Scaling of the distribution of fluctuations of financial market indices

We study the distribution of fluctuations of the S&P 500 index over a time scale ?t by analyzing three distinct databases. Database ?i ? contains approximately 1 200 000 records, sampled at 1-min intervals, for the 13-year period 1984–1996, database ?ii ?

PHYSICALREVIEWEVOLUME60,NUMBER5NOVEMBER1999

Scalingofthedistributionof uctuationsof nancialmarketindices

´sA.NunesAmaral,1MartinMeyer,1andH.EugeneStanley1ParameswaranGopikrishnan,1VasilikiPlerou,1,2Lu

1

CenterforPolymerStudiesandDepartmentofPhysics,BostonUniversity,Boston,Massachusetts02215

2

DepartmentofPhysics,BostonCollege,ChestnutHill,Massachusetts02167

Received20May1999

Westudythedistributionof uctuationsoftheS&P500indexoveratimescale tbyanalyzingthreedistinctdatabases.Database i containsapproximately1200000records,sampledat1-minintervals,forthe13-yearperiod1984–1996,database ii contains8686dailyrecordsforthe35-yearperiod1962–1996,anddatabase iii contains852monthlyrecordsforthe71-yearperiod1926–1996.Wecomputetheprobabilitydistributionsofreturnsoveratimescale t,where tvariesapproximatelyoverafactorof104—from1minuptomorethanonemonth.We ndthatthedistributionsfor tр4d 1560min areconsistentwitha

´vyregimepower-lawasymptoticbehavior,characterizedbyanexponent 3,welloutsidethestableLe

0 2.TotesttherobustnessoftheS&Presult,weperformaparallelanalysisontwoother nancialmarketindices.Database iv contains3560dailyrecordsoftheNIKKEIindexforthe14-yearperiod1984–1997,anddatabase v contains4649dailyrecordsoftheHang-Sengindexforthe18-yearperiod1980–1997.We ndestimatesof consistentwiththosedescribingthedistributionofS&P500dailyreturns.Onepossiblereasonforthescalingofthesedistributionsisthelongpersistenceoftheautocorrelationfunctionofthevolatility.Fortimescaleslongerthan( t) 4d,ourresultsareconsistentwithaslowconvergencetoGaussianbehavior. S1063-651X 99 11211-X PACSnumber s :05.40.Fb,05.45.Tp,89.90. nI.INTRODUCTIONANDBACKGROUND

Theanalysisof nancialdatabymethodsdevelopedforphysicalsystemshasalongtradition 1–4 ,andhasrecentlyattractedtheinterestofphysicists 5–28 .Amongtherea-sonsforthisinterestisthescienti cchallengeofunderstand-ingthedynamicsofastrongly uctuatingcomplexsystemwithalargenumberofinteractingelements.Inaddition,itispossiblethattheexperiencegainedbystudyingcomplexphysicalsystemsmightyieldnewresultsineconomics.Financialmarketsarecomplexdynamicalsystemswithmanyinteractingelementsthatcanbegroupedintotwocat-egories: i thetraders—suchasindividualinvestors,mu-tualfunds,brokerage rms,andbanks—and ii theassets—suchasbonds,stocks,futures,andoptions.Interactionsbetweentheseelementsleadtotransactionsmediatedbythestockexchange.Thedetailsofeachtransactionarerecordedforlateranalysis.Thedynamicsofa nancialmarketaredif culttounderstandnotonlybecauseofthecomplexityofitsinternalelementsbutalsobecauseofthemanyintractableexternalfactorsactingonit,whichmayevendifferfrommarkettomarket.Remarkably,thestatisticalpropertiesofcertainobservablesappeartobesimilarforquitedifferentmarkets 29–43 ,consistentwiththepossibilitythattheremayexist‘‘universal’’results.

Themostchallengingdif cultyinthestudyofa nancialmarketisthatthenatureoftheinteractionsbetweenthedif-ferentelementscomprisingthesystemisunknown,asisthewayinwhichexternalfactorsaffectit.Therefore,asastart-ingpoint,onemayresorttoempiricalstudiestohelpuncovertheregularitiesor‘‘empiricallaws’’thatmaygovern nan-cialmarkets.

Theinteractionsbetweenthedifferentelementscompris-ing nancialmarketsgeneratemanyobservablessuchasthetransactionprice,thesharevolumetraded,thetradingfre-1063-651X/99/60 5 /5305 12 /$15.00

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quency,andthevaluesofmarketindices Fig.1 .Anumberofstudiesinvestigatedthetimeseriesofreturnsonvaryingtimescales tinordertoprobethenatureofthestochasticprocessunderlyingit 10,13–15,29–43 .ForatimeseriesS(t)ofpricesormarketindexvalues,thereturnG(t) G t(t)overatimescale tisde nedastheforwardchangeinthelogarithmofS(t) 44 ,

G t t lnS t t lnS t .

1

ForsmallchangesinS(t),thereturnG t(t)isapproxi-matelytheforwardrelativechange,

G t t

S t t S t

.

S t

2

In1900,Bachelierproposedthe rstmodelforthesto-chasticprocessofreturns—anuncorrelatedrandomwalkwithindependent,identicallydistributed i.i.d. Gaussianrandomvariables 1 .Thismodelisnaturalifoneconsidersthereturnoveratimescale ttobetheresultofmanyindependent‘‘shocks,’’whichthenleadbythecentrallimittheoremtoaGaussiandistributionofreturns 1 .However,empiricalstudies 4,10,13–15,29–43 showedthatthedistri-butionofreturns 45 haspronouncedtailsinstrikingcon-trasttothatofaGaussian.Toillustratethisfact,weshowinFig.2the10-minreturnsoftheS&P500marketindex 46 for1986and1987,andcontrastitwithasequenceofi.i.d.Gaussianrandomvariables.Botharenormalizedtohaveunitvariance.Clearly,largeeventsareveryfrequentinthedata,afactlargelyunderestimatedbyaGaussianprocess.Despitethisempiricalfact,theGaussianassumptionforthedistribu-tionofreturnsiswidelyusedintheoretical nancebecauseofthesimpli cationsitprovidesinanalyticalcalculation;

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©1999TheAmericanPhysicalSociety

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