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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 o⁄er a chance to make pro ts. If it would be possible to forecast future
price developments, short or long positions should yield high pro ts. However,
this possibility collides with the theoretical assessments. If …nancial markets re-
allyre‡ectthewholeinformation,thenpricescannotbeforecastedbecauseonly
1newinformationalterstheprices. Recentyearswitnessedalivelydebateabout
models of …nancial markets that are somewhat in-between these con icting po-
sitions. Empirical and theoretical challenges to the e¢ cient view have come up
1i.e. future prices are unknown. Principally, one can make a forecast based e.g. on pure
intuition. But this is not the meaning of unforcastable above.
4with competing views about traders who do not act in the fully rational way
assumed by the protagonists of the e¢ cient view. On the other side, evidence
for a possibility to employ technical analysis (charts) in order to forecast prices
is-atleast-verysparse. Sothesearchto…ndarealisticpictureoftheprocesses
in …nancial markets is still ongoing.
One promising advance is made by the introduction of psychological expla-
nation for indiduals action in …nancial markets. Here, peoples motivation is
analysed andexperimentalas wellas theoreticalresults are then transformedto
explain the dynamcis within e.g. stock markets. Other lines of …nance theory
focusonthespeci cmicrostructureofmarkets,ortrytorationalisecommonbe-
haviourbyintroducingsomeformofprivateinformationthatonlysomemarket
participants posess. However, while these directions are able to explain some
speci cempiricalfeaturesof…nancialmarkets,theycannotaccountforthemore
general behaviour of asset prices.
The following work is inspired by the ideas of several theoretical physicists.
They developed propositions about …nancial markets so as to interpret them
as examples of complex self-organising buildings, similar to many other natural
systems. They stress the fact that some systems, physical, social or …nancial,
display similar statistical properties, which cannot completely explained by ex-
ogenous factors. Historical events like the famous Tulipmania bubble or the
south see bubble feed the assumption of an endogenous reason for large price
‡uctuations, because they show no signs of fundamental exogenous reasons. To
be more precise, physicist claim to have found some universal statistical fea-
tures that prevail in every system that consists of a large number of interacting
members. For …nancial markets, the members are the people who trade assets
andtheinteractionisusuallyinterpretedasthecommunicationthattakesplace
between them. These members, so the hypothesis, build a network that in few
casesworksoastoalignalltraderstobehaveinthesamemanner,thuscreating
a herd that produces bubbles and crashes.
This work in divided into three parts. The …rst shortly summarises the E¢ -
cient Market Hypothesis and its principal empirical shortcomings as well as the
competingtheoreticallinecalledBehaviouralFinanceTheory. Thepresentation
of a new idea based on the theory of complex systems completes part one. The
second part analyses the main statistical facts of …nancial markets. Because
these empirical characteristics are the yardstick with which proposed new mod-
els have to be compared, it is essential to have a precise picture of what should
betargeted. Thelastpartpresentsanewframeworktomodelstockmarkets. It
is based on the idea that these markets consist of many heterogenous interact-
ing traders. These traders determine through their actions the price dynamics.
The simulations of part three will try to use this concept in order to convert it
into numerical models that reproduce the facts of part two. It must be stressed
that the provision of some new empirical estimations and two new variants of
simulations are not the sole contribution of this work. It also aims to give an
5overview of the whole concept of statistical physics and its application to eco-
nomicproblems. Therearebynowsomenotetablyintroducorybookspublished
(Mantegna and Stanley (2000), Bouchaud and Potters (2000) and LØvy, LØvy
and Solomon (2000) among others), but none of these tries to give a complete
picture that connects the empirical facts with the numerical simulations. They
focus either on the statistical featurs of …nancial markets or its simulation.
6Part I
E¢ cient Markets and other
Concepts
7Chapter 2
The E¢ cient Market
Hypothesis and its
Challenges
The notion of e¢ ciency in …nancial markets has a long tradition. The idea be-
hind the term, originally coined by Harry Roberts (1967), goes back to Gibson
1(1889)andBachelier(1900)givesa…rstmathematicaltreatmentofthesubject.
The concept of the E¢ cient Market Hypothesis (henceforth EMH) in its most
generalformclaimsthatpricesof…nancialassetsre ectallrelevantinformation,
or as Mandelbrot (1971, p. 225) explains: ”Roughly speaking, a competitive
market of securities, commodities or bonds may be considered e¢ cient if every
price already re‡ects all the relevant information that is available. The ar-
rival of new information causes imperfection, but it is assumed that every such
imperfection is promptly arbitraged away. As this e¢ ciency concept involves
the modellation of information, the expression of information e¢ ciency is also
frequently used to characterise e¢ cient …nancial markets (as opposed to other
familiar notions of economic e¢ ciency like, for example, Pareto-e¢ ciency).
In its strongest interpretation, individuals do not have di⁄erent comparative
advantages in information acquisition. All people trade on the same complete
information set that even includes inside information. Because this reading
demands an ability of information gathering that is rarely met in reality Fama
(1970) divides the EMH into three categories depending on the information set:
1SeeShiller(1998). In fact, BacheliersthesisTheorie de la Speculation already included the
idea of a martingale measure for the evalutation of assets. He explicitely modelled the markets
prices as a continous Markov process. Bachelier was also the …rst who developed many of the
mathematical properties of Brownian Motion - …ve years prior to Einstein’s famous work on
the same subject (1905). For a short review of Bacheliers work see Courtault et al. (2000).
Other early works on the topic include Williams (1938) and Graham and Dodd (1934, 1996).
8(i) Strong form of market e¢ ciency
There is no public or even private information that will allow an investor
to earn abnormal returns based on that information. It is assumed that
all information is available to everyone at the same time cost-free, i.e., a
perfect market exists.
(ii) Semistrong form of market e¢ ciency
Thereisnopublicinformationthatwillallowaninvestortoearnabnormal
returns based on that information. Public information includes all stock
marketinformationplus allpubliclyavailable…nancial,economic,orother
type of information on the speci c company, the national economy, the
world, etc. Security prices react immediately to all new information.
(iii) Weak form of market e¢ ciency
There is no information in past stock prices (of a particular asset) which
willallowaninvestortoearnabnormalreturns(fromthatparticularasset)
basedonthatinformation. Stockmarketinformationincludesstockprices
2as well as relevant macroeconomic and …rm speci…c data.
The concept of e¢ cient markets as stated above is an appealing idea since it
is di¢ cult for an economist to sustain the case that agents in …nancial markets
do not behave rationally and maximise their pro…ts by processing all available
information. Ontheothersideisithardtoimaginethattradersliveinaperfect
rational world where psychology plays no role at all. Actually, even strong sup-
porters of the rational behaviour paradigm would accept some irrational beliefs
as a factor of in uence at least to some of the market participants. But the
problem with these often called noise traders is that they are buying overpriced
while selling underpriced assets. As a consequence, their pro ts are lower than
thoseofsmarttraderswhomakegreaterpro tsbyexploitingarbitragedeals. As
Friedman (1953) noticed, this is not a situation that can last forever, because
noise traders will eventually leave the market because of permanently losing
against the rational actors. Through this process, the EMH should be restored
at least in the middle-run.
The concept of informational e¢ cient markets is closely associated with a
probabilistic handling of the subject. Economists use the concept of martingale
theory to formalise the idea of an informational e¢ cient market in an elegant
3and compact manner.
De nition 2.1 (Martingale):
Let
= ( ) be a family of information subsets of T up tot t2T
the time index t < T, and let E[x j
] be the expectation of xt s t
2In this form, informational e¢ cient market requires that the costs for gathering informa-
tion and trading are zero (Grossmann and Stieglitz (1980)).
3A rigorous treatment can be found in Doob (1953) or Billingsley (1976). The …rst who
used martingales as a description of asset prices where Samuelson (1965) and Mandelbrot
(1966).
9

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