Agent-based Models of Financial Markets

arXiv:physics/0701140v1[physics.soc-ph]11Jan2007Agent-basedModelsofFinancialMarketsE.Samanidou1,E.Zschischang2,D.Stauffer3,andT.Lux41DeutscheBundesbank,ReferentBankgesch¨aftlichePr¨ufungen,BerlinerAllee40,D-40212D¨usseldorf2HSHNordBank,PortfolioMngmt.&Inv.,Martensdamm6,D-24103Kiel3InstituteforTheoreticalPhysics,CologneUniversity,D-50923K¨oln4DepartmentofEconomics,UniversityofKiel,Olshausenstrasse40,D-24118KielAbstractThisreviewdealswithseveralmicroscopic(“agent-based”)modelsoffinancialmarketswhichhavebeenstudiedbyeconomistsandphysi-cistsoverthelastdecade:Kim-Markowitz,Levy-Levy-Solomon,Cont-Bouchaud,Solomon-Weisbuch,Lux-Marchesi,Donangelo-SneppenandSolomon-Levy-Huang.Afteranoverviewofsimulationapproachesinfinancialeconomics,wefirstgiveasummaryoftheDonangelo-Sneppenmodelofmonetaryexchangeandcompareitwithrelatedmodelsineconomicsliterature.Ourselectivereviewthenoutlinesthemainingredientsofsomeinfluentialearlymodelsofmulti-agentdynamicsinfinancialmarkets(Kim-Markowitz,Levy-Levy-Solomon).Aswillbeseen,thesecontributionsdrawtheirinspirationfromthecomplexappearanceofinvestors’interactionsinreal-lifemarkets.Theirmainaimistoreproduce(and,thereby,providepossibleexplana-tions)forthespectacularbubblesandcrashesseenincertainhistori-calepisodes,buttheylack(likealmostalltheworkbefore1998orso)aperspectiveintermsoftheuniversalstatisticalfeaturesoffinancialtimeseries.Infact,awarenessofasetofsuchregularities(power-lawtailsofthedistributionofreturns,temporalscalingofvolatility)onlygraduallyappearedoverthenineties.Withthemoreprecisedescrip-tionoftheformerlyrelativelyvaguecharacteristics(e.g.movingfromthenotionoffattailstothemoreconcreteoneofapower-lawwithindexaroundthree),itbecameclearthatfinancialmarketsdynamicsgiverisetosomekindofuniversalscalinglaws.Showingsimilaritieswithscalinglawsforothersystemswithmanyinteractingsub-units,anexplorationoffinancialmarketsasmulti-agentsystemsappearedtobeanaturalconsequence.Thistopicwaspursuedbyquiteanumberofcontributionsappearinginboththephysicsandeconomicsliter-aturesincethelatenineties.Fromthewealthofdifferentflavorsofmulti-agentmodelsthathaveappearedbynow,wediscusstheCont-Bouchaud,Solomon-Levy-HuangandLux-Marchesimodels.Openre-searchquestionsarediscussedinourconcludingsection.11IntroductionPhysicistsnotonlyknoweverything,theyalsoknoweverythingbetter.Thisindis-putabledogmadoesnotexclude,however,thatsomeeconomistspublishedworksimilartowhatphysicistsnowcelebrateas“econophysics”,onlymuchearlier,likeNobellaureateStigler[181](whichwasnotexactlyagentbased;norareallecono-physicsmodelsagent-based).AreeconophysicistslikeChristopherColumbus,re-discoveringsomethingwhichothershadfoundearlier,andalsogettingthingssome-whatwrong,butneverthelesschanginghumanhistory?Astheteamofauthorsofthissurveycollectsscientistsfrombothdisciplines,wedonotattempttogiveadefiniteanswertothisquestion,butsimplyreviewsomeinfluentialmodelsbybothphysicistsandeconomists,toallowafaircomparison.Longago,accordingto[94],economistslikeWalrasandParetowereinspiredbyNewtonianmechanics.Stylizedfactsistheeconomist’snameforuniversalpropertiesofmarkets,in-dependentofwhetherwelookatNewYork,Tokyo,orFrankfurt,orwhetherweareconcernedwithsharemarkets,foreignexchangemarketsorderivativemarkets.Thefollowingisacollectionofthose“stylizedfacts”thatarenowalmostuniver-sallyacceptedamongeconomistsandphysicists:(i)Thereiswidespreadagreementthatwecannotpredictwhetherthepricetomorrowwillgoupordown,onthebaseofpastpricetrendsorothercurrentinformation.(ii)Iftodaythemarkethadbeenveryvolatile,thentheprobabilityforobservingalargechange(positiveornegative)tomorrowisalsohigherthanonaverage(volatilityclustering).(iii)Theprobabilitytohavealargechangeinthemarket,byatleastx%,decayswithapowerlawin1/x.Fact(iii)hasfirstbeendiscoveredbyMandelbrot[138]whoproposedtheLevystablemodelforfinancialreturns.Overtherecentyears,themajorityopinion(see[196,72]fordissent)amongresearchersinthefieldhas,how-ever,convergedtotheviewthatthetailsofthecumulativedistributionofreturnsarecharacterizedbyapower-lawwithexponentaround3.Theunderlyingdatawould,hence,possessfinitevarianceincontradictiontotheLevystablemodel.(iv)Theq-thmomentsofthedistributionofpricechangesaremultifractal,i.e.,theirexponentisnotalinearfunctionofthisindexq(arathernewobservation).Facts(i)to(iii)canbefoundinsurveysontheeconometricsoffinancialmarkets,cf.deVries[189]andPagan[148].Fact(iv)hasbeenfirstpartiallydocumentedinDingetal.[61]andhasmeanwhilealsoobtainedthestatusofanuniversalfeatureofallmarketsintheempiricalfinanceliterature(LobatoandSavin[125]).Similarresearchonmultiscaling(multifractality),albeitwithdifferentanalyticaltools,wasconductedinnumerouseconophysicspapers,startingwithMandelbrotetal.[139],VandewalleandAusloos[186].Althoughresearchinagent-basedmodelsstartedfromadiverserangeofinten-tions(seebelow),muchofthephysics-inspiredliteratureconsideredinthissurvey2aimsatbehavioralexplanationsoftheabovestylizedfacts.Themoresuccessfulones,infact,generateevennumericallyaccurateandrobustscalingexponents.Itappearsworthwhiletopointoutthatwiththeseempi