanism of …nding trading part-
ners. In short, …nancial markets among all markets should be the place where
prices come nearest to fully re ect the opinions of the participants. These are
moreover supposed to have perfect knowledge about the intrinsic values of each
equity. The E¢ cient Market Hypothesis is a logical consequence of these cir-
cumstances.
However, the importance of …nancial markets does not only come from this
more theoretical statement. Financial markets are also important in …nancial
intermediation. For example, stock markets allow an e¢ cient risk sharing as
stressed by Diamond (1967). They also provide incentives to gather informa-
tion, which drives stock prices more closely to its true values. These market
prices then provide signals for an e¢ cient allocation of …nancial capital (see
e.g. Diamond and Verrecchia (1981)). A more practical point is that …nancial
markets