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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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