Wavelet methods in (financial) time-series processing

Centrum voor Wiskunde en Informatica

REPORT

RAPPORT

Wavelet Methods in (Financial) Time-series Processing

Z.R. Struzik

Information Systems (INS)

CWI

P.O. Box 94079

1090 GB Amsterdam

The Netherlands

CWI is the National Research Institute for Mathematics

and Computer Science. CWI is part of the Stichting

Mathematisch Centrum (SMC), the Dutch foundation

for promotion of mathematics and computer science

and their applications.

SMC is sponsored by the Netherlands Organization for

Scientific Research (NWO). CWI is a member of

ERCIM, the European Research Consortium for

Informatics and Mathematics.

Copyright © Stichting Mathematisch Centrum

P.O. Box 94079, 1090 GB Amsterdam (NL)

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Telephone +31 20 592 9333

Telefax +31 20 592 4199

ZbigniewR.Struzik

CWI

P.O.Box94079,1090GBAmsterdam,TheNetherlands

email: Zbigniew.Struzik@cwi.nl

ABSTRACT

Webrie ydescribethemajoradvantagesofusingthewavelettransformfortheprocessingof nancialtime

seriesontheexampleoftheS&Pindex. Inparticular,weshowhowtouncoverlocalthescaling(correlation)

characteristicsoftheS&PindexwiththewaveletbasedeectiveHolderexponent[1,2 ]. Weuseittodisplay

the local spectral (multifractal)contents of the S&Pindex. In addition to this, we analyse thecollective

propertiesofthelocalcorrelationexponentasperceivedbythetrader,exercisingvarioustimehorizonanalyses

oftheindex.Weobservedanintriguinginterplaybetweensuch(dierent)timehorizons. Heavyoscillationsat

shortertimehorizonswhichseemtobeaccompaniedbyasteadydecreaseofcorrelationlevelforlongertime

horizons,seemtobecharacteristicpatternsbeforethebiggestcrashesoftheindex. Wendthatthiswayof

localpresentationofscalingpropertiesmaybeofeconomicimportance.

2000MathematicsSubjectClassication: 28A80,68T10,68P10,82D99

1999ACMComputingClassicationSystem: H1,I5,Jm,J2, E2

KeywordsandPhrases: econophysics,wavelettransform,Holderexponent,localcorrelation

Note: fullcolourversionofthispapercanbedownloadedfrom www.cwi.nl/~zbyszek

1. Introduction

Economicshasbeenproducingmoreandmorecomplicatedmodels,tryingtocapturedeviationsofthe

modelsituationfromreality. Butpatchingamodeltoincreaseitscomplexitymaynotbeanoptimal

wayofmodelling. Anyeconomicsystemisextremely complicatedbut, to alargedegree,thisis due

to the enormousnumber of degreesof freedom. Theinteractions betweeneconomic entities donot

needtobeverycomplicated. Approachesaimingat improvingthedescriptionofasystembymeans

ofhighordercorrectionsmay,therefore,havearatherslowconvergencetothesatisfactorymodel.

On the contrary, statistical physicsuses verysimple models of interactions between components

of a huge ensemble. Concepts from statistical physicshaveproven quite successful in the work of

econophysicists. Thepowerofmodern computersallowsnotonlythetesting ofsuchmodelsbutalso

theanalysis ofdatafor thepresenceofmultiscalingcharacteristics,in particularlocally in thedata.

Ananalysisofcorrelationscanbelinkedtothemodelsofinteractionbetweenthesystemcomponents,

andsocanthemultiscaledistributionanalysis.

Physicshasalongtraditionofdealingwithsystemswithanextremenumberofdegreesoffreedom,

where entities are coupled with very simpleinteraction rules. This is probably why economicshas

recentlyexperienced great interestamong physicists, followingthe pioneering work of Gene Stanley

et al[3],RosarioMantegna,[5],AlainArneodoet al,[4],MarcelAuslooset al[6] and,nottoforget,

theearlyworkofBenoitMandelbrot[7].

Whereas thephysicists derivethe nalmodelfrom thecharacteristicsofthe dataanalysed, using

theirlearnedphysicsknowledgeandscienticintuition, thelatestcomputersciencetrendistoassist

thedata analyserin model discovery. Data miningis thename forthis recentdirection in machine

data mining/model discoveryapproachesin theeconomicalsciences willgrowrapidlyin the coming

decade.

Therefore,workingwithoutanaprioriassumedmodelischaracteristicofthemodernapproachto

economics. Instead,themodelistobeinferredfromthedata. Thedataisanalysedintermsofvery

genericanalysismethodslike,forexample,waveletdecomposition. Thewavelettransformcomponents

are then analysed and, of course, simple or complex models can be tted to such decomposition

components. Thescalingofmomentsordistributionscanbetestedandahypothesisdrawn,butthe

rststepis dataanalysisinthemostgenericterms.

Thepossibilityofdoinganalysislocallyisanotherveryattractiveoption. Thestationarityofalmost

any statisticalcharacteristicfailswhen appliedto thenancialdata. Asimilar situationpertains in

other complex phenomena (take, for example, the human heartbeat [9, 10]). But where the

non-stationarities occur, interestingness begins, and with tools like the wavelet transform, capable of

tamingthenon-stationarities(trends),interesting(local)patterns canbediscoveredinthedata.

1

In section 2, we brie y introduce the wavelet transformation in its continuous form, we describe

therequirementsforthewaveletinsection3anddiscusstheadvantagesfortimeseriesprocessing. In

particular,wefocusin section4ontheabilityofthewavelettransformationtocharacterisescale-free

behaviourthroughtheHolderexponent. Wedescribeinbriefatechnicalmodelenablingustoestimate

thescale-freecharacteristic(theeectiveHolderexponent)forthebranchesofamultiplicativeprocess.

Amoreextensivecoverageofthismethodisavailablein[1,2]. Insection5,weusethederivedeective

HolderexponentforthelocaltemporaldescriptionoftheS&Pindex. Section6providesanextension

to the uctuation analysis of the eective Holderexponent of theS&P index. Section 7 closesthe

paperwithconclusions.

2. Why Wavelets?

Figure1: ContinuousWaveletTransformrepresentationoftherandomwalk(Brownianprocess)time

series. The wavelet used is the Mexican hat - the second derivative of the Gaussian kernel. The

coordinateaxesare: positionx,scalein logarithmlog(s),andthevalueofthetransformW(s;x).

1

Thewavelettransformis aconvolutionproductofthesignalwiththescaledandtranslatedkernel

-thewavelet (x).[11,12]Thescalingandtranslationactionsareperformedbytwoparameters;the

scaleparameters`adapts'thewidthofthewaveletkerneltotheresolutionrequiredandthelocation

oftheanalysingwaveletisdeterminedbytheparameterb:

(Wf)(s;b)= 1 s Z dxf(x) ( x b s ) (2.1)

wheres;b2Rand s>0forthecontinuousversion(CWT).

The3Dplotingure1showshowthewavelettransformrevealsmoreandmoredetailwhilegoing

towardssmallerscales,i.e. towardssmallerlog(s)values. Thewavelettransformissometimesreferred

toasthe`mathematicalmicroscope'[4],duetoitsabilitytofocusonweaktransientsandsingularities

inthetimeseries. Thewaveletuseddeterminestheopticsofthemicroscope;itsmagnicationvaries

withthescalefactors.

WhetherwewanttousecontinuousordiscreteWT,seegure2,islargelyamatterofapplication.

Forcoding purposes, onewantsto usethesmallest numberofcoeÆcients which canbecompressed

bythresholdinglowvaluesorusingcorrelationproperties. Forthispurposeadiscrete(forexamplea

dyadic)schemeofsamplingthescales,positionb spaceisconvenient. Suchsampling oftenspansan

orthogonalwaveletbase.

For analysis purposes, oneis not somuch concernedwith numerical ortransmissioneÆciency or

representationcompactness,but ratherwithaccuracy andadaptivepropertiesoftheanalysing tool.

Therefore, in analysis tasks, continuouswavelet decomposition is mostlyused. Thespace of scales

andpositionb,isthensampledsemi-continuously,usingthenestdataresolution available.

b

s

b

s/2

s/4

s/8

b/8

Figure2: Continuoussamplingoftheparameterspace(left)versusdiscrete(dyadic)sampling(right)

For decomposition,a simplebasis function is used. The wavelet , see Eq. 2.1 took its name from

its wave-likeshape. It has to cross the zero value line at least once since its mean value must be

zero. Thecriterion of zeromean is referredto asthe admissibility ofthe wavelet, and is related to

thefact thatonewantstohavethepossibilityofreconstructingtheoriginalfunctionfromitswavelet

decomposition.

signalbeingdecomposed. Thisistheso-calledbandpasslteringofthesignal. Onlyacertainband

offrequencies(levelofdetail)iscapturedbythewaveletsworkingatonescale. Ofcourse,atanother

scaleadierentset ofdetails (band offrequencies) iscaptured. Butother frequencies(inparticular

zerofrequency)arenottakeninto thecoeÆcients. Thisistheideaofdecomposition.

Kernels like the Gaussian smoothing kernel are low-pass lters, which means they evaluate the

entireset of frequenciesup to the currentresolution. This is the ideaof approximationat various

resolutions.

The reader may rightly guess here that it is possible to get band pass information

(wavelet-coeÆcients) from subtracting two low-pass approximations at various levels of resolution. This is,

in fact,aso-calledmulti-resolutionschemeof decomposition intoWT components. Butback tothe

admissibility-reconstructionfrommultipleresolutionapproximationswouldnotbepossiblesincethe

samelowfrequency detailwould bedescribed in several coeÆcientsof the low passdecomposition.

This, ofcourse,is notthecaseforwavelets;theyselectonlyanarrowband ofdetailwithverylittle

overlap(in theorthogonal casenooverlapat all!). Inparticular, if one requires that thewavelet is

zeroforfrequencyzero,i.e. itfullyblockszerofrequencycomponents,thiscorrespondswiththezero

meanadmissibilitycriterion.

3. The Wavelet

Theonlyadmissibility requirementforthewavelet isthatithaszeromean-itisawavefunction,

hencethenamewavelet.

Z 1

1

x (x)dx=0 (3.1)

However, in practice, wavelets are often constructed with orthogonality to a polynomial of some

degreen. Z 1 1 x n (x)dx=0 (3.2) Mexican Hat

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scaling function

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Figure 3: Left: thesmoothingfunction: theGaussian. Centre: waveletwithonevanishingmoment,

therstderivativeoftheGaussian. Right: waveletwithtwovanishingmoments,thesecondderivative

oftheGaussian.

Thispropertyofthewavelets-orthogonalitytopolynomialsofdegreen-hasaveryneapplication

insignalanalysis. Itisreferredtobythenameofthenumberofvanishingmoments. Ifthewaveletis

degreeP 0

. This canbe done, forexample, by the rstderivative of theGaussian kernelplotted in

gure3. SimilarlythesecondderivativeofthesameGaussiankernel,whichisoftenusedupsidedown

and thenappropriatelycalled theMexicanhatwavelet,hastwovanishing momentsandin addition

to constantscan also lterlinear trendsP

1

. Ofcourse, if thewavelethas m vanishingmoments, it

canlterpolynomialsofdegreem 1,m 2,... 0.

4. The

H

older Exponent

Theuseofvanishingmomentsbecomesapparentwhenweconsiderlocalapproximationstothe

func-tiondescribingourtimeseries. Supposewecanlocallyapproximatethefunctionwithsomepolynomial

P n

,buttheapproximationfailsforP n+1

. OnecanthinkofthiskindofapproximationastheTaylor

se-riesdecomposition. InfacttheargumentstobegivenaretrueevenifsuchTaylorseriesdecomposition

doesnotexist,butitcanserveasanillustration.

For the sakeof illustration, let us assume that the function f can be characterised by the Holder

exponenth(x

0 )inx

0

,andf canbelocally describedas:

f(x) x0 =c 0 +c 1 (x x 0 )+


+c n (x x 0 ) n +Cjx x 0 j h(x 0 ) =P n (x x 0 )+Cjx x 0 j h(x 0 ) : Theexponenth(x 0

) iswhat `remains' after approximatingwith P

n

andwhat doesnotyet `t' into

anapproximationwithP

n+1

. Moreformally,ourfunctionortimeseries f(x)islocally describedby

thepolynomialcomponentP

n

andtheso-calledHolderexponenth(x

0 ) . jf(x) P n (x x 0 )jCjx x 0 j h(x0) : (4.1)

It is traditionally considered to beimportant in economics to capture trend behaviourP

n . It is,

however,widelyrecognisedinothereldsthatitisnotnecessarilytheregularpolynomialbackground

but quite often the transient singular behaviour which can carry important information about the

phenomena/underlying system`producing'thetimeseries.

One of the main reasons for the focus on the regular component was that until the advent of

multi-scaletechniques(likeWT)capableoflocallyassessingthesingularbehaviour,itwaspractically

impossibletoanalysesingularbehaviour. Thisisbecausetheweaktransientexponentshareusually

completelymaskedbythemuch strongerP

n .

However,waveletsprovidea remedyin this case! The reader hasperhapsalready notedthe link

with thevanishing moments ofthe wavelets. Indeed, if thenumberof the vanishing momentsis at

leastashigh as thedegree ofP

n

, thewavelet coeÆcientswill capturethe localscaling behaviourof

thetimeseriesasdescribedbyh(x

0 ).

Infact,thephrase`ltering'withreferencetothepolynomialbiasisnotentirelycorrect. Theactual

lteringhappens onlyfor waveletsthe support of which is fully incorporatedin thebiased interval.

Ifthewaveletisattheedgeofsuchaninterval(wherebiasbegins)orifthecurrentresolutionofthe

waveletissimplytoolargewithrespecttothebiasedinterval,thewaveletcoeÆcientswillcapturethe

informationpertinenttothebias. Thisisunderstandablesincetheinformationdoesnotget`lost'or

`gained'in theprocessof WTdecomposition. The entire decomposed function canbereconstructed

from the wavelet coeÆcients,including the trendswithin the function. Whatwaveletsprovidein a

uniquewaytotameand managetrendsin alocalfashion, throughlocalisedwaveletscomponents.

Above, we have suggested that the function can locally be described with Eq. 4.1. Its wavelet

transformW

(n)

f withthewaveletwithat leastnvanishingmomentsnowbecomes:

W (n) f(s;x 0 )= 1 s Z Cjx x 0 j h(x 0 ) ( x x 0 s )dx=Cjsj h(x 0 ) Z jx 0 j h(x 0 ) (x 0 )dx 0 :

Therefore,wehavethefollowingpowerlawproportionalityforthewavelettransformofthe(Holder)

W (n) f(s;x 0 )jsj h(x0) :

From the functional form of the equation, one can immediately attempt to extract the value of

thelocalHolderexponentfromthescalingofthewavelettransformcoeÆcientsinthevicinityofthe

singularpointx

0

. This is indeed possiblefor singularities which are isolatedoreectivelyisolated,

that isthat canbeseenasisolatedfrom thecurrentresolution oftheanalysingwavelet. Acommon

approachtotracesuchsingularitiesandtorevealthescalingofthecorrespondingwaveletcoeÆcients

istofollowtheso-calledmaximalinesoftheCWTconvergingtowardstheanalysedsingularity. This

approachwasrstsuggestedbyStefanMallatet al[13]andlaterusedandfurtherdevelopedamong

othersinRefs[14, 4,15].

Ingure4,weplottheinputtimeserieswhichis apartoftheS&Pindex containingthecrashof

'87. In thesame gure,weplot correspondingmaxima derivedfrom theCWT decomposition with

the Mexican hatwavelet. The maxima correspondingto the crash stand outboth in the top view

(they are the longestones) and in theside log-logprojectionof allmaxima (they havea valueand

slopedierentfromtheremainingbulkofmaxima). Theonlymaximahigherin valuearetheendof

thesamplenitesizeeectmaxima. Theseobservationsindicatethat thecrashof'87canbeviewed

asan isolatedsingularity in theanalysed recordof theS&P index for practicallytheentire wavelet

rangeused.

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minima

maxima

crash ’87 maxima

log(s)

x

input time series(x)

S&P 500

-7

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0

1

2

3

4

5

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7

8

9

edge singularity maxima

crash singularity maxima

Wf(log(s))

log(s)

Figure4: Left: theinputtimeserieswiththeWTmaximaaboveitinthesamegure. Thestrongest

maximacorrespondtothecrashof'87. Theinputtimeseriesisde-biasedandL1normalised. Right:

weshowthesamecrashrelatedmaximahighlightedintheprojectionshowingthelogarithmicscaling

ofallthemaxima.

This is, however,(luckily)an unusual event andin general in time serieswehavedensely packed

singularities which cannot be seenasisolatedcasesfor awiderrangeof wavelet scales. Therelated

Holder exponent can then be measured either by selecting smaller scales or by using some other

approach. A possibility we would like to suggest is using the multifractal paradigm in order to

estimatewhatwecall theeectiveHolderexponent. Thedetailed discussionofthisapproachcanbe

foundin[1,2],butletusquicklypointoutthattheeectiveHolderexponentcaptureslocaldeviations

fromthemeanscalingexponentofthedecompositioncoeÆcientsrelatedtothesingularityinquestion.

Densesingularitiescanbeseenasevolvingfromamultiplicativecascadingprocesswhichtakesplace

acrossscales. TheCWThasbeensuccessfullyused in revealingsuchaprocess andin recoveringits

characteristics. Inshort,allthatis thenneededtoevaluatethelocaleectiveHolderexponentfora

singularity ataparticular scaleis thegainin process density acrossscales withrespectto thescale

gain. ^ h shi s lo = log(Wf! pb (s lo )) log(Wf! pb (s hi )) log(s lo ) log(s hi ) ; where Wf! pb

(s) is the value of the wavelet transform at the scale s, along the maximum line !

pb

correspondingto thegivenprocessbranch bp. Scales

lo

correspondswithgenerationF

max

,whiles hi

correspondswithgenerationF

0

,(simplythelargestavailable scaleinourcase).

5. Employing theLocalEffective

H

olderExponentinthe Characterisationof Time

Series

Suchanestimatedlocal

^ h(x

0

;s)canbedepictedinatemporalfashion,forexamplewithcolourstripes,

aswe havedone in gure 5. The colour of the stripesis determined by the value of the exponent

^ h(x

0

;s)anditslocationissimplythex 0

locationoftheanalysedsingularity(inpracticethisamounts

tothelocationofthecorrespondingmaximumline). Colourcodingisdonewithrespecttothemean

value,whichis setto thegreencolourcentralto ourrainbowrange. All exponentvalueslowerthan

themeanvaluearegivencoloursfromthe`warmer'sideoftherainbow,allthewaytowardsdarkred.

Allhigherthanaverageexponentsget`colder'colours,downtodarkblue.

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histogram eff. h of S&P index

h

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histogram eff. h of fBm H=0.6

h

Figure 5: Left: example time series with local Hurst exponent indicated in colour: the record of

healthy heartbeat intervals andwhite noise. The backgroundcolour indicates the Holder exponent

locally,centredattheHurstexponentatgreen;thecolourgoestowardsblueforhigher

^

handtowards

redforlower ^

h. Right: thecorrespondinglog-histogramsofthelocalHolderexponent.

bothsmootherandroughercomponents. Inparticular,onecanobserveanextremalredvalueatthe

crash of'87 coordinate, followedbyveryroughbehaviour(a rather obviousfact, but to thebest of

our knowledge notreported to date in therapidly growingcoverage of this time series record), see

e.g.[16].

The second example time series is a computer generated sample of fractional Brownian motion

with H = 0:6. It shows almost monochromatic behaviour, centred at H =0:6; thecolour green is

dominant. There are, however, several instances of darker green and light blue, indicating locally

smoothcomponents.

It isimportantto notice that h=H

BrownianWalk

, theHolderexponentvalueequalto theHurst

exponentofanuncorrelatedBrownianwalk,correspondswithnocorrelationintimeseries.

2

Anideal

randomwalkwouldhaveonlymonochromaticcomponentsofthisvalue. Ofcourse,anideal,innitely

long record of fractional Brownian motion of H = 0:6 is correlated, but this correlation would be

stationaryinsuchanidealcaseandno uctuationsincorrelationlevel(incolour)wouldbeobserved.

Bythesameargument,wecaninterpretthevariationsin hasthelocal uctuationsofcorrelationin

theS&Pindex. Themoreredthecolour,themoreunstable,themoreanti-correlatedtheindex. And

themoreblue,themorestableandcorrelated.

Totherightofgure5,thelog-histogramsareshownoftheHolderexponentdisplayedinthecolour

panels. Theyaremadebytakingthelogarithmofthemeasureineachhistogrambin. Thisconserves

the monotonicity of the original histogram, but allows us to compare the log-histograms with the

spectrumofsingularities D(h). Thelog-histogramsare actuallyclosely relatedto the(multifractal)

spectraof the Holderexponent[2]. Themultifractal spectrum of the Holderexponent isthe `limit

histogram' D

s!0

(h) of the Holderexponentin the limitof innite resolution. Of coursewecannot

speakofsuchalimitotherthantheoreticallyand,therefore,alimithistogram(multifractalspectrum)

hastobeestimatedfrom theevolutionofthelog-histogramsalongscale. Fordetailssee [2].

Letuspointoutthatthewidthofthespectraaloneisarelativelyweakargumentinfavourofthe

hypothesis of the multifractality of the S&P index. The log-histogram of the S&P is only slightly

wider than the log-histogram of arecord of fractional Brownian motion of comparable length, see

gure5. An interestingobservationis,however,thatthecrashof'87isclearlyanoutlierinthesense

ofthelog-histogramofHolderexponent(andthereforein thesenseoftheMF spectrum). Theissue

ofcrashesasoutliershasbeenextensivelydiscussedbyJohansen[17]andrecentlybyL'vovetal[18].

Herewesupport thisobservationfrom anotherpointofview.

3

6. Discovering Structure Through the Analysis of Collective Properties of

Non-stationary Behaviour

Non-stationarities are usually seen asthe curse of the exact sciences, economics notexcluded. Let

us herepresentadierentopinion: where thenon-stationarities occur, interestingnessbegins!

Non-stationaritiescanbeseenasadeparturefrom some(usually)simple`model'. Forexample,this can

bethefailureofthestationarityoftheeectiveHolderexponent(seegure5).

Insomesense,theyindicatethatthesimplemodelusedisnotadequate,butthisdoesnotnecessarily

meanthatoneneedstopatchorreplacethislowlevelmodel. Onthecontrary,theinformationrevealed

bysuchalowlevelmodelmaybeused todetecthigher orderstructures. Inparticular,correlations

in thenon-stationaritiesmayindicate theexistenceof interestingstructures. An intriguingexample

of such an approach in the nancial domain is the work by A. Arneodo et al, where a correlation

structurein S&Pindexhasbeenrevealed[19].

The simplest way of detecting structure, we suggest, is detecting uctuations or the collective

behaviour of the local eective h. This has already been successfully applied in human heartbeat

analysis.[10]. Here wewillpresentsomepreliminary resultsfortheS&Pindex.

2

Theoreticallythisish=0:5,butnitesizesampleeectsusuallyaddsomedegreeofcorrelation,slightlyincreasing

thisvalue. 3

Thenon-stationarybehaviourinhcanbequantied,andforthispurposeweusealowpassmoving

average lter (MA) to detect/enhance trends. This processing is, of course, done on the Holder

exponentvaluesetfh

i

(f(x))g, notontheinputsignalf(x). An-MAlteringofnbaseisdenedas

follows: h MAn (i)= 1 n i=n X i=1 h i (f(x)); (6.1) whereh i

(f)arethesubsequentvaluesoftheeectiveHolderexponentof thetimeseriesf.

Let us now go back to the S&P index and its eective Holder exponent description. Dierent

windowlengthsin ourMA lterrepresentdierenthorizonsforthetrader. Iftheindex isallthatis

available,in orderto evaluatetheriskassociatedwiththetrading(orin otherwords,topredictthe

riskofanindexcrash),thetradermightwanttoknowhow`stable'themarket/indexisonadailyor

monthlytimescale. Infact acomparisonbetweenthetwoindicatorsofstabilitymightbeevenmore

indicative.

This is exactly what wehave done using twodierent time scales (twotradinghorizons) for the

MAsmoothing,seegure6. ThesmoothedinputistheeectiveHolderexponentoftheS&P index.

Itcorrespondscloselywiththelogarithmofthelocal volatilityandassuchitre ectsthestabilityof

themarket.

Wemadethefollowingobservationsfromthisexperiment: theshorttimehorizonMAshowsastrong

oscillatorypatternin collectivebehaviouroftheh. These oscillationshavealreadybeenobservedby

Y.Liu et al [16] and by N.Vandewalle et al. [6] This is, however, not log-periodic behaviour in our

resultsanditdoesnotconvergetoamomentofcrash. Whatcanperhapsbeusedinordertohelpthe

trader in evaluatingthe growingriskis the interplayof the various timehorizons. Thesecond MA

lterhasatimehorizontentimeslongerand itshowspracticallynooscillations. However,its value

decaysalmost monotonically, in themomentjust before thecrash,reachingthelevelofcorrelations

characteristicfor therandomwalk(seegure6rightinserts). Notethat the crashesthemselvesare

notvisible intheinsertplots. Letusrecallthat themain advantageoftheeectiveHolderexponent

abovesometraditional measuresof volatility isthat it describesthe locallevelof correlationin the

timeseries. Ifthevalueofhisbelowh=H

BrownianWalk

,thismeanswehaveananti-correlatedtime

serieswhichintuitivelycorrespondswitharatherunstableprocess. Thehaboveh=H

BrownianWalk

indicates presence of correlations and generallycan be associated with `stability'. Pleasenote that

theoscillationsinMA50beforethecrashesbringthecollectivehupanddownbetweenthecorrelated

andtheanti-correlatedregimes. SimilarlyMA500steadily decaystowardstheanti-correlatedregime

just beforethecrashes.

7. Conclusions

The local eective Holder exponent has been applied to evaluate the correlation level of the S&P

index locallyat anarbitraryposition (time)andresolution (scale). In additionto this, weanalysed

collective properties of the local correlation exponent as perceived by the trader exercising various

time horizonanalyses ofthe index. A movingaveragelteringof Holderexponentbasedvariability

estimateswasused to mimicthevarious timehorizon analysisof theindex. We observedintriguing

interplay betweendierent timehorizons before thebiggest crashesof theindex. Wend that this

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and rescaled

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slope of decay

h level of Brownian Walk

h(BW)

h

0.49

0.5

0.51

0.52

0.53

0.54

0.55

0.56

0.57

0.58

0.59

0.6

3500

3600

3700

3800

3900

4000

4100

MA50(h)

MA500(h)

slope of decay

h level of Brownian Walk

h(BW)

h

Figure 6: TheeectiveHolderexponentsmoothedwithtwowindows(MA50and MA500)is shown.

The two plots to the right show windows on the smoothed eective Holder exponent just before

crash #1and crash#2. (Thecrashesare visible in theleftgure but notin thewindows). Visible

oscillationsofMA50anddecayof MA500characteriseprecursorsofbothcrashes. Theaveragelevel

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