INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS CONDENSED MATTER

J.Phys.:Condens.Matter17(2005)S1259–S1268

JOURNALOFPHYSICS:CONDENSEDMATTER

doi:10.1088/0953-8984/17/14/015

Heterogeneityandfeedbackinanagent-basedmarketmodel

Fran¸coisGhoulmie1,RamaCont2andJean-PierreNadal1

LaboratoiredePhysiqueStatistique,EcoleNormale,Sup´erieure24rueLhomond,75231ParisCedex,France

2CMAP-EcolePolytechnique,F91128Palaiseau,FranceE-mail:ghoulmie@santafe.edu

1

Received3September2004,infinalform17November2004Published24March2005

Onlineatstacks.iop.org/JPhysCM/17/S1259

Abstract

Weproposeanagent-basedmodelofasingle-assetfinancialmarket,describedintermsofasmallnumberofparameters,whichgeneratespricereturnswithstatisticalpropertiessimilartothestylizedfactsobservedinfinancialtimeseries.Ouragent-basedmodelgenericallyleadstotheabsenceofautocorrelationinreturns,self-sustainingexcessvolatility,mean-revertingvolatility,volatilityclusteringandendogenousburstsofmarketactivitynon-attributabletoexternalnoise.Theparsimoniousstructureofthemodelallowstheidentificationoffeedbackandheterogeneityasthekeymechanismsleadingtotheseeffects.

(Somefiguresinthisarticleareincolouronlyintheelectronicversion)

1.Introduction

Thestudyofstatisticalpropertiesoffinancialtimeserieshasrevealedawealthofuniversalstylizedfactswhichseemtobecommontoawidevarietyofmarkets,instrumentsandperiods[6,15].Agent-basedmarketmodels,whicharebasedonastylizeddescriptionforthebehaviourofagents,attempttoexplaintheoriginsoftheobservedbehaviourofmarketpricesintermsofsimplebehaviouralrulesofmarketparticipants:inthisapproachafinancialmarketismodelledasasystemofheterogeneous,interactingagentsandseveralexamplesofsuchmodelshavebeenshowntogeneratepricebehaviourwithstatisticalpropertiessimilartothoseobservedinrealmarkets[1,4,5,13,17,19–21,18,12,26].Howevermostagent-basedmodelsareformulatedinacomplexmannerand,duetotheircomplexity,itisoftennotclearwhichaspectofthemodelisresponsibleforgeneratingthestylizedfactsandwhetheralltheingredientsofthemodelareindeedrequiredforexplainingempiricalobservations.Thiscomplexityalsodiminishestheexplanatorypowerofsuchmodels.

0953-8984/05/141259+10$30.00

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PrintedintheUK

S1259

S1260FGhoulmieetal

Wereporthereourfindings[8]onaparsimoniouslyparameterizedagent-basedmodelofasingle-assetfinancialmarket,whichgeneratesreturnswithstatisticalpropertiessimilartothestylizedfactsobservedinfinancialtimeseries.Ouragent-basedmodelgenericallyleadstotheabsenceofautocorrelationinreturns,self-sustainingexcessvolatility,volatilityclusteringandendogenousburstsofmarketactivitynon-attributabletoexternalnoise.Theparsimoniousstructureofthemodelallowstheidentificationofheterogeneityofstrategiesandfeedbackgeneratedbythepriceimpactoforderflowasthekeymechanismsleadingtotheseeffects.Inparticular,weshowthatdirectinteractionorherdingeffectsarenotneededtogeneratestylizedfacts.Ourmodelpresentsanexamplewhereheterogeneityisendogenousandfollowsastochasticevolutionintime,leadingtoadynamicdisorderedsystemwherethedisorderdevelopswithtimeinahistorydependentmanner.2.Statisticalpropertiesofassetreturns

Timeseriesofstockreturnsexhibitinterestingstatisticalfeatureswhichseemtobecommontoawiderangeofmarketsandtime-periods[6]:

(i)Excessvolatility.By‘excessvolatility’onereferstotheobservationthatthelevelofvariabilityinmarketpricesismuchhigherthancanbeexpectedbasedonthevariabilityoffundamentaleconomicvariables[25]andtheoccurrenceoflarge(negativeorpositive)returnsisnotalwaysexplainablebythearrivalofnewinformationonthemarket[25,9].(ii)Heavytails.The(unconditional)distributionofdailyandhourlyreturnsdisplaysaheavy

tailwithpositiveexcesskurtosis.

(iii)Absenceofautocorrelationsinreturns.(Linear)autocorrelationsofassetreturnsareoften

insignificant,exceptforverysmallintradaytimescales(󰀇20min)forwhichmicrostructureeffectscomeintoplay.

(iv)Volatilityclustering.AsnotedbyMandelbrot[22],‘largechangestendtobefollowed

bylargechanges,ofeithersign,andsmallchangestendtobefollowedbysmallchanges’.Inquantitativeterms,whilereturnsthemselvesareuncorrelated,absolutereturns|rt(󰀋)|displayapositive,significantandslowlydecayingautocorrelationfunction:corr(|rt|,|rt+󰀋|)>0for󰀋rangingfromafewminutestoaseveralweeks.

(v)Volume/volatilitycorrelation.Tradingvolumeispositivelycorrelatedwithmarket

volatility.Thefactthattheseempiricalpropertiesarecommontoawiderangeofmarketsandtimeperiodssuggeststhattheirorigincanberetracedtosomesimplemarketmechanisms,commontomanymarketsandthuslargelyindependentoftheir‘microstructure’[5].Thisisthebasisforthedevelopmentofagent-basedmarketmodels,whicharebasedonastylizeddescriptionforthebehaviourofagents,andattempttoexplaintheoriginsoftheobservedbehaviourofmarketpricesasemergingfromsimplebehaviouralrulesofalargenumberofheterogeneousmarketparticipants.Severalexamplesofsuchmodelshavebeenshowntogeneratepricebehaviourwithstatisticalpropertiessimilartothoseobservedinrealmarkets[1,3,4,17,19,21,18,12,26].Numericalsimulationsofmanyofthemodelsaboveleadtotimeseriesof‘returns’whichhavepropertiesconsistentwith(someof)theempiricalstylizedfactsobservedabove.However,duetothecomplexityofsuchmodelsitisoftennotclearwhichaspectofthesemodelsisresponsibleforgeneratingthestylizedfactsandwhetheralltheingredientsofthemodelareindeedrequiredforexplainingempiricalobservations.Thiscomplexityalsodiminishestheexplanatorypowerofsuchmodels.

Akeypointin[7]whichleadstoheavytailsinthedistributionoforderflowistoallowforinvestorinertia—thefactthatmostmarketparticipantstradeveryinfrequently.Apossible

Heterogeneityandfeedbackinanagent-basedmarketmodelS1261

mechanismforgeneratinginvestorinertiaisthresholdresponseinthebehaviourofmarketparticipants[14]:theriskaversionofagentswhichleadsthemtobeinactiveifuncertainabouttheiraction.Basedontheseremarks,weformulateamodelofasingle-assetmarketretainingtheingredientsabove.3.Descriptionofthemodel

Ourmodeldescribesamarketwhereasingleasset,whosepriceisdenotedbypt,istradedbyNagents.Tradingtakesplaceatdiscreteperiodst=0,1,2,....Wewillseethat,providedtheparametersofthemodelarechoseninacertainrange,wewillbeabletointerprettheseperiodsas‘tradingdays’.Ateachperiod,agentshavethepossibilityofsendinganordertothemarketforbuyingorsellingaunitofasset:denotingbyφi(t)thedemandoftheagent,wehaveφi(t)=1forabuyorderandφi(t)=−1.Weallowthevalueφi(t)tobezero;theagentistheninactiveatperiodt.TheinflowofpublicinformationismodelledbyasequenceofIIDGaussianrandomvariables(󰀁t,t=0,1,2,...)with󰀁t∼N(0,D2).󰀁trepresentsthevalueofacommonsignalreceivedbyallagentsatdatet−1.Thesignal󰀁tisaforecastofthefuturereturnrtandeachagenthastodecidewhethertheinformationconveyedby󰀁tissignificant,inwhichcaseshewillplaceabuyorsellanorderaccordingtothesignof󰀁t.

Thetradingruleofeachagenti=1,...,Nisrepresentedbya(time-varying)decisionthresholdθi(t).Thethresholdθi(t)canbeviewedastheagent’s(subjective)viewonvolatility.Thetradingrulewestudymaybeseenasastylizedexampleofthresholdbehaviour:withoutsufficientexternalstimulus(|󰀁t|󰀁θi(t)),anagentremainsinactiveφi(t)=0andiftheexternalsignalisaboveacertainthreshold,theagentwillact:if󰀁t>θi(t),φi(t)=1,if󰀁t<−θi(t),φi(t)=−1.Thecorrespondingdemandgeneratedbytheagentisthereforegivenby:

φi(t)=1󰀁t>θi(t)−1󰀁t<−θi(t).

TheexcessdemandisthengivenbyZt=changeinthepricegivenby

󰀂󰀃

ptZt

=grt=ln

pt−1N

󰀄N

i=1

(1)

φi(t).Anon-zerovalueofZproducesa

(2)

wherethepriceimpactfunctiong:R→Risanincreasingfunctionwithg(0)=0.We

definethe(normalized)marketdepthλby:g󰀐(0)=1/λ.Whilemostoftheanalysisbelowholdsforageneralpriceimpactfunctiong,insomecasesitwillbeusefultoconsideralinearpriceimpact:g(z)=z/λ.

Initially,westartfromapopulationdistributionF0ofthresholds:θi(0),i=1...NarepositiveIIDvariablesdrawnfromF0.Updatingofstrategiesisasynchronous:ateachtimestep,anyagentihasaprobability0󰀁s󰀁1ofupdatingherthresholdθi(t).Thus,inalargepopulation,srepresentsthefractionofagentsupdatingtheirviewsatanyperiod;1/srepresentsthetypicaltimeperiodduringwhichanagentwillholdagivenviewθi(t).Ifperiodsaretobeinterpretedasdays,sistypicallyasmallnumbers󰀇10−1–10−3.Whenanagentupdatesherthreshold,shesetsittobeequaltotherecentlyobservedabsolutereturn,whichisanindicator

t

|.IntroducingIIDrandomvariablesui(t),i=1...N,t󰀂0ofrecentvolatility|rt|=|lnppt−1

uniformlydistributedon[0,1],whichindicatewhetheragentiupdatesherthresholdornot:

θi(t)=1ui(t)(3)

Thiswayofupdatingcanbeseenasastylizedversionofvariousestimatorsofvolatilitybasedonmovingaveragesofabsoluteorsquaredreturns.Itisalsocorroboratedbyarecentempirical

S1262FGhoulmieetal

studybyZovkoandFarmer[27],whoshowthattradersuserecentvolatilityasasignalwhenplacingorders.

Theasynchronousupdatingschemeproposedhereavoidsintroducinganartificialorderingofagentsasinsequentialchoicemodels.Therandomnatureofupdatingisalsoaparsimoniouswaytointroduceheterogeneityintimescales,afeaturebelievedtobeimportant[19],withoutintroducingextraparameters.Giventhisrandomupdatingscheme,evenifwestartfromaninitiallyhomogeneouspopulationθi(0)=θ0,heterogeneitycreepsintothepopulationthroughtheupdatingprocessandevolvesinarandommanner,leadingtoahistory-dependentdisorderedsystem.

Letusrecallthemainingredientsofthemodel.Ateachtimeperiod:(i)Agentsreceiveacommonsignal󰀁(t)∼N(0,D2).(ii)Eachagenticomparesthesignaltoherthresholdθi(t).

(iii)If|󰀁(t)|>θi(t)theagentconsidersthesignalassignificantandgeneratesanorderφi(t)

accordingto(1).

(iv)Themarketpriceisimpactedbytheexcessdemandandmovesaccordingto(2).(v)Eachagentupdates,withprobabilitys,herthresholdaccordingto(3).

Comparedtomostagent-basedmodelsconsideredintheliterature,thereisnoexogenous‘fundamentalprice’processandwedonotdistinguishbetween‘fundamentalist’and‘chartist’traders.Also,thesameinformationisavailabletoallagentsbuttheydifferinthewaytheyprocesstheinformation.Wedonotintroduceany‘socialinteraction’amongagents:nonotionoflocality,latticeorgraphstructureisintroduced.Themodelhasveryfewparameters:sdescribestheaverageupdatingfrequency,Dthestandarddeviationofthenoiserepresentingthenewsarrivalprocess,themarketdepthλandthenumberofagentsNwhichistypicallylarge.Wewillobserveneverthelessthatthissimplemodelgeneratestimeseriesofreturnswithinterestingpropertiessimilartoempiricallyobservedpropertiesofassetreturns.4.Simulationresults

Inorderforadirectcomparisonwithempiricalstylizedfactstobemeaningful,wehavetoconsiderthatinthecaseofempiricaldataonlyasinglesamplepathofthepriceisavailableand(unconditional)momentsarecomputedbyaveragingoverthe(single)samplepath.Wethereforeadoptasimilarapproachhere:aftersimulatingasamplepathofthepriceptforT=104periods,wecomputethetimeseriesofreturnsrt=ln(pt/pt−1),t=1...T,theirhistogram,amovingaverageestimatorofthestandarddeviationofreturns(‘volatility’),thesampleautocorrelationfunctionofreturnsandthesampleautocorrelationfunctionofabsolutereturns.Inordertodecreasethesensitivityofresultstoinitialconditions,weallowforatransitoryregimeanddiscardthefirst103periodsbeforeaveraging.

Inordertointerpretthetradingperiodsas‘days’andcomparetheresultswithpropertiesofdailyreturns,wenotethatwhengislinear|rt|󰀁1/λandchoose5󰀁λ󰀁20whichallowsa(maximal)rangeofdailyreturnsbetween5%and20%.Also,theamplitudeDoftheinputnoisecanbechosensuchastoreproducearealisticrangeofvaluesforthe(annualized)volatility:thisleadstochoosingDintherange10−3–10−2.Letusemphasizethatwearediscussingthecalibrationoftheorderofmagnitudeofparameters,notfine-tuningthemtoasetofcriticalvalues.Theresultsdiscussedinthesequelaregenericwithinthisrangeofparameters.Figures1and2illustratetypicalsamplepathsobtainedwithdifferentparametervalues:theyallgenerateseriesofreturnswithrealisticrangesandrealisticvaluesofannualizedvolatility.Foreachseries,werepresentthehistogramofreturnsbothinlinearandlogarithmic

Heterogeneityandfeedbackinanagent-basedmarketmodelS1263

(a)120

11010090memory8070605040302020

30

40

50

60

70

80

90

100

updating period 1/s

(b)110

10510095πi(t)90858075700

10002000300040005000600070008000900010000

t

Figure1.Numericalsimulationofthemodelwithupdatingfrequencys=0.01(averageupdatingperiod:100‘days’)N=1000agents,D=0.001andλ=10.

scales,theACFofreturnsCr,theACFofabsolutereturnsC|r|.Thereturnseriesobtainedpossessregularitieswhichmatchthepropertiesoutlinedinsection2:

(i)Excessvolatility.Thesamplestandarddeviationofreturnscanbemuchlargerthanthestandarddeviationoftheinputnoiserepresentingnewsarrivalsσˆ(t)󰁽D.

(ii)Mean-revertingvolatility.Themarketpricefluctuatesendlesslyandthevolatility,as

measuredbythemovingaverageestimatorσˆ(t),goesneithertozeronortoinfinityanddisplaysamean-revertingbehaviour.

(iii)Thesimulatedprocessgeneratesaleptokurticdistributionofreturnswith(semi-)heavy

tails,withanexcesskurtosisaroundκ󰀇7.Asshowninthelogarithmichistogramplotsinfigures1,2,thetailsexhibitanapproximatelyexponentialdecay,asobservedinvariousstudiesofdailyreturns[10].

(iv)Thereturnsareuncorrelated.Thesampleautocorrelationfunctionofthereturnsexhibits

aninsignificantvalue(verysimilartothatofassetreturns)atalllags,indicateabsenceoflinearserialdependenceinthereturns.

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Market Activity

1500100050000.10.050–0.05–0.15001500100050001500100010

2FGhoulmieetal

Trading volume

43210500010000Distribution of returns

010

4Price

05000Log returns

100000500010000Distribution of returns

00500010000–0.100.1Annualized Moving average volatilityAuto– correlation of returns0.30.210–0.100.1Auto– correlation of absolute returns0.60.40.2000.250.10– 0.10.20500010000– 0.20200400600– 0.202040Figure2.Numericalsimulationofthemodelwithupdatingfrequencys=0.1(averageupdating

period:10‘days’)N=1500agents,D=0.001andλ=10.

(v)Volatilityclustering.Theautocorrelationfunctionofabsolutereturnsremainssignificantly

positiveovermanytimelags,correspondingtopersistenceoftheamplitudeofreturnsatimescale󰀇1/s.5.Somelimitingcases

(i)Feedbackwithoutheterogeneity.Inthecasewheres=1,allagentssynchronouslyupdatetheirthresholdateachperiod.Consequently,theagentshavethesamethresholds,givenbythelastperiodsabsolutereturn:θi(t)=|rt−1|andwillthereforegeneratethesameorder:Zt=Nφ1(t)∈{0,N,−N}.So,thereturnrtdependsonthepastonlythroughtheabsolutereturn|rt−1|:

rt=f(|rt−1,󰀁t|)=g(N)1󰀁t>|rt−1|+g(−N)1󰀁t<−|rt−1|,

adependencestructuretypicalofARCHmodels[11],leadingtouncorrelatedreturnsandvolatilityclustering.Inthiscase,thedistributionofrtconditionalon|rt−1|isactuallyatrinomialdistribution:rt∈{0,g(N),g(−N)},whichisnotrealistic.Simulationstudiesshowthatasimilarbehaviourpersistsfor1−s󰁼1,leadingtotri-modaldistributions.Thisconfirmsourintuitionthattheupdatingprobabilitysshouldbechosensmall.

(ii)Heterogeneitywithoutfeedback.Inthecasewheres=0,noupdatingtakesplaces:the

tradingstrategies,givenbythethresholdsθi,areunaffectedbythepricebehaviourand

Heterogeneityandfeedbackinanagent-basedmarketmodelS1265

thefeedbackeffectisnolongerpresent.Heterogeneityisstillpresent:thedistributionofthethresholdsremainsidenticaltowhatitwasatt=0.Thereturnrtdependsonlyon󰀁t:

󰀃󰀂󰀅

1N

rt=g1󰀁>θ−1󰀁t<−θi=F(󰀁t).

Ni=1tiWeconcludethereforethatthereturnsareIIDrandomvariables,obtainedbytransformingtheGaussianIIDsequence(󰀁t)bythenonlinearfunctionFgivenin(ii),whosepropertiesdependonthe(initial)distributionofthresholds(θi,i=1...N).Thelog-pricethenfollowsa(non-Gaussian)randomwalkandthemodeldoesnotexhibitvolatilityclustering.6.Behaviourofpricesandvolatility

Thetwolimitingcasesaboveshowthat,inordertoobtaintheinterestingstatisticalpropertiesobservedinthesimulatedexamplesshownabove,itisnecessarytohave0•Markoviandynamics.Thethresholds[θi(t),i=1...N]followaMarkovchainin{g(k),k=0...N}.Wehaveθi(t+1)=θi(t)withprobability1−sand

󰀁󰀂󰀃󰀁N󰀅󰀁󰀁1󰀁withprobabilitys.[1󰀁>θ−1󰀁t<−θi]󰀁θi(t+1)=|rt|=󰀁g󰀁Ni=1tiInfactgiventhatagentsareindistinguishableandonlytheempiricaldistributionof

󰀄

thresholdvaluesaffectsthereturns,definingNk(t)=iN=11[0,ak[(θi(t))then(Nk(t),k=0...N−1)t=0,1,...evolvesasaMarkovchainin{0,...,N}N.N(t)=(Nk(t),k=0...N−1)isnoneotherthanthe(cumulative)populationdistributionofthethresholds.ThefactthatN(t)itselffollowsaMarkovchainmeansthatthepopulationdistributionofthresholdsisarandommeasureon{0,...,N},whichischaracteristicofdisorderedsystems[23],evenifwestartfromadeterministicsetofvaluesfortheinitialthresholds(evenidenticalones).Herethedisorderisendogenousandisgeneratedbytherandomupdatingmechanism.

•Excessvolatility.Inthismodel,thevolatilityofthenewsarrivalprocessisquantifiedbyDwhichisthestandarddeviationoftheexternalnoise󰀁t,whereasthevolatilityofthereturnscanbemeasuredaposterioriasthe(conditionalorunconditional)standarddeviationofrt.Asseenfromthenonlinearrelationbetween󰀁tandrt,

󰀂󰀄N󰀃

1−1󰀁t<−θi(t)i=1󰀁t>θi(t)

rt=g

λNevenafterconditioningonthecurrentstatesofagentsθi(t),i=1...N,equation(6)yieldsanonlinearrelationbetweentheinputnoise󰀁tandthereturnswhichcanhavetheeffectofamplifyingthenoisebyanorderofmagnitudeormore.Inthesimulationexampleshowninfigure1,D=10−3whichcorrespondstoanannualizedvolatilityof1.6%,whiletheannualizedvolatilityofreturnsisintherangeof20%,anorderofmagnitudelarger;theorderofmagnitudeofthevolatilityofreturnsmaybequitedifferentfromthatoftheinputnoise.

•Absenceofautocorrelation.Fromthedynamicequationsofthemodel

NN1󰀅1󰀅Zt=φi(t)=[1󰀁>θ−1󰀁t<−θi]

Ni=1Ni=1ti

(4)

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Market Activity

60040020000.0540030020010005000Log returns

1000000500010000Distribution of returnsTrading volume

21.510.50101010

3FGhoulmieetalPrice

0500010000Distribution of returns

400300202001100 –0.0500500010000 –0.100.1Annualized Moving average volatilityAuto–correlation of returns 0.40.20.150.30.20.10.10.05005000100000

200

400

600

10

–0.100.1Auto–correlation of absolute returns0.30.20.10 –0.102004006000Figure3.Left:correlationtimescaleτcofabsolutereturns,asafunctionoftheupdatingperiod1/s.Right:evolutionoftheportfolioofatypicalagent,withlongperiodsofinactivitypunctuatedbyburstsofactivity.

󰀃N

1󰀅

rt=g(Zt)=g[1󰀁>θ−1󰀁t<−θi]

Ni=1ti

󰀂

(5)

onecandeducethat,ifgisanoddfunction(inparticularifgislinear),thenassetreturns(rt)t󰀁0areuncorrelated;cov(rt,rt+1)=0.Thisisduetothefactthatthetrading/nontradingdecisionisbasedonlyontheamplitudeofthesignal,notitssign.Thesignofthereturnisdeterminedbythesignofthecommonsignal,whichisindependentacrossperiods.

•Investorinertia.Exceptintimesofcrisisormarketcrash,atagivenpointintimeonlyasmallproportionofstockholdersareactuallytradinginthemarket.Asaresult,the(daily)orderflowforatypicalstockcanbemuchsmallerthanthemarketcapitalization.Thisphenomenon,sometimesreferredtoasinvestorinertia,isagenericoutcomeinourmodelduetothresholdbehaviourofagents.Startingfrom󰀄taninitialholdingofπi(0),thequantityofassetheldbyagentiisgivenbyπi(t)=τ=0φi(τ).Figure3displaystheevolutionoftheportfolioπi(t)ofatypicalagent;shortperiodsofactivity(trading)areseparatedbylongperiodsofinertia,wheretheportfolioremainsconstant.This‘inertia’increasesinperiodsofhighvolatility,aneffectsimilartothebehaviourofrisk-averseagent.

•Clusteringandmean-reversioninvolatility.Manymarketmicrostructuremodels—especiallythosewithlearningorevolution—convergeoverlargetimeintervalstoan

Heterogeneityandfeedbackinanagent-basedmarketmodelS1267

equilibriumwherepricesandotheraggregatequantitiesceasetofluctuaterandomly.Bycontrast,inthepresentmodel,pricesfluctuateendlesslyandthevolatilityexhibitsmean-revertingbehaviour.Supposeweareinaperiodof‘lowvolatility’;theamplitude|rt|ofreturnsissmall.Agentswhoupdatetheirthresholdswillthereforeupdatethemtosmallvalues,becomemoresensitivetonewsarrivals,thusgeneratinghigherexcessdemandandthusincreasingtheamplitudeofreturns.Conversely,inaperiodofhighvolatility,agentswillupdatetheirthresholdvaluestohighvaluesandbecomelessreactivetotheincomingsignal:thisincreaseininvestorinertiawillthusdecreasetheamplitudeofreturns.Themeanreversiontimeinthevolatilityisthereforethetimeittakesforagentstoadjusttheirthresholdstocurrentmarketconditions,whichisoftheorderofτc=1/s.

Whentheamplitudeofthenoiseissmallitcanbeshown[8]thatvolatilitydecaysexponentiallyintimeandincreasesthroughupward‘jumps’.ThisbehaviourisactuallysimilartothatofaclassofstochasticvolatilitymodelsintroducedbyBarndorff-NielsenandShephard[2]andsuccessfullyusedtodescribevariouseconometricpropertiesofreturns.7.Conclusion

Wehavepresentedaparsimoniousagent-basedmodelcapableofreproducingthemainempiricalstylizedfactsdescribedinsection2,basedonthreemainingredients:

(i)Thresholdbehaviourofagents.

(ii)Heterogeneityofagentstrategies,generatedendogenouslythroughrandomasynchronous

updatingofthresholds.

(iii)Feedbackofrecentpricebehaviouronagentsbehaviour.

Numericalsimulationsofthemodelgenericallyproducetimeseriesthatcapturethestylizedfactsobservedinassetreturns.Duetothesimplestructureofthemodel,thesesimulationresultscanbeexplainedbyatheoreticalanalysisofthepriceprocessinthemodel.Theseobservationsillustratethatthesethreeingredientssufficeforreproducingseveralempiricalstylizedfactssuchasheavytails,absenceofautocorrelationinreturnsandvolatilityclustering,withrealisticvaluesinthetimescalesinvolvedandwithoutanyexogenous‘fundamental’price,directinteractionbetweenagentsordistinctionbetween‘chartist’or‘fundamentalist’traders.Theseresultsquestionsomepreviousconclusionsregardingtheoriginsofstylizedpropertiesofassetreturnspreviouslydrawnfromsimulationofagent-basedmodelsandcallforacloser,criticallookatthisissuethroughthestudyofawidervarietyofagent-basedmarketdesigns.Thesepointsarefurtherdevelopedin[8].References

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