Volatility Clustering in Financial Markets

October27,20009:38WSPC/104-IJTAF00082InternationalJournalofTheoreticalandAppliedFinanceVol.3,No.4(2000)675{702cWorldScienticPublishingCompanyVOLATILITYCLUSTERINGINFINANCIALMARKETS:AMICROSIMULATIONOFINTERACTINGAGENTSTHOMASLUXDepartmentofEconomics,UniversityofKiel,Olshausenstr.40,24118Kiel,GermanyE-mail:[e-mail]entofElectricalandElectronicEngineering,UniversityofCagliari,piazzaD'Armi,09123Cagliari,ItalyE-mail:[e-mail]4October1999ThendingofclusteredvolatilityandARCHeectsisubiquitousinnancialdata.Thispaperpresentsapossibleexplanationforthisphenomenonwithinamulti-agentframe-workofspeculativeactivity.Inthemodel,bothchartistandfundamentaliststrategiesareconsideredwithagentsswitchingbetweenbothbehaviouralvariantsaccordingtoobserveddierencesinpay-os.Pricechangesarebroughtaboutbyamarketmakerre-actingtoimbalancesbetweendemandandsupply.Mostofthetime,astableandecientmarketresults.However,itsusualtranquilperformanceisinterspersedbysuddentran-sientphasesofdestabilisation.Anoutbreakofvolatilityoccursifthefractionofagentsusingchartisttechniquessurpassesacertainthresholdvalue,butsuchphasesarequicklybroughttoanendbystabilisingtendencies.Formally,thispatterncanbeunderstoodasanexampleofanewtypeofdynamicbehaviourknownas\on-ointermittencyinphysicsliterature.Statisticalanalysisofsimulatedtimeseriesshowsthatthemainstylisedfacts(unitrootsinlevelstogetherwithheteroscedasticityandleptokurtosisofreturns)canbefoundinthis\articialmarket.Keywords:Volatilityclustering;interactingagents;on-ointermittency.1.IntroductionBothforeignexchangemarketsandnationalstockmarketsshareanumberofstylisedfactsforwhichasatisfactoryexplanationisstilllackinginstandardthe-oriesofnancialmarkets.Pagan[35]providesanauthoritativesurveyofthosesalientfeaturesthatappeartobecommoncharacteristicsofallnancialmarketstogetherwiththeeconometrictechniquesfordealingwiththem.Asconcernsfor-eignexchangemarkets,hisdescriptioncanbesupplementedbyrecentreviewsoftheirempiricalregularitiesbydeVries[39]andGuillaumeetal.[20].Comparingthesepapers,themaindierenceseemstobethataremarkablenumberoffactspresentedbydeVriesconcernnegativeresults,liketherejectionofuncoveredin-terestrateparityorpurchasingpowerparityaswellasothertheoreticallysensible675October27,20009:38WSPC/104-IJTAF00082676T.Lux&M.Marchesibutempiricallydoubtfulrelationshipsbetweenexchangeratesandothereconomicvariables.However,hispositiveresultsaremostlystrikinguni-variatestatisticalfeaturesofthedatawhichalsoplayaprominentroleintheothersurveysandap-peartobeextremelyuniformacrossvariousassets,nationsandsamplinghorizons.aAsitappearsfromtheempiricalliterature,thefeatureshighlightedbelowarealsoroutinelyfoundinthepricesandreturnsofnancialandcommodityfuturesaswellasinpricesforpreciousmetals.Inthispaper,wewillconcentrateonthosethreeuni-variatepropertieswhichap-peartobethemostimportantandpervasive,andwilltrytoprovideanexplanationusingamulti-agentmodelofspeculativeactivity.Beginningwiththecharacteris-ticsofsharepricesandforeignexchangeratesthemselves(ortheirlogarithms),weencounterthefollowingempiricalregularity:Fact1.Unitrootpropertyofassetpricesandspotexchangerates(ortheirlogs).Moreformally,denotingbyptthepriceattimet,Fact1initsmostelemen-taryformimpliesthatptfollowsanautoregressiveprocess:pt=pt−1+twithstationaryincrementst,andthatoneisusuallyunabletorejectthehypothesis=1usingstandardstatisticalproceduressuchastheDickey-Fullertest.Ex-pressedsomewhatdierently,oneisunabletorejectthehypothesisthatnancialpricesfollowarandomwalkormartingale.bWhiletheimpliednon-stationarityandlackofpredictabilityofspotratesappearstobeatoddswithtraditionalmodelsofexchangeratedetermination,itsquareswellwiththeecientmarketviewofstockpricedeterminationandservedasastartingpointformodelsthatproposeaviewofforexmarketsasarbitrage-freenancialmarkets.Iflevels(orlogs)obeyaunitrootdynamics,returnsordierencesoflogsshouldbestationary.Infact,thishasbeenconrmedthroughouttheliterature.However,certaindistributionalcharacteristicsofreturnsalsocountaswell-establishedfactswhich|inthewordsofdeVries|\haveasoundstatisticalbasisbutforwhichnoconvincingeconomicexplanationhasbeenestablished.Therstoftheseis:Fact2.Fattailphenomenon.Returnsatweekly,dailyandhigherfrequenciesexhibitmoreprobabilitymassinthetailsandinthecentreofthedistributionthandoesthestandardNormal.Itisperhapsalsoremarkablethat,besidesthisdeviationfromtheGaussian,theshapeofthedistributionusuallyappearswell-behaved:namely,histogramsofstockpriceorexchangeratereturnsmostlyshowauni-modalbellshapewith,inmostcases,onlymodestlevelsofskewness.Intheearlyliterature,Fact2hasbeenidentiedwithexcessivefourthmoments(leptokurtosis).Kurtosisis,however,averylimitedmeasureofdeviationsfromGaussianshape.Fortunately,recentliteratureprovidesaHarrison[21]showsthat18thcenturynancialdataalsosharemostofthecharacteristicsoftoday'snancialmarkets.Inparticular,theyalsoexhibitleptokurtosisandvolatilityclustering.bTheformernotionappliesifincrementstareindependentlyandidentical