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Endogenousversusexogenousoriginsof nancialralliesandcrashesinanagent-basedmodelwithBayesianlearningandimitationGeorgesHarras&DidierSornetteChairofEntrepreneurialRisksDepartmentofManagement,TechnologyandEconomicsETHZurich,CH-8001Zurich,SwitzerlandAbstractWepresentasimpleagent-basedmodeltostudyhowtheproximatetriggeringfactorofacrashorarallymightrelatetoitsfundamentalmechanism,andviceversa.Ouragentsformopinionsandinvest,basedonthreesourcesofinformation,(i)publicinformation,i.e.news,(ii)informationfromtheir\friendshipnetwork,promotingimitationand(iii)privateinformation.AgentsuseBayesianlearningtoadapttheirstrategyaccordingtothepastrelevanceofthethreesourcesofinfor-mation.We ndthatralliesandcrashesoccurasampli cationsofrandomluckyorunluckystreakofnews,duetothefeedbackofthesenewsontheagents'strategiesintocollectivetransientherdingregimes.Theseingredientsprovideasimplemech-anismfortheexcessvolatilitydocumentedin nancialmarkets.Paradoxically,itistheattemptforinvestorstolearnthelevelofrelevanceofthenewsonthepriceformationwhichleadstoadramaticampli cationofthepricevolatilityduetotheircollectivesearchforthe\truth.Apositivefeedbackloopiscreatedbythetwodom-inatingmechanisms(Bayesianlearningandimitation)which,byreinforcingeachother,resultinralliesandcrashes.Themodelo ersasimplereconciliationofthetwoopposite(herdingversusfundamental)proposalsfortheoriginofcrasheswithinasingleframeworkandjusti estheexistenceoftwopopulationsinthedistribu-tionofreturns,exemplifyingtheconceptthatralliesandcrashesarequalitativelydi erentfromtherestofthepricemoves.Keywords:stockmarket,crash,rallies,bubble,herdingPreprintsubmittedtoElsevier18June2008arXiv:0806.2989v1[physics.soc-ph]18Jun20081IntroductionStockmarketcrashesaremomentous nancialeventsthatarefascinatingtoacademicsandpractitionersalike.Accordingtotheconsecratedacademicviewthatmarketsareecient,onlytherevelationofadramaticpieceofinformationcancauseacrash,yetinrealityeventhemostthoroughpost-mortemanalysesaretypicallyinconclusiveastowhatthispieceofinformationmighthavebeen.Itisoftenobservedthatcrashesoccursoonafteralongrun-upofprices,re-ferredtoasabubble.Acrashisthusoftentheburstofthebubble.Accordingtotherationalexpectationtheoryofbubbles(Blanchard,1979;BlanchardandWatson,1982),bubblesarespontaneousdeviationsofthepricefromfunda-mentalvalue,andtheassociatedapparentanomalousreturnsarenothingbuttherequiredremunerationforrationalinvestorstostayinvestedinthemarketinthepresenceofgrowingcrashrisks.Thus,thepricehastoincreasewiththecrashhazardrateandviceversa,sothatitbecomesmoreandmoreprob-ableforacrashtooccurtowardstheendwhenthebubblepricepeaks.Forinstance,inthespeci cationofJohansenetal.(Johansenetal.,1999c),itistheincreasingcrashhazardratedeterminedbythedynamicsofnoisetraderswhichdrivesthepricerun-up.Contrarily,inthespeci cationofSornetteetal.(SornetteandAndersen,2002),positivefeedbacksoperatingonthepricebyagentsfollowingmomentumstrategiesdriveitup,thereforeconcomitantlypushingupthecrashhazardrate.Thereisavastempiricalliteratureaimingmoregenerallyatdiagnosingthepresenceofbubbles(see(Camerer,1989;AdamandSzafarz,1992)forsur-veysofthisliterature)andatcharacterizingtheirunderlyingorigin(s)andmechanism(s)(seee.g.(Kaufman,2001;She rin,2005;Shiller,2000;Sornette,2003)).However,thereisstillnoconsensusintheacademiccommunityonwhatisreallyabubbleandwhatareitscharacteristicproperties.Bubblesdonotseemtobefullyexplainedbyboundedrationality(LevineandZajac,2007),speculation(Leietal.,2001)ortheuncertaintyinthemarket(Smithetal.,1988).Itisalsonotclearwhatisthenatureandlimitsofthepredictabil-ityofitsbursttime,ifany.Finally,thereisnoreallysatisfactorytheoryofbubbles,whichbothencompassesitsdi erentpossiblemechanismsandad-herestoreasonableeconomicprinciples(noorlimitedarbitrage,equilibriumorquasi-equilibriumwithonlytransientdeviations,boundedrationality,andsoforth).Indeed,theliterature,whichistoolargetobeexplicatedhere,isstilluncertainonevenhowtode neabubble,becauseanexponentiallygrowingpricecanalwaysbearguedtoresultfromsomefundamentaleconomicfactor.Thisisrelatedtotheproblemthatthefundamentalpriceisnotdirectlyob-servable,givingnostronganchortounderstandobservedprices.Howthenisitpossibletoascertainwithsomelevelofcon dencethatamarketisorisnotinabubbleregime?Howdoesthisquestionimpacttheexpectationsand2anticipationofinvestors,andasaresultthestabilityofthe nancialsystem?Comingbacktocrashes,mostapproachestoexplainingcrashessearchforpossiblemechanismsore ectsthatoperateatveryshorttimescales(hours,days,orweeksatmost).Here,webuildontheradicallydi erenthypothesissummarizedin(Sornette,2003)thattheunderlyingcauseofthecrashshouldbefoundintheprecedingmonthsandyears,intheprogressivelyincreasingbuild-upofacharacteristicthatwerefertoas\marketcooperativity,whichexpressesthegrowthofthecorrelationbetweeninvestors'decisionsleadingtostrongere ectiveinteractionsbetweenthemasaresultofseveralpositivefeedbackmechanisms.Thehypothesisisthatthisincreasedmarketcoopera-tivityistranslatedintoacceleratingascentofthemarketprice(thebubble).Accordingto