FRACTAl MARKET ANALYSIS Applying Chaos Theory to Investment and Economics Edgar E. Peters IOHN WILEY & SONS, INe. New York. Chichester· Brisbant: • Toronto · Singapore l Copyri,ht C 1994 by John Wiky &. Soni, I~, All ri&hti reserved. Published .imullaneo'II,ly in Clnlda, ReprodUClion Or tr.n51alion of ~ny put oIthi~ wOfk beyond th,. permiued by Seclion 107 Of 108 of I~ 1976 Uniled Stales Copyri,hl ACI withOlllthe permission of the copyri,bt owpCr ii unlawful. Requesls for permission or fUJI~r information lihould be Iddrencd 10 lhe PermissioM DcJnflment. John Wiley &: Som, Inc., 605 Third Aw: .... c, New York, NY 101511-0012. Thili publication is dui,ned 10 pl"0¥idc: ICCU,ale and aUlhorit"ivoe informalion in .eprd 10 the subject malter covercd. It is ~d with the unlk.uandinglhal Ihe publisher is not en,lted in rcooeringie,al, accOUnling, I)T other professional ..,rvi«J. If le&ll advice or other upert assistance is required, the services of a competent prof~sional pcr:oon should be tough!. from" Dtcloro,;OtI of PrltfCiplts join/ly tuicpltd by" C_iIfH Df ,ltt Amtril'On BD, A.uoci,,"''''' "Nl" CommifUt of PubiisltcrJ. , Libror, of Conlms C"talogilfl-itl-P~tJlicatilHl I)Qta: Ptle,s, &1,1' E., 19S2- Fractll market lnalysis: applying ch:aa.: IheOfY to invutmentlnd ccOflQmics I &I,ar E. Pt:leu. p. cm. Inc:ludu indu, ISBN 0-471·58S24-6 I. Inw:stmcnu-Mllhematic:s. 2. Frlol;l.ls. ). Chaotic behavior in s)'litemli, I. Title. 11. Tille: Chaos theory, HG4S1S.).P47 1994 ))2.6'01 'S 14 74....-«20 9)·2IIS98 Printed in the: United Statu of Americl 10 9 11 7 6 S 4 ) 2 • To SI/er)'/ L , l Preface In 1991. I finished writing a book entitled. Chaos and Order in the Capital Markets. It was published in the Fall of,that year (Peters, 1991 a). My goal was to write a conceptual inlroduction, for the investment community, 10 chaos the ory and fraclal statistics. I also wanted to present some preliminary evidence thai, contrary 10 accepted theory. markets are nOI well-described by the ran dom walk model. and the widely laught Efficient Market Hypothesis (EMH) is nol well-supported by empirical evidence. I have received, in general. a very positive response [0 that book. Many readers have communicated their approval-and some. their disapproval- and have asked detailed questions. The questions fell into two categories: (I) tech nical. and (2) conceptual. In the technical category were the requests for more detail about the analysis. My book had not been intended to be a textbook, and I had glossed over many technical details involved in the analysis. This ap proach improved the readability of the book, but it left many readers wonder ing how to proceed. In the second category were questions concerned with conceptual issues. If the EMH is flawed, how can we fix it? Or better still, what is a viable replace ment? How do chaos theory and fractals fit in with trading strategies and with the dichotomy between technical and fundamental analysis? Can these seem ingly disparate theories be united? Can traditional theory become nonhnear? In this book, I am addressing both categories of questions. This book is differ ent from the previous one, but it reflects many similar features. Fraclal Market Analysis is an attempt 10 generalize Capital Market Theory (CMT) and 10 ac count for the diversity of the investment community. One of the failings oflradi lional theory is its attempt to simplify" the market" into an average prototypical vii L --g Cl =n (-Ill oI 3 .., • P,~face Part Four: Fractal Noise Having used RIS analysis IQ find evidence 10 support the Fractal Market Hy pothesis, I supply models to explain those findings. Part Four approaches market act ivity from the viewpoint of slochastic processes; as such, it concentrates on fractal noise. [n Chapter 13, using RIS analysis, different "colored" noises are analyzed and compared 10 the market analysis. The findings are remarkably similar. In addition. the behavior of vola! ility is given a significant explanat ion. Chapler 14 discusses the statistics of fraclal noise processes, and offers them as an alternative 10 the traditional Gaussian normal distribution. The impact of fraclal distributions on market models is discussed. Chapler 15 shows the im pact of fractal statistics on the portfolio selection problem and option pricing. Methods for adapting those models for fractal distributions are re ... ie-.....ed. Part Four is a ...e ry detaited section and will not be appropriate for all readers. Howe ...e r, because the application oflraditional CMT has become ingrained into most of the in ...e stment community, I belie ...e that most readers should read the summary seclions of each chapter, if nothing else, in Part Four. Chapter 13, with its sludy of the nature of volalility, should be of particular interes!. Part Five: Noisy Chaos Part Fi ...e offers a dynamical systems alternative 10 the slochaslic processes of Part Four. In particular, it offers noisy chaos as a possible e)(planation of the frac lal structure of markets. Chapter 16, which gives RIS analysis of chaotic sys tems. re. .. eals remarkable similarities with market and other time series. A particular emphasiS is placed on distinguishing between fractal noise and noisy chaos. A review is given of the BOS (Brock-Dechert-Scheinkman) test, which, when used in conjunction wilh RIS analysis, can give conclusive evidence Qne way or the other. Chapter 17 applies fractal statistics 10 noisy chaos, reconciling the two approaches. An explanation is offered for why evidence of both fractal noise and noisy chaos can appear simultaneously. The result is closely tied to the Fraclal Market Hypothesis and the theory of multiple investment horizons. Chapter 18 is a review of the findings on a conceptual level. This final chapler unites the Fractal Market Hypothesis with the empirical work and theoretical models presented tMoughout the book. For readers who under stand a problem better when they know the solution, it may be appropriate 10 read Chapter 18 first. The appendices offer software that can be used for analYSis and reproduce tables of the fractal distributions. J. ,; While reading the book. many of you will wonder, where is Ihis leading? Will this htdp me make money? This book does 001 offer new trading tech niques or find pockets of inefficiency that the savvy inveSlor can profit from. It is not a book of strategy for making beller predictions. Instead. il offers a new view of how markets work and how 10 leSI lime series for predictability. More importantly. it gives additional information about the risks investors lake, and how those risks change over lime. If knowledge is power. as the old cliche goes, then the information here should be conduci\'c. if nol 10 polNer, at least 10 beller profits. EOGAR E. PETERS L ,