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The Variation of Certain Speculative PricesAuthor(s): Benoit MandelbrotSource: The Journal of Business, Vol. 36, No. 4 (Oct., 1963), pp. 394-419Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/2350970

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THE VARIATION OF CERTAIN SPECULATIVE PRICES*

BENOIT MANDELBROTt

I. INTRODUCTION

Ename of Louis Bachelieris oftenmentioned in books on diffusionprocess.Until very recently, how-

ever, few people realized that his early(1900) and path-breaking contributionwas the constructionof a random-walkmodel for securityand commoditymar-kets.' Bachelier's simplestand most im-portantmodelgoes as follows: et Z(t) be

the priceof a stock,or of a unit of a com-

modity, at the end of time period t. Thenit is assumed that successive differencesof the form Z(t + T) - Z(t) are inde-pendent, Gaussian or normally distrib-uted, random variables with zero meanand variance proportionalto the differ-encing interval T.2

Despite the fundamental importanceof Bachelier'sprocess,which has cometobe called "Brownianmotion," it is nowobvious that it does not account for theabundant data accumulated since 1900by empiricaleconomists, simply becausethe empiricaldistributions f price changesare usually too "peaked" o be relative osamples romGaussianpopulations.3That

*The theorydevelopedn this paper s a naturalcontinuationof my study of the distributionof in-come.I wasstill working n the latterwhenHendrikS. Houthakkerdirected my interest toward thedistribution fpricechanges.Thepresentmodelwasthus suggestedby Houthakker'sdata; it was dis-cussed with him all along and was first publiclypresentedat hisseminar. therefore wehim agreatdebt of gratitude.

The extensive computationsrequired by thisworkwereperformed n the 7090 computerof theI.B.M. Research Center and were mostly pro-

grammedbyN.

J. Anthony,R. Coren,and F. L.

Zarnfaller.Many of the datawhichI haveusedweremostkindlysuppliedby F. Lowenstein ndJ. Don-ald of the EconomicStatisticssection of the UnitedStates Departmentof Agriculture.Some stages ofthe present work were supported n part by theOfficeof Naval Research,under contract numberNonr-3775(00),NR-047040.

t HarvardUniversityand ResearchCenterof theInternationalBusinessMachinesCorporation.

1The materialsof this paperwill be included ngreaterdetail n mybook entatively itledStudies nSpeculation, Economics, and Statistics, to be pub-lishedwithin a year by JohnWileyand Sons.

The present ext is a modified ersionof my "Re-searchNote," NC-87,issuedon March26, 1962bythe ResearchCenter of the InternationalBusinessMachinesCorporation. havebeen careful o avoidany changein substance,but certainparts of thatexpositionhave been clarified,and I have omittedsome less essential sections, paragraphs,or sen-tences. Sections I and II correspondroughly tochaps. i and ii of the original,SectionsIII and IVcorrespondo chaps. v andv, SectionsV and VI, tochap.vi, and SectionVII, to chap.vii.

2 The simple Bacheliermodel implicitlyassumesthat the varianceof the differences (t + T) - Z(t)is independent f the level of Z(t). There s reason oexpect, however, that the standard deviation ofAZ(t)will be proportionalo the price evel,andforthis reason many modernauthors have suggested

that the originalassumptionof independent ncre-ments of Z(t) be replacedby the assumption f inde-pendentand Gaussian ncrementsof log. Z(t).

Since Bachelier'soriginalwork is fairly inacces-sible, it is good to mentionmorethan one reference:"Theoriede la speculation" Paris DoctoralDisser-tationin Mathematics,March29, 1900) Annales del'Ecole Normale Superieure, ser. 3, XVII (1900), 21-86; "Theoriemathematiquedu jeu," Annales del'Ecole Normale Superieure, ser. 3, XVIII (1901),143-210; Calcul des probabilites (Paris: Gauthier-Villars, 1912); Le jeu, la chance et le hasard (Paris,1914[reprinted p to 1929 at least]).

3 To the best of my knowledge, he first to notethis fact wasWesleyC. Mitchell,"TheMakingandUsing of Index Numbers,"Introductionto IndexNumbers and Wholesale Prices in the United Statesand Foreign Countries (published in 1915 as BulletinNo. 173of the U.S. Bureauof Labor Statistics,re-printed n 1921as BulletinNo. 284). But unques-tionableproofwas only given by MauriceOlivier n"Les Nombres indices de la variation des prix"(Parisdoctoraldissertation, 926),andFrederickC.Mills in The Behavior of Prices (New York: NationalBureau of EconomicResearch, 1927). Other evi-

394



VARIATIONOF CERTAIN SPECULATIVEPRICES 395

is, the histogramsof price changesarein-deed unimodal and their central "bells"remindone of the "Gaussianogive." But

there are typically so many "outliers"that ogives fitted to the mean square ofprice changes are much lower and flatterthan the distribution of the data them-selves (see, e.g., Fig. 1). The tails of thedistributionsof price changes are in factso extraordinarily ong that the samplesecond moments typically vary in anerratic fashion. For example, the secondmoment reproduced n Figure 2 does notseem to tend to any limit even thoughthe samplesize is enormousby economicstandards,and even though the series towhich it applies is presumably station-ary.

It is my opinionthat these facts war-rant a radically new approach to theproblemof price variation.4The purposeof this paperwill be to presentand testsuch a new model of price behavior inspeculative markets. The principalfea-

ture of this model is that starting fromthe Bachelierprocessas applied to log,Z(t) instead of Z(t), I shall replace theGaussian distributions throughout byanotherfamily of probability aws, to bereferredto as "stable Paretian," whichwerefirstdescribedn PaulLevy's classic

Calculdesprobabilites 1925). In a some-what complex way, the Gaussian is alimiting case of this new family, so the

new model is actually a generalizationofthat of Bachelier.Since the stable Paretian probability

laws are relatively unknown, I shall be-gin with a discussionof some of the more

important mathematical properties of

these laws. Following this, the resultsofempirical tests of the stable Paretianmodel will be examined.The remainingsectionsof the paperwill thenbe devotedto a discussion of some of the moresophisticatedmathematical and descrip-tive properties of the stable Paretianmodel. I shall, in particular,examineitsbearingon the very possibility of imple-mentingthe stop-loss rulesof speculation

(Section VI).

FIG. 1.-Two histograms llustratingdeparturefrom normalityof the fifth and tenth difference fmonthly wool prices, 1890-1937.In each case, thecontinuousbell-shapedcurve represents he Gaus-sian "interpolate" aseduponthe samplevariance.Source: Gerhard Tintner, The Variate-DifferenceMethod Bloomington, nd., 1940).ence, referringeither to Z(t) or to log,Z(t) and

plottedon variouskindsof coordinates, an be foundin ArnoldLarson,"Measurement fa RandomProc-ess in FuturePrices,"FoodResearchnstituteStudies,I (1960), 313-24; M. F. M. Osborne, "BrownianMotion in the Stock Market,"Operations esearch,VII (1959), 145-73, 807-11; S. S. Alexander,"PriceMovements n SpeculativeMarkets:Trendsof RandomWalks?" ndustrialManagement eviewof M.I.T., II, pt. 2 (1961),7-26.

4Such an approach asalso beennecessary-andsuccessful-in othercontexts; orbackgroundnfor-mation and many additionalexplanations ee my"NewMethods n StatisticalEconomics," ournalofPoliticalEconomy,Vol. LXXI (October,1963).

I believe,however, hat each of the applicationsshouldstand on their own feet andhave minimized

the numberof crossreferences.



396 THE JOURNALOF BUSINESS

II. MATHIEMATICAL TOOLS: PAUL LEVY'S

STABLE PARETIAN LAWS

A. PROPERTY OF ?CSTABILITYYYF THE GAUSSIAN

LAW AND ITS GENERALIZATION

Oneof the principalattractionsof themodified Bachelier process is that thelogarithmicrelative

L(t, T) = loge Z(t + T) - log, Z(t)

is a Gaussian randomvariable for everyvalue of T; the only thing that changeswith T is the standard deviation of L(t,T). This feature s the consequenceof thefollowing fact:

LetG' andG" betwo ndependentGaus-sian randomvariables,of zero meansand

of mean squares respectively qual to o-J2

and o0-12. Then,the sum G' + G" is also aGaussianvariable,of meansquareequal o0)2 + 0u"2. In particular, the "reduced"

Gaussian variable, with zero mean andunit mean square, s a solutionto

(S) s'U + s"U = sU,

where is a function of s' and s" given bytheauxiliaryrelation

(A2) s2 =S-2 + S"2

It should be stressedthat, fromthe view-point of equation (S) and relation (A2),

the quantities s', s", and s are simplyscale factors that "happen"to be closelyrelated to the root-mean-square n theGaussiancase.

The property (S) expresses a kind ofstability or invariance under addition,which is so fundamentalin probabilitytheory that it came to be referred tosimply as "stability." The Gaussian isthe only solution of equation (S) for

which the second moment is finite-orfor which the relation (A2) is satisfied.

,When the variance is allowed to be in-finite, however, (S) possessesmany othersolutions.This was shown constructivelyby Cauchy,who consideredthe randomvariable U for which

Pr(U > u) = Pr(U <-u)

= a-(1/7r) tan-l (u),

so that its density is of the formdPr(U < u) = [7r(l + u2)]f.

For this law, integral moments of allordersareinfinite,and the auxiliaryrela-tion takes the form

(A1) s = si + si/

where the scale factors s', s", and s arenot definedby any moment.

As to the general solution of equation

FIG. 2.-Both graphsarerelativeto the sequen-tial samplesecondmomentof cottonpricechanges.Horizontal cale represents ime in days, with twodifferentorigins T?: on the upper graph, T? wasSeptember21, 1900; on the lower graph T? wasAugust 1, 1900.Vertical inesrepresenthe value ofthe function

(T-7T?)-1j [L0t11) 12 X

t= TO

whereL(t, 1) = log. Z(t + 1) - log. Z(t) and Z(t)is the closing potpriceof cotton onday 1,asprivate-ly reported by the United States DepartmentofAgriculture.



VARIATION OF CERTAIN SPECULATIVE PRICES 397

(S), discoveredby Paul Levy,5the loga-rithm of its characteristicunction takesthe form

co(PL)log exp(iuz)dPr( U< u)=i6z

-yZ a [1+j ( Z/ Z I )tan(axr/2)]

It is clear that the Gaussian aw andthe law of Cauchy are stable and thatthey correspond o the cases(a = 2) and(a = 1; A = 0), respectively.

Equation (PL) determines a familyof distribution and density functionsPr(U < u) and dPr(U < u) that depend

continuouslyuponfourparameterswhichalso happento play the roles usuallyas-sociated with the first four moments ofU, as, for example, in Karl Pearson'sclassification.

First of all, the a is an indexof "peak-edness"that variesfrom0 (excluded)to2 (included); if a = 1, A must vanish.This a will turn out to be intimatelyrelatedto Pareto'sexponent.The A is anindex of "skewness" hat can vary from- 1 to + 1. If - 0, the stable densitiesare symmetric.

Onecan say that a and f togetherde-termine the "type" of a stable randomvariable, and such a variable can becalled "reduced" f y =- 1 and a = 0. Itis easy to see that, if U is reduced,sU isa stable variablehaving the same valuesfor a, f and a and having a value of yequal to sa: this means that the third

parameter, y, is a scale factor raised tothe power a. Supposenow that U' andU" are two independentstable variables,reducedand having the same values fora and $; sincethe characteristicunction

of s'U' + s"U" is the product of thoseof s'U' and of s"U", the equation (S) isreadily seen to be accompaniedby the

auxiliaryrelation(A) S s=S + S"t

If on the contraryU' and U" are stablewith the same values of a, a and of8 = 0, but with differentvalues of oy re-spectively,oy' nd oy"), he sum U' + U"is stable with the parametersa, (, 7 =

7' + y" and 8 = 0. Thus the familiaradditivity property of the Gaussian"variance" (definedby a mean-square)

is now played by either oyor by a scalefactor raisedto the power a.

The final parameterof (PL) is 8;strict-ly speaking, equation (S) requiresthat8 = 0, but wehave addedthe term i8z to(PL) in order to introduce a locationparameter.If 1 < a ? 2 SO hat E(U) isfinite,one has 8 = E(U); if A = 0 so thatthe stable variablehas a symmetricden-sity function, 8 is the median or modal

valueof U; but 8hasno obvious nterpre-tation when 0 < a < 1 with A 0 0.

B. ADDITION OF MORE THAN TWO

STABLE RANDOM VARIABLES

Let the independentvariablesU. sat-isfy the condition (PL) with values of a,

A, y,and 8equalfor all n. Then, the loga-rithm of the characteristic unction of

SN = Ul +U2 + * *Un + * * UN

is N timesthe logarithmof the character-istic functionof Un,and it equals

i 8Nz Ny zIa [1+ iO(z/Izf) tan(ar/2)],

so that SNis stable with the same a and( as Un, and with parameters8 and ymultipliedby N. It readily follows that

NU. - a andN-'/a E (US-6)

n=1

have identical characteristic functions

and thus are identicallydistributedran-

Paul L6vy, Calcul des probabilitds Paris:Gauthier-Villars,925);PaulLevy, Thdoriee 'addi-lion des variables l.atoires Paris: Gauthier-Villars,1937 [2d ed., 1954]). The most accessible ource onthese problems s, however,B. V. GnedenkoandA. N. Kolmogoroff,Limit Distributionsor Sums ofIndependent andomVariables, rans. K. L. Chung

(Reading,Mass.: Addison-Wesley ress, 1954).




"stable non-Gaussian"by the less nega-tive one of "stableParetian." The twonumbersa' and a" sharethe role of thestandard deviation of a Gaussian vari-ableand will be designatedas the "stand-ard positive deviation"and the "stand-ard negative deviation."

In the extreme cases whereA = 1 andhence C" = 0 (respectively,where ( =

-1 and C' = 0), the negative tail (re-spectively, the positive tail) decreasesfaster than the law of Pareto of index a.In fact, one can prove6that it withersaway even fasterthan the Gaussianden-

sity so that the extreme cases of stablelaws arepracticallyJ-shaped. They playan important role in my theory of thedistributions of personal income or ofcity sizes.A numberof furtherpropertiesof stable laws may thereforebe foundinmy publicationsdevotedto these topics.7

D. STABLE VARIABLES AS THE ONLY POSSIBLE LIM-

ITS OF WEIGHTED SUMS OF INDEPENDENT

IDENTICALLY DISTRIBUTED ADDENDS

The stability of the Gaussian aw maybe consideredas being only a matter ofconvenience,and it is often thought thatthe following property is more impor-tant.

Let the Un be independent, denticallydistributed,andomvariables,withafinitea2 = E[Un - E(U)]2. Then the classicalcentral imit theorem sserts hat

N

NlimN-1 -1E [ Un-E (U)]N -+co

n=1

is a reducedGaussianvariable.This resultis of course the basis of the

explanation of the presumedoccurrenceof the Gaussian law in many practicalapplicationsrelativeto sums of a varietyof random effects. But the essentialthing in all-theseaggregativeargumentsis not that [U - E(U)] is weightedbyany special factor, such as N-1/2, butratherthat the following is true:

There exist two functions, A(N) and

B(N), such that, as N -? o, the weightedsum

N

(L) A (N) E~ Un B (N)n=1

has a limit that s finite and is notreducedto a non-randomonstant.

If the varianceof Un s not finite,how-ever, condition (L) may remainsatisfiedwhilethe limit ceases to be Gaussian.Forexample, if Un is stable non-Gaussian,the linearly weighted sum

N-l/af2 ( Un - )

was seen to be identical n law to Un,sothat the "limit" of that expression s al-ready attained for N = 1 and is a stablenon-Gaussian law. Let us now supposethat U. is asymptotically Paretian with0 < a < 2, but not stable;one can showthat the limit exists in a real sense, and

that it is the stable Paretianlaw havingthe same value of a. Again the functionA (N) can be chosen equal to N-/ .These results are crucialbut I had betternot attempt to rederive hem here.Thereis little sensein copying the readilyavail-able full mathematical arguments, andexperience howsthat what was intendedto be an illuminating heuristicexplana-tion often looks like anotherinstance inwhichfar-reaching onclusionsare based

6A. V. Skorohod, "Asymptotic Formulas for

Stable Distribution Laws," Dokl. Ak. Nauk SSSR,XCVIII (1954), 731-35, or Sdect. Tranl. Math. Stat.Proba. Am. Math. Soc. (1961), pp. 157-61.

7 Benoit Mandelbrot, "The Pareto-L6vy Lawand the Distribution of Income," International Eco-nomic Review, I (1960), 79-106, as amended in "TheStable Paretian Income Distribution, When the Ap-parent Exponent Is nearTwo," International Econom-ic Review, IV (1963), 111-15; see also my "StableParetian Random Functions and the MultiplicativeVariation of Income," Econometrica,XXIX (1961),517-43, and "Paretian Distributions and IncomeMaximization,"QuarterlyJournal of Economics,LXXVI (1962), 57-85.



400 THE JOURNAL OF BUSINESS

on loose thoughts.Let me therefore ustquote the facts:

The problemof the existenceof a limitfor A(N)iUn - B(N) can be solved byintroducing the following generalizationof the asymptotic law of Pareto:8

The conditions of Pareto-Doeblin-Gnedenko.-Introducehenotations

Pr(U > u) = Q(u)m-a ;

Pr(U <-u) = Q"(u)ua-,.

Theconditionsof P-D-G require hat (a)whenu -* c, Q'(u)/Q"(u) tends to a limit

C'/C", (b) thereexists a valueof a > 0such that or everyk > 0, andfor u -*

one has

Q (u) +Q ( ) -Q'(ku) +Q"(ku)

These conditionsgeneralize he law ofPareto,forwhich Q'(u)and Q"(u)them-selves tend to limits as u -> c. With

their help, and unless a = 1, the prob-lem of the existenceof weightingfactors

A(N) and B(N) is solved by the follow-ing theorem:

If the Un,are independent, denticallydistributedrandom variables,there mayexist no functions A(N) and B(N) suchthatA(N) Z Un- B(N) tendsto a properlimit. But, if suchfunctions A(N) andB(N) exist, oneknows hatthe limit is oneofthesolutionsofthestabilityequation S).Moreprecisely,he imitis Gaussian f and

only if theUnhasfinite variance;helimitis stablenon-Gaussian f and only if theconditions fPareto-Doeblin-Gnedenkoresatisfied or some 0 < a < 2. Then A =

(C' - C")/(C' + C") and A(N) is de-terminedbytherequirementhat

N Pr[U > u A-'(N)] -* C'u-a

(Whicheverthe value of a, the P-D-Gcondition(b) also plays a central role inthe study of the distribution of the ran-dom variable max Us.)

As an application of the above defini-tion and theorem, let us examine theproduct of two independent,identicallydistributed Paretian (but not stable)variables U' and U". First of all, foru > 0, one can write

Pr(U'UI" > u) = Pr(U' > 0; U" > 0

and log U' + log U" > log u)

+ Pr(U' <; IU"< Oand

log IU'I + log U"I > logu).

But it followsfromthe lawof Pareto that

Pr(U > ez) -' C' exp(- az) and

Pr(U < - ez) -,' C" exp(- az),

where U is either U' or U". Hence, thetwo terms P' and P" that add up toPr(U'U" > u) satisfy

PI,, C'2 az exp(- az) andPI' , C"2 az exp(- az).

Therefore

Pr(U'U" > u) -,' a(C'2 + C"2) (log, U) Ua.

Similarly

Pr(U'U" < - u) --' a2C'C"(loge u) U-a.

It is obvious that the Pareto-Doeblin-Gnedenkoconditionsare satisfiedfor the

functions Q'(u) (CI2+ C"2)a loge U

and Q"(u) -,' 2C'C"a log, u. Hence theweighted expression

(N logN) 1/a2S U' U"nn=1

convergestoward a stable Paretian limitwith the exponenta and the skewness

= (C'2 + C"2 - 2C'C"l)/(C'2 + C"12

+ 2C'C")= [(C' C")/(C' + C"I)]2> 0.

8See Gnedenkoand Kolmogoroff, p. cit., n. 4,p. 175,who usea notationthat doesnot emphasize,as I hope to do, the relationbetweenthe law ofParetoand its presentgeneralization.



VARIATIONOF CERTAINSPECULATIVEPRICES 401

In particular, he positive tail shouldal-ways be bigger than the negative.

E. SHAPE OF STABLE PARETIAN DISTRIBUTIONS

OUTSIDE ASYMPTOTIC RANGE

The result of Section IIC should nothide the fact that the asymptotic be-havior is seldom the main thing in theapplications.For example, if the samplesize is N, the ordersof magnitudeof thelargestand smallest item are given by

N Pr[U > u+(N)] = 1,and

NPr[U <-u-(N)= 1,

and the interesting values of u lie be-tween -u- and u+. Unfortunately,ex-cept in the cases of Gaussand of Cauchyand the case (a = 2; = 1), there are noknown closed expressionsfor the stabledensities and the theory only says thefollowing: (a) the densities are alwaysunimodal; (b) the densities depend con-tinuously upon the parameters; (c) if

A > 0, the positive tail is the fatter-hence, if the mean is finite (i.e., if 1 <a < 2), it is greaterthan the most prob-able value and greaterthan the median.

To go further, I had to resort to nu-merical calculations. Let us, however,begin by interpolativearguments.

The symmetriccases, fi = O.-For a =

1,onehasthe Cauchy aw, whosedensity[ir(1+ u2)]-l is always smaller than theParetian density 1/wXu2 oward which ittends in relative value as u --> . There-

fore, Pr (U> u) <l/lr u,

and it follows that for a = 1 the doublylogarithmicgraphof loge [Pr(U > u)] isentirely on the left side of its straightasymptote. By continuity, the sameshape must apply when a is only a littlehigherora little lower than 1.

For a = 2, the doubly logarithmicgraph of the Gaussian oge [Pr(U > u)]

dropsdownveryfast to negligiblevalues.

Hence, again by continuity, the graphfora = 2 - e must also begin by a rapiddecrease.But, since its ultimate slope isclose to 2, it must have a point of inflec-tion corresponding o a maximumslopegreater than 2, and it must begin by"overshooting" its straight asymptote.

Interpolatingbetween 1 and 2, we seethat thereexists a smallest value of a, sayao, for -which the doubly logarithmicgraph begins by overshooting ts asymp-tote. In the neighborhood of a?, theasymptotic a can be measuredas a slopeeven if the sampleis small. If a < a?, the

asymptotic slope will be underestimatedby the slope of small samples;for a > a0it will be overestimated.The numericalevaluationof the densities yields a valueof a0 n the neighborhood f 1.5. A graph-ical presentation of the results of thissection is given in Figure3.

The skew cases.-If the positive tail isfatter than the negative one, it may wellhappen that its doublylogarithmicgraphbegins by overshooting its asymptote,whilethe doublylogarithmicgraphof thenegative tail does not. Hence, there aretwo criticalvalues of a0, onefor each tail;if the skewnessis slight, a is between thecritical values and the samplesize is notlarge enough,the graphs of the two tailswill have slightly different over-all ap-parent slopes.

F. JOINT DISTRIBUTION OF INDEPENDENT

STABLE PARETIAN VARIABLES

Let p1(u1)and p2(u2)be the densitiesof Ui and of U2. If both u1 and u2 arelarge, the joint probability density isgiven by

p0(Uli U2) = aC1'u ( a+1) aC2'u2 (a+1)

= a2C 'C2' (U1U2)( a+l)

Hence, the lines of equal probabilityareportionsof the hyperbolas

Ulu2= constant.




In the regions where either U1 or U2islarge (but not both), these bits of hyper-bolas are linked together as in Figure 4.That is, the isolines of small probability

have a characteristic"plus-sign"shape.On the contrary, when both U1 and U2are small, logepl(ul) and logep2(u2) arenear their maximaand thereforecan belocally approximated by a, - (ul/bl)2 and

a2 (U2/b2)2. Hence, the probability iso-lines are ellipses of the form

(ul/b1)2 + (u2/b2)2= constant.

The transition between the ellipsesand the "plus signs" is, of course, con-tinuous.

G. DISTRIBUTION OF U1, WHEN U1 AND U2 ARE

INDEPENDENT STABLE PARETIAN VARI-

ABLES AND U1 + U2 = U IS KNOWN

This conditional distribution can beobtainedas the intersectionbetween thesurfacethat representsthe joint densitypo(ul, u2) and the plane ul + u2 = u.

Hence the conditional distribution isunimodal for small u. For large u, it hastwo sharplydistinctmaxima ocated nearul = 0 and near u2 = 0.

More precisely, the conditional den-sity of Ui is given by p1(u1)p2(u - u)/

q(u), where q(u) is the density of U =

U1+ U2. Let u be positive and verylarge; if ul is small, one can use theParetian approximations or p2(u2) and

q(u), obtainingP1(u1)P2(u -ui)/q(u)

[Cl'/(Cl' + C2')]pl(ul).

If U2 iS small,onesimilarly btains

P1(U1)P2(U -ui)/q(u)

[C2'/C1' + C2')]p2(U - U1)

In other words, the conditional den-sity pl(ul)p2(u -ul)/q(u) looks as if

two unconditioned distributions scaleddown in the ratios Cl'/(Cl' + C2') and

C2'/(C1' + C2') had been placed near

ul = 0 and ul = u. If u is negative but

Jul is very large, a similar result holdswith Ci" and C2" replacingCl' and C2'.

For example, for a = 2 - and Cl' =

C2', the conditioneddistribution s made

up of two almost Gaussianbells, scaleddown to one-half of their height. But, as

a tends toward2, these two bellsbecome

FIG. 3.-The various lines are doubly logarithmicplots of the symmetric stable Paretian probabilitydistributions with a = 0, y = 1, B = 0 and variousvalues of a. Horizontally, log, u; vertically, log,Pr(U > u) = log, Pr(U < - u). Sources: unpub-lished tables based upon numerical computationsperformed at the author's request by the I.B.M.Research Center.




smaller and a third bell appears nearUl = u/2. Ultimately, the two side bellsvanish and one is left with a central bellwhich corresponds o the fact that whenthe sum U1 + U2 is known, the condi-tional distribution of a Gaussian UI isitself Gaussian.

III. EMPIRICAL TESTS OF THE STABLE

PARETIAN LAWS: COTTON PRICES

This section will have two main goals.First, from the viewpoint of statisticaleconomics, its purpose is to motiVateand develop a model of the variation of

speculative prices based on the stableParetian laws discussed in the previoussection. Second, from the viewpoint ofstatistics consideredas the theory of dataanalysis, I shall use the theorems con-cerningthe sums22Uno build a new testof the lawof Pareto. Beforemovingon tothe main points of the section, however,let us examine two alternative ways oftreating the excessive numbers of largeprice changes usually observed in thedata.

A. EXPLANATION OF LARGE PRICE CHANGES BY

CAUSAL OR RANDOM "CONTAMINATORS"

Onevery commonapproach s to notethat, a posteriori, arge pricechangesareusually traceable to well-determined"causes" that should be eliminated be-fore one attempts a stochasticmodel ofthe remainder.Such preliminarycensor-ship obviously brings any distributioncloser to the Gaussian. This is, for ex-ample, what happenswhen one restrictshimselfto the study of "quietperiods"ofprice change.There need not be any ob-servablediscontinuitybetweenthe "out-liers" and the rest of the distribution,however, and the above censorship isthereforeusuallyundeterminate.

Another popular and classical proce-dure assumesthat observationsare gen-

erated by a mixture of two normal dis-

tributions, one of which has a smallweight but a large variance and is con-sidered as a random"contaminator." norderto explain the sample behaviorofthe moments, it unfortunatelybecomesnecessary o introducea largernumberofcontaminators,and the simplicity of themodel is destroyed.

B. INTRODUCTION OF THE LAW OF PARETO

TO REPRESENT PRICE CHANGES

I propose to explain the erratic be-havior of sample moments by assumingthat the populationmoments areinfinite,an approachthat I have used with suc-cess in a number of other applicationsand which I have explainedand demon-strated in detail elsewhere.

This hypothesis amounts practicallyto the law of Pareto. Let us indeed as-sume that the increment

L(t, 1) = loge Z(t + 1) - logeZ(t)

FIG. 4.-Joint distribution of successive pricerelativesL(t, 1) and L(t + 1, 1) undertwo alterna-tive models.If L(t, 1) andL(t + 1, 1) are independ-ent, they shouldbe plotted alongthe horizontal ndverticalcoordinate xes.If L(t, 1)andL(t + 1, 1)arelinked by the model in SectionVII, they shouldbeplotted along the bisectrixes, or else the abovefigure should be rotated by 45? before L(t, 1) andL(t + 1, 1) are plotted along the coordinateaxes.




is a randomvariablewith infinitepopu-lation moments beyond the first. Thisimplies that its density p(u) is such thatfp(u) u2du diverges but fp(u) udu con-

verges (the integrals being taken all theway to infinity). It is of coursenatural,at least in the first stage of heuristic mo-tivating argument,to assume that p(u)is somehow "well behaved" for large u;if so, our two requirementsmean that asU -> cx, p(u)u3 tends to infinity andp(u)u2tends to zero.

In words: p(u) must somehow de-crease faster than u-2 and slower than

u-3. From the analytical viewpoint, thesimplest expressions of this type arethose with an asymptotically Paretianbehavior. This was thefirst motivation ofthe present study. It is surprising that I

couldfindno recordof earlierapplicationof the law of Pareto to two-tailed phe-nomena.

My furthermotivationwasmoretheo-retical. Grantedthat the facts impose arevision of Bachelier'sprocess, it would

be simple indeed if one could at leastpreserve the convenient feature of theGaussianmodel that the various incre-ments,

L(t, T) = log, Z(t + T) - log, Z(t),

depend upon T only to the extent of hav-ing differentscale parameters.From allother viewpoints, price incrementsoverdays, weeks, months, and years wouldhave the same distribution,which wouldalso rule the fixed-base relatives. Thisnaturallyleads directly to the probabil-ists' concept of stability examined inSection II.

In other terms, the facts concerningmoments, togetherwith a desire to havea simple representation, suggested acheckas to whether the logarithmicpricerelatives for unsmoothed and unproc-essed time series relative to very active

speculativemarkets are stable Paretian.

Cottonprovideda good example,and thepresent paper will be limited to the ex-amination of that case. I have, however,also established that my theory appliesto many other commodities (such aswheat and other edible grains), tomany securities (such as those of therailroadsn theirnineteenth-centuryhey-day),andto interestrates suchas those ofcall or time money.9On the other hand,there areunquestionablymany economicphenomena for which much fewer "out-liers" are observed, even though theavailable series are very long; it is natu-

ral in these cases to favor Bachelier'sGaussianmodel-known to be a limitingcase in my theory as well as its proto-type. I must, however,postponea discus-sion of the limits of validity of my ap-proachto the study of prices.

C. PARETO'S GRAPHICAL METHOD APPLIED

TO COTTON-PRICE CHANGES

Let us begin by examining n Figure5the doubly logarithmicgraphsof various

kinds of cotton price changes as if theywere independent of each other. Thetheoretical log Pr(U > u), relative toa = 0, a = 1.7, and A = 0, is plotted(solidcurve)on the same graph for com-parison.If the various cotton pricesfol-lowedthe stable Paretian awwith a = 0,a = 1.7 and A = 0, the various graphsshould be horizontal translates of eachother, and a cursory examinationshowsthat the data are in close conformity

with the predictions of my model. Acloser examination suggests that thepositive tails contain systematicallyfewer data than the negative tails, sug-

9 These examples were mentioned in my 1962"Research Note" (op. cit., n. 1). My presentation,however, was too sketchy and could not be improvedupon without modification of the substance of that"Note" as well as its form. I prefer to postpone ex-amination of all the other examples as well as thesearch for the point at which my model of cottonprices ceases to predict the facts correctly. Both will

be taken up in my forthcoming book (op. cit., n. 1).




gesting that A actually takes a smallneg-ative value. This is also confirmedby thefact that the negative tails alonebegin by

slightly "overshooting" their asymp-totes, creating the bulge that should beexpected whena is greater than the criti-cal value a' relative to one tail but notto the other.

D. APPLICATION OF THE GRAPHICAL METHOD

TO THE STUDY OF CHANGES IN THE

DISTRIBUTION ACROSS TIME

Let us now look more closely at thelabels of the various series examinedin

the previoussection. Two of the graphsrefer to daily changes of cotton prices,near 1900and near 1950, respectively.It

is clear that these graphs do not coincidebut are horizontal translates of eachother. This implies that between 1900and 1950 the generating process haschanged only to the extent that its scale,y has become much smaller.

Ournext test willbe relativeto month-ly pricechanges over a longer time span.It would be best to examine the actualchanges between, say, the middle of one

FIG. 5.-Composite of doubly logarithmic graphs of positive and negative tails for three kinds of cottonprice relatives, together with cumulated density function of a stable distribution. Horizontal scale u of

lines la, lb, and Ic is marked only on lower edge, and horizontal scale it of lines 2a, 2b, and 2c is markedalong upper edge. Vertical scale gives the following relative frequencies: (la) Fr[loge Z(t + one day) -

loge Z(t) > u], (2a) Fr[loge Z(t + one day) - loge Z(t) < - u], both for the daily closing prices of cottonin New York, 1900-1905 (source: private communication from the United States Department of Agricul-ture).

(lb) Fr[logeZ(t + one day) - log, Z(t) > u], (2b) Fr[logeZ(t + one day) - log, Z(t) < - u], both foran index of daily closing prices of cotton in the United States, 1944-58 (source: private communicationfrom Hendrik S. Houthakker).

(1c) Fr[loge Z(t + one month) - loge Z(t) > u], (2c) Fr[loge Z(t + one month) - log, Z(t) < -u

both for the closing prices of cotton on the 15th of each month in New York, 1880-1940 (source: privatecommunication from the United States Department of Agriculture).

The reader is advised to copy on a transparency the horizontal axis and the theoretical distribution andto move both horizontally until the theoretical curve is superimposed on either of the empirical graphs;the only discrepancy is observed for line 2b; it is slight and would imply an even greater departure from

normality.




monthto the middle of the next. A longersample is available, however, when onetakes the reported monthly averagesofthe price of cotton; the graphsof Figure6 were obtainedin this way.

If cotton prices were indeed generatedby a stationary stochastic process, ourgraphs should be straight, parallel, anduniformly spaced. However, each of the15-year subsamples contains only 200-odd months, so that the separate graphscannot be expected to be as straight asthose relative to our usual samples of1,000-odd tems. The graphs of Figure6

are, indeed,not quite as neat as those re-lating to longer periods; but, in the ab-sence of accurate statistical tests, theyseem adequately straight and uniformlyspaced, except for the period 1880-96.

I conjecture herefore hat, since 1816,the process generatingcotton priceshaschanged only in its scale, with the pos-sible exceptionof the Civil War and ofthe periods of controlled or supportedprices. Long series of monthly pricechanges should therefore be representedby mixturesof stable Paretian laws; suchmixtures remain Paretian.10

E. APPLICATION OF THE GRAPHICAL METHOD

TO STUDY EFFECTS OF AVERAGING

It is, of course, possible to derivemathematically the expected distribu-tion of the changes between successivemonthly means of the highest and lowestquotation; but the result is so cumber-

some as to be useless. I have, however,ascertainedthat the empirical distribu-tion of these changes does not differ sig-nificantly from the distribution of thechanges between the monthly means ob-tained by averagingall the daily closingquotations within months; one maythereforespeak of a single averagepricefor each month.

Wethen see on Figure7 that the great-er part of the distribution of the aver-ages differsfrom that of actual monthlychanges by a horizontal translation tothe left, as predicted in Section IIC(actually, in orderto apply the argumentof that section, it wouldbe necessarytorephrase t by replacingZ(t) by log, Z(t)throughout;however, the geometric andarithmetic averages of daily Z(t) do notdiffer much in the case of medium-sizedover-allmonthly changesof Z(t)).

However, the largest changes betweensuccessive averagesaresmaller than pre-

dicted. This seems to suggest that thedependence between successive dailychanges has less effect upon actualmonthly changes than upon the regu-larity with which these changesare per-formed.

IF. A NEW PRESENTATION OF THE EVIDENCE

Let me now show that my evidenceconcerningdaily changes of cotton pricestrengthens my evidence concerningmonthly changesand conversely.

The basicassumptionof my argumentis that successive daily changes of log(price)are independent.(This argumentwill thus have to be revised when the as-sumption is improvedupon.) Moreover,the population second moment of L(t)seems to be infinite and the monthly oryearly price changes are patently notGaussian.Hence the problemof whether

any limit theoremwhatsoeverappliestolog, Z(t + T) - log, Z(t) can alsobe an-swered in theoryby examining whetherthe daily changes satisfy the Pareto-Doeblin-Gnedenkoconditions. In prac-tice, however, it is impossible to everattain an infinitelylarge differencingn-terval T or to ever verify any conditionrelativeto an infinitely arge value of therandomvariable u. Hence one must con-

siderthat a month or a year is infinitely

10See my "New Methods n StatisticalEconom-

ics," Journal of Political Economy, October,1963.



FIG. 6.-A rough test of stationarity for the process of change of cotton prices between 1816 and 1940.Horizontally, negative changes between successive monthly averages (source: Statistical Bulletin No. 99 ofthe Agricultural Economics Bureau, United States Department of Agriculture.) (To avoid interferencebetween the various graphs, the horizontal scale of the kth graph from the left was multiplied by 2k-1.)

Vertically, relative frequencies Fr(U < - u) corresponding respectively to the following periods (fromleft to right):1816-60,1816-32, 1832-47,1847-61,1880-96,1896-1916,1916-31,1931-40,1880-1940.

FIG. 7.-These graphs illustrate the effect of averaging. Dots reproduce the same data as the lines 1cand 2c of Fig. 5. The x's reproduce distribution of loge Z?(t + 1) - loge ZO(t),where ZO(t) is the averagespot price of cotton in New York during the month t, as reported in the Statistical Bulletin No. 99 of theAgricultural Economics Bureau, United States Department of Agriculture.




long, and that the largestobserved dailychanges of loge Z(t) are infinitely large.Under these circumstances, one can

make the followinginferences.Inferencerom aggregation.-The cot-ton price data concerningdaily changesof loge Z(t) surely appear to follow theweaker condition of Pareto-Doeblin-Gnedenko. Hence, from the propertyofstability and according to Section IID,one should expect to find that, as T in-creases,

7-l/a { oge Z(t + T) - loge Z(t)

-T E[L(t, 1)])tends toward a stable Paretianvariablewith zero mean.

Inference from disaggregation.-Dataseem to indicate that price changes overweeks and months follow the same lawup to a change of scale. This law musttherefore be one of the possible non-Gaussian limits, that is, it must be astable Paretian.As a result, the inverse

partof the theoremof SectionIID showsthat the daily changes of log Z(t) mustsatisfy the conditionsof Pareto-Doeblin-Gnedenko.

It is pleasant to see that the inversecondition of P-D-G, which greatly em-barrassedme in my work on the distribu-tion of income,can be put to use in thetheory of prices.

A few of the difficulties involved inmaking the above two inferences will

now be discussed.Disaggregation.-The P-D-G condi-

tions areweakerthan the asymptotic lawof Pareto becausethey requirethat lim-its exist for Q'(u)/Q"(u) andfor [Q'(u) +

Q"(u)]/[Q'(ku)+ Q"(ku)], but not forQ'(u) and Q"(u) taken separately. Sup-pose, however,that Q'(u)and Q"(u)stillvary a great deal in the useful range oflarge daily variations of prices. If so,

A(N)2Un-

B(N) will not approach ts

own limit until extremelyarge values ofN are reached.Therefore, f one believesthat the limit is rapidly attained, the

functions Q'(u) and Q"(u) of dailychangesmust vary very little in the re-gionsof the tails of the usualsamples.Inother words,it is necessaryafter all thatthe asymptotic law of Pareto apply todaily price changes.

Aggregation.-Here, the difficultiesareof a differentorder.Fromthe mathemati-cal viewpoint, the stable Paretian lawshould become ncreasinglyaccurateas Tincreases.Practically, however, there is

no sense in even consideringvalues of Tas long as a century,becauseone cannothope to get samples sufficientlylong tohave adequately inhabited tails. Theyear is an acceptable span for certaingrains,but only if one is not worriedbythe fact that the long available series ofyearly pricesare ill known and variableaveragesof small numbersof quotations,not pricesactually quotedon some mar-

ket on a fixed day of each year.From the viewpoint of economics,there are two much more fundamentaldifficultieswith very large T. First of all,the model of independentdailyL's elimi-nates from considerationevery "trend,"except perhapsthe exponentialgrowthordecay due to a non-vanishing 8. Manytrends that are negligible on the dailybasis would, however, be expectedto bepredominanton the monthly or yearly

basis. For example,weathermight haveupon yearly changes of agriculturalpricesan effectdifferentfromthe simpleadditionof speculativedaily pricemove-ments.

The second difficulty lies in the"linear"characterof the aggregationofsuccessiveL's used in my model. Since Iuse natural ogarithms,a smalllog, Z(t +T) - log, Z(t) will be undistinguishable

from the relative price change [Z(t +




T) - Z(t)]/Z(t). The addition of smallL's is therefore related to the so-called"principle of random proportionate ef-

fect"; it also means that the stochas-tic mechanism of prices readjusts itselfimmediately to any level that Z(t) mayhave attained. This assumption is quiteusual, but very strong. In particular, Ishall show that, if one finds that logZ(t + one week) - log Z(t) is very large,it is very likely that it differs ittle fromthe change relative to the single day ofmost rapid price variation (see SectionVE); naturally, this conclusion onlyholds for independentL's. As a result,the greatest of N successive daily pricechanges will be so large that one mayquestionboth the use of log, Z(t) and theindependenceof the L's.

There are other reasons (see SectionIVB) to expect to find that a simplead-dition of speculative daily price changespredicts values too high for the pricechanges over periods such as whole

months.Given all these potential difficulties,Iwas franklyastonishedby the quality ofthe predictionsof my model concerningthe distributionof the changesof cottonpricesbetween the fifteenthof one monthand the fifteenth of the next. The nega-tive tail has the expected bulge,and eventhe most extremechangesarepreciseex-trapolates from the rest of the curve.Even the artificial excisionof the Great

Depressionand similarperiodswouldnotaffect the generalresultsvery greatly.

It was thereforeinteresting to checkwhether the ratios between the scale co-efficients,C'(T)/C'(1) and C"(T)/C"(1),were both equalto T, as predictedby mytheory whenever the ratios of standarddeviations o'(T)/o-'(s) and o"(T)/I"(s)followthe TI/ageneralizationof the "TT1/2

Law" referredto in Section IIB. If the

ratios of the C parameter are different

from T, theirvalue may serveas a meas-ure of the degreeof dependencebetweensuccessiveL(t, 1).

The above ratios were absurdlylargein my original comparisonbetween thedaily changes near 1950 of the cottonprices collectedby Houthakkerand themonthly changesbetween1880and 1940of the prices communicated by theUSDA. This suggested that the sup-ported prices around 1950 varied lessthan their earlier counterparts.There-fore I repeatedthe plot of daily changesfor the period near 1900, chosen hap-

hazardly but not actually at random.The new values of C'(T)/C'(1) andC"(T)/C"(1) became quite reasonable,equal to each other and to 18. In 1900,there were seven tradingdays per week,but they subsequently decreased to 5.Besides, one cannot be too dogmaticabout estimatingC'(T)/C'(1). Thereforethe behaviorof this ratio indicatedthatthe "apparent"numberof tradingdays

per month was somewhat smaller thanthe actual number.

IV. WHY ONE SHOULD EXPECT TO FIND

NONSENSE MOMENTS AND NONSENSE

PERIODICITIES IN ECONOMIC TIME SE-

RIES

A. BEHAVIOR OF SECOND MOMENTS AND

FAILURE OF THE LEAST-SQUARES

METHOD OF FORECASTING

It is amusing to note that the first

known non-Gaussian stable law, namely,Cauchy'sdistribution,was introduced nthe course of a study of the method ofleast squares.In a surprisingly ively ar-gument following Cauchy's 1853 paper,J. Bienayme"lstressed that a methodbased upon the minimizationof the sum

"J. Bienayme,"Considerations I'appuide lad6couvertede Laplacesur la loi de probabilit6 ansla m6thodedes moindrescarres,"Comptes endus,Acad6tmiees Sciencesde Paris, XXXVII (August,

1853), 309-24 (esp.321-23).




of squaresof sampledeviations cannotbereasonably used if the expected value ofthis sum is known to be infinite. Thesameargumentapplies fully to the prob-lem of least-squares smoothing of eco-nomic time series,when the "noise"fol-lows a stable Paretian law other thanthat of Cauchy.

Similarly, consider the problem ofleast-squares forecasting, that is, of theminimizationof the expectedvalue of thesquare of the errorof extrapolation. Inthe stable Paretian case this expectedvalue will be infiniteforevery forecast,so

that the method is, at best, extremelyquestionable.One can perhaps apply amethod of "least c-power" of the fore-casting error, where r < a, but such anapproach would not have the formalsimplicity of least squares manipula-tions; the most hopeful case is that of

= 1, which correspondsto the mini-mizationof the sum of absolutevalues ofthe errorsof forecasting.

B. BEHAVIOR OF THE KURTOSIS AND ITS FAILUREAS A MFASURE OF "PEAKEDNESS"

Pearson's index of "kurtosis" is de-fined as

_3 ? fourthmomentsquareof the secondmoment'

If 0 < a < 2, the numerator and thedenominatorboth have an infinite ex-pected value. One can, however, showthat the kurtosis behaves proportion-ately to its "typical" value given by

(1/N) (mostprobablealueof 2 L4)

[(1/N) (mostprobable alueof2 L2)]'

const.N-1+4/a

[const.N"1+2/a]2 const.N.

In other words, the kurtosis is expectedto increase without bound as N -.* .

For small N, things are less simple butpresumablyquite similar.

Let me examinethe work of Cootner nthis light.'2He has developed the tempt-ing hypothesisthat pricesvary at randomonly as long as they do not reach eitheran upperor a lower bound, that are con-sidered by well-informed speculators todelimit an interval of reasonable valuesof the price. If and when ill-informedspeculatorslet the price go too high ortoo low, the operations of the well-in-formed speculatorswill induce this priceto come back within a "penumbra"a laTaussig. Under the circumstances,theprice changes over periods of, say, four-

teen weeks shouldbe smallerthan wouldbe expected if the contributing weeklychangeswere independent.

This theory is very attractive a prioribut could not be generallytrue because,in the case of cotton, it is not supportedby the facts. As for Cootner's own jus-tification, it is based upon the observa-tion that the price changesof certain se-curities over periods of fourteen weekshave a much smaller kurtosis than one-week changes. Unfortunately,his samplecontains 250-odd weekly changes andonly 18fourteen-weekperiods.Hence,onthe basis of generalevidenceconcerningspeculativeprices,I would have expecteda priorito find a smaller kurtosisfor thelonger time increment, and Cootner'sevidence is not a proof of his theory;other methods must be used in ordertoattack the still very open problemof the

possible dependence between successiveprice changes.

C. METHOD OF SPECTRAL ANALYSIS

OF RANDOM TIME SERIES

Applied mathematicians are fre-quently presented these days with thetask of describingthe stochastic mecha-

Paul H. Cootner, "Stock Prices: RandomWalksvs. Finite Markov Chains," ndustrialMan-agementReview f M.I.T., III (1962), 24-45.




nism capableof generatinga given timeseriesu(t), known orpresumed o be ran-dom. The responseto such questions is

usually to investigate first what is ob-tained by applying the theory of the"second-orderrandom processes."Thatis, assuming that E(U) = 0, one formsthe sample covariance

U(t)U(t+ r),

which is used, somewhat indirectly, toevaluate the population covariance

R(T)=

E[U(t)U(t + r)] .Of course, R(r) is always assumed to befinite for all; its Fouriertransformgivesthe "spectraldensity" of the "harmonicdecomposition" of U(t) into a sum ofsine and cosine terms.

Broadly speaking, this method hasbeen very successful, though manysmall-sampleproblemsremain unsolved.Its applicationsto economicshave, how-ever, been questionable even in thelarge-samplecase. Within the context ofmy theory, there is unfortunatelynoth-ing surprising n such a finding. The ex-pression2E[U(t)U(t + r)] equalsindeedE[U(t) + U(t + 7)]2 - E[U(t)]2 -

E[U(t + r)]2; these three variances areall infinite for time series coveredby mymodel, so that spectralanalysis loses itstheoreticalmotivation. I must, however,postponea more detailedexaminationof

this fascinating problem.

V. SAMPLE FUNCTIONS GENERATED BY

STABLE PARETIAN PROCESSES; SMALL-

SAMPLE ESTIMATION OF THE MEAN

"DRIFT"1 OF SUCH A PROCESS

The curvesgeneratedby stable Pare-tian processes present an even largernumber of interesting formations thanthe curves generated by Bachelier'sBrownian motion. If the price increase

over a long period of time happens aposteriori o have beenusually large,in astable Paretian market, one should ex-pect to find that this changewas mostlyperformedduringa few periodsof espe-cially high activity. That is, one will findin most cases that the majority of thecontributing daily changes are distrib-uted on a fairly symmetriccurve,whileafew especially high values fall well out-side this curve. If the total increaseis of the usual size, the only differencewill be that the daily changeswill showno "outliers."

In this section these results will beused to solve onesmall-sample tatisticalproblem, that of the estimation of themean drift b, when the other parametersare known. We shall see that there is no"sufficient statistic" for this problem,and that the maximumlikelihoodequa-tion does not necessarily have a singleroot. This has severe consequences romthe viewpoint of the very definitionofthe concept of "trend."

A. CERTAIN PROPERTIES OF SAPLE

PATHS OF BROWNIAN MOTION

As noted by Bachelierand (independ-ently of him and of each other) by sev-eral modern writers,'3 he sample pathsof the Brownian motion very much"look like" the empirical curves of timevariationof pricesor of priceindexes. Atcloserinspection,however,one sees very

well the effect of the abnormalnumberof13 See esp. HolbrolkWorking,"A Random-Dif-

ference Seriesfor Use in the Analysisof Time Se-ries," Journal of the American Statistical Association,XXIX (1934), 11-24;MauriceKendall,"TheAnal-ysis of Economic Time-Series-Part I: Prices,"Journal of she Royal Statistical Society, Ser. A,CXVI (1953), 11-34;M. F. M. Osborne, BrownianMotion in the Stock Market," op. cit.; Harry V.Roberts, "Stock-Market Patterns'and FinancialAnalysis:MethodologicalSuggestions,"Journal ofFinance,XIV (1959), 1-10; and S. S. Alexander,"PriceMovements n SpeculativeMarkets:Trendsor Random

Walks,"op. cit., n.3.




large positive and negative changes ofloge Z(t). At still closer inspection, onefinds that the differences oncernsome of

the economically most interesting fea-tures of the generalizedcentral-limit he-orem of the calculusof probability. It isthereforenecessaryto discuss this ques-tion in detail, beginningwith a reminderof some classicalfacts concerning Gaus-sian randomvariables.

Conditionaldistributionof a GaussianL(t), knowing L(t, T) = L(t, 1) +. . . + L(t + T - 1, 1).-Let the prob-

ability density of L(t, T) be

(2 ro-2T)-112xp[- (u - 5T)2/2To2]

It is then easy to see that-if one knowsthe valueu of L(t, T)-the densityofanyof the quantities L(t + r, 1) is given by

[27ro2(T-1)/TI]1/2

r -uIT-/)21exL2 a2(T- 1) IT

We see that each of the contributingL(t + r, 1) equals u/T plus a Gaussianerror erm.Forlarge T, that term has thesame varianceas the unconditionedL(t,1); one canin fact prove that the value ofu has little influenceupon the size of thelargest of those "noise terms." One canthereforesay that, whichever ts value,uis roughly uniformlydistributedover theT time intervals, each contributingneg-ligibly to the whole.

Sufficiencyof ufor the estimationof themean drift 6 from the L(t + r, 1).-Inparticular,a has vanished from the dis-tribution of any L(t + r, 1) conditionedby the value of u. This fact is expressedin mathematicalstatistics by saying thatu is a "sufficient tatistic"for the estima-tion of a from the values of all the L(t +r, 1). That is, whichevermethod of esti-mation a statistician may favor, his esti-

mate of a must be a function of u alone.

The knowledge of intermediatevalues ofloge Z(t + r) is of no help to him. Mostmethods recommend estimating a by

u/T and extrapolating he future linearlyfrom the two known points, loge Z(t) andloge Z(t + T).

Since the causes of any price move-ment can be traced back only if it isample enough, the only thing that can beexplained,in the Gaussian case is themean drift interpreted as a trend, andBachelier'smodel, which assumes a zeromean for the price changes, can only rep-resent the movement of prices once the

broad causal parts or trends have beenremoved.

B. SAMPLE FROM A PROCESS OF INDEPENDENT

STABLE PARETIAN INCREMENTS

Returning to the stable Paretian case,supposethat one knows the values of 'yand ,B (or of C' and C") and of a. Theremainingparameter s the mean driftb,which one must estimate starting fromthe known L(t, T) =

log,Z(t + T) -

log, Z(t).The unbiased estimate of 3 is L(t,

T)/T, while the maximum ikelihood es-timate matches the observed L(t, T) toits a priori most probable value. The"bias" of the maximum likelihood isthereforegiven by an expressionof theform zyi/af(3), where the function f(,)must be determinedfrom the numericaltables of the stable Paretian densities.

Since : is mostly manifestedin the rela-tive sizes of the tails, its evaluation re-quires very large samples,and the qual-ity of one's predictions will dependgreatly upon the quality of one's knowl-edge of the past.

It is, of course, not at all clear thatanybodywouldwish the extrapolation obe unbiased with respectto the mean ofthe change of the logarithmof the price.

Moreover, he bias of the maximum ike-




lihood estimate comes principally froman underestimateof the size of changesthat are so large as to be catastrophic.The forecaster may thereforevery wellwishto treat suchchangesseparatelyandto take account of his private feelingsabout many things that are not includedin the independent-incrementmodel.

C. TWO SAMPLES FROM A STABLE

PARETIAN PROCESS

Supposenow that T is even and thatone knows L(t, T/2) and L(t + T/2,T/2) and their sum L(t, T). We haveseenin SectionII G that, when the value

u = L(t, T) is given, the conditional dis-tribution of L(t, T/2) depends verysharply upon u. This means that the to-tal change u is not a sufficientstatisticfor the estimation of 8; in other words,the estimatesof 8will be changed by theknowledgeof L(t, T/2) and L(t + T/2,T/2).

Considerfor example the most likelyvalue 8. If L(t, T/2) and L(t + T/2,

T/2) areof the same orderof magnitude,this estimate will remain close to L(t,T)/T, as in the Gaussian case. But sup-pose that the actuallyobserved alues ofL(t, T/2) and L(t + T/2, T/2) are veryunequal,thus implying that at least oneof these quantitiesis very differentfromtheircommonmeanand median. Such anevent is most likely to occur when 8 isdose to the observed value either ofL(t + T/2, T/2)/(T/2) or of L(t, T/2)/

(T/2).We see that as a result, the maximum

likelihood equationfor 8 has two roots,respectively close to 2L(t, T/2)/T andto 2L(t + T/2, T/2)/T. That is, themaximum-likelihood rocedure ays thatone should neglect one of the availableitems of information,any weightedmeanof the two recommendedextrapolationsbeingworse than either;but nothing says

which item one shouldneglect.

It is clearthat few economists will ac-cept such advice. Some will stress thatthe most likely value of a is actuallynothing but the most probablevalue inthe case of a uniform distribution of apriori probabilities of 8. But it seldomhappens that a priori probabilities areuniformlydistributed.It is also true, ofcourse,that they areusuallyvery poorlydetermined; in the present problem,however, the economist will not need todetermine these a priori probabilitieswith any precision: t will be sufficient ochoose the most likelyfor him of the two

maximum-likelihood stimates.An alternative approach to be pre-

sented later in this paperwill arguethatsuccessive ncrementsof loge Z(t) are notreally independent, so that the estima-tion of 8 depends upon the order of thevalues of L(t, T/2) and L(t + T/2, T/2)as well as upontheir sizes.This may helpeliminate the indeterminacyof estima-tion.

A third alternative consists in aban-doningthe hypothesisthat 8 is the samefor both changes L(t, T/2) and L(t +T/2, T/2). For example, if these changesarevery unequal,onemay be temptedtobelieve that the trend 8 is not linearbutparabolic.Extrapolationwould then ap-proximatelyamount to choosing amongthe two maximum-likelihoodestimatesthe one which is chronologically he lat-est. This is an example of a variety of

configurationswhichwouldhave been sounlikely in the Gaussiancase that theyshould be consideredas non-randomandwould be of help in extrapolation.In thestable Paretian case, however, theirprobability may be substantial.

D. THREE SAMPLES FROM A STABLE

PARETIAN PROCESS

The number of possibilities increasesrapidly with the sample size. Assume

now that T is a multiple of 3, and con-




sider L(t, T/3), L(t + T/3, T/3), andL(t + 2T/3, T/3). If these three quanti-ties are"of comparablesize, the knowl-edge of log Z(t + T/3) and log Z(t +2T/3) willagainbring ittle changeto theestimate basedupon L(t, T).

But suppose that one datum is verylarge and the other are of much smallerand comparablesizes. Then, the likeli-hood equationwill have two local maxi-mums, having very different positionsand sufficientlyequalsizes to makeit im-possible to dismiss the smaller one. Theabsolute maximum yields the estimate

a = (3/2T) (sumof the two small data);the smaller local maximum yields theestimate 6 = (3/T) (the large datum).

Supposefinallythat the three data areof very unequal sizes. Then the maxi-mumlikelihoodequationhas threeroots.

This indeterminacyof maximum like-lihood can again be lifted by one of thethree methods of Section VC. For ex-ample, if the middledatumonly is large,the method of non-linear extrapolationwill suggesta logistic growth.If the dataincreaseor decrease-when taken chron-ologically-one will rather try a para-bolic trend. Again the probability ofthese configurationsarisingfrom chanceunder my model will be much greaterthan in the Gaussiancase.

E. A LARGE NUMBER OF SAMPLES FROM

A STABLE PARETIAN PROCESS

Let us now jump to a very largenum-

ber of data. In order to investigate thepredictionsof my stable Paretianmodel,we must first re-examine he meaningtobe attached to the statement that, inorder that a sum of random variablesfollow a centrallimit of probability,it isnecessary that each of the addends benegligiblerelative to the sum.

It is quite true,of course,that one canspeak of limit laws only if the value of

the sum is not dominatedby any single

addend known in advance. That is, tostudy the limit of A(N)2Un- B(N),one must assume that (for every n)Pr|A(N) Un- B(N)/NI > e) tends to

zero with 1/N.As each addend decreases with 1/N,

their number ncreases,however, and theconditionof theprecedingparagraphdoesnot by itself insure that the largest of the

IA N) U4 - B(N)/N is negligible incomparisonwith their sum. As a matterof fact, the last condition is true only ifthe limit of the sum is Gaussian.In theParetian case, on the contrary, the fol-

lowing ratios,max fA(N) Un B(N)/N f

A(N)ZU, - B(N)and

sum of k largest IA(N) Un - B(N)/NIA(N)2 Un--B (BN)

tend to non-vanishinglimits as N in-creases.14If one knowsmoreover hat thesum A (N)2J; n- B(N) happens to be

large,onecanprovethat the above ratiosshouldbe expectedto be close to one.

Returning to a process with independ-ent stable Paretian L(t), we may saythe following:If, knowing a, f, 'y,and 6,one observes that L(t, T = one month)is not large, the contribution of the dayof largest pricechange s likely to be non-negligible-n relativevalue, but it will re-main small in absolute value. For largebut finiteN, this will not differtoo muchfrom the Gaussianpredictionthat eventhe largestaddendis negligible.

Suppose however that L(t, T= onemonth)is very arge.The Paretiantheory

14 Donald Darling,"The Influenceof the Maxi-mum Term n the Additionof IndependentRandomVariables," Transactionsof the American Mathemati-cal Society,LXX (1952),95-107; and D. Z. Arovand A. A. Bobrov, "The Extreme Members ofSamplesand Their Role in the Sum of IndependentVariables," Theory of Probability and Its Applica-

tions, V (1960),415-35.




then predicts that the sum of a few larg-est daily changeswill be very closeto thetotal L(t, T); if one plots the frequenciesof various values of L(t, 1), conditionedby a known and very large value forL(t, T), one should expect to find thatthe law of L(t + r, 1) contains a fewwidely "outlying" values. However, ifthe outlying values are taken out, theconditioned distribution of L(t + -r, 1)shoulddepend ittle uponthe value of theconditioning L(t, T). I believe this lastprediction to be very well satisfied byprices.

Implications concerning estimation.-Supposenow that a is unknownand thatone has a large sample of L(t + r, 1)'s.The estimationprocedure onsists in thatcase of plotting the empirical histogramand translatingit horizontallyuntil onehas optimized its fit to the theoreticaldensity curve. One knows in advancethat this best value will be very little in-fluenced by the largest outliers. Hence"rejectionof the outliers" is fully justi-fied in the present case, at least in itsbasic idea.

F. CONCLUSIONS CONCERNING ESTIMATION

The observationsmade in the preced-ing sections seemto confirmsome econo-mists' feeling that predictionis feasibleonly if the samplesize is both very largeand stationary, or if the sample size issmall but the sample values are of com-

parablesizes. One can also predictwhenthe sample size is one, but here theunicity of the estimator is only due toignorance.

G. CAUSALITY AND RANDOMNESS IN

STABLE PARETIAN PROCESSES

We mentionedin SectionV A that, inorder to be "causally explainable," aneconomic change must at least be largeenough to allow the economist to trace

back the sequenceof its causes.As a re-

sult, the only causal part of a Gaussianrandom unction is the mean drift8.Thiswill also apply to stable Paretian random

functionswhen their changes happen tobe roughlyuniformly distributed.Things are different when loge Z(t)

varies greatly between the times t ant + T, changing mostly during a few ofthe contributing days. Then, these larg-est changesare sufficientlyclear-cut,andare sufficiently separated from "noise,"to be traced back and explainedcausally,just as well as the mean drift.

In others words,a carefulobserverof a

stable Paretian random function will beable to extract causalparts from it. But,if the total change of loge Z(t) is neithervery largenorvery small, there will be alarge degree of arbitrariness n this dis-tinction between causal and random.Hence one couldnot tell whether the pre-dicted proportionsof the two kinds ofeffects are empiricallycorrect.

To sum up, the distinction betweenthe causaland the randomareasis sharpin the Gaussiancase and very diffuseinthe stable Paretian case. This seems tome to be a strong recommendation nfavor of the stable Paretianprocessas amodel of speculativemarkets. Of course,I have not the slightest idea why thelarge price movements should be repre-sentedin this way by a simple extrapola-tion of movements of ordinary size. Icame to believe, however,that it is very

desirablethat both "trend"and "noise"be aspects of the same deeper "truth,"whichmay not be explainable oday, butwhichcan be adequatelydescribed.I amsurely not antagonistic to the ideal ofeconomics:eventually to decomposeeventhe "noise" ntopartssimilar o the trendand to link various series to each other.But, until we can approximate his ideal,we can at least representsome trendsas

being similarto "noise."




H. CAtJSALITYAND RANDOMNESS IN

AGGREGATION "IN PARALLEL"

Borrowing a term from elementaryelectricalcircuit theory, the addition ofsuccessive daily changes of a price maybe designated by the term "aggrega-tion in series," the term "aggregation nparallel"applying to the operation

I

L(t, T) = EL(i, t, T),i=l

I T-1

'= E L(i,tT 1)i=1 T0?

where i refers to "events" that occursimultaneouslyduringa given time inter-val such as T or 1.

In the Gaussian case, one should, ofcourse, expect any occurrenceof a largevalue for L(t, T) to be traceable to arareconjunctionof largechanges n all ormost of the L(i, t, T). In the stable Pare-tian case, one shouldon the contrary ex-

pect largechangesL(t, T) to be traceableto one or a small number of the con-tributingL(i, t, T). It seemsobvious thatthe Paretian predictionis closer to thefacts.

To add up the two types of aggrega-tion in a Paretianworld,a largeL(t, T) islikely to be traceable to the fact thatL(i, t + r, 1) happens to be very largefor one or a few sets of values of i and ofr. These contributionswould stand outsharplyandbe causally explainable.But,aftera while, they should of courserejointhe "noise"madeup by the otherfactors.The next rapidchangeof logeZ(t) shouldbe due to other "causes."If a contribu-tion is "trend-making"n the above senseduring a large number of time-incre-ments, one will, of course,doubt that itfalls underthe same theoryas the fluctu-ations.

VI. PRICE VARIATION IN CONTINUOUS TIME

AND THE THEORY OF SPECULATION

The main point of this section is to

show that certainsystems of speculation,which would have been advantageousifone could implement them, cannot inreality be followed in the case of priceseries generated by a Paretian process.

A. INFINITE DIVISIBILITY OF

STABLE PARETIAN LAWS

Whichever N, it is possible to con-sider that a stable Paretian increment

L(t, 1) = logeZ(t + 1) - logeZ(t)is the sum of N independent, identicallydistributed,randomvariables, and thatthose variables differ from L(t) only bythe value of the constants y, C' and C",which are N times smaller.

In fact, it is possible to interpolate theprocess of independent stable Paretianincrementsto continuoustime, assumingthat L(t, dt) is a stable Paretianvariable

with a scale coefficient y(dt) = dt y(l).This interpolatedprocess is a very im-portant "zeroth"orderapproximation othe actualpricechanges.That is, its pre-dictions are surelymodifiedby the mech-anisms of the market,but they are veryilluminatingnonetheless.

B. PATH FUNCTIONS OF A STABLE

PROCESS IN CONTINUOUS TIME

It is almost universally assumed, in

mathematical models of physical or ofsocial sciences, that all functions cansafely be consideredas being continuousand as havingas many derivativesas onemay wish. The functions generated byBachelierareindeed continuous("almostsurely almost everywhere,"but we mayforget this qualification); although theyhave no derivatives ("almost surely al-most nowhere"), we need not be con-cerned because price quotations are al-




ways rounded to simple fractions of theunit of currency.

In the Paretian case things are quitedifferent.If my process s interpolated ocontinuoust, the paths which it gener-ates become everywhere discontinuous

(or rather, they become "almost surelyalmosteverywherediscontinuous").Thatis, most of their variation is performedthrough non-infinitesimal"jumps," thenumberof jumps largerthan u and locat-ed within a time increment T, beinggiven by the law C'T Id(w-a) .

Let us examine a few aspects of thisdiscontinuity. Again, very small jumpsof loge Z(t) could not be perceived, since

price quotations are always expressed nsimple fractions. It is moreinterestingtonote that there is a non-negligibleprob-ability that a jump of price is so largethat "supply and demand"cease to bematched.In otherwords,the stablePare-tian modelmay be consideredas predict-ing the occurrenceof phenomenalikelyto force the market to close. In a Gaus-sian model such large changesare so ex-tremely unlikely that the occasional

closureof the markets must be explainedby non-stochasticconsiderations.

The most interestingfact is, however,the large probability predicted for me-dium-sizedjumps by the stable Paretianmodel. Clearly, if those medium-sizedmovements were oscillatory, they couldbe eliminated by market mechanismssuch as the activities of the specialists.But if the movement is all in one direc-

tion, market specialists could at besttransforma discontinuityinto a changethat is rapid but progressive. On theotherhand, very few transactionswouldthen be expected at the intermediatesmoothing prices.As a result, even if theprice Z?is quoted transiently,it may beimpossibleto act rapidly enough to sat-isfy more thana minute fractionof orders

to "sell at ZO." n other words, a largenumberof intermediateprices are quotedeven if Z(t) performs a large jump in ashort time; but they are likely to be sofleeting, and to apply to so few transac-tions, that they are irrelevant from the

viewpoint of actually enforcinga "stoploss order" of any kind. In less extremecases-as, for example, when borrowingsare oversubscribed-the market mayhave to resort to special rules of alloca-tion.

These remarksare the crux of my criti-cism of certainsystematic methods: theywould perhaps be very advantageous ifonly they could be enforced,but in fact

they can only be enforcedby very fewtraders. I shall be content here with adiscussionof one exampleof this kind ofreasoning.

C. THE FAIRNESS OF ALEXANDER'S GAME

S. S. Alexanderhas suggestedthe fol-lowingruleof speculation:"if the marketgoes up 5%,go long and stay long until itmoves down 5%, at which time sell andgo short until it again goes up 5%."15

This procedure is motivated by thefact that, according o Alexander's nter-pretation, data would suggest that "inspeculative markets, price changes ap-pear to follow a randomwalk over time;but ... if the markethas moved up x%,it is likely to move up more than x%further before it moves down x%." Hecalls this phenomenonthe "persistenceof moves." Since thereis no possible per-sistenceof moves in any "randomwalk"with zero mean, we see that if Alexan-der's interpretation of facts were con-firmed, one would have to look at a veryearly stage for a theoretical mprovementover the random walk model.

In order to follow this rule, one mustof course watch a price series continu-

15 S. S. Alexander, op. cit. n. 3.




ously in time andbuy or sellwhenever tsvariationattains the prescribedvalue. Inotherwords, this rule can be strictly fol-lowedif and only if the process Z(t) gen-erates continuous path functions, as forexample in the original Gaussianprocessof Bachelier.

Alexander'sprocedurecannot be fol-lowed, however, in the case of my ownfirst-approximation model of pricechange in which there is a probabilityequal to one that the first move notsmallerthan 5 per cent is greater han 5per cent and notequalto 5 per cent. It is

therefore mandatory to modify Alex-ander'sscheme to suggestbuyingor sell-ing whenmoves of 5 per cent arefirst ex-ceeded.One can prove that the stableParetian theory predicts that this gamealso is fair. Therefore,the evidence-asinterpretedby Alexander-would againsuggest that one must go beyond thesimple model of independent incrementsof price.

But Alexander's nferencewas actuallybased upon the discontinuousseriescon-stituted by the closing prices on succes-sive days. He assumed that the inter-mediate prices could be interpolated bysome continuous function of continuoustime-the actual formof which need notbe specified.That is, whenever therewasa differenceof over5 percentbetween theclosing price on day F' and day F",Alexander mplicitly assumed that there

was at least one instance between thesemoments when the price had gone upexactly5 per cent. He recommendsbuy-ing at this instant, and he computestheempiricalreturns to the speculatoras ifhe were able to follow this procedure.

For price series generated by myprocess,however, the priceactually paidfor a stock will almost always be greaterthan that correspondingo a 5 per centrise; hence the speculatorwill almost al-

ways have paid more than assumed inAlexander's evaluation of the returns.On the contrary,the price received willalmost always be less

than suggested byAlexander. Hence, at best, Alexanderoverestimates the yield corresponding ohis method of speculation and, at worst,the very impression hat the yield is posi-tive may be a delusion due to overopti-mistic evaluationof what happensduringthe few most rapid price changes.

One can of course imagine contractsguaranteeing hat the brokerwill charge(or credit) his client the actual price

quotationnearestby excess (or default)to a price agreed upon, irrespective ofwhether the broker was able to performthe transactionat the price agreed upon.Such a system would make Alexander'sprocedureadvantageous to the specula-tor; but the money he would be makingon the average would come from hisbroker and not from the market; andbrokerage ees would have to be such asto make the game at best fair in the longrun.

VII. A MORE REFINED MODEL

OF PRICE VARIATION

Broadly speaking, the predictions ofmy mainmodel seem to me to be reason-able. At closer inspection, however, onenotes that large price changes are notisolated betweenperiods of slow change;they rather tend to be the result of sev-

eral fluctuations, some of which "over-shoot" the final change. Similarly, themovement of prices in periods of tran-quillity seem to be smoother than pre-dicted by my process. In other words,large changes tend to be followed bylarge changes-of either sign-and smallchanges tend to be followed by smallchanges,so that the isolinesof low prob-ability of [L(t, 1), L(t - 1, 1)] are X-shaped.In the case of daily cotton prices,




Hendrik S. Houthakkerstressed this factin several conferencesand private con-versation.

Such an X shape can be easily ob-

tained by rotation from the "plus-signshape" which was observedin Figure 4to be applicable when L(t, 1) and L(t -

1, 1) are statistically independent andsymmetric.The necessaryrotationintro-duces the two expressions:

S(t) = (1/2)[L(t, 1) + L(t - 1, 1)]

= (1/2) [logeZ(t + 1) - logeZ(t - 1)1

and

D(t) = (1/2) [L(t, 1) -L(t- 1, 1)]

= (1/2) [logeZ(t+ 1) - 2 logeZ(t)

+ loge Z(t - 1)].

It follows that in order to obtain the Xshape of the empiricalisolines, it wouldbe sufficientto assume that the first andsecond finite differencesof loge Z(t) aretwo stable Paretian random variables,independent of each other and naturally

of logeZ(t) (see Fig. 4). Sucha process sinvariantby time inversion.

It is interesting to note that the dis-tribution of L(1, 1), conditionedby theknown L(t - 1, 1), is asymptoticallyParetian with an exponentequalto 2a +1.1' Since,for the usualrangeof a, 2a + 1

is greaterthan 4, it is clearthat no stableParetian law can be associated with theconditioned LQ, 1). In fact, even the

kurtosis is finite for the conditionedL(t, 1).Let us then consider a Markovian

process with the transition probabilityI have just introduced. If the initialL(TO, 1) is small, the first values of

L(t, 1) will be weakly Paretian with ahigh exponent 2a + 1, so that loge Z(t)will begin by fluctuatingmuch less rap-idly than in the case of independent L(t,

1). Eventually, however,a large L(t0, 1)will appear. Thereafter,L(t, 1) will fluc-tuate forsometime between values of theorders of magnitude of L(t0, 1) and-L(t0, 1). This will last long enough tocompensate fully for the deficiency oflarge values during the period of slowvariation. In otherwords, the occasionalsharp changesof L(t, 1) predicted by themodelof independentL(t, 1) are replacedby oscillatory periods, and the periodswithout sharp changeare less fluctuatingthan when the L(t, 1) are independent.

We see that, for the correct estimationof a, it is mandatoryto avoid the elimina-tion of periodsof rapidchangeof prices.One cannot argue that they are "caus-ally" explainableand ought to be elimi-nated before the "noise" is examinedmore closely. If one succeeded in elimi-nating all large changesin this way, one

would have a Gaussian-like remainderwhich, however,would be devoid of anysignificance.

16 Proof: Pr[L(t, 1) > u, when w < L(t - 1,1) < w + dw] is the product by (l/dw) of the in-tegral of the probability density of [L(t - 1, l)L(t,1)], over a strip that differs infinitesimally from thezone defined by

S(t) > (u + w)/2;

w + S(t < D(t) < w + S(t) + dw.

Hence, if u is large as compared to w, the conditionalprobability in question is equal to the followingintegral, carried from (u + w)/2 to .

fC'as-(a+l)C'a(s + w)-(a+l) ds

- (2a + 1)-i (C')2a22-(2a+l) U-(2a+l)

mandelbrot variations

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