MAR KET ANALYSISCharles

Chaos Theory to InvestmentANALYZING

EconomicsCHAOS

E.

E. PetersAnthony

WILEY &

INC.GENENTIC

ChichesterBrisbaneTorontoSingaporeThis text is printed on acid-free paper.Copyright 1994 by John Wiley & Sons,Inc.All rights reserved. Publishedsimultaneously in Canada.Reproduction or translation of any partof this work beyondthat permitted by Section 10701108 of the 1976 UnitedStates Copyright Act without thepermission of the copyrigbtowner is unlawful. Requestsfor permission or furtherinformation should be addressed to thePermissions Department,John Wiley & Sons, Inc., 605 ThirdAvenue, New York, NY10158-0012.This publication is designed to provide accurateandauthoritative information in regard to thesubjectmatter covered. It is sold withthe understanding thatthe publisher is not engaged inrendering legal, accounting,or other professional services.If legal advice or otherexpert assistance is required, theservices of a competentprofessional person should be sought. From aDeclarationof

Edgar E., 1952Fractal market analysisapplying chaos theory to investment andeconomics / Edgar E. Peters.p.cm.Includes index.IS8N 0-471-58524-6I. InvestmentsMathematics.2. Fractals.3. Chaotic behavior insystems.I. Title.II. Title: Chaos theory.HG45I5.3.P471994332.6015 l474dc2O93-28598Printed in the United States of America10 9 8 7 6 5 4 3 2

PrefaceIn

and Order in the CapitalMarkets. It

viiiPrefacePrefaceixrational investor. The reasons for setting out on this route were noble. In the tradi-tion of Western science, the founding fathers of CMT attempted to learn some-thing about the whole by breaking down the problem into its basic components.That attempt was successful. Because of the farsighted work of Markowitz,Sharpe, Fama, and others, we have made enormous progress over the past 40 years.However, the reductionist approach has its limits, and we have reached them.It is time to take a more holistic view of how markets operate. In particular, it istime to recognize the great diversity that underlies markets. All investors do notparticipate for the same reason, nor do they work their strategies over the sameinvestment horizons. The stability of markets is inevitably tied to the diversityof the investors. A mature" market is diverse as well as old. If all the partici-pants had the same investment horizon, reacted equally to the same information,and invested for the same purpose, instability would reign. Instead, over the longterm, mature markeis have remarkable stability. A day trader can trade anony-mously with a pension fund: the former trades frequently for short-term gains;the latter trades infrequently for long-term financial security. The day trader re-acts to technical trends; the pension fund invests based on long-term economicgrowth potential. Yet, each participates simultaneously and each diversifies theother. The reductionist approach, with its rational investor, cannot handle thisdiversity without complicated multipart models that resemble a Rube Goldbergcontraption. These models, with their multiple limiting assumptions and restric-tive requirements, inevitably fail. They are so complex that they lack flexibility,and flexibility is crucial to any dynamic system.The first purpose of this book is to introduce the Fractal Market Hypothesisa basic reformulation of how, and why, markets function. The second purpose ofthe book is to present tools for analyzing markets within the fractal framework.Many existing tools can be used for this purpose. I will present new tools to add tothe analyst's toolbox, and will review existing ones.This book is not a narrative, although its primary emphasis is still concep-tual. Within the conceptual framework, there is a rigorous coverage of analyti-cal techniques. As in my previous book, I believe that anyone with a firmgrounding in business statistics will find much that is useful here. The primaryemphasis is not on dynamics, but on empirical statistics, that is, on analyzingtime series to identify what we are dealing with.THE STRUCTURE OF THE BOOKThe book is divided into five parts, plus appendices. The final appendix con-tains fractal distribution tables. Other relevant tables, and figures coordinatedto the discussion, are interspersed inthe text. Each part builds on the previousparts, but the book can be readnonsequentially by those familiar with the con-cepts of the first book.Part One: Fractal Time SeriesChapter 1 introduces fractal time series and definesboth spatial and temporalfractals. There is a particular emphasis on whatfractals are, conceptually andphysically. Why do they seem counterintuitive, even thoughfractal geometry ismuch closer to the real world than the Euclidean geometry weall learned inhigh school? Chapter 2 is a brief review of CapitalMarket Theory (CMT) andof the evidence of problems with the theory. Chapter3 is, in many ways, theheart of the book: I detail the Fractal MarketHypothesis as an alternative tothe traditional theory discussed in Chapter 2.As a Fractal

Hypothesis,it combines elements of fractals from Chapter 1with parts of traditional CMTin Chapter 2. The Fractal Market Hypothesis setsthe conceptual frameworkfor fractal market analysis.Part Two: Fractal (R/S) AnalysisHaving defined the problem in Part One, I offertools for analysis in PartTwoin particular, rescaled range (RIS)

Many of the technicalquestions I received about the first book dealt withR/S analysis and re-quested details about calculations and significance tests.Parts Two andThree address those issues. R/S analysis is a robustanalysis technique for un-covering long memory effects, fractal statistical structure,and the presenceof cycles. Chapter 4 surveys the conceptual backgroundof R/S analysis anddetails how to apply it. Chapter 5 gives both statistical testsfor judging thesignificance of the results and examples of how R/S analysis reacts toknownstochastic models. Chapter 6 shows how R/S analysis can beused to uncoverboth periodic and nonperiodic cycles.Part Three: Applying Fractal AnalysisThrough a number of case studies, Part Three details howR/S

tech-niques can be used. The studies, interesting in their ownright, have been se-lected to illustrate the advantages and disadvantages ofusing RIS analysis ondifferent types of time series and different markets. Along the way,interestingthings will be revealed about tick data, market volatility, andhow currencies aredifferent from other markets.PrefacePrefacePart Four: Fractal NoiseHaving

Concord, Massachusetts

Part Five: Noisy ChaosPart

LAcknowledgmentsI

LContentsPART ONE FRACTAL TIME SERIESIntroduction to Fractal Time Series3Fractal

of the Gaussian Hypothesis18Capital

A Fractal Market Hypothesis39Efficient

xviContentsContentsXviiPART TWO FRACTAL (R/S) ANALYSIS10Volatility: A Study in Antipersistence143Realized

Measuring MemoryThe Hurst Process and R/S Analysis53Implied

Problems with Undersampling: Gold and U.K. Inflation151Randomness

R/S Analysis65Summary,

A True Hurst Process159Stochastic

Finding Cycles: Periodic and Nonperiodic86Pound/Dollar,

FOUR FRACTAL NOISEPART THREE APPLYING FRACTAL ANALYSIS13Fractional Noise and R/S Analysis1697Case Study Methodology107The Color

H

1.0,

Jones Industrials, 18881990: An Ideal Data Set112Summary, 196Number

Statistics197

S&P 500 Tick Data, 19891992: Problems withOversampling13215Applying Fractal Statistics217The

xviiiContentsPART FIVENoisy Chaos16Noisy Chaos and R/S Analysis235Information

TIME SERIES17

Statistics, Noisy Chaos, and the FMH252Frequency

Markets271

1:

Chaos Game277

2: GAUSS Programs279Appendix 3: Fractal Distribution Tables287Bibliography296Glossary306Index313L1 Introduction to FractalTime SeriesWestern

4Introduction to Fractal Time SeriesFractal Time5FRACTAL SPACEFractal

TIMEThis

6Great

Introduction to Fractal Time SeriesFractal Time78Introduction to Fractal Time Seriesfractal Mathematics-

MATHEMATICSAll

10Introduction to Fractal Time SeriesThe Chaos Game11

CHAOS GAMEThe

and Order in theCapitalMarkets (1991a),

Game.To

times.

FIGURE 1.1The Chaos Game. (a) Start with three points, anequal distance apart,and randomly draw a point within theboundaries defined by the points. (b) Assum-ing you roll a fair die that comes upnumber 6, you go halfway to the pointmarlcedC(5,6). (c) Repeat step (b) 10,000 timesand you have the Sierpinski triangle.and

structure. Appendix

for creating the Sierpinski

You are encouraged to try thisyourself.The Chaos Game shows us that localrandomness and global determinism cancoexist to create a stable, self-similar structure,which we have called a fractal.Prediction of

sequence of points is impossible.Yet, the

of plot-

point are not equal. The empty spaceswithin each triangle have a zeropercent probability

The edges outlining

triangle have ahigher

local randomness does not equate withequal probability

all possible solutions. It also does not equatewith indepen-dence.

the

point

dependent on

current

B(3,4)C

12which

IS A FRACTAL?We

to the Galaxy by

diameter

2-1/3.

ln(q)

Introduction to Fractal Time SeriesWhat Is a Fractal?1314Introduction to Fractal Time SeriesThe Fractal Dimension15BFIGURE 1.3Log/Log plot.FIGURE 1.2The lung with exponential scaling. (From West and Goldberger(1987); reproduced with permission from American Scientist.)scaling

\J)Cl-THE FRACTAL DIMENSIONTo

0 a,a,E (00io-2-J

0.20.40.6O.$1.021.4L6i.e2.02.22.4262.63.032LOG GENERATION (LOG 2)16Introduction to Fractal Time Series

the

radius

the

Fractal Market Analysis17

d

2,

MARKET ANALYSISThis

Capital Market Theory192 Failure of the GaussianHypothesisWhen

L

MARKET THEORYTraditional

20Failure of the Gaussian HypothesisIt has long been conventional to view security prices and their associatedreturns from the perspective of the speculatorthe ability of anindividual toprofit on a security by anticipating its future value before other speculatorsdo. Thus, a speculator bets that the current price of a security is above/belowits future value and sells/buys it accordingly at the current price. Speculationinvolves betting, which makes investing a form of gambling. (Indeed, probabil-ity was developed as a direct result of the development of gamblingusing"bones," an early form of dice.) Bachelier's "Theory of Speculation" (1900)does just that. Lord Keynes continued this view by his famous comment thatmarkets are driven by "animal spirits." More recently, Nobel Laureate HarryMarkowitz (1952, 1959) used wheels of chance to explain standard deviationto his audience. He did this in order to present his insight thatstandard devia-tion is a measure of risk, and the covariance of returns could be used to explainhow diversification (grouping uncorrelated or negatively correlated stocks) re-duced risk (the standard deviation of the portfolio).Equating investment with speculation continued with the BlackScholes op-tion pricing model, and other equilibrium-based theories. Theories of specula-tion, including Modern Portfolio Theory (MPT), did not differentiate betweenshort-term speculators and long-term investors. Why?Markets were assumed to be "efficient"; that is, prices already reflected allcurrent information that could anticipate future events. Therefore,only thespeculative, stochastic component could be modeled; the change in prices dueto changes in value could not. If market returns are normallydistributed"white" noise, then they are the same at all investment horizons. This isalent to the "hiss" heard on a tape player. The sound is the same regardless ofthe speed of the tape.We are left with a theory that has assumed away the differentiating featuresof many investors trading over many investment horizons. The risks tothe same. Risk and return grow at a commiserative rate over time. There is noadvantage to being a long-term investor. In addition, price changes are deter-mined primarily by speculators. By implication, forecasting changes in eco-nomic value would not be useful to speculators.This uncoupling of changes in the value of the underlying security from theeconomy and the shifting of price changes mostly to speculators havereinforcedthe perception that investing and gambling are equivalent, no matter what theinvestment horizon. This stance is most clearly seen in the common practice ofactuaries to model the liabilities of pension funds by taking short-term returns(annual returns) and risk (the standard deviation of monthly returns), and ex-trapolating them out over 30-year horizons. It is also reflected in the tendency ofindividuals and the media to focus on short-term trends and values.Statistical Characteristics of Markets21If markets do not follow a random walk,it is possible that we may be over-or understating our risk and returnpotential from investing versus speculating.In the next section, we will examine thestatistical characteristics of marketsmore closely.STATISTICAL CHARACTERISTICS OF MARKETSIn general, statistical analysis requires thenormal distribution, or the familiarbell-shaped curve. It is well known that market returns arenot normally dis-tributed, but this information has been downplayed orrationalized away overthe years to maintain the crucial assumptionthat market returns follow a ran-dom walk.Figure 2.1 shows the frequency distributionof 5-day and 90-day Dow JonesIndustrials returns from January 2, 1888, throughDecember 31, 1991, some103 years. The normal distribution is alsoshown for comparison. Both returndistributions are characterized by a high peak at the meanand fatter tails thanthe normal distribution, and the two Dowdistributions are virtually the sameshape. The kink upward at four standarddeviations is the total greater than(less than) four (4) standard deviations above(below) the mean. Figure 2.2shows the total probability contained withinintervals of standard deviation for1412108 6 4 2 02345FIGURE 2.1Dow Jones Industrials, frequency distribution of returns:18881991.-5-4-3-2-101Standard DeviationsC.)I)

222Failure of the Gaussian HypothesisI Ii,Ii

5040

10 0-90-Day

5-Day

2.2Dow Jones Industrials, frequency within intervals.the two Dow investment

fatter.

-4-2024-3-113Standard

normal

3

5 4

2.4aDow Jones

normal

r4433U221.E1U00-11-2-2-5-4-3-2-1012345Standard

2.4bDow Jones Industrials, 10-day returns normal frequency.55443>'3U a.)u221.E

000

-1

-2-3-3-5-4-3-2-1012345Standard

2.4cDow Jones Industrials, 20-day returns normal frequency.-5-4-3-2-10

2.4dDow Jones Industrials, 30-day returns normal frequency.-5-4-3-2-1012345Standard

Dow Jones Industrials, 90-day returns normal frequency.26Failure of the Gaussian_Hypothesis

FIGURE 2.5Yen/Dollar exchangerate,frequency distributionof returns:19711990.IThe Term Structure of VolatilityFIGURE19791992.2715105 02.6Twenty-year U.S. T-Bond yields, frequency distribution of returns:

TERM STRUCTURE OF VOLATILITYAnother

-5-4-3-2-101234Standard

28Failure of the Gaussian Hypothesisfor

The

1,000

The Term Structure of Volatility-29Table 2.1Dow jones Industrials, Term Structure ofVolatility:

DaysStandardDeviationNumber ofDaysStandardDeviation

0.135876

2000.196948

5

2500.21 3792

260

0.5 0-0.5-1.500-2-2.5FIGURE 2.7Dow jones Industrials, volatility term structure: 18881990.Table 2.2Dow Jones Industrials, Regression Results,Term Structure of Volatility: 188819900123Log(Number

5N = 1,000 DaysRegression output:Constant1.96757

Standard errorof Y (estimated)R squared0.0268810.9960320.107980.61 2613Number ofobservations3010Degrees offreedomX coefficient(s)280.53471 380.347383Standard errorof coefficient0.0063780.09766630Failure of the Gaussian Hypothesis

2.3Dow

Number ofDaysSharpeRatioNumber ofDaysSharpeRatioI1.289591301.1341621.2176652000.83051341.2892892080.86430651.2063572500.88181.1901432600.978488101.1724283251.150581131.201 3724000.650904161.0861075000.838771201.1786975200.919799251.1634496501.173662

501.0618511,3002.437258521.0851091,6251.124315

The Term Structure of Volatility31

Despite

For

32Failure of the Gaussian_HypothesisThe Term Structure of Volatility33FIGURE 2.8Daily bond yields, volatility term structure: January 1, 1979

30, 1992.we

2.4Long T-Bonds, Term Structure of Volatility:January 1, 1978June 30, 1990N = 1,000 DaysRegression output:Constant4.08912.26015Standard errorof Y (estimated)0.0538740.085519R squared0.9850350.062858Number ofobservations213Degrees offreedom191X coefficient(s)0.5481 020.07547Standard errorolcoellicient0.0154990.29141-2-2.5g-3

-4515100.511.522.5Log(Number

53.5

45

DeviationsFIGURE 2.9aMark/Dollar, frequency distribution of returns.To

a bounded

this

we

in

2.7. It remains bctunded. Table 2.5

results

Therefore, either

have

interval than stocks, or they have no bounds. The latter wouldimplythat exchange rate risk grows at a faster rate than the normaldistribution but neverstops growing. Therefore, long-term holdersof currency face ever-increasing20

0-5-4-3-2

4Standard

FIGURE 2.9bPound/Dollar, frequency distribution of returns.I__I

2.lObPound/Dollar exchange rate, volatility term structure.II.

Pound/Dollar

Deviations

Yen/Pound, frequency distribution of returns.Traditional Scaling

-3

-4.5

0.5

2.lOaMark/Dollar exchange rate, volatility term structure.Traditional ScalingYen/Pound

-3-3.5

Scaling-4.50

of

2.lOcYen/Poundexchange rate, volatility term structure.3536The Bounded Set37of risk as their investment horizon widens.Unlike stocks and bonds, curren-NNcies offer no investment incentive to a buy-and-hold strategy.NIn the short term, stock, bond, and currencyspeculators face similar risks,d dbut in the long term, stock and bond investorsface reduced risk.THE BOUNDED SETThe appearance of bounds for stocks and bonds, but notfor currencies, seemspuzzling at first. Why should currencies be adifferent type of security thanstocks and bonds? That question contains its own answer.In mathematics, paradoxes occur when anassumption is inadvertently for-0N..0NN NNUIgotten. A common mistake is to divideby a variable that may take zero as avalue. In the above paragraph, the question called a currencya "security."QCurrencies are traded entities, but they are notsecurities. They have no in-vestment value. The only return one can getfrom a currency is by speculatingEon its value versus that of another currency.Currencies are, thus, equivalentto the purely speculative vehicles that arecommonly equated with stocks andbonds.Stocks and bonds are different. They do haveinvestment value. Bonds earninterest, and a stock's value is tied to the growthin its earnings through eco-nomic activity. The aggregate stock market istied to the aggregate economy.N.CICurrencies are not tied to the economic cycle. In the1950s and 1960s, we hadan expanding economy and a strongdollar. In the 1980s, we had an expandingeconomy and a falling dollar. Currenciesdo not have a "fundamental" valuethat is necessarily related to economic activity, though it may betied to eco-nomic variables like interest rates.r'lWhy are stocks and bonds bounded sets? Amean-reverting stochastic pro-cess is a possible explanation ofboundedness, but it does not explain the faster-growing standard deviation. Bounds and fast-growingstandard deviations areusually caused by deterministic systems with periodic ornonperiodic cycles.Figure 2.11 shows the term structure of volatility for asimple sine wave. Wecan clearly see the bounds of the systemand the faster-growing standard devi-ation. But we know that the stock and bond markets are notperiodic. Granger2-2(1964) and others have performed extensive spectralanalysis and have foundno evidence of periodic cycles.>-SHowever, Peters (1991b) and Cheng and Tong(1992) have found evidence00of nonperiodic cycles typically generated bynonlinear dynamical systems,OOUUIor "chaos."38Failure of (he Gaussian Hypothesis00.511.5Log(Time)

2.11Sine wave, volatility

structure.

point,

In

3 A FractalMarket HypothesisWe

New York Observer, "It's

MARKETS REVISITEDThe

C 00.50 -0.51-1.5-222.5L40A Fractal

HypothesisStable Markets versus

41

MARKETS VERSUS EFFICIENT MARKETSThe

to

42A Fractal Market HypothesisStatistical Characteristics of Markets,Revisited43

THE SOURCE OF LIQUIDITY

.000284

ercent,

L

SETS AND INVESTMENTHORIZONSIn

CHARACTERISTICS OF MARKETS, REVISITEDIn

44A Fractal Market Hypothesis

FRACTAL MARKET HYPOTHESISThe

The Fractal Market Hypothesis45Table 3.1Frequency distributions (%) of intraday returnsStandard198919901989199019891990Deviations60-Minute30-Minute5-Minute5-MinuteLess than4.000.40%0.3 7%0.52%0.47%3.800.050.110.080.083.600.000.050.110.083.400.050.150.150.093.200.100.120.120.153.000.070.160.170.132.800.100.270.18

2.600.250.130.230.232.400.500.300.350.282.200.690.410.480.352.000.790.460.510.411.800.890.660.650.581.600.870.940.760.671.401.461.180.890.781.201.611.751.210.991.002.702.271.341.620.803.053.212.272.160.604.614.303.603.850.406.497.196.717.150.208.459.1811.7513.770.0016.1115.2216.4419.580.2013.2815.1419.9216.260.409.529.5710.8010.600.607.788.376.286.040.805.635.253.653.001.004.614.082.522.131.203.022.481.861.421.401.811.631.251.431.601.161.391.001.181.800.990.860.820.952.000.820.730.670.652.200.570.580.500.472.400.550.360.430.452.600.350.270.260.322.80

0.28

3.20

0.150.243.400.070.070.130.153.600.050.080.120.143.800.050.010.060.08Greater than4.000.150.200.530.4746A Fractal Market Hypothesishas a longer

1963.

The Fractal Market Hypothesis47

48A Fractal Market_HypothesisSummary-then liquidity can also be affected. For instance, on April 1, 1993, Phillip Mor-ris announced price cuts on Marlboro cigarettes. This, of course, reduced thelong-term prospects for the company, and the stock was marked down accord-ingly. The stock opened at $48, 17V8 lower than its previous close of $55V8.However, before the stock opened, technical analysts on CNBC, the cable fi-nancial news network, said that the stock's next resistance level was 50. PhillipMorris closed at 49. It is possible that 49 was Phillip Morris' "fair" value,but it is just as likely that technicians stabilized the market this time.Even when the market has achieved a stable statistical structure, market dy-namics and motivations change as the investment horizon widens. The shorter theterm of the investment horizon, the more important technical factors, tradingactivity, and liquidity become. Investors follow trends and one another. Crowdbehavior can dominate. As the investment horizon grows, technical analysisgradually gives way to fundamental and economic factors. Prices, as a result, re-flect this relationship and rise and fall as earnings expectations rise and fall.Earnings expectations rise gradually over time. If the perception is a change ineconomic direction, earnings expectations can rapidly reverse. If the market hasno relationship with the economic cycle, or

that relationship is very weak, thentrading activity and liquidity continue their importance, even at long horizons.If the market is tied to economic growth over the long term, then risk willdecrease over time because the economic cycle dominates. The economic cycleis less volatile than trading activity, which makes long-term stock returns lessvolatile as well. This relationship would cause variance to become bounded.Economic capital markets, like stocks and bonds, have a short-term fractalstatistical structure superimposed over a long-term economic cycle, which maybe deterministic. Currencies, being a trading market only, have only the fractalstatistical structure.Finally, information itself would not have a uniform impact on prices; in-stead, information would be assimilated differently by the different investmenthorizons. A technical rally would only slowly become apparent or important toinvestors with long-term horizons. Likewise, economic factors would changeexpectations. As long-term investors change their valuation and begin trading,a technical trend appears and influences short-term investors. In the shortterm, price changes can be expected to be noisier because general agreementon fair price, and hence the acceptable band around fair price, is a larger com-ponent of total return. At longer investment horizons, there is more time to di-gest the information, and hence more consensus as to the proper price. As aresult, the longer the investment horizon, the smoother the time series.SUMMARYThe Fractal Market Hypothesis proposesthe following:1.The market is stable when it consists ofinvestors covering a large num-ber of investment horizons. This ensures thatthere is ample liquidity fortraders.2.The information set is more related tomarket sentiment and technicalfactors in the short term than in the longer term.As investment hori-zons increase, longer-termfundamental information dominates. Thus,price changes may reflect informationimportant only to that invest-ment horizon.3.If an event occurs that makes the validityof fundamental informationquestionable, long-term investors either stopparticipating in the marketor begin trading based on theshort-term information set. When the over-all investment horizon of the market shrinks to auniform level, the mar-ket becomes unstable. There are no long-terminvestors to stabilize themarket by offering liquidity to short-terminvestors.4.Prices reflect a combination of short-termtechnical trading and long-term fundamental valuation. Thus,short-term priceare likely tobe more volatile, or "noisier," thanlong-term trades. The underlyingtrend in the market is reflective of changes inexpected earnings, based onthe changing economic environment.Short-term trends are more likelythe result of crowd behavior. There is no reasonto believe that the lengthof the short-term trends is related to thelong-term economic trend.5.If a security has no tie to the economiccycle, then there will be no long-term trend. Trading, liquidity, andshort-term information will dominate.Unlike the EMH, the Fractal MarketHypothesis (FMH) says that informa-tion is valued according to the investmenthorizon of the investor. Because thedifferent investment horizons value informationdifferently, the diffusion of in-formation will also be uneven. At any onetime, prices may not reflect all avail-able information, but only the informationimportant to that investment horizon.The FMH owes much to the CoherentMarket Hypothesis (CMI-() of Vaga(1991) and the K-Z model of Larrain (1991).I discussed those models exten-sively in my previous book, Like the CMH,the FMH is based on the premisethat the market assumes different statesand can shift between stable and un-stable regimes. Like the K-Z model, theFMH finds that the chaotic regimeL50A Fractal Market Hypothesisoccurs

RAc1A I (R IS)tunately,

4 Measuring MemoryThe HurstProcess andR/S

Standard statistical analysis begins

that

process

created the

Limit Theorem (or the

54Measuring MemoryThe Hurst Process and R/S

RJS Analysis

the

a

and Order in

Capital Markets, I

DEVELOPMENT OF k/S ANALYSISH.

Hurst

to

the

1

of

(x1

.. +

56Measuring Memory_The Hurst Process and R/S Analysiswhich is

r

I

Zr))

2

Because

T,

c

a

IBackground: Development of R/SAnalysis57

H*log(n)

0.50.

0.91!

Background: Development of K/SAnalysis59*N +1 II+1NN.opdoL 000N.cO IJo odd'.0LI)0oddIf) NN. 0 '.00) II)0)0)N. LI)N.p000 0000 000 0

doN."dILI)0 0N dddN.N.NN'.0N.N.N.N.N. N. N.N.dddd ddd ddddda'.00) '.0 aIf)00LI)LI)'0NN.LII N N 0LI)'0N.0)-0)NN'-LI)CII LI) N0,-Na 0.aU000I000LIII I=CC10El31

Swnmc/ioi'Dev,c/*'a

0 0 LI)00 0 0 LI)0 0 0 r 058FIGURE 4.1Hurst (1951) RIS analysis.(Reproduced with permission of theAmer-ican Society of Civil Engineers.)60Measuring MemoryThe Hurst Process and R/S Analysis

JOKER EFFECTBefore

random

7,

0.72,

K/S Analysis: A Step-by-StepGuide61these

AND PERSISTENCE:INTERPRETINGTHE HURST EXPONENTAccording

0.50

analysis

H 1.00

0.50

ANALYSIS: A STEP-BY-STEPGUIDER/S

62Measuring Memory.The Hurst Process and R/S AnalysisAn Example: The YenJDoIIarExchange Rate

MI

i

I)

1,

1,

(1/n)*

average

e5)

min(Xka)

= ((1/n)*

value

= (l/A)*

(4.13)7.

1)/n

(M1)12.

EXAMPLE: THE YEN/DOLLAREXCHANGE RATEAs

(a

new

a,b = constants

Regression output, Daily yen:ConstantStandard error of Y (estimated)0.187R squaredo.oi 2Hurst exponent0.642Standard error of coefficientSignificance5.848

0.64.

1.55 Testing R/SAnalysisWe

64Measuring MemoryThe Hurst Process and R/S AnalysisTable 4.2R/S Analysis2:0.5 o0.511.522.533.54Log(Number

4.2R/S analysis, daily yen: January 1972 through December 1990.I-66Testing R/S AnalysisThis

null

THE RANDOM NULL HYPOTHESIS

hypothesis.

the

The Random Null Hypothesis67E(R(n))

lr/2)*n

5,000

independent

Carlo SimulationsThe

values

is

The

FIGURE 5.1

[f{o.5*(nI))

=r)I r

Testing R/S AnalysisThe Random Null Hypothesis

Table 5.1

(RIS) Value

70Testing R/S AnalysisThe Random Null Hypothesis71

20.

0.5 0FIGURE 5.2RIS values,

Carlo simulation versus

Lloyd's equation.Table 5.2Log (R/S) Value EstimatesNumber ofObservationsScrambledS&P 500Monte Carlo

200.64740.7123250.8891500.88121.0577100

with

0.5)

r)

Expected Value of the HurstExponentUsing

0.511.522.533,54Log(Number

72Testing R/S AnalysisThe Random Null Hypothesis73FIGURE 5.3R/S values, Monte Carlo simulation versus corrected Anis and Lloydequation.economics, we will begin with n = 10. The final

of n will depend on thesystem under study. in Peters (199 Ia), the monthly returns of the S&P 500were found to have persistent scaling for n U1'E(R/S)1.41.31.2

0.6

Observations)3FIGURE 8.2V statistic, Dow Jones Industrials: 20-day returns.Table 8.2RegressionResults: DowJonesIndustrials,20-Day ReturnsDow JonesDow JonesIndustrials,E(R/S)Industrials,10 0, there also exists b > 0 suchthat:(14.1)This relationship exists for all distributionfunctions. F(x) is a general char-acteristic of the class of stable distributions,rather than a property of any onedistribution.The characteristic functions of F canbe expressed in a similar manner:(14.2)Therefore, f(b1*t), f(b2*t), and f(b*t) all havethe same shaped distribution,despite their being products of one another.This accounts for their "stability."The actual representation of thestable distributions is typically done inthemanner of Mandeibrot (1964), usingthe log of their characteristic functions:(14.3)The stable distributions have four parameters: a,c, and & Each has itsown function, althoughonly two are crucial.First, consider the relatively unimportantc and &is the loca-

the distribution can have different meansthan 0 (thestandard normal mean), depending on & In most cases,the distribution understudy is normalized, and= 0; that is, the meanof the distribution is set to 0.

0

I

0,

is

= 0,

(I)205Fractal Statistics1.31.25

1.151.11.05

0I23456Thousands

0.9FIGURE 14.2bSequential standard deviation, DowJones Industrials, five-dayreturns: 18881990.IOO6H-1005DowRat1000C,)998997

114900 4950 5000 5050 5100 5150 5200 5250 53005350Number

Convergence of

standard deviation, Dow

returns.The Special Cases: Normal and CauchyEmbedded

13, c,

=(if212)*t2

= the

2*ix2. Ifwe also have

0,

206_________________________________Fractal StatisticsStability under Addition207

c*J

Tails and the Law of ParetoWhen

(u/U)a

UNDER ADDITIONFor

= E(e*t*b*o)

stable

+b2*x?)) =

d

means

+tt

ba

CHARACTERISTICS OF FRACTALDISTRIBUTIONSStable

Why

remain

Thus,

can

although

Unfortunately,

2.0

208Fractal Statistics

Additive PropertiesMeasuring aDiscontinuities: Price JumpsThe

a

aFama

u))

log(Ui))

u))

_a*(loglog(U2))

210Fractal Statistics

a.

AnalysisMandeibrot

we

where

Log/log plot for various values ol a. (From MandelbrOt (l%4).

produced with permission of M.I.T. Press.)S211Measuring a

times

log(R/S)

R1,

H,

AnalysisWe

Equation

= 2.45

1.73,

212Fractaj StatisticsMeasurings213-2-1.5-I-0.500.511.52Log(Pr(U>u))

214__________________________________________Fractal StatisticsIInfinite Divisibility and IGARCH215MEASURING PROBABILITIESAs

stated before, the major problemwith the family

is that they do not lend themselvesto closed-form solutions,except in thespecial cases of the normal and Cauchydistributions. Therefore, theprobabil-ity

functions cannot be solved for

probabilities can

for only numerically,whichis a bit tedious. Luckily,

number of re-searchers have already accomplishedsolutions for some common values.Holt and

solved for

probability

functions for

0.25to 2.00 and= 1.00 to + 1.00, both in increments of0.25. The methodologythey used interpolated between theknown distributions, suchas the Cauchy andnormal, and an integral representationfrom Zolotarev (1964/1966).Produced forthe former National Bureau ofStandards, the tables remain themost completerepresentation of the probability densityfunctions of stable distributions.Some readers may find theprobability density functionuseful; most aremore interested in the cumulative distributions,which can be compared di-rectly to frequency distributions,as in Chapter 2. Fama and Roll (1968,1971)

cumulative distribution tables fora wide range of alphas. However,they concentrated on symmetricstable distributions, thusconstrainingto 0.Markets have been shownnumerous times to be skewed, but theimpact of thisskewness on market risk isnot obvious. We can assume that thesesymmetricvalues will

for most applications.Appendix 3 reproduces thecumulative distributions of the Famaand Rollstudies. The appendix also brieflydescribes the estimation methodology.INFINITE DIVISIBILITy ANDIGARCHThe ARCH

The reason is obvious: ARCH

the only

to the fam-ily of fractal

fit

empirical

are characterized by

that

... +

g1

... +

1.

1,

1)

216SUMMARYFractal StatisticsI

15Applying Fractal StatisticsIn

the traditional

I218Applying Fractal Statisticsdo just that. In addition, we will examine work by McCulloch (1985),who devel-oped

2.0,

SELECTIONMarkowitz

++

the

fPortfolio Selection219

a1

b*I

d,

the

error

(15.2)

220The

Xr*Cd +

weight

dispersion

= dispersion

dispersion

= sensitivity

rApplying Fractal StatisticsPortfolio Selection

221

2.0,

p,

1

0.

x28)

1/N,

1/N

1.0,

1.0,

1,

222Applying Fractal StatisticsPortfolio Selection223not

fore,

Table 15.1The Effects of Diversification: Nonmarket Risk0.350.3

NAlpha(a)2.001.751.501.251.000.50100.10000.17780.31620.56231.00003.1623200.05000.10570.22360.47291.00004.4721300.03330.07800.18260.42731.00005.4772400.02500.06290.15810.39761.00006.3246500.02000.05320.14140.37611.00007.0711600.01670.04640.12910.35931.00007.7460700.01430.04130.11950.34571.00008.3666800.01250.03740.11180.33441.00008.9443900.01110.03420.10540.32471.00009.48681000.01000.03160.10000.31621.000010.00001100.00910.02940.09530.30881.000010.48811200.00830.02760.09130.30211.000010.95451300.00770.02600.08770.29621.000011.40181400.00710.02460.08450.29071.000011.83221500.00670.02330.08160.28571.000012.24741600.00630.02220.07910.28121.000012.64911700.00590.02120.07670.27691.000013.03841800.00560.02030.07450.27301.000013.41641900.00530.01950.07250.26931.000013.78402000.00500.01880.07070.26591.000014.14212500.00400.01590.06320.25151.000015.81143000.00330.01390.05770.24031.000017.32053500.00290.01240.05350.23121.000018.70834000.00250.01120.05000.22361.000020.00004500.00220.01020.04710.21711.000021.21325000.00200.00950.04470.21151.000022.36075500.00180.00880.04260.20651.000023.45216000.00170.00820.04080.20211.00006500.00150.00780.03920.19801.000025.49517000.00140.00730.03780.19441.000026.45757500.00130.00700.03650.19111.000027.38618000.00130.00660.03540.18801.000028.28438500.00120.00640.03430.18521.000029.15489000.00110.00610.03330.18261.000030.00009500.00110.00580.03240.18011.000030.82211,0000.00100.00560.03160.17781.000031.6228

of

FIGURE 15.1Diversification.224Applying Fractal StatisticsOPTION VALUATIONIn

Option Valuation225Those

ApproachMcCulloch

can

I

I.

+l(l),

the

and Forward PricesWe

log(U2/U1)

I226Applying Fractal StatisticsOption Valuation---227

62,

1

ci'+

Cr

cl=lI *c\2

e5+

e=

paid

=*c

equal

1;

u1u3

u1(15.12)

(15.15)

Pricing Options

I (U2X0*U1)dP(U1,U2)C*enl*T *

U1dP(U1,U2)

228Applying Fractal Statistics

and

*

____*

maximally

/c2*z

( c,*z

12

and

show

Option Valuation229

C

F)*e_YT(15.27)

RatioMcCulloch

9(C*en'T)

230Applying Fractal StatisticsNumerical Option ValuesMcCulloch

1

1

Option Valuation231Using

(15.33)

0.1,

2.0,

T 9Table 15.2Fractal Option Prices: c = 0.1,

= 1.0232Applying Fractal StatisticsTableFractal Option Prices: a= 1.5,= 0.0cXO/F0.51.01.12.00.0150.0070.7870.0790.0140.0350.0382.2400.4580.0740.1050.2406.7843.4660.4810.3051.70417.69414.0643.4081.0064.13145.64243.06528.262PART FIVENOISY CHAOSA Final WordI

not

This

Chaos andR/S

In

VERTICAL LINES REPRESENTBULL MARKET PEAKSFIGURE 16.1Stock market and peak rates of economic growth.(Used with per-mission of Boston Capital Markets Group.)Information and Investors237The

AND INVESTORSThere

236Noisy Chaos and k/S Analysisjl10 8 SW2015.10239238Noisy Chaos and R/S AnalysisChaos_________________________________

Chaotic

SpaceA

241240Noisy Chaos and R/S Analysis(Applying R/S Analysisbe a closed circle. However, if there is friction,or damping,

each time thependulum swings back and forth, it goes a little slower, and itsamplitude de-creases until it eventually stops. The corresponding phase plot will spiral into theorigin, where velocity and position become zero.The phase space of the pendulum tells us allwe need to know about the dy-namics of the system, but the pendulum is nota very interesting system. If wetake a more complicated process and study its phasespace, we will discover anumber of interesting characteristics.We have already examined one such phasespace, the Lorenz attractor(Chapter 6). Here, the phase plot never repeats itself, although it isbounded bythe "owl eyes" shape. It is "attracted" to that shape, which is oftencalled its"attractor." If we examine the lines within the attractor,we find a self-similarstructure of lines, caused by repeated folding of the attractor. The noninter-secting structure of lines means that the process willnever completely fill itsspace. Its dimension is, thus, fractional. The fractal dimension of the Lorenzattractor is approximately 2.08. This means that its structure is slightlymorethan a two-dimensional plane, but less thana three-dimensional solid. It is,therefore, also a creation of the Demiurge.In addition, the attractor itself is bounded toa particular region of space,because chaotic systems are characterized by growth anda decay factor. Eachtrip around the attractor is called an orbit. Two orbits thatare close togetherinitially will rapidly diverge, even if theyare extremely close at the outset. Butthey will not fly away from one another indefinitely. Eventually,as each orbitreaches the outer bound of the attractor, it returns toward thecenter. The di-vergent points will come close together again, althoughmany orbits may beneeded to do so. This is the property of sensitive dependenceon initial condi-tions. Because we can never measure current conditionsto an infinite amountsof precision, we cannot predict where theprocess will go in the long term. Therate of divergence, or the loss in predictive power,can be characterized bymeasuring the divergence of nearby orbits in phasespace. A rate of divergence(called a "Lyapunov exponent") is measured for each dimensionin phasespace. One positive rate means that there are divergent orbits. Combinedwitha fractal dimension, it means that the system is chaotic. In addition, theremustbe a negative exponent to measure the foldingprocess, or the return to the at-tractor. The formula for Lyapunov exponents isas follows:Llim[( lit) * log2(pI(t)/p(O))Jwhere L, = the Lyapunov exponent for dimension ip(t)position in the ith dimension, at time(16.1)Equation (16.1) measures how the volume of a sphere grows over time, t, bymeasuring the divergence of two points, p(t) and p(O), in dimension i. The dis-tance is similar to a multidimensional range. By examiningequation (16.1), wecan see certain similarities to R/S analysis and tothe fractal dimension calcu-lation. All are concerned with scaling. However, chaotic attractors haveorbitsthat decay exponentially rather than through power laws.APPLYING K/S ANALYSISWhen we studied the attractor of Mackey and Glass (1988) briefly in Chapter6, we were concerned with finding cycles. In this chapter, we will extendthatstudy and will see how R/S analysis can distinguish between noisy chaosandfractional noise.The Noise IndexIn Chapter 6, we did not disclose the value of the Hurst exponent. ForFigure 6.8,H = 0.92. As would be expected, the continuous, smooth nature of thechaoticflow makes for a very high Hurst exponent. It is not equal tobecause of thefolding mechanism or the reversals that often occur in the time trace ofthisequation. In Figure 6.11, we added one standard deviation of white, uniformnoise to the system. This brought the Hurst exponent down to 0.72 and illus-trated the first application of R/S analysis to noisy chaos: Use the Hurst expo-nent as an index of noise.Suppose you are a technical analyst who wishes to test a particular type ofmonthly momentum indicator, and you plan to use the MackeyGlass equationto test the indicator. You know that the Hurst exponentfor monthly data has avalue of 0.72. To make the simulation realistic, one standard deviation of noiseshould be added to the data. In this manner, you can see whether your techni-cal indicator is robust with respect to noise.Now suppose you are a scientist examining chaotic behavior. You have a par-ticular test that can distinguish chaos from random behavior. To make the testpractical, you must show that it is robust with respect to noise. Because mostobserved time series have values of H close to 0.70 (as Hurst found; see Table5.1), you will need enough noise to make your test series have H = 0.70.Or,you could gradually add noise and observe thelevel of H at which your testbecomes uncertain.Figure 16.2 shows values of H as increasing noise is added to the MackeyGlass equation. The Hurst exponent rapidly drops to 0.70 and then gradually

1.31.21.10.9

0.30.2Noisy Chaos and R/S Analysis

FIGURE 16.2MackeyGlass equation, Hurst exponent sensitivity to noise.

0.70,

which

Applying R/S Analysis243

1.20.8

0.20

FIGURE 16.3MackeyGlass equation, Hurst exponent sensitivity tonoise.050100150200Observational

244Noisy Chaos and K/S AnalysisApplying K/S Analysis2451.42.

phase

nod

length

1.5FIGURE 16.4R/S analysis, MackeyGlass equation withsystem noise.1.41.3Cycles1.2We

1.1cycle

0.72),

0.6

FIGURE 16.5V statistic,

1.52

N=50

246Noisy Chaos and R/S AnalysisDISTINGUISHING NOISY CHAOS FROMFRACTIONAL NOISEThe

H

1.0)

BDS TestThree

Distinguishing Noisy Chaos Ironi Fractional Noise247test

lx XiI),i

(16.2)

1

lxi Xj

0;

the

distance

correlation

fl248Noisy Chaos and R/S Analysis

CN(e,T)C1(e)Nwith

probability(16.3)

wN(e,T) = CN(e,T)

the

percent

0.50

6.

Iijistinguishing Noisy Chaos from Fractional Noise249

6

0.72,

In

6.Table 16.1BDS StatistBDSIc: Simulated ProcessesEmbeddingNumber ofProcessMackeyGlassNo noiseObservational noiseSystem noiseFractional noise (H0.72)GARCHStatistic56.881 3.073.1213.856.23Epsilon0.120.060.080.070.01Dimension6 6 6 6 61,0001,0001,0001,4007,500MarketBDSStatisticEpsilonEmbeddingDimensionNumber ofObservationsDowfive-day28.720.0165,293Dow20-day14.340.0361,301Yen/Dollardaily116.050.0364,459

28.72

TestsIn

for the FMHFor

SummarySUMMARYWe

analysis

250Noisy Chaos and R/S AnalysisTable 16.2BDS Statistic: Market Time SeriesL17Fractal Statistics,NoisyChaos, and theFMHIn

We

Frequency253

The

68FIGURE 17.1MackeyGlass equation: no noise.7 6 5 2 0-8-6-4-2024Standard

Frequency Distributions255now

FIGURE 1 7.2bMackeyGlass equation:system noise.FIGURE 1 7.3aMackeyGlass equation: no noisenormal.I254Fractal Statistics, Noisy Chaos,and the EMIl8 7 6 53-4-202Standard

FIGURE 17.2aMackeyGlass equation: observationalnoise.9 8I.6 5 4I2-4-202Standard

462 0 I-20I23456

256Fractal Statistics, NoisyChaos, and the FMHII4FIGURE 17.3bMackeyGlass equation:observational noisenormal.FIGURE 17.3c

5Mackey_Glass equation:system noise__normalVolatility Term Structure257

different.

TERM STRUCTUREIn

0.92

50

30I2Standard

3454 3 256

-2.3

259I have done similar

Rosseler attractors. I encour-age readers to try the analysis for themselves, using the final programsuppliedin Appendix 2 or a program of their own manufacture. The volatility term struc-ture of these chaotic systems bears a striking resemblance to similar plots of thestock and bond markets, supplied in Chapter 2. Currencies do not have thisbounded characteristica further evidence that currencies are not "chaotic"but are, instead, a fractional noise process. This does not mean that currenciesdo not have runs; they clearly do, but there is no average length to these runs. Forcurrencies, the joker truly appears at random; for U.S. stocks and bonds, thejoker has an average appearance frequency of four years.SEQUENTIAL STANDARD DEVIATION AND MEANIn Chapter 14, we examined the sequential standard deviation and mean of theU.S. stock market, and compared it to a time series drawn from the Cauchy dis-tribution. We did so to see the effects of infinite variance and mean on a timeseries. The sequential standard deviation is the standard deviation of the timeseries as we add one observation at a time. If the series were from a Gaussianrandom walk, the more observations we have, the more the sequential standarddeviation would tend to the population standard deviation. Likewise, if the meanis stable and finite, the sample mean will eventually converge to the populationmean. For the Dow Jones Industrials file, we found scant evidence of conver-gence after about 100 years of data. This would mean that, in shorter periods,the process is much more similar to an infinite variance than to a finite variance

The sequential mean converged more rapidly, and looked more sta-ble. A fractal distribution would, of course, be well-described by an infinite orunstable variance, and a finite and stable mean. After studying the Dow, weseemed to find the desired characteristics.It would now be interesting to study the sequential statistics of chaotic sys-tems. Do they also have infinite variance and finite mean? They exhibit fat-taileddistributions when noise is added, but that alone is not enough to account for themarket analysis we have already done.Without noise, it appears that the MackeyGlass equation is persistent withunstable mean and variance. With noise, both observational and system, the sys-tem is closer to matket series, but

this study, as in Chapter 15,all series have been normalized to a mean of 0 and a standard deviation of 1. Thefinal value in each series will always have a mean of 0.Figure 17.5(a) shows the sequential standard deviation of 1,000 iterationsof the MackeyGlass equation without noise. The system is unstable, with

Statistics, Noisy Chaos, and the FMHSequential Standard Deviation and Mean

0 I)I -0.5

of

FIGURE 17.4aMackeyGlass equation: volatility termstructure.

of Observat ions)FIGURE 1 7.4bMackeyGlass equation with noise: volatilityterm structure.4

500600700

17.5a

equation: sequential standard deviation.discrete

260Fractal Statistics, Noisy Chaos, andthe FMH

1.03

1.01

0.981100 5)FIGURE 17.5bMackeyGlass equation with observational noise: sequentialstandard

1.03

C1.0201.015

1.0050.9950.990.985

FIGURE 17.5cMackeyGlass equation with system noise: sequential standarddeviation.261Measuring a263like

aThe

Graphical MethodUsing

0.64.

-3-2-t0Log(Pr(U>u))FIGURE 1 7.bbMackeyGlass

FIGURE 17.7

T200300400500600700800Number

262Fractal Statistics, Noisy Chaos, and the FMH0.050 -0.05-0.1-0.1590010001100FIGURE 17.6a

0.020 -0.02-0.04-0.06-0.0810001100-2-3-4 -5-7500600700800900Number

-8-9-1012

0.72,

they

LIKELIHOOD OF NOISY CHAOSThe

entirely

ORBITAL CYCLESA

_if*)( +=

Y

= x*Y

8/3,

28

264R/S AnalysisFractal Statistics, Noisy Chaos, and the FMHOrbital Cycles265266Fractal Statistics, Noisy Chaos, and theFMI-I20.7 Second0.5 SecondOrbital Cycles267E(R/S)0 U,1/)>1.41.31.21.10.9

3

0.5 0.5

of

1 7.8aLorenz attractor: R/S analysis.30.52.5

1.511.5

2

MackeyGlass equation with observational noise: V statistic.n=33

1 7.8b

1.81.71.61.50 U,1.41.31.21.10.5

1.5

17.lOaMackeyGlass equation, sampled every three intervals: V statistic.268Fractal Statistics, Noisy Chaos, and theFMHresult

Noisy

269Self-Similarity

n=33

0.9>0.8

FIGURE 17.11MackeyGlaSs equation, sampled every three intervals: no noise.20t5

0.51

of

FIGURE 17.lObMackeyGlass equation with noise,sampled every three intervals:V statistic.

Deviations

FIGURE 17.12MackeyGlass equations sampled every three intervals: observationalnoise.270Fractal Statistics, Noisy Chaos, andthe FMIIsystems.

PROPOSAL: UNITINGGARCH, FBM, AND CHAOSThe

18Understanding MarketsThis

IINFORMATION AND INVESTMENTHORIZONSWe

The

Risk273horizons.

Each

2.0.272Understanding Markets274The

0.0,

In

There

275nonperiodic

A MORE COMPLETE MARKETTHEORYMuch

Understanding MarketsToward a More Complete Market TheoryTI276Understanding Markets

T IAppendix 1The Chaos GameThis

r

278Appendix 1it

the

lower

iterations.

leaves

(x+640)

TAppendix 2GAUSS ProgramsIn

and Order in the Capital Markets, I

...,I

280Appendix 2CALCULATING THERESCALED RANGECalculating the Rescaled Range281

500 observations,

@Open

dat

282Appendix 2

calculating Sequential Standard Deviation and Mean283CALCULATING THE E(R/S)

CALCULATING SEQUENTIAL STANDARDDEVIATION AND MEANThe

rows(datx);@

prices.prn;

rows

ln(datx[2:obvl./datx[l:obvlfl;

rows

1;

19;

x

floor

reshape

n,

v

284Appendix 2CALCULATING THE TERMSTRUCTURE OF VOLATIUTyICalculating The Term Structure of Volatility285

2

+2

0;x

1];

ITable A2.1cxpected Value of R/S, Gaussian RandomVariable: Representative ValuesNE(R/S)Iog(N)Log(E(R/S))102.87221.00000.4582153.75181.17610.5742204.49581.30100.6528255.15251.39790.7120305.74691.47710.7594356.29391.54410.7989Appendix 3406.80341.60210.8327457.28221.65320.8623507.73521.69900.8885Fractal Distribution Tables558.16621.74040.9120608.57811.77820.9334658.97331.81290.9530709.35371.84510.9710759.72071.87510.98778010.07581.90311.00338510.42001.92941.01799010.75421.95421.03169511.07931.97771.0445This

11.39602.00001.056820016.57982.30101.219630020.55982.47711.31301.It presents tables that

0.

110.92773.90312.0450GENERATING THE TABLES8,500114.37793.92942.05839,000

ized

287xSUThe

I

As

+

+R(u)

F

Appendix 3

289As

+

(2*k1)!

F(a*k)

fR(u)du

That

'C C T a b l e A 3 . 1 C u m u l a t i v e Di s t r i b u t i o n Fu n c t i o n s fo r S t a n da r di ze d S y m m e t r i c S t a b l e Di s t r i b u t i o n s , F( u ) 1 . 6 0 1 . 7 0 1 . 8 0 1 . 9 0 1 . 9 5 2 . 0 0 0 . 0 5 0 . 5 1 5 9 0 . 5 1 5 3 0 . 5 1 5 0 0 . 5 1 4 7 0 . 5 1 4 5 0 . 5 1 4 4 0 . 5 1 4 3 0 . 5 1 4 2 0 . 5 1 4 2 0 . 5 1 4 1 0 . 5 1 4 1 0 . 5 1 4 1 0 . 1 0 0 . 5 3 7 1 0 . 5 3 0 6 0 . 5 2 9 9 0 . 5 2 9 4 0 . 5 2 9 0 0 . 5 2 8 7 0 . 5 2 8 5 0 . 5 2 8 4 0 . 5 2 8 3 0 . 5 2 8 2 0 . 5 2 8 2 0 . 5 2 8 2 0 . 1 5 0 . 5 4 7 4 0 . 5 4 5 8 0 . 5 4 4 7 0 . 5 4 3 9 0 . 5 4 3 4 0 . 5 4 3 0 0 . 5 4 2 7 0 . 5 4 2 5 0 . 5 4 2 4 0 . 5 4 2 3 0 . 5 4 2 3 0 . 5 4 2 2 0 . 2 0 0 . 5 6 2 8 0 . 5 6 0 8 0 . 5 5 9 4 0 . 5 5 8 4 0 . 5 5 7 7 0 . 5 5 7 2 0 . 5 5 6 8 0 . 5 5 6 6 0 . 5 5 6 4 0 . 5 5 6 3 0 . 5 5 6 3 0 . 5 5 6 2 0 . 2 5 0 . 5 7 8 0 0 . 5 7 5 6 0 . 5 7 4 0 0 . 5 7 2 8 0 . 5 7 1 9 0 . 5 7 1 3 0 . 5 7 0 9 0 . 5 7 0 6 0 . 5 7 0 4 0 . 5 7 0 2 0 . 5 7 0 2 0 . 5 7 0 2 0 . 3 0 0 . 5 9 2 8 0 . 5 9 0 2 0 . 5 8 8 3 0 . 5 8 6 9 0 . 5 8 6 0 0 . 5 8 5 3 0 . 5 8 4 8 0 . 5 8 4 4 0 . 5 8 4 2 0 . 5 8 4 1 0 . 5 8 4 0 0 . 5 8 4 0 0 . 3 5 0 . 6 0 7 2 0 . 6 0 4 4 0 . 6 0 2 4 0 . 6 0 0 9 0 . 5 9 9 8 0 . 5 9 9 1 0 . 5 9 8 5 0 . 5 9 8 2 0 . 5 9 7 9 0 . 5 9 7 8 0 . 5 9 7 8 0 . 5 9 7 7 0 . 4 0 0 . 6 2 1 1 0 . 6 1 8 3 0 . 6 1 6 2 0 . 6 1 4 6 0 . 6 1 3 5 0 . 6 1 2 7 0 . 6 1 2 2 0 . 6 1 1 8 0 . 6 1 1 5 0 . 6 1 1 4 0 . 6 1 1 4 0 . 6 1 1 4 0 . 4 5 0 . 6 3 4 6 0 . 6 3 1 8 0 . 6 2 9 7 0 . 6 2 8 1 0 . 6 2 7 0 0 . 6 2 6 2 0 . 6 2 5 6 0 . 6 2 5 2 0 . 6 2 5 0 0 . 6 2 4 9 0 . 6 2 4 8 0 . 6 2 4 8 0 . 5 0 0 . 6 4 7 6 0 . 6 4 4 9 0 . 6 4 2 8 0 . 6 4 1 3 0 . 6 4 0 2 0 . 6 3 9 4 0 . 6 3 8 9 0 . 6 3 8 5 0 . 6 3 8 3 0 . 6 3 8 2 0 . 6 3 8 2 0 . 6 3 8 2 0 . 5 5 0 . 6 6 0 1 0 . 6 5 7 6 0 . 6 5 5 7 0 . 6 5 4 2 0 . 6 5 3 2 0 . 6 5 2 4 0 . 6 5 1 9 0 . 6 5 1 6 0 . 6 5 1 4 0 . 6 5 1 3 0 . 6 5 1 3 0 . 6 5 1 3 0 . 6 0 0 . 6 7 2 0 0 . 6 6 9 8 0 . 6 6 8 1 0 . 6 6 6 8 0 . 6 6 5 8 0 . 6 6 5 1 0 . 6 6 4 7 0 . 6 6 4 4 0 . 6 6 4 3 0 . 6 6 4 3 0 . 6 6 4 3 0 . 6 6 4 3 0 . 6 5 0 . 6 8 3 5 0 . 6 8 1 7 0 . 6 8 0 2 0 . 6 7 9 0 0 . 6 7 8 2 0 . 6 7 7 6 0 . 6 7 7 2 0 . 6 7 7 0 0 . 6 7 7 0 0 . 6 7 7 0 0 . 6 7 7 0 0 . 6 7 7 1 0 . 7 0 0 . 6 9 4 4 0 ,6 9 3 0 0 . 6 9 1 9 0 . 6 9 0 9 0 . 6 9 0 2 0 . 6 8 9 8 0 . 6 8 9 5 0 . 6 8 9 4 0 . 6 8 9 4 0 . 6 8 9 5 0 . 6 8 9 6 0 . 6 8 9 7 0 . 7 5 0 . 7 0 4 8 0 . 7 0 3 9 0 . 7 0 3 1 0 . 7 0 2 5 0 . 7 0 2 0 0 . 7 0 1 7 0 . 7 0 1 5 0 . 7 O 1 5 0 . 7 0 1 6 0 . 7 0 1 8 0 . 7 0 1 9 0 . 7 0 2 1 0 . 8 0 0 . 7 1 4 8 0 . 7 1 4 4 0 . 7 1 4 0 0 . 7 1 3 6 0 . 7 1 3 4 0 . 7 1 3 3 0 . 7 1 3 3 0 . 7 1 3 4 0 . 7 1 3 6 0 . 7 1 3 9 0 . 7 1 4 0 0 . 7 1 4 2 0 . 8 5 0 . 7 2 4 2 0 . 7 2 4 4 0 . 7 2 4 4 0 . 7 2 4 4 0 . 7 2 4 4 0 . 7 2 4 5 0 . 7 2 4 7 0 . 7 2 5 0 0 . 7 2 5 3 0 . 7 2 5 7 0 . 7 2 5 9 0 . 7 2 6 1 0 . 9 0 0 . 7 3 3 3 0 . 7 3 4 0 0 . 7 3 4 5 0 . 7 3 4 8 0 . 7 3 5 1 0 . 7 3 5 5 0 . 7 3 5 8 0 . 7 3 6 3 0 . 7 3 6 7 0 . 7 3 7 2 0 . 7 3 7 5 0 . 7 3 7 7 0 . 9 5 0 . 7 4 1 8 0 . 7 4 3 2 0 . 7 4 4 1 0 . 7 4 4 9 0 . 7 4 5 5 0 . 7 4 6 1 0 . 7 4 6 7 0 . 7 4 7 2 0 . 7 4 7 9 0 . 7 4 8 5 0 . 7 4 8 8 0 . 7 4 9 1 1 . 0 0 0 . 7 5 0 0 0 . 7 5 1 9 0 . 7 5 3 4 0 . 7 5 4 5 0 . 7 5 5 5 0 . 7 5 6 3 0 . 7 5 7 2 0 . 7 5 7 9 0 . 7 5 8 7 0 . 7 5 9 5 0 . 7 5 9 9 0 . 7 6 0 2 1 . 1 0 0 . 7 6 5 1 0 . 7 6 8 2 0 . 7 7 0 7 0 . 7 7 2 7 0 . 7 7 4 4 0 . 7 7 5 9 0 . 7 7 7 2 0 . 7 7 8 4 0 . 7 7 9 5 0 . 7 8 0 6 0 . 7 8 1 1 0 . 7 8 1 7 1 . 2 0 0 . 7 7 8 9 0 . 7 8 3 1 0 . 7 8 6 5 0 . 7 8 9 4 0 . 7 9 1 9 0 . 7 9 4 0 0 . 7 9 5 9 0 . 7 9 7 6 0 . 7 9 9 1 0 . 8 0 0 6 0 . 8 0 1 3 0 . 8 0 1 9 1 . 3 0 0 . 7 9 1 3 0 . 7 9 6 5 0 . 8 0 1 0 9 . 8 0 4 8 0 . 8 0 8 0 0 . 8 1 0 8 0 . 8 1 3 3 0 . 8 1 5 5 0 . 8 1 7 5 0 . 8 1 9 3 0 . 8 2 0 2 0 . 8 2 1 0 1 . 4 0 0 . 8 0 2 6 0 . 8 0 8 8 0 . 8 1 4 2 0 . 8 1 8 8 0 . 8 2 2 8 0 . 8 2 6 3 0 . 8 2 9 4 0 . 8 3 2 2 0 . 8 3 4 6 0 . 8 3 6 9 0 . 8 3 7 9 0 . 8 3 8 9 1 . 5 0 0 . 8 1 2 8 0 . 8 1 9 4 0 . 8 2 6 1 0 . 8 3 1 6 0 . 8 3 6 4 0 . 8 4 0 6 0 . 8 4 4 3 0 . 8 4 7 5 0 . 8 5 0 5 0 . 8 5 3 1 0 . 8 5 4 4 0 . 8 5 5 6 1 . 6 0 0 . 8 2 2 2 0 . 8 3 0 0 0 . 8 3 7 0 0 . 8 4 3 3 0 . 8 4 8 7 0 . 8 5 3 6 0 . 8 5 7 9 0 . 8 6 1 7 0 . 8 6 5 1 0 . 8 6 8 2 0 . 8 6 9 7 0 . 8 7 1 1 1 . 7 0 0 . 8 3 0 7 0 . 8 3 9 3 0 . 8 4 7 0 0 . 8 5 3 9 0 . 8 6 0 0 0 . 8 6 5 5 0 . 8 7 0 3 0 . 8 7 4 7 0 . 8 7 8 6 0 . 8 8 2 1 0 . 8 8 3 8 0 . 8 8 5 3 1 . 8 0 0 . 8 3 8 6 0 . 8 4 7 7 0 . 8 5 6 0 0 . 8 6 3 5 0 . 8 7 0 2 0 . 8 7 6 3 0 . 8 8 1 7 0 . 8 8 6 5 0 . 8 9 0 9 0 . 8 9 4 9 0 . 8 9 6 7 0 . 8 9 8 5 1 . 9 0 0 . 8 4 5 8 0 . 8 5 5 4 0 . 8 6 4 3 0 . 8 7 2 3 0 . 8 7 9 5 0 . 8 8 6 1 0 . 8 9 2 0 0 . 8 9 7 3 0 . 9 0 2 1 0 . 9 0 6 5 0 . 9 0 8 5 0 . 9 1 0 4 2 . 0 0 0 . 8 5 2 4 0 . 8 6 2 5 0 . 8 7 1 9 0 . 8 8 0 2 0 . 8 8 7 9 0 . 8 9 5 0 0 . 9 0 1 3 0 . 9 0 7 1 0 . 9 1 2 3 0 . 9 1 7 0 0 . 9 1 9 2 0 . 9 2 1 4 2 . 2 0 0 . 8 6 4 2 0 . 8 7 5 0 0 . 8 8 5 0 0 . 8 9 4 1 0 . 9 0 2 5 0 . 9 1 0 3 0 . 9 1 7 4 0 . 9 2 3 8 0 . 9 2 9 8 0 . 9 3 5 2 0 . 9 3 7 7 0 . 9 4 0 1 2 . 4 0 0 . 8 7 4 3 0 . 8 8 5 6 0 . 8 9 6 1 0 . 9 0 5 7 0 . 9 1 4 6 0 . 9 2 2 8 0 . 9 3 0 4 0 . 9 3 7 4 0 . 9 4 3 8 0 . 9 4 9 7 0 . 9 5 2 5 0 . 9 5 5 2 2 . 6 0 0 . 8 8 3 1 0 . 8 9 4 8 0 . 9 0 5 5 0 . 9 1 5 5 0 . 9 2 4 6 0 . 9 3 3 1 0 . 9 4 0 9 0 . 9 4 8 2 0 . 9 5 5 0 0 . 9 6 1 2 0 . 9 6 4 2 0 . 9 6 7 0 2 . 8 0 0 . 8 9 0 8 0 . 9 0 2 7 0 . 9 1 3 6 0 . 9 2 3 6 0 . 9 3 2 9 0 . 9 4 1 5 0 . 9 4 9 5 0 . 9 5 6 9 0 . 9 6 3 8 0 . 9 7 0 2 0 . 9 7 3 2 0 . 9 7 6 1 3 . 0 0 0 . 8 9 7 6 0 . 9 0 9 6 0 . 9 2 0 5 0 . 9 3 0 6 0 . 9 3 9 9 0 . 9 4 8 4 0 . 9 5 6 4 0 . 9 6 3 8 0 . 9 7 0 7 0 . 9 7 7 1 0 . 9 8 0 1 0 . 9 8 3 1 3 . 2 0 0 . 9 0 3 8 0 . 9 1 5 6 0 . 9 2 6 5 0 . 9 3 6 5 0 . 9 4 5 7 0 . 9 5 4 2 0 . 9 6 2 0 0 . 9 6 9 2 0 . 9 7 6 0 0 . 9 8 2 3 0 . 9 8 5 3 0 . 9 8 8 2 3 . 4 0 0 . 9 0 8 9 0 . 9 2 0 9 0 . 9 3 1 8 0 . 9 4 1 7 0 . 9 5 0 7 0 . 9 5 9 0 0 . 9 6 6 6 0 . 9 7 3 6 0 . 9 8 0 2 0 . 9 8 6 2 0 . 9 8 9 1 0 . 9 9 1 9 3 . 6 0 0 . 9 1 3 8 0 . 9 2 5 7 0 . 9 3 6 5 0 . 9 4 6 2 0 . 9 5 5 0 0 . 9 6 3 1 0 . 9 7 0 4 0 . 9 7 7 1 0 . 9 8 3 4 0 . 9 8 9 2 0 . 9 9 1 9 0 . 9 9 4 5 3 . 8 0 0 . 9 1 8 1 0 . 9 2 9 9 0 . 9 4 0 6 0 . 9 5 0 1 0 . 9 5 8 7 0 . 9 6 6 5 0 . 9 7 3 6 0 . 9 8 0 0 0 . 9 8 5 9 0 . 9 9 1 4 0 . 9 9 3 9 0 . 9 9 6 4 4 . 0 0 0 . 9 2 2 0 0 . 9 3 3 8 0 . 9 4 4 2 0 . 9 5 3 6 0 . 9 6 1 9 0 . 9 6 9 4 0 . 9 7 6 2 0 . 9 8 2 3 0 . 9 8 7 9 0 . 9 9 3 0 0 . 9 9 5 4 0 . 9 9 7 7 4 . 4 0 0 . 9 2 8 9 0 . 9 4 0 3 0 . 9 5 0 4 0 . 9 5 9 3 0 . 9 6 7 2 0 . 9 7 4 2 0 . 9 8 0 4 0 . 9 8 5 9 0 . 9 9 0 8 0 . 9 9 5 1 0 . 9 9 7 2 0 . 9 9 9 1 4 . 8 0 0 . 9 3 4 6 0 . 9 4 5 8 0 . 9 5 5 5 0 . 9 6 4 0 0 . 9 7 1 4 0 . 9 7 7 8 0 . 9 8 3 4 0 . 9 8 8 3 0 . 9 9 2 7 0 . 9 9 6 4 0 . 9 9 8 1 0 . 9 9 9 7 5 . 2 0 0 . 9 3 9 5 0 . 9 5 0 4 0 . 9 5 9 7 0 . 9 6 7 8 0 . 9 7 4 7 0 . 9 8 0 7 0 . 9 8 5 8 0 . 9 9 0 2 0 . 9 9 3 9 0 . 9 9 7 2 0 . 9 9 8 6 0 . 9 9 9 9 5 . 6 0 0 . 9 4 3 8 0 . 9 5 4 3 0 . 9 6 3 3 0 . 9 7 0 9 0 . 9 7 7 4 0 . 9 8 3 0 0 . 9 8 7 6 0 . 9 9 1 6 0 . 9 9 4 9 0 . 9 9 7 7 0 . 9 9 8 9 1 . 0 0 0 0 6 . 0 0 0 . 9 4 7 4 0 . 9 5 7 6 0 . 9 6 6 3 0 . 9 7 3 6 0 . 9 7 9 7 0 . 9 8 4 8 0 . 9 8 9 1 0 . 9 9 2 7 0 . 9 9 5 6 0 . 9 9 8 0 0 . 9 9 9 1 1 . 0 0 0 0 7 . 0 0 0 . 9 5 4 8 0 . 9 6 4 3 0 . 9 7 2 1 0 . 9 7 8 6 0 . 9 8 3 9 0 . 9 8 8 2 0 . 9 9 1 8 0 . 9 9 4 6 0 . 9 9 6 9 0 . 9 9 8 6 0 . 9 9 9 4 1 . 0 0 0 0 8 . 0 0 0 . 9 6 0 4 0 . 9 6 9 2 0 . 9 7 6 4 0 . 9 8 2 1 0 . 9 8 6 8 0 . 9 9 0 5 0 . 9 9 3 5 0 . 9 9 5 8 0 . 9 9 7 6 0 . 9 9 9 0 0 . 9 9 9 5 1 . 0 0 0 0 1 0 . 0 0 0 . 9 6 8 3 0 . 9 7 6 0 0 . 9 8 2 0 0 . 9 8 6 8 0 . 9 9 0 5 0 . 9 9 3 4 0 . 9 9 5 6 0 . 9 9 7 2 0 . 9 9 8 5 0 . 9 9 9 4 0 . 9 9 9 7 1 . 0 0 0 0 1 5 . 0 0 0 . 9 7 8 8 0 . 9 8 4 7 0 . 9 8 9 1 0 . 9 9 2 3 0 . 9 9 4 7 0 . 9 9 6 5 0 . 9 9 7 7 0 . 9 9 8 6 0 . 9 9 9 3 0 . 9 9 9 7 0 . 9 9 9 9 1 . 0 0 0 0 2 0 . 0 0 0 . 9 8 4 1 0 . 9 8 8 8 0 . 9 9 2 3 0 . 9 9 4 7 0 . 9 9 6 5 0 . 9 9 7 7 0 . 9 9 8 6 0 . 9 9 9 2 0 . 9 9 9 6 0 . 9 9 9 8 0 . 9 9 9 9 1 . 0 0 0 0 Fr o m Fa m a a n d R o l l ( 1 9 7 1 ) . R e p r o du c e d wi t h p e r m i s s i o n o f t h e A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n . T a b l e A 3 . 2 Fr a c t i l e s o f S t a n da r di ze d S y m m e t r i c S t a b l e Di s t r i b u t i o n s , u A l p h a ( a ) F 1 . 0 0 1 . 1 0 1 . 2 0 1 . 3 0 1 . 4 0 1 . 5 0 1 . 6 0 1 . 7 0 1 . 8 0 1 . 9 0 1 . 9 5 2 . 0 0 0 . 5 2 0 0 0 . 0 6 3 0 . 0 6 5 0 . 0 6 7 0 . 0 6 8 0 . 0 6 9 0 . 0 7 0 0 . 0 7 0 0 . 0 7 0 0 . 0 7 1 0 . 0 7 1 0 . 0 7 1 0 . 0 7 1 0 . 5 4 0 0 0 . 1 2 6 0 . 1 3 1 0 . 1 3 4 0 . 1 3 6 0 . 1 3 8 0 . 1 3 9 0 . 1 4 0 0 . 1 4 1 0 . 1 4 1 0 . 1 4 2 0 . 1 4 2 0 . 1 4 2 0 . 5 6 0 0 0 . 1 9 1 0 . 1 9 7 0 . 2 0 2 0 . 2 0 5 0 . 2 0 8 0 . 2 1 0 0 . 2 1 1 0 . 2 1 2 0 . 2 1 3 0 . 2 1 3 0 . 2 1 4 0 . 2 1 4 0 . 5 8 0 0 0 . 2 5 7 0 . 2 6 5 0 . 2 7 1 0 . 2 7 5 0 . 2 7 9 0 . 2 8 1 0 . 2 8 3 0 . 2 8 4 0 . 2 8 5 0 . 2 8 6 0 . 2 8 6 0 . 2 8 6 0 . 6 0 0 0 0 . 3 2 5 0 . 3 3 4 0 . 3 4 1 0 . 3 4 7 0 . 3 5 0 0 . 3 5 3 0 . 3 5 5 0 . 3 5 7 0 . 3 5 7 0 . 3 5 8 0 . 3 5 8 0 . 3 5 8 0 . 6 2 0 0 0 . 3 9 6 0 . 4 0 6 0 . 4 1 4 0 . 4 2 0 0 . 4 2 4 0 . 4 2 7 0 . 4 2 9 0 . 4 3 0 0 . 4 3 2 0 . 4 3 2 0 . 4 3 2 0 . 4 3 2 0 . 6 4 0 0 0 . 4 7 1 0 . 4 8 1 0 . 4 8 9 0 . 4 9 5 0 . 4 9 9 0 . 5 0 2 0 . 5 0 4 0 . 5 0 6 0 . 5 0 6 0 . 5 0 7 0 . 5 0 7 0 . 5 0 7 0 . 6 6 0 0 0 . 5 5 0 0 . 5 6 0 0 . 5 6 7 0 . 5 7 3 0 . 5 7 7 0 . 5 8 0 0 . 5 8 1 0 . 5 8 3 0 . 5 8 3 0 . 5 8 3 0 . 5 8 3 0 . 5 8 3 0 . 6 8 0 0 0 . 6 3 5 0 . 6 4 3 0 . 6 4 9 0 . 6 5 4 0 . 6 5 8 0 . 6 6 0 0 . 6 6 1 0 . 6 6 2 0 . 6 6 2 0 . 6 6 2 0 . 6 6 1 0 . 6 6 1 0 . 7 0 0 0 0 . 7 2 7 0 . 7 3 2 0 . 7 3 6 0 . 7 3 9 0 . 7 4 2 0 . 7 4 3 0 . 7 4 4 0 . 7 4 4 0 . 7 4 3 0 . 7 4 3 0 . 7 4 2 0 . 7 4 2 0 . 7 2 0 0 0 . 8 2 7 0 . 8 2 8 0 . 8 2 9 0 . 8 3 0 0 . 8 3 0 0 . 8 3 0 0 . 8 3 0 0 . 8 2 9 0 . 8 2 8 0 . 8 2 6 0 . 8 2 5 0 . 8 2 4 0 . 7 4 0 0 0 . 9 3 9 0 . 9 3 2 0 . 9 2 8 0 . 9 2 6 0 . 9 2 4 0 . 9 2 1 0 . 9 1 9 0 . 9 1 7 0 . 9 1 5 0 . 9 1 2 0 . 9 1 1 0 . 9 1 0 0 . 7 6 0 0 1 . 0 6 5 1 . 0 4 8 1 . 0 3 7 1 . 0 3 0 1 . 0 2 4 1 . 0 1 8 1 . 0 1 4 1 . 0 1 0 1 . 0 0 6 1 . 0 0 3 1 . 0 0 1 0 . 9 9 9 0 . 7 8 0 0 1 . 2 0 9 1 . 1 7 9 1 . 1 5 8 1 . 1 4 3 1 . 1 3 1 1 . 1 2 2 1 . 1 1 5 1 . 1 0 8 1 . 1 0 2 1 . 0 9 7 1 . 0 9 5 1 . 0 9 2 1 . 2 6 8 1 . 2 4 9 1 . 2 3 5 1 . 2 2 3 1 . 2 1 3 1 . 2 0 4 1 . 1 9 7 1 . 1 9 4 1 . 1 9 0 1 . 4 0 9 1 . 3 8 0 1 . 3 5 8 1 . 3 4 1 1 . 3 2 6 1 . 3 1 4 1 . 3 0 4 1 . 2 9 9 1 . 2 9 5 1 . 5 7 1 1 . 5 2 8 1 . 4 9 6 1 . 4 7 1 1 . 4 5 0 1 . 4 3 3 1 . 4 1 9 1 . 4 1 3 1 . 4 0 7 1 . 7 6 2 1 . 7 0 0 1 . 6 5 3 1 . 6 1 6 1 . 5 8 7 1 . 5 6 4 1 . 5 4 4 1 . 5 3 6 1 . 5 2 8 1 . 9 9 6 1 . 9 0 5 1 . 8 3 7 1 . 7 8 5 1 . 7 4 4 1 . 7 1 1 1 . 6 8 4 1 . 6 7 2 1 . 6 6 2 2 . 2 9 7 2 . 1 6 1 2 . 0 6 1 1 . 9 8 5 1 . 9 2 7 1 . 8 8 0 1 . 8 4 3 1 . 8 2 7 1 . 8 1 3 2 . 7 0 8 2 . 5 0 3 2 . 3 5 1 2 . 2 3 7 2 . 1 5 0 2 . 0 8 4 2 . 0 3 0 2 . 0 0 7 1 . 9 8 8 3 . 3 3 1 3 . 0 0 2 2 . 7 6 3 2 . 5 8 1 2 . 4 4 4 2 . 3 4 1 2 . 2 6 1 2 . 2 2 8 2 . 1 9 9 3 . 7 9 8 3 . 8 6 9 3 . 0 5 3 2 . 8 1 6 2 . 6 3 8 2 . 5 0 5 2 . 4 0 4 2 . 3 6 3 2 . 3 2 7 4 . 4 5 3 3 . 8 8 2 3 . 4 4 8 3 . 1 2 7 2 . 8 8 7 2 . 7 0 8 2 . 5 7 6 2 . 5 2 2 2 . 4 7 7 5 . 4 7 6 4 . 6 5 9 4 . 0 4 9 3 . 5 7 7 3 . 2 3 4 2 . 9 8 0 2 . 7 9 5 2 . 7 2 2 2 . 6 6 1 6 . 2 5 1 5 . 2 4 0 4 . 4 8 5 3 . 9 0 1 3 . 4 7 8 3 . 1 6 0 2 . 9 3 3 2 . 8 4 6 2 . 7 7 2 7 . 3 5 9 6 . 0 6 3 5 . 0 9 9 4 . 3 5 7 3 . 7 9 9 3 . 3 9 4 3 . 1 0 4 2 . 9 9 6 2 . 9 0 5 9 . 1 0 0 7 . 3 4 1 6 . 0 4 3 5 . 0 5 6 4 . 2 8 3 3 . 7 2 8 3 . 3 3 0 3 . 1 9 1 3 . 0 7 0 1 2 . 3 1 3 9 . 6 5 9 7 . 7 3 7 6 . 2 8 5 5 . 1 6 6 4 . 2 9 1 3 . 6 7 0 3 . 4 6 1 3 . 2 9 0 2 0 . 7 7 5 1 5 . 5 9 5 1 1 . 9 8 3 9 . 3 3 2 7 . 2 9 0 5 . 6 3 3 4 . 3 7 5 3 . 9 4 7 3 . 6 4 3 1 2 0 . 9 5 2 7 9 . 5 5 6 5 4 . 3 3 7 3 7 . 9 6 7 2 6 . 6 6 6 1 8 . 2 9 0 1 1 . 3 3 3 7 . 7 9 0 4 . 6 5 3 0 . 8 0 0 0 1 . 3 7 6 1 . 3 2 7 1 . 2 9 3 0 . 8 2 0 0 1 . 5 7 6 1 . 5 0 5 1 . 4 4 7 0 . 8 4 0 0 1 . 8 1 9 1 . 7 0 9 1 . 6 2 8 0 . 8 6 0 0 2 . 1 2 5 1 . 9 6 4 1 . 8 4 7 0 . 8 8 0 0 2 . 5 2 6 2 . 2 9 0 2 . 1 2 2 0 . 9 0 0 0 3 . 0 7 8 2 . 7 2 9 2 . 4 8 0 0 . 9 2 0 0 3 . 6 9 5 3 . 3 6 6 2 . 9 8 4 0 . 9 4 0 0 5 . 2 4 2 4 . 3 7 9 3 . 7 7 4 0 . 9 5 0 0 6 . 3 1 4 5 . 1 6 5 4 . 3 7 0 0 . 9 6 0 0 7 . 9 1 6 6 . 3 1 9 5 . 2 3 0 0 . 9 7 0 0 1 0 . 5 7 9 8 . 1 8 9 6 . 5 9 6 0 . 9 7 5 0 1 2 . 7 0 6 9 . 6 5 1 7 . 6 4 5 0 . 9 8 0 0 1 5 . 8 9 5 1 1 . 8 0 2 9 . 1 6 4 0 . 9 8 5 0 2 1 . 2 0 5 1 5 . 3 0 0 1 1 . 5 8 9 0 . 9 9 0 0 3 1 . 8 2 0 2 2 . 0 7 1 1 6 . 1 6 0 0 . 9 9 5 0 6 3 . 6 5 7 4 1 . 3 4 8 2 8 . 6 3 0 0 . 9 9 9 5 6 3 6 . 6 0 9 3 3 4 . 5 9 5 1 9 3 . 9 8 9 Fr o m Fa m a a n d R o l l ( 1 9 7 1 ) . R e p r o du c e d wi t h p e r m i s s i o n o f t h e A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n . Description of the Tables295ALTERNATIVE METHODSThere

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10,

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OF THE TABLESTable

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3.29.

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BibliographyAkgiray,

of Business and Economic Statistics7,

Random Character of Stock MarketPrices. Cam-

63,

Pareto Distributions Fairland,

Random Character ofStock Market Prices. Cambridge,

Singapore:

"Self-Organized

American January

Review A38,

Everywhere San

for Dynamic Modeling. Boston:

nomjc Studies 48,

fur Mate,natik 2,

of Business45,

Model:

in the Theory of Capital Markets.New

of Political Economy, May/June

ofEconometrics 31,

and

Mirror. New York:

of Eco-nomic Theory 40,

Econometric Mod-eling. Cambridge,

of the Royal Statistical Society 54, 1992.

Today 27,1974.

nal of the Royal Statistical Society 54, 1991.

DynamicsReview 4,

of the Royal Statistical Society 54,

of Business andEconomic Statistics Il,

of Business and Economic Statistics 11,

Random Character of Stock Market Prices. Cambridge,

Random Character of Stock Market Prices. Cambridge,

of Financial Economics 3,

Markets. Englewood

74,

Journal of Economics 98,

Economic Review, June

31,

I298BibliographyDeLong,

Rational

of Finance 45, 1990.

Introduction to Chaotic Dynamical Systems.Menlo

Critique,"

ofthe American Statistical Association 68,

of the American Statistical Association 70,

The Economist, "Back

in

of Physics322,

Gruber,

Portfolio Theory and Investment Analysis. New

50,

Reviews 5,

55, 1987.

of Business 38,

of Finance 25,

Random Character of Stock Market Prices.Cambridge,

Science 11,

E. F.,

nal of Finance 47,

Theory of Finance. New

of the American Statistical Association 63,

of the American Statistical Association 66,

Introduction to Spatial and Tempo-ral FractaJs. Berlin:

New

7D,

of Mathematics and Statistics 22,

Transactions onInformation Theory 35,

Papers on Economic Activity 2,

American 238,

of Mathematical Biology 7,

Clocks to Chaos. Princeton,

Making a New Science. New

53,

of Economic Theory 40,

Analysis of Economic Time Series. Princeton,

of the American Statistical Association 67,1972.

Review Letters 48,1983.

Research, 1979.

of Financial Economics 4,

of the Royal Statisti-cal Society 54,1992.

of Modern Physics 47,

in Mathematical Physics 50,

in Economics. New

Themas.New

Princeton,

of Research of the National Bureau of Standards 77B,

tions in Pure and Applied Mathematics 1,

68,

of Finance 46,

of Business 62,

Music,"

of the NationalAcademy of Sciences 87,

I&iblwgraphy300BibliographyHsu,

ofthe

15,

and Difference

Experiments. New

Society 6,

of Bifurcation and Chaos2,

T301

1986.

Resources Research 5,I

Resources Research 5,l969b.

Resources Research 5, l969c.

Resources Research 5,l969d.

of Finance 7,

Selection: Efficient Diversificationof Investments, New

261,

England EconomicReview, November/December1987.

England EconomicReview, July/August

England Economic Review,July/August

England Economic Review,November/December

Journal of Economics, May

Random Character of StockMarket Prices. Cambridge,

Infinite. London:

34,

Biology. Berlin:

of the Royal Statistical Society54, 1992.

Random Character of StockMarket Prices. Cambridge,

Review Letters 45,1980.

d'Economie Politique.Lausanne,

Analysts Journal, July/

and Order in the CapitalMarkets. New

Sons,

Analysts Journal, March/

An-alysts Journal, November/December

Pattern, Chaos and Beauty. New

Signals and Noise. New

Row,

Eco-nomic Journal 96,

and Method. New

Complexity. New

Out of Chaos. New

of Economic Issues 24,

Random Character of Stock Market Prices. Cambridge,

of Finance, 1959.)

of Financialand Quantitative Analysis 4, 1969.

of Financial Economics 4,1977.

of Finance 35,1980.

of Economic Theory13,

Portfolio Theory. Homewood,

Evolution and Strange Attractors. Cambridge,

7D,

of Financial and Quantitative Analysis, June

of Business 62,

of

Studies 48,

ReviewLetters 68,

Chaos, Power Laws. New

Analysts Journal. May/June

in Economics. Westport,

304BibliographyShannon, C.

Weaver, W. The

University ofIllinois,1963.Sharpe, W. F. "Capital

A

Equilibrium

Condi-tions

the

BibliographY305Wilson,

GlossaryAlphaThe measure of the peakedness of the probability density function.

Glossary307Autoregressive

IL309308GlossaryGlossarytogether

heteroskedas-Econometrics

0.50

H

Glossarycondjt

noise.Glossaryf

motion.IndexAlpha,

314Fama,

analysis,

315Technical

IndexIndex