new economic windows - download.e-bookshelf.de€¦ · new economic windows series editor massimo...

New Economic Windows

Series EditorMASSIMO SALZANO

Series Editorial Board

Jaime Gil AlujaDepartament d’Economia i Organització d’Empreses, Universitat de Barcelona, Spain

Fortunato ArecchiDipartimento di Fisica, Università di Firenze and INOA, Italy

David Colander Department of Economics, Middlebury College, Middlebury, VT, USA

Richard H. Day Department of Economics, University of Southern California, Los Angeles, USA

Mauro GallegatiDipartimento di Economia, Università di Ancona, Italy

Steve KeenSchool of Economics and Finance, University of Western Sydney, Australia

Giulia Iori Department of Mathematics, King’s College, London, UK

Alan KirmanGREQAM/EHESS, Université d’Aix-Marseille III, France

Marji Lines Dipartimento di Science Statistiche, Università di Udine, Italy

Alfredo MedioDipartimento di Scienze Statistiche, Università di Udine, Italy

Paul OrmerodDirectors of Environment Business-Volterra Consulting, London, UK

J. Barkley RosserDepartment of Economics, James Madison University, Harrisonburg, VA, USA

Sorin SolomonRacah Institute of Physics, The Hebrew University of Jerusalem, Israel

Kumaraswamy (Vela) Velupillai Department of Economics, National University of Ireland, Ireland

Nicolas VriendDepartment of Economics, Queen Mary University of London, UK

Lotfi ZadehComputer Science Division, University of California Berkeley, USA

Editorial AssistantsMaria Rosaria AlfanoMarisa Faggini Dipartimento di Scienze Economiche e Statistiche, Università di Salerno, Italy

STR1988-SP-R@001-004# 13-06-2005 17:50 Pagina I

Marisa Faggini Dipartimento di Scienze Economiche e Statistiche, Università di Salerno, Italy

Series Editorial Board

Jaime Gil AlujaDepartament d’Economia i Organització d’Empreses, Universitat de Barcelona, Spain

Fortunato ArecchiDipartimento di Fisica, Università di Firenze and INOA, Italy

David ColanderDepartment of Economics, Middlebury College, Middlebury, VT, USA

Richard H. DayDepartment of Economics, University of Southern California, Los Angeles, USA

Mauro GallegatiDipartimento di Economia, Università di Ancona, Italy

Steve KeenSchool of Economics and Finance, University of Western Sydney, Australia

Alan KirmanGREQAM/EHESS, Université d’Aix-Marseille III, France

Marji LinesDipartimento di Science Statistiche, Università di Udine, Italy

Thomas LuxDepartment of Economics, University of Kiel, Germany

Alfredo MedioDipartimento di Scienze Statistiche, Università di Udine, Italy

Paul OrmerodDirectors of Environment Business-Volterra Consulting, London, UK

Peter RichmondSchool of Physics, Trinity College, Dublin 2, Ireland

J. Barkley RosserDepartment of Economics, James Madison University, Harrisonburg, VA, USA

Sorin SolomonRacah Institute of Physics, The Hebrew University of Jerusalem, Israel

Pietro TernaDipartimento di Scienze Economiche e Finanziarie, Università di Torino, Italy

Kumaraswamy (Vela) VelupillaiDepartment of Economics, National University of Ireland, Ireland

Nicolas VriendDepartment of Economics, Queen Mary University of London, UK

Lotfi ZadehComputer Science Division, University of California Berkeley, USA

Editorial AssistantsMarisa FagginiDipartimento di Scienze Economiche e Statistiche, Università di Salerno, Italy

Arnab Chatterjee • Sudhakar Yarlagadda •Bikas K Chakrabarti (Eds.)

Econophysics of Wealth Distributions

STR1988-SP-R@001-004# 13-06-2005 17:50 Pagina III

Arnab Chatterjee • Bikas K Chakrabarti (Eds.)

Econophysicsof Stock and other MarketsProceedings of the Econophys-Kolkata II

1 3

ARNAB CHATTERJEE

SUDHAKAR YARLAGADDA

BIKAS K CHAKRABARTI

Theoretical Condensed Matter Physics Division and Centre for Applied Mathematics and Computational ScienceSaha Institute of Nuclear Physics, Kolkata, India

The publication of this book has been made possible by a partial support from a fund provided by the Centre for Applied Mathematics and Computational Science, Saha Institute ofNuclear Physics, Kolkata, India

Library of Congress Control Number: 2006928743

ISBN-10 88-470-0501-9 Springer Milan Berlin Heidelberg New York

ISBN-13 978-88-470-0501-3

This work is subject to copyright. All rights are reserved, whether the whole of part of thematerial is concerned, specifically the rights of translation, reprinting, re-use of illustrations,recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data-banks. Duplication of this pubblication or parts thereof is only permitted under the provisionsof the Italian Copyright Law in its current version, and permission for use must always beobtained from Springer-Verlag. Violations are liable for prosecution under the ItalianCopyright Law.

Springer is a part of Springer Science+Business Media

springer.com

© Springer-Verlag Italia 2006

Printed in Italy

Cover design: Simona Colombo, MilanoTypeset by the authors using a Springer Macro packagePrinting and binding: Signum srl, Bollate (MI)

Printed on acid-free paper

IV A. Achiron et al.

STR1988-SP-R@001-004# 13-06-2005 17:50 Pagina IV

BIKAS K CHAKRABARTI

Theoretical Condensed Matter Physics Division and Centre for Applied Mathematics and Computational ScienceSaha Institute of Nuclear Physics, Kolkata, India

Preface

Successful or not, we all (have to?) go to various markets and participate intheir activities. Yet, so little is understood about their functionings. Efforts tomodel various markets are now substantial. Econophysicists have also comeup recently with several innovative models and their analyses.

This book is a proceedings of the International Workshop on “Econo-physics of Stock Markets and Minority Games”, held in Kolkata during Febru-ary 14-17, 2006, under the auspices of the Centre for Applied Mathemat-ics and Computational Science, Saha Institute of Nuclear Physics, Kolkata.This is the second event in the Econophys-Kolkata series of meetings; theEconophys-Kolkata I was held in March 2005 (Proceedings: Econophysics ofWealth Distributions, published in the same New Economic Windows seriesby Springer, Milan in 2005). We understand from the enthusiastic responseof the participants that the one-day trip to the Sunderbans (Tiger Reserve; aworld heritage point) along with the lecture-sessions on the vessel had beenhugely enjoyable and successful. The concluding session had again very livelydiscussions on the workshop topics as well as on econophysics in general, ini-tiated by J. Barkley Rosser, Matteo Marsili, Rosario Mantegna and RobinStinchcombe (Chair). We plan to hold the next meeting in this series, on“Econophysics and Sociophysics: Debates on Complexity Issues in Economicsand Sociology” early next year.

We are very happy that several leading economists and physicists engagedin these recent developments in the econophysics of markets, their analysis andmodelling could come and participate. Although a few of them (Fabrizio Lillo,Thomas Lux and Rosario Mantegna) could not contribute to this proceedingsvolume due to shortage of time (we again try to get this proceedings publishedwithin six months from the workshop), we are indeed very happy that most ofthe invited participants could contribute in this book. The papers on marketanalysis and modellings are very original and their timely appearance here willrender the book extremely useful for the researchers. The two historical notesand the Comments and Discussions section will give the readers two examplesof personal perspectives regarding the new developments in econophysics, and

VI Preface

some ‘touch’ of the lively debates taking place in these Econophys-Kolkataseries of workshops.

We are extremely grateful to Mauro Gallegati and Massimo Salzano ofthe editorial board of this New Economic Windows series of Springer fortheir encouragement and support, and to Marina Forlizzi for her efficientmaintenance of publication schedule.

Kolkata, Arnab ChatterjeeJune 2006 Bikas K. Chakrabarti

Contents

Part I Markets and their Analysis

On Stock-Price Fluctuations in the Periods of Booms andStagnationsTaisei Kaizoji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

An Outlook on Correlations in Stock PricesAnirban Chakraborti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

The Power (Law) of Indian Markets: Analysing NSE and BSETrading StatisticsSitabhra Sinha, Raj Kumar Pan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

A Random Matrix Approach To Volatility In An IndianFinancial MarketV. Kulkarni, N. Deo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Why do Hurst Exponents of Traded Value Increase as theLogarithm of Company Size?Zoltan Eisler, Janos Kertesz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Statistical Distribution of Stock Returns RunsHonggang Li, Yan Gao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Fluctuation Dynamics of Exchange Rates on Indian FinancialMarketA. Sarkar, P. Barat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Noise Trading in an Emerging Market: Evidence and AnalysisDebasis Bagchi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

VIII Contents

How Random is the Walk: Efficiency of Indian Stock andFutures MarketsUdayan Kumar Basu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Part II Markets and their Models

Models of Financial Market Information EcologyDamien Challet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Estimating Phenomenological Parameters in Multi-AssetsMarketsGiacomo Raffaelli, Matteo Marsili . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Agents Play Mix-gameChengling Gou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Triangular Arbitrage as an Interaction in Foreign ExchangeMarketsYukihiro Aiba, Naomichi Hatano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Modelling Limit Order Financial MarketsRobin Stinchcombe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Two Fractal Overlap Time Series and Anticipation of MarketCrashesBikas K. Chakrabarti, Arnab Chatterjee, Pratip Bhattacharyya . . . . . . . . . 153

The Apparent Madness of Crowds: Irrational CollectiveBehavior Emerging from Interactions among Rational AgentsSitabhra Sinha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Agent-Based Modelling with Wavelets and an EvolutionaryArtificial Neural Network: Applications to CAC 40ForecastingSerge Hayward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Information Extraction in Scheduling Problems withNon-Identical MachinesManipushpak Mitra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Modelling Financial Time SeriesP. Manimaran, J.C. Parikh, P.K. Panigrahi S. Basu, C.M. Kishtawal,M.B. Porecha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Random Matrix Approach to Fluctuations and Scaling inComplex SystemsM. S. Santhanam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Contents IX

The Economic Efficiency of Financial MarketsYougui Wang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Regional InequalityAbhirup Sarkar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Part III Historical Notes

A Brief History of Economics: An Outsider’s AccountBikas K Chakrabarti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

The Nature and Future of EconophysicsJ. Barkley Rosser, Jr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Part IV Comments and Discussions

Econophys-Kolkata II Workshop SummaryJ. Barkley Rosser, Jr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Econophysics: Some Thoughts on Theoretical PerspectivesMatteo Marsili . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Comments on “Worrying Trends in Econophysics”: IncomeDistribution ModelsPeter Richmond, Bikas K. Chakrabarti, Arnab Chatterjee, John Angle . . 244

List of Invited Speakers and Contributors

Yukihiro AibaDepartment of PhysicsUniversity of Tokyo, KomabaMeguro, Tokyo 153-8505 [email protected]

John AngleInequality Process InstituteP.O. Box 429 Cabin JohnMaryland 20818-0429 [email protected]

Debasis BagchiThe Institute of Cost and WorksAccountants of India,12, Sudder StreetKolkata 700 016 [email protected]

P. BaratVariable Energy Cyclotron Centre1/AF Bidhan NagarKolkata 700064 [email protected]

J. Barkley Rosser, Jr.Department of EconomicsJames Madison UniversityHarrisonburg, VA, [email protected]

S. BasuSpace Applications CentreAhmedabad 380 015 India

Udayan Kumar BasuFuture Business SchoolKolkata, Indiaudayan [email protected]

Pratip BhattacharyyaPhysics DepartmentGurudas College, NarkeldangaKolkata 700 054 [email protected]

Damien ChalletNomura Centre for QuantitativeFinanceMathematical InstituteOxford University24–29 St Giles’Oxford OX1 3LB, [email protected]

Bikas K. ChakrabartiTheoretical Condensed MatterPhysics Division and Centre forApplied Mathematics and Computa-tional ScienceSaha Institute of Nuclear Physics1/AF BidhannagarKolkata 700064 [email protected]

List of Invited Speakers and Contributors XI

Anirban ChakrabortiDepartment of PhysicsBanaras Hindu UniversityVaranasi 221005 [email protected]

Arnab ChatterjeeTheoretical Condensed MatterPhysics Division and Centre forApplied Mathematics and Computa-tional ScienceSaha Institute of Nuclear Physics1/AF BidhannagarKolkata 700064 [email protected]

N. DeoDepartment of Physics and Astro-physicsUniversity of DelhiDelhi 110007 [email protected]

Zoltan EislerDepartment of Theoretical PhysicsBudapest University of Technologyand EconomicsBudafoki ut 8H-1111 Budapest [email protected]

Yan GaoDepartment of Systems ScienceSchool of ManagementBeijing Normal UniversityBeijing 100875 [email protected]

Chengling GouPhysics DepartmentBeijing University of Aeronauticsand Astronautics37 Xueyuan Road, Heidian DistrictBeijing 100083 [email protected]

Naomichi HatanoInstitute of Industrial ScienceUniversity of TokyoKomaba, MeguroTokyo 153-8505 [email protected]

Serge HaywardDepartment of FinanceEcole Superieure de Commerce deDijon,29, rue Sambin21000 Dijon [email protected]

Taisei KaizojiDivision of Social SciencesInternational Christian UniversityTokyo 181-8585 [email protected]

Janos KerteszDepartment of Theoretical PhysicsBudapest University of Technologyand EconomicsBudafoki ut 8H-1111 Budapest, [email protected]

C. M. KishtawalSpace Applications CentreAhmedabad 380 015 India

V. KulkarniDepartment of Physics and Astro-physicsUniversity of DelhiDelhi 110007 Indiavarsha s [email protected]

Honggang LiDepartment of Systems ScienceSchool of ManagementBeijing Normal UniversityBeijing 100875 [email protected]

XII List of Invited Speakers and Contributors

Fabrizio LilloDipartimento di Fisica e TecnologieRelativeviale delle Scienze - Edificio 18I-90128 Palermo [email protected]

Thomas LuxDepartment of EconomicsUniversity of KielOlshausenstr. 40D-24118 Kiel [email protected]

P. ManimaranPhysical Research LaboratoryAhmedabad 380 009 India

Rosario N. MantegnaDip. di Fisica e Tecnologie RelativeUniversita’ di PalermoViale delle ScienzeI-90128 Palermo [email protected]

Matteo MarsiliThe Abdus Salam InternationalCentre for Theoretical PhysicsStrada Costiera 1434014 Trieste [email protected]

Manipushpak MitraEconomic Research UnitIndian Statistical Institute203 B. T. RoadKolkata-700108 [email protected]

Raj Kumar PanThe Institute of MathematicalSciencesC. I. T. Campus, TaramaniChennai 600 113 [email protected]

P. K. PanigrahiPhysical Research LaboratoryAhmedabad 380 009 India

J. C. ParikhPhysical Research LaboratoryAhmedabad 380 009 [email protected]

M. B. PorechaPhysical Research LaboratoryAhmedabad 380 009 India

Giacomo RaffaelliDipartimento di FisicaCNR-ISC and INFM-SMCUniversita di Roma “La Sapienza”p.le A. Moro 2, 00185 Roma Italy

Peter RichmondSchool of PhysicsUniversity of DublinTrinity College Dublin 2 [email protected]

M. S. SanthanamPhysical Research LaboratoryNavarangpuraAhmedabad 380 009 [email protected]

Abhirup SarkarEconomic Research UnitIndian Statistical Institute203 B. T. RoadKolkata 700108 [email protected]

Apu SarkarVariable Energy Cyclotron Centre1/AF Bidhan NagarKolkata 700064 [email protected]

Sitabhra SinhaThe Institute of MathematicalSciencesC. I. T. Campus, TaramaniChennai 600 113 [email protected]

List of Invited Speakers and Contributors XIII

Robin StinchcombeRudolf Peierls Centre for TheoreticalPhysicsOxford University1 Keble RoadOXFORD, OX1 3NP, [email protected]

Yougui WangDepartment of Systems ScienceSchool of ManagementBeijing Normal UniversityBeijing 100875People’s Republic of [email protected]

Part I

Markets and their Analysis

On Stock-Price Fluctuations in the Periods ofBooms and Stagnations

Taisei Kaizoji

Division of Social Sciences, International Christian University, Tokyo 181-8585,Japan. [email protected]

1 Introduction

The statistical properties of the fluctuations of financial prices have beenwidely researched since Mandelbrot [1] and Fama [2] presented an evidencethat return distributions can be well described by a symmetric Levy stablelaw with tail index close to 1.7. Many empirical studies have shown that thetails of the distributions of returns and volatility follow approximately a powerlaw with estimates of the tail index falling in the range 2 to 4 for large valueof returns and volatility. (See, for examples, de Vries [3]; Pagan [4]; Longin[5], Lux [6]; Guillaume et al. [7]; Muller et al. [8]; Gopikrishnan et al. [9],Gopikrishnan et al. [10], Plerou et al. [11], Liu et al. [12]). However, there isalso evidence against power-law tails. For instance, Barndorff-Nielsen [13], andEberlein et al. [14] have respectively fitted the distributions of returns usingnormal inverse Gaussian, and hyperbolic distribution. Laherrere and Sornette[15] have suggested to describe the distributions of returns by the stretched-exponential distribution. Dragulescu and Yakovenko [16] have shown that thedistributions of returns have been approximated by exponential distributions.More recently, Malevergne, Pisarenko and Sornette [17] have suggested thatthe tails ultimately decay slower than any stretched exponential distributionbut probably faster than power laws with reasonable exponents as a resultfrom various statistical tests of returns.

Thus opinions vary among scientists as to the shape of the tail of the dis-tribution of returns (and volatility). While there is fairly general agreementthat the distribution of returns and volatility has fat tails for large values ofreturns and volatility, there is still room for a considerable measure of dis-agreement about universality of the power-law distributions. At the momentwe can only say with fair certainty that (i) the power-law tail of the distri-bution of returns and volatility is not an universal law and (ii) the tails ofthe distribution of returns and volatility are heavier than a Gaussian, and arebetween exponential and power-law. There is one other thing that is impor-tant for understanding of price movements in financial markets. It is a fact

4 Taisei Kaizoji

that the financial market has repeated booms (or bull market) and stagna-tions (or bear market). To ignore this fact is to miss the reason why pricefluctuations are caused. However, in most empirical studies, which have beenmade on statistical properties of returns and volatility in financial markets,little attention has been given to the relationship between market situationsand price fluctuations. Our previous work [18] investigates this subject us-ing the historical data of the Nikkei 225 index. We find that the volatility inthe inflationary period is approximated by an power-law distribution whilethe volatility distribution in the deflationary period is described by an ex-ponential distribution. The purpose of this paper is to examine further thestatistical properties of volatility distribution from this viewpoint. We use thedaily data of the four stock price indices of the three major stock markets inthe world: the Nikkei 225 index, the DJIA. SP500, and FT100, and comparethe shape of the volatility distribution for each of the stock price indices inthe periods of booms with that in the period of stagnations. We find that (i)the tails of the distribution of the absolute log-returns are approximated bya power-law function with the exponent close to 3 in the periods of boomswhile the distribution is described by an exponential function with the scaleparameter close to unity in the periods of stagnations. These indicate that sofar as the stock price indices we used are concerned, the same observation onthe volatility distribution holds in all cases.

The rest of the paper is organized as follows: the next section analyzes thestock price indices and shows the empirical findings. Section 3 gives concludingremarks.

2 Empirical analysis

2.1 Stock price indices

We investigate quantitatively the four stock price indices of the three majorstock markets in the world1, that is, (a) the Nikkei 225 index (Nikkei 225),which is the price-weighted average of the stock prices for 225 large companieslisted in the Tokyo Stock Exchange, (b) the Dow Jones Industrial Average(DJIA), which is the price-weighted average of 30 significant stocks tradedon the New York Stock Exchange and Nasdaq, (c) Standard and Proor’s 500index (SP 500), which is a market-value weighted index of 500 stocks chosenfor market size, liquidity, and industry group representation, and (d) FT 100,which is similar to SP 500, and a market-value weighted index of shares ofthe top 100 UK companies ranked by market capitalization. Figure 1(a)-(d)show the daily series of the four stock price indices: (a) the Nikkei 225 fromJanuary 4, 1975 to August 18, 2004, (b) DJIA from January 2, 1946 to August

1 The prices of the indices are close prices which are adjusted for dividends andsplits.

On Stock-Price Fluctuations in the Periods of Booms and Stagnations 5

18, 2004, (c) SP 500 from November 22, 1982 to August 18, 2004, and (d) FT100 from April 2, 1984 to August 18, 2004.

After booms of a long period of time, the Nikkei 225 reached a high ofalmost 40,000 yen on the last trading day of the decade of the 1980s, andthen from the beginning trading day of 1990 to mid-August 1992, the indexhad declined to 14,309, a drop of about 63 percent. A prolonged stagnation ofthe Japanese stock market started from the beginning of 1990. The time seriesof the DJIA and SP500 had the apparent positive trends until the beginning of2000. Particularly these indices surged from the mid-1990s. There is no doubtthat this stock market booms in history were propelled by the phenomenalgrowth of the Internet which has added a whole new stratum of industryto the American economy. However, the stock market booms in the US stockmarkets collapsed at the beginning of 2000, and the descent of the US marketsstarted. The DJIA peaked at 11722.98 on January 14, 2000, and dropped to7286.27 on October 9, 2002 by 38 percent. SP500 arrived at peak for 1527.46on March 24, 2000 and hit the bottom for 776.76 on October 10, 2002. SP500dropped by 50 percent. Similarly FT100 reached a high of 6930.2 on December30, 2000 and the descent started from the time. FT100 dropped to 3287 onMarch 12, 2003 by 53 percent.

From these observations we divide the time series of these indices in thetwo periods on the day of the highest value. We define the period a perioduntil it reaches the highest value as the period of booms and the period afterthat as stagnations, respectively. The periods of booms and stagnations foreach index of the four indices are collected into Table 1.

Table 1. The periods of booms and stagnations.

Name of Index The Period of Booms The Period of Stagnations

Nikkei225 4 Jan 1975-30 Dec 1989 4 Jan. 1990-18 Aug 2004DJIA 2 Jan 1946-14 Jan 2000 18 Jan 2000-18 Aug 2004SP500 22 Nov 1982-24 Mar 2000 27 Mar 2000-18 Aug 2004FT100 3 Mar 1984-30 Dec 1999 4 Jan 2000-18 Aug 2004

2.2 Comparisons of the distributions of absolute log returns

In this paper we investigate the shape of distributions of absolute log returnsof the stock price indices. We concentrate to compare the shape of the distri-bution of volatility in the period of booms with that in the period of stagna-tions. We use absolute log return, which is a typical measure of volatility. Theabsolute log returns is defined as |R(t)| = |lnS(t) − lnS(t − 1)|, where S(t)denotes the index at date t. We normalize the absolute log-return |R(t)| usingthe standard deviation. The normalized absolute log return V (t) is defined as

6 Taisei Kaizoji

�

� ��

� � ��

��

��

� � ��

� � ��

� ��

� � � ��

��

��

� ��

��

� ��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

� ��

� ��

��

��

��

� � ��

��

��

!��

!��

! � ��

��

��

"��

"��

" � ��

��

#

$�#%#

&�#%#

'�#%#

(�#%#

) #*#%#

)*$ #%#

) &*#%#

)*' #%#

)*( #%#

+,- ..- ,,

+/- ..- ,,

+0- ..- ,,

++- ..- ,,

12- ..- ,,

1,- ..- ,,

1/- ..- ,,

10- ..- ,,

1+- ..- ,,

22- ..- ,,

2,- ..- ,,

3

4 3 3 3

5 3 3 3

6 3 3 3

7 3 3 3

8 3 3 3

9 3 3 3

: 3 3 3

; 3 3 3

<=> ?> @

<A> ?> @

<<> ?> @

BC> ?> @

BD> ?> @

B=> ?> @

BA> ?> @

B<> ?> @

CC> ?> @

C D> ?> @

C => ?> @

(a) Nikkei 225 (b) DJIA

(c) SP500 (c) FT100

Fig. 1. The movements of the stock price indices: (a) Nikkei 225 (b) DJIA, (c)SP500, (d) FT100.


V (t) = |R(t)|/σ where σ denotes the standard deviation of |R(t)|. The panels(a)-(h) of Figure 2 show the semi-log plots of the complementary cumulativedistribution of the normalized absolute log-returns V for each of the four in-dices: Nikkei225, DJIA, SP500, and FT100 in the period of booms and thatin the period of stagnations, respectively. In the all panels it follows that thetail of the volatility distribution of V is heavier in the period of booms thanin the period of stagnations. The solid lines in all panels represent the fits ofthe exponential distribution,

P (V > x) ∝ exp(−xβ

) (1)

where the scale parameter β is estimated from the data using a leastsquared method. In all cases of the period of stagnations, which are panels(b), (d), (f) and (h), the exponential distribution (1) describes very well thedistributions of V over a whole range of values of V2. The scale parameter βis estimated from the data except for these two extreme values using a leastsquared method is collected in Table 2. In all cases the estimated values β arevery close to unity.

Table 2. The scale parameter β of an exponential function (1) estimated from thedata using the least squared method. R2 denotes the coefficient of determinant.

Name of Index The scale parameter β R2

Nikkei225 1.02 0.995DJIA 1.09 0.995SP500 0.99 0.997FT100 0.99 0.999

On the other hand, the panels (a), (c), (e), and (g) of Figure 2 show thecomplementary cumulative distribution of V in the period of booms for eachof the four indices in the semi-log plots. The solid lines in all panels representthe fits of the exponential distribution estimated from the data of only thelow values of V using a least squared method. In these cases the low valuesof V are only approximately well described by the exponential distribution(1), but completely fails in describing the large values of V . Apparently, anexponential distribution underestimates large values in the complementarycumulative distribution of V in the period of booms.

Finally the panels (a) and (b) of Figure 3 show the complementary cumu-lative distributions of V for the four indices in the period of booms in a log-log

2 Note that we exclude the tow extreme values of V: (i) the extreme value in theNikkei 225 on September 28, 1990, and (ii) the extreme value in the DJIA onSeptember 10, 2001. The jump of Nikkei 225 perhaps was caused by investorsspeculation on the 1990 Gulf War. The extreme value of DJIA was caused byterror attack in New York on September 10, 2001.

8 Taisei Kaizoji

� � � � � ��

� � � � ��

� � � ��

� � ��

� � �

�

� � � � � � � � � � ��

� � �

� �

�

� � � � �

(a) Booms in Nikkei225 (b) Stagnations in Nikkei225

(c) Booms in DJIA (d) Stagnations in DJIA

( )P V x>

( )P V x> ( )P V x>

x

� � � � � �

� � � � �

� � � �

� � �

�

� � � � � � � ��

� � � � �

� � � �

� � �

�

� � � � � � �x

( )P V x>

x

� � � � � �

� � � � �

� � � �

� � �

�

� ! " # � �

(e) Booms in SP500 (f) Stagnations in SP500

( )P V x> ( )P V x>

x x$ % $ $ $ &

$ % $ $ &

$ % $ &

$ % &

&

$ '(& $)& '+* $,* '.- $,- '

/ 0 / / / 1

/ 0 / / 1

/ 0 / 1

/ 0 1

1

/ 2 3 4 5 1 /

(g) Booms in FT100 (h) Stagnations in FT100

( )P V x> ( )P V x>

x x6 7 6 6 6 8

6 7 6 6 8

6 7 6 8

6 7 8

8

6 9 8 6 8 9 : 6 : 9

x

Fig. 2. The panels (a), (c), (e) and (g) indicate the complementary cumulativedistribution of absolute log returns V for each of the four stock price indices in theperiod of booms, and the panels (b), (d), (f) and (h) indicate that in the period ofstagnations. These figures are shown in a semi-log scale. The solid lines representfits of the exponential distribution.


scale, and those in the period of stagnations in a semi-log scale. The two fig-ures confirm that the shape of the fourth volatility distributions in the periodsof booms and of stagnations is almost the same, respectively. Furthermore, thecomplementary cumulative distribution of V in the period of booms for eachof the four indices are approximately described by the power-law distributionin the large values of V ,

P (V > x) ∝ x−α (2)

The power-law exponent α is estimated from the data of the large values of Vusing the least squared method. The best fits succeed in describing approxi-mately large values of V. Table 3 collect the power-law exponent α, which isestimated. The estimated value α are in the range from 2.83 to 3.69.

Table 3. The power-law exponent α of a power-law function (2) estimated from thedata using the least squared method. R2 denotes the coefficient of determinant.

Name of Index The power-law parameter α R2

Nikkei225 2.83 0.992DJIA 3.69 0.995SP500 3.26 0.986FT100 3.16 0.986

3 Concluding remarks

In this paper we focus on comparisons of shape of the distributions of absolutelog returns in the period of booms with those in the period of stagnations forthe four major stock price indices. We find that the complementary cumula-tive distribution in the period of booms is very well described by exponentialdistribution with the scale parameter close to unity while the complemen-tary cumulative distribution of the absolute log returns is approximated bypower-law distribution with the exponent in the range of 2.8 to 3.8. The lat-ter is complete agreement with numerous evidences to show that the tail ofthe distribution of returns and volatility for large values of volatility followapproximately a power law with the estimates of the exponent falling in therange 2 to 4. We are now able to see that the statistical properties of volatilityfor stock price index are changed according to situations of the stock markets.Our findings make it clear that we must look more carefully into the relation-ship between regimes of markets and volatility in order to fully understandprice fluctuations in financial markets. The question, which we must considernext, is the reasons why and how the differences are created. That tradersherd behavior may help account for it would be accepted by most people.Recently we have proposed a stochastic model [19] that may offer the key

10 Taisei Kaizoji

x

x

��

��

��

��

�

� � � ��

��

( )P V x>

(b) Stagnations

��

��

��

��

��

�

�� !�

"�# $%$'&!#(�)�* +,�-.�/

(a) Booms

( )P V x>

Fig. 3. The panels (a) and (b) show the complementary cumulative distributionsof V for the four indices in the period of booms in a log-log scale, and those in theperiod of stagnations in a semi-log scale.


to an understanding of the empirical findings we present here. The resultsof the numerical simulation of the model suggest the following: in the periodof booms, the noise traders’ herd behavior strongly influences to the stockmarket and generate power-law tails of the volatility distribution while in theperiod of stagnations a large number of noise traders leave a stock market andinterplay with the noise traders become weak, so that exponential tails of thevolatility distribution is observed. However it remains an unsettled questionwhat causes switching from boom to stagnation3. Our findings may provide astarting point to make a new tool of risk management of index fund in finan-cial markets, but to apply the rule which we show here to risk management, weneed to establish the framework of analysis and refine the statistical methods.We began with a simple observation on the stock price indices, and dividedthe price series into the two periods: booms and stagnations. However, thereis room for further investigation on how to split the price series into periodsaccording to the situations of markets.

Acknowledgements

My special thanks are due to Prof. Thomas Lux and Prof. Enrico Scalasfor valuable comments, and the organizeer of Econophysics-Kolkata II, Prof.Bikas K Chakrabarti.

An earlier version of this paper was presented at the 8th Annual Workshopon Economics with Heterogeneous Interacting Agents (WEHIA2003) held atInstitute in World Economy, Kiel, Germany, May 29-31, 2003. Financial sup-port by the Japan Society for the Promotion of Science under the Grant-in-Aidis gratefully acknowledged.

References

1. Mandelbrot B (1963) Journal of Business 36:392-4172. Fama EF (1965) Journal of Business 38:34-1053. de Vries CG (1994) in The Handbook of International Macroeconomics, F. van

der Ploeg (ed.) pp.348-389 Blackwell4. Pagan A (1996) Journal of Empirical Finance 3:15-1025. Longin FM (1996) Journal of Business 96:383-408

3 Yang, et.al. [20] also studies the log-return the dynamics of the log-return dis-tribution of the Korean Composition Stock Price Index (KOSPI) from 1992 to2004. As a result of the empirical study using intraday data of the index, theyfound that while the index during the late 1990s showed a power-law distribu-tion, the distribution in the early 2000s was exponential. To explain this changein distribution shape, they propose a spin like model of financial markets. Theyshow that changing the shape of the return distribution was caused by changingthe transmission speeds of the information that flowed into the market.

12 Taisei Kaizoji

6. Lux T (1996) Applied Financial Economics 6:463-4757. Guillaume DM, Dacorogna MM, Dave RR, Muller UA, Olsen RB, Pictet OV

(1997) Finance and Stochastics 1:95-1308. Muller UA, Dacarogna MM, Picktet OV (1998) in R.J.Adler, R.E.Feldman,

M.S.Taqqu, Birkhauser, Boston (eds.), A Practical Guide to Heavy Tails, pp.55-78

9. Gopikrishnan P, Meyer M, Amaral LAN, Stanley HE (1998) European PhysicalJournal B 3:139-140

10. Gopikrishnan P, Plerou V, Amaral LAN, Meyer M, Stanley HE (1999) Phys.Rev. E 60:5305-5316

11. Plerou V, Gopikrishnan P, Amaral LAN, Meyer M, Stanley HE (2000) Phys.Rev. E 60:6519-6529

12. Liu Y, Gopikrishnan P, Cizeau P, Meyer M, Peng C-K, Stanley HE (1990)Physical Review E 60(2):1390-1400

13. Barndorff-Nielsen OE (1997) Scandinavian Journal of Statistics 24:1-1314. Eberlein E, Keller U, Prause K (1998) Journal of Business 71:371-40515. Laherrere J, Sornette D (1999) European Physical Journal B 2:525-53916. Dragulescu AA, Yakovenko VM (2002) Quantitative Finance 2(6):443-45317. Malevergne Y, Pisarenko VF, Sornette D (2005) Quantitative Finance 5(4):379-

40118. Kaizoji T (2004), Physica A343:662-668.19. Kaizoji T (2005) in T. Lux, S. Reitz, and E. Samanidou (eds.): Nonlinear

Dynamics and Heterogeneous Intercting Agents: Lecture Notes in Economicsand Mathematical Systems 550, Springer-Verlag, Berlin- Heidelberg, pp.237–248

20. Yang J-S, Chae S, Jung W-S, Moon H-T (2006) Physica A 363:377-382

An Outlook on Correlations in Stock Prices

Anirban Chakraborti

Department of Physics, Banaras Hindu University, Varanasi-221005, [email protected]

Summary. We present an outlook of the studies on correlations in the price time-series of stocks, discussing the construction and applications of ”asset tree”. Thetopic discussed here should illustrate how the complex economic system (financialmarket) enrichens the list of existing dynamical systems that physicists have beenstudying for long.

“If stock market experts were so expert, they would be buying stock,not selling advice.”– Norman Augustine, US aircraft businessman (1935 - )

1 Introduction

The word “correlation” is defined as “a relation existing between phenom-ena or things or between mathematical or statistical variables which tend tovary, be associated, or occur together in a way not expected on the basis ofchance alone” (see http://www.m-w.com/dictionary/correlations). As soon aswe talk about “chance”, the words “probability”,“random”, etc come to ourmind. So, when we talk about correlations in stock prices, what we are reallyinterested in are the nature of the time series of stock prices, the relation ofstock prices with other variables like stock transaction volumes, the statis-tical distributions and laws which govern the price time series, in particularwhether the time series is random or not. The first formal efforts in this di-rection were those of Louis Bachelier, more than a century ago [1]. Eversince,financial time series analysis is of prevalent interest to theoreticians for mak-ing inferences and predictions though it is primarily an empirical discipline.The uncertainty in the financial time series and its theory makes it speciallyinteresting to statistical physicists, besides financial economists [2, 3]. One ofthe most debatable issues in financial economics is whether the market is“efficient” or not. The “efficient” asset market is one in which the informa-tion contained in past prices is instantly, fully and continually reflected in

14 Anirban Chakraborti

the asset’s current price. As a consequence, the more efficient the market is,the more random is the sequence of price changes generated by the market.Hence, the most efficient market is one in which the price changes are com-pletely random and unpredictable. This leads to another relevant or pertinentquestion of financial econometrics: whether asset prices are predictable. Twosimplest models of probability theory and financial econometrics that dealwith predicting future price changes, the random walk theory and Martin-gale theory, assume that the future price changes are functions of only thepast price changes. Now, in Economics the “logarithmic returns” is calculatedusing the formula

r(t) = lnP (t) − lnP (t− 1), (1)

where P (t) is the price (index) at time step t. A main characteristic of therandom walk and Martingale models is that the returns are uncorrelated.

In the past, several hypotheses have been proposed to model financial timeseries and studies have been conducted to explain their most characteristicfeatures. The study of long-time correlations in the financial time series is avery interesting and widely studied problem, especially since they give a deepinsight about the underlying processes that generate the time series [4]. Thecomplex nature of financial time series (see Fig. 1) has especially forced thephysicists to add this system to their existing list of dynamical systems thatthey study. Here, we will not try to review all the studies, but instead givea brief outlook of the studies done by the author and his collaborators, andthe motivated readers are kindly asked to refer the original papers for furtherdetails.

2 Analysing correlations in stock price time series

2.1 Financial correlation matrix and constructing asset trees

In our studies, we used two different sets of financial data for different pur-poses. The first set from the Standard & Poor’s 500 index (S&P500) of the NewYork Stock Exchange (NYSE) from July 2, 1962 to December 31, 1997 con-taining 8939 daily closing values, which we have already plotted in Fig. 1(d).In the second set, we study the split-adjusted daily closure prices for a totalof N = 477 stocks traded at the New York Stock Exchange (NYSE) over theperiod of 20 years, from 02-Jan-1980 to 31-Dec-1999. This amounts a totalof 5056 price quotes per stock, indexed by time variable τ = 1, 2, . . . , 5056.For analysis and smoothing purposes, the data is divided time-wise into Mwindows t = 1, 2, ..., M of width T , where T corresponds to the number ofdaily returns included in the window. Several consecutive windows overlapwith each other, the extent of which is dictated by the window step lengthparameter δT , which describes the displacement of the window and is alsomeasured in trading days. The choice of window width is a trade-off between

An Outlook on Correlations in Stock Prices 15

too noisy and too smoothed data for small and large window widths, re-spectively. The results presented in this paper were calculated from monthlystepped four-year windows, i.e. δT = 250/12 ≈ 21 days and T = 1000 days.We have explored a large scale of different values for both parameters, and thecited values were found optimal [5]. With these choices, the overall number ofwindows is M = 195.

In order to investigate correlations between stocks we first denote theclosure price of stock i at time τ by Pi(τ) (Note that τ refers to a date, nota time window). We focus our attention to the logarithmic return of stock i,given by ri(τ) = lnPi(τ) − lnPi(τ − 1) which for a sequence of consecutivetrading days, i.e. those encompassing the given window t, form the returnvector r

ti. In order to characterize the synchronous time evolution of assets,

0 2000 4000 6000 8000 10000time

-0.3-0.2-0.1

00.1

0 1000 2000 3000-0.25

0

0.25

0 1000 2000 3000

-1

0

1

0 500 1000 1500 2000 2500 3000-4

-2

02

4 (a)

(b)

(c)

(d)

Fig. 1. Comparison of several time series which are of interest to physicists andeconomists: (a) Random time series (3000 time steps) using random numbers froma Normal distribution with zero mean and unit standard deviation. (b) Multivariatespatio-temporal time series (3000 time steps) drawn from the class of diffusivelycoupled map lattices in one-dimension with sites i = 1, 2...n′ of the form: yi

t+1 =(1− ε)f(yi

t) + ε2(f(yi+1

t ) + f(yi−1t )), where f(y) = 1− ay2 is the logistic map whose

dynamics is controlled by the parameter a, and the parameter ε is a measure ofcoupling between nearest-neighbor lattice sites. We use parameters a = 1.97, ε = 0.4for the dynamics to be in the regime of spatio-temporal chaos. We choose n = 500and iterate, starting from random initial conditions, for p = 5×107 time steps, afterdiscarding 105 transient iterates. Also, we choose periodic boundary conditions,x(n + 1) = x(1). (c) Multiplicative stochastic process GARCH(1,1) for a randomvariable xt with zero mean and variance σ2

t , characterized by a Gaussian conditionalprobability distribution function ft(x): σ2

t = α0+α1x2t−1+β1σ

2t−1, using parameters

α0 = 0.00023, α1 = 0.09 and β1 = 0.01 (3000 time steps). (d) Empirical Returntime series of the S&P500 stock index (8938 time steps).

new economic windows - download.e-bookshelf.de€¦ · new economic windows series editor massimo...

Documents