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A financial market model with confirmation bias

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ContentslistsavailableatScienceDirect

Structural

Change

and

Economic

Dynamics

jo u r n al h om ep a g e :w w w . e l s e v i e r . c o m / l o c a t e/ s c e d

A

financial

market

model

with

confirmation

bias

Alessia

Cafferata

a

,

Fabio

Tramontana

b,∗

aUniversityofGenova,Italy

bCatholicUniversityofSacredHeart(Milano),Italy

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received12March2019

Receivedinrevisedform12August2019 Accepted12August2019

Availableonline1October2019 JELclassification: G12 D83 D84 D91 C61 Keywords: Financialmarkets Heterogeneousagents Piecewise-definedmaps MonteCarlosimulations

a

b

s

t

r

a

c

t

Wedevelopafinancialmarketmodelwithheterogeneousagentswhocanbeaffectedbyconfirmation bias.Inparticularweconsideroptimisticandpessimisticagentswhoadjusttheirbeliefsgivingmore attentionandconsiderationtoevidencessupportingtheirpriorbeliefs.Thesekindsoftraderscoexist withfundamentalistsandchartists.Weshowthatthispsychologicalbiasmakesbeliefsmoreandmore distantastimepasses,andpermitstobetterexplainsomeimportantstylizedfactsoffinancialmarkets. ©2019ElsevierB.V.Allrightsreserved.

1. Introduction

“Onceahumanintellecthasadoptedanopinion(eitheras some-thing it likes or as something generally accepted), it draws everythingelseintoconfirmandsupportit”LordFrancisBacon (1620)

ThisquotefromLordFrancisBaconisanexcellentdefinitionof whatnowadayspsychologistscallConfirmation(orConfirmatory) bias.Humanbeingsdisplay thetendencytoselectthe informa-tiontheyreceiveand togivemoreimportancetonewevidence confirmingtheirpriorbeliefsandless(orevennothingatall)to thoseagainstthem.Thisisanerrorintheupdatingofbeliefsinthe lightofnewevidence,thatshouldrespecttheso-calledBayesian updatingtobeconsideredrational.Initialbeliefsshouldonlybea startingpoint,becominglessandlessrelevantasthenumberof evidenceaccumulates(aprocessknownaswashing-outthe pri-ors).Instead,accordingtothedefinitionofconfirmationbias,our

夽 TheauthorswanttothankGianItaloBischi,FrankWesterhoffandtwo anony-mousrefereesfortheirusefulcommentsandsuggestionsonapreviousversionof thispaper.

∗ Correspondingauthor.

E-mailaddress:fabio.tramontana@unicatt.it(F.Tramontana).

firstimpressionswillinfluencehowweupdateourbeliefsabout someuncertainthing,distortingourreasoningprocess.

Thefirstattempttocollectandreviewevidenceofconfirmation biasinseveralcontextscanbeprobablyfoundinNickerson(1998). Heidentifiedconfirmationbiasinmedicine,judicialreasoningand otherfields,buthedidnottalkaboutfinanceandaboutitsinfluence intheformationofassetprices.Quitesurprisingly,onlyrecently confirmationbiashasbeenempiricallyfoundinfinancialmarkets, forinstancebyParketal.(2013)inthefield(inSouthKorea)orby Bisiereetal.(2014)inthelab.Theroleplayedbysuchabiasinthe persistenceofmispricings(i.e.pricesdifferentfromtheir funda-mentalvalues)isintuitive.Themoreanassetpriceincreases,the morebullishtradersarereinforcedintheirbeliefs,makingtheir assumptionscorrectwiththeirownbehavior.Thesamehappens forbearishtraderswhenpricesdrop.Confirmationbiasmaythus playaroleintheemergenceand/orotherfeatures(duration, fre-quency,etc.)ofspeculativebubblesandotherimportantstylized factsinfinancialmarkets.

Besidesexperimentsand empirical evidenceof confirmation bias,there isalsoabranchofresearchdealingwiththe formal-izationofa beliefsupdatingconsistentwiththis bias.Themost knownattemptistheonemadebyRabinandSchrag(1999),who correctedtheclassicalBayesianupdatingmechanism,provingthat undersomecircumstancesalmostallagentsmaycometobelievein https://doi.org/10.1016/j.strueco.2019.08.004

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awronghypothesis.Amongthehugeamountoffollow-uppapers, onlyrecentlyPougetetal.(2017)adaptedandappliedto finan-cialmarketstheRabinandSchragmechanism,replicatingseveral stylizedfactssuchasexcessvolatility,excessvolumeand momen-tum.Inadditiontothem,alsoCharnessandDave(2017),adapted theRabinandSchragmechanismtotestbackgroundstrategiesto correctormitigatethenegativeeffectsofthebias,andformerly Bowden(2015)insertedthismechanismtoanagent-based frame-work,findingthatthebiasmayhaveambiguousconsequenceson thevolatilityandkurtosis.Toourknowledgetheonlypaperwhere theconfirmationbiasismodeledinanalternativewaywithrespect totheoneproposedbyRabinandSchragisAldashevetal.(2011), whointroduceconfirmationbiasinthewaysocialinteractiontakes place,andcombineitwithadaptiveexpectations,showingthatthe twobiasesmaybothincreaseordecreaseinformationalefficiency. Afirstgoalofthepresentworkistobuildasimplefinancial marketmodelwheretraders(oraportionofthem)areendowed withconfirmationbiasinordertobetterunderstandhowthisbias influencesthecharacteristicsoffinancialbubblesandotherstylized facts.InthissensetheworkissimilartoBowden(2015)but differ-entlyfromhim,weintroduceanalternative,simpler,mechanism toinsertconfirmationbias,thatleadstoapiecewisedefinitionof thedynamicalsystemregulatingthetimeevolutionofassetprice andbeliefs.Themodelisatleastpartiallyanalyticallytractablein itsfullydeterministicversionwhilebyintroducingsomenoisewe replicateimportantfeaturesofrealfinancialmarkets.

Ourmodelisinsertedintotheso-calledheterogeneousagents models(HAM)literature,thathasprovedtobeaquitegood frame-worktointroduceandstudytheeffectsofthebehavioralbiasesof traders(seeChiarellaetal.,2009;HommesandWagener,2009;Lux, 2009;Westerhoff,2009forcompletedsurveys).Usuallythe inter-actionsbetweenfundamentalandtechnicaltradingrulesareable toreproducethefeaturesoffinancialmarketsthatarehardly recon-cilablewiththeassumptionofrationalityoftradersandefficiency ofthemarkets.Bothlaboratoryexperiments(Hommes,2011)and otherempiricalevidences(likethosesurveyedbyMenkhoffand Taylor(2007))supportthehypothesis thattradersrely on sim-plerules. Startingfromthepioneering contributionofDay and Huang(1990),severalpapershaveprovedthatcomplicatedprice dynamicscanbeobtainedbysimpledeterministicmodels(seefor instanceKirman,1991;DeGrauweetal.,1993;Lux,1995;Brock andHommes,1998;Chiarellaetal.,2002;Westerhoff,2004).In thisworkwemovefromthesubdivisionoftechnicaltradersin pessimistic(orbearish)andoptimistic(orbullish),similarlytoLux (1995,1998)andLuxandMarchesi(1999,2000),andweintroduce confirmationbiasinthewaytheyinterpretandusecurrent infor-mation.Apositivetrendwillbeconsideredbybullishtradersand ignoredbypessimisticones(if itisnottoopositive)and,atthe opposite,anegativepricetrendwillbeconsideredbypessimistic tradersandignoredbyoptimistictraders(ifitisnottoonegative). Thedynamicalsystemsarisingin HAMareusuallysmooth, but nonlinear.Thenonlinearitymayoriginatefromthetradingruleof thetradersorfromtheswitchingmechanismfromonestrategyto anotherone,inevolutivemodels.OnlyafewpapersintheHAM literaturedealwithdiscontinuous(oratleastnotdifferentiable) dynamicalsystems.AmongthemwehaveHuangandDay(1993), Huang et al. (2010),Tramontana et al. (2010)and Tramontana etal.(2011).Ourmodelisdiscontinuousbecauseoftheintrinsic dichotomyoftheconfirmationbiasitself.Tradersaffectedbythis biasdisplayasortofcognitivedissonance,oscillatingbetweena certainbehaviorwhentheyfaceevidencesupportingtheircurrent hypothesisandatotallydifferentonewhendealingwithevidence againstwhattheyactuallybelieve.

Inordertomimicmorequalitativefeaturesoffinancial mar-kets(suchasexcessvolatility,fattails,volatilityclustering,etc.) someresearchersstudyastochasticversionofadeterministicHAM,

obtainingquiteinterestingresults(seeLuxandMarchesi,1999; WesterhoffandDieci,2006;GaunersdorferandHommes,2007or HeandLi,2007).Inthepresentworkwealsopresenta stochas-ticversionofthemodel,replicatingsomefeaturesofrealfinancial markets.Asecondgoalofthemodelisthustotrytoidentifythe degreeofconfirmationbiasthatbetterpermitstoreplicatesuch stylizedfacts.

Thepaperisorganizedasfollows:inSection2weintroducethe financialmarketmodelformalizingthebehavioralassumptions.In Section3westudythedeterministicskeletonofthemodel,inorder tounderstandtheroleofthebehavioralparameters.InSection4 weperformananalysisofastochasticversionofthemodelbyusing theMonteCarlomethod,replicatingsomestylizedfactsoffinancial markets.Section5includesfinalconsiderations.

2. Themodel

Ourfinancialmarketmodelconsistsofastandardbuildingblock whichformalizesthebehaviorofamarketmakerandoffour dif-ferentgroupsoftraders.Onenoveltywehaveintroducedinour modelisthat,inadditiontoagroupoffundamentaliststraders,i.e. investorswhobelievethatthepriceoftheassetwillfollowits fun-damental,andtoagroupofchartiststraders,weconsidertwoother kindsoftraders.Theselasttwogroupsdifferaccordingtotheway theyformtheexpectedpriceor,moreprecisely,accordingtotheir initialguess:theycanbeoptimisticorpessimistic.While funda-mentalistsbelievethatthepriceoftheassetwillbeclosetoits fundamentalandchartiststhatthecurrentpricetrendwillpersist, theothertwogroupsoftradersconsiderthedifferencebetween thelastrealizedreturn(orlog-return)andtheexpectedreturn(or log-return)forthenextperiodandtheyadjusttheirexpectations throughamechanismthatisaffectedbyconfirmationbias. 2.1. Ourmodel’sbuildingblock

Inourmodelweadoptalog-linearpriceadjustmentmechanism withmarketmaker,inamarketwithonlyasingleriskyasset.In ourframeworkthefirstplayerwehaveisthemarketmakerwho quotesthelogoftheprice(p)accordingtothefollowingequation:

pt+1=pt+˛Dt,

whereDtisthetotalexcessdemandattimetmadeupbytheexcess

demandsofthedifferentkindsoftradersand˛isapositivescaling coefficientwhichcalibratesthelog-priceadjustmentspeed.The primaryfunctionofthemarketmaker,infact,istomediate trans-actionsoutofequilibrium,whendemandexceedsthesupply(asin thiscase),orviceversa.Heactsbyprovidingorabsorbing liquid-ity,accordingtowhetherthereisapositiveornegativeexcessof demand.Weconsideraclassicgroupoftraderscalled fundamen-talists,whobehavelikethat:

Dft=f(F−pt)3,

wheref≥0istheirspeedofreaction.Theybelievethepriceofthe assetmustfollowitsfundamentalvalue(whoselogisF),sothey buyanassetwhenitisundervalued,(thepriceisbelowits fun-damentalvalue),drivenbyahigherprobabilityofacapitalgain, andtheysellanovervaluedone(thepriceishigherthanthe fun-damental),becauseinthiscaseitismorelikelytoincurinacapital loss.Thecubicfunctionimpliesthatfundamentalistsbecomemore andmoreactivewhenthemispricingbecomeslarge.This nonlin-earformulationoftheexcessdemandoffundamentalistspermitsto capturetheincreasingprofitopportunitiesthatbecomeavailable whenthepriceismoreandmoredistantfromthefundamental value,assuggestbyDayandHuang(1990).

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Chartistsbehaveinanoppositewaywithrespectto fundamen-talists.Thebuytheassetwhenitspriceisabovethefundamental valuewhiletheysellitwhenitisbelow,bettingonthepersistence ofthecurrenttrend:

Dc

t =c(pt−F),

wherec0measuresthereactivityofchartists.

Wealsoconsidertwofurthergroupsoftraders,whohave dif-ferentexpectationsaboutthefuturepriceoftheasset:theycan thereforebeoptimisticorpessimistic.Denotingbyrtthelog-return

characterizingtheassetintheperiodbetweentandt−1,wehave that:

rt=pt−pt−1.

Theseinvestorstradeonthebasisoftheirexpectationsaboutthe log-returnoftheassetinthenextperiod(Ert+1)accordingtothis

behavioralrule:

Dopt/pest =dopt/pesEropt/pest+1 .

Again,d≥0isanotnegativereactionparameter.Optimistsand pessimistsadjusttheirexpectationaffectedbyconfirmationbias. Thustheytendtogivedifferentimportancetofactsthatsupport theirinitialguessratherthannewsthatgoesintheopposite direc-tion.

The group of optimist traders thinks the price is going to increase:

Eroptt+1≥0.

Moreprecisely,theythinkthepriceisgoingtogrowatan endoge-nousgrowthratet+1:

Eroptt+1=t+1.

Theythenadjusttheexpectedgrowthratebyconsideringthelast log-return(rt)andaccordingtothreedifferentrules.

OptimistsRule1:Ifthelastlog-returnhasbeenpositiveand largerthantheexpectone(rt>t),theoptimistsadjusttheirbeliefs

asfollows: t+1=min



ˇ1t+(1−ˇ1)rt,¯



,

where0≤ˇ1≤1measurestheanchoringtothepreviousbelief.We

assumethatoptimisticexpectationscannotexcessacertainvalue ¯>0.

OptimistsRule2:Whenthelastlog-returnhasbeenpositivebut lowerthantheexpectedornegative,butnotexcessivelynegative (−wrt≤t,withw>0),optimistsbasicallyignorethesignal

andsticktothecurrentbelief: t+1=t.

Thepositiveparameterwmeasuresthestrengthofthe confir-mationbias,sohowmuchinvestorsareanchoredtotheircurrent beliefs.Thisparameterisveryimportantforourmodelbecauseit quantifiestheroleofconfirmationbiasontheformationof fluc-tuationsinthepricepattern.Theboundarycasew=0permitsto modelaclassicfundamentalists/chartistsmodelwithoutthe con-firmationbias,andwewilluseitasabenchmarkcaseinthestudy.1

AswearegoingtodemonstrateinSection2,themorethestrength ofconfirmationbiasincreases,themorethedifferentexpectations arefar.

1 Actuallywecouldconsiderasbenchmarkanevensimplermodelwhere

opti-mistsandpessimistsarenotconsidered,thatisbyassumingd1=d2=0.Nevertheless

weprefertonotuseitasreferencebecauseprefertostayinaclassofmodelwith thesamegroupsoftraders.

OptimistsRule3:Finally,ifthelastlog-returnhasbeennegative andgreaterthanthethresholdw(rt<−w),theinvestorsofthis

groupreducetheirgrowthexpectationsinthisway: t+1=max



ˇ2t+(1−ˇ2)rt,0



.

Alsohere0≤ˇ2≤1measurestheanchoringtothepreviousbelief.

We will assume ˇ2 not lower than ˇ1. They realize that their

expectationsweretoooptimisticandreluctantlytheyreducethey expectationsbutnotwiththesamestrengthusedformakingthem evenmoreoptimistic.Insomesensoalsohereconfirmationbias couldbeconsideredatwork.Byavoidingnegativevaluesofwe preventthescenariowhereoptimistsradicallychangetheirinitial expectationsandbecomepessimists.

Pessimists,ofcourse,haveanoriginalexpectationthatinvolves adecreasingprice:

Erpest+1≤0.

Again,theythinkthepriceisgoingtodropatanendogenousgrowth ratet+1<0:

Erpest+1=t+1.

Similartooptimists,theyadjusttheexpectedgrowthrateby consideringthelastrealizedpricevariation,accordingtothe fol-lowingrules.

PessimistsRule1:Ifthelastlog-returnhaseffectivelybeen neg-ativeandevenlowerthantheirexpectations(rt<t),thepessimists

adjusttheirbeliefsinthisway: t+1=max



ˇ1t+(1−ˇ1)rt,¯



,

wherealsointhiscase0≤ˇ1≤1measurestheanchoringtothe

previousbeliefand,again,negativeexpectationscannotcrossthe (negative)levelgivenby¯.

PessimistsRule2:Ifthelastlog-returnhasbeennegativebut lessthanexpectedorpositive,butnotexcessivelypositive(t≤

rt≤w,withw>0),pessimistsignorethesignalanddonotadjust

theirbeliefsatall: t+1=t.

PessimistsRule3:Ifthelastlog-returnhasbeenpositiveand greaterthanthethresholdw,pessimistscannottotallyignorethe signalandthentheyadjusttheirnegativeexpectationsaccording tothisrule: t+1=min



ˇ2t+(1−ˇ2)rt,0



.

Again, 0≤ˇ1≤ˇ2≤1 measures the anchoring to the previous

belief.Asinthepreviouscase,itisstrongerwhenthemarketsignal isopposite(denotingsomekindofreluctance).Weavoidapositive valueofbecausepessimistscannotbecomeoptimists.2

Ifweputeverythingtogetherandconsidertwopotentially dif-ferentreactivities(d1foroptimistsandd2forpessimists),weobtain

thefollowingdynamicsystemregulatingtheassetprice,thereturn andtheexpectations’dynamics:

2Inamoregeneralversionofthemodelwecouldconsidertheanchoringandthe

confirmationbiasparameterwdifferentforpessimistsandoptimistsbutthiskind ofasymmetryisoutofthescopeofthispaper.

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T:

[c]lpt+1=pt+˛



f(F−pt)3+c(pt−F)+d1t+1+d2t+1



, t+1=

min



ˇ1t+(1−ˇ1)rt,¯



if rt>t t if −w≤rt≤t max



ˇ2t+(1−ˇ2)rt,0



rt<−w









, t+1=

max



ˇ1t+(1−ˇ1)rt,¯



if rt<t t if t≤rt≤w min



ˇ2t+(1−ˇ2)rt,0



rt>w









, rt+1=˛



f(F−pt)3+c(pt−F)+d1t+1+d2t+1



, (1) thatisasystemofdifferenceequationsofthefirstorder,thatallows ustomanagethesystemmoreeasily.

3. Studyofthedeterministicmodel

Map(1)regulatesthedynamicsoftheassetpriceandofthe beliefsofoptimisticandpessimistictraders.Thesystemadmitsat leastthreeequilibria(EF,E+andE−),allofthemcharacterizedbya log-returnequalto0(r*=0)andoptimistsandpessimistsconverge

intheirexpectationsaboutnopricemovements(*=*=0).The

equilibriadifferintheequilibriumvalueofthelog-price3:

EF−→p∗F=F, E+−→p∗+=F+

c f, E−−→p∗−=F−

c f. (2)

Thethreedifferentlevelsoftheequilibriumlog-pricerevealthat attheequilibriumthepricecanbecorrect(EF),overvalued(E+)or

undervalued(E).Ifthepriceisequaltoitsfundamentalvaluethen notraderisactiveinthemarket(alltheexcessdemandsarenull), whileintheothertwoequilibria,thepositiveexcessdemandsof somegroupsoftradersareperfectlycompensatedbythenegative excessdemandsoftheothergroups.So,theyareequilibriawith transactionswhileEFisanequilibriumwithoutanytransaction.

Thelocalstabilitypropertiesoftheequilibriamustbestudiedby takingintoaccountthepiecewisedefinitionofthemap.Ina neigh-borhoodoftheequilibria,dynamicsareregulatedbythefollowing dynamicalsystem:

pt+1=pt+˛



f(F−pt)3+c(pt−F)+d1t+1+d2t+1



, t+1=t, t+1=t, rt+1=˛



f(F−pt)3+c(pt−F)+d1t+1+d2t+1



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associatedwiththefollowingJacobianmatrix:

J:

1+˛



−3f (F−p)2+c



˛d1 ˛d2 0 0 1 0 0 0 0 1 0 ˛



−3f (F−p)2+c



˛d1 ˛d2 0

, (4)

3Thereexistalsootherequilibria,oftheform:r*=0,p*=Fand=d2

d1

but

wedonottakethemintoconsiderationbecausedynamicallystructurally unsta-bleandmeaningless froman economicpoint ofview. Infact,even ifatthe equilibriumthepricestaysfixed,optimists/pessimistsdonotneverchangetheir increasing/decreasingexpectations,andthisisnotrealistic.

whoseeigenvaluesare: (1,2,3,4)=



0,1,1,1+˛



−3f (F−p)2−c



. (5) Consideringthefundamentalequilibrium(EF),wehavethatthe

fourtheigenvalueis: F

4=1+˛c,

whichisstrictlypositivegiventhepositivityoftheparameters˛ andc.So,thefundamentalequilibriumisalwaysunstable.

Fortheothertwoequilibriatheanalysisismorecomplicated: infactforboththefourtheigenvalueis:

±4 =1−2˛c,

anditishigherthan-1providedthat: c< 1

˛. (6)

So,ifchartistsreactstoostrongly,alsotheothertwo equilib-riaareunstable.Algebraically,nothingcanbesaidwhenc<1

˛.In

fact,whilethefourtheigenvalueisinthestabilityregion(



±4



<1)

westillhavetwoeigenvaluesequalto1,sowecannoteasilytalk aboutalocallystableequilibrium.Insuchcasesitisquite compli-catedtoknowwhichkindofdynamicsaregoingtooccur,andonly numericalsimulationsmayhelptounderstandit.

Fig. 1 represents bifurcation diagrams obtained by keeping alltheparameters fixedwiththeexception ofc.Weused˛=1, d1=d2=1,ˇ1=0.95.ˇ2=0.98,f=200,¯=0.05,¯=−0.05,F=1and

w=0, that is withnoconfirmation bias.We assume thesame reactivitiesforoptimistsandpessimistsinordertoavoidthatone groupbecomesmoreimportantthantheother.Thevaluesofall theotherparameters,ifsuchthatˇ2≥ˇ1,donotaffect

qualita-tivelytheshapeofthebifurcationdiagrams.Wecanseethatwhile thevalueofcislowerthan1,condition(6)holdsandanequilibrium (inthiscaseE)isreached,eveniftwoeigenvaluesareequalto1. Afterthat,whencishigherthan1andcondition(6)isviolated,the fourtheigenvaluebecomeslowerthan−1,andacascadeofperiod doublingbifurcationsoccurs,leadingtochaoticdynamicswhenc isaround1.3.Inthislastcaseendogenousfluctuationscharacterize themotionofthedynamicvariables.

Thisnumericalresultofourbenchmarkmodelwithout confir-mationbiasisnotsurprisingbecauseitistypicalofthisbranchof theliteratureonheterogeneousagentsmodels.

Theintroductionofconfirmationbias(thatisstrictlypositive valuesofw)hasnoteffectonthestabilityoftheequilibria,infact parameterwdoesnotappearneitherintheeigenvalues(5)norin theJacobianmatrix(4).Itdoesnotimplythatconfirmationbias hasnoeffectsonthedynamicsand inthepointsofthechaotic attractormoreorlessvisitedbythetrajectories.Inorderto bet-terinvestigatetheroleofconfirmationbiasinthenextsectionwe buildastochasticversionofourmodelandwewillseethatsame featuresofrealfinancialtimeseriescanbebetterreplicatedtaking intoconsiderationthiscognitivebias.

4. Thestochasticmodel

Inprevioussections,wehavediscussedhowendogenous fluc-tuationsofpricesandreturnsemergeaslongasthereactivityof chartistsbecomeshigherthanthethresholdidentifiedin(6). Nev-ertheless,inordertodeepentheroleplayedbyconfirmationbias weshouldperformsomecomplicatedanalysisonthefeaturesof thechaoticattractorwhenwispositive.

Itisalsopossibletoenlightentheroleofconfirmationbiasby introducingsomenoiseinourmodel.Inparticular,weintroduce twostochasticelementstooursetting:

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Fig.1. Bifurcationdiagrams.Parameters:d1=d2=1,ˇ1=0.95.ˇ2=0.98,f=200,¯=0.05,¯=−0.05,F=1andw=0.Reactivityofchartistscvariesbetween0and1.7.The

asymptoticvaluesoflog-returns,log-prices,optimistsexpectationsandpessimistsexpectationsareinpanels(a),(b),(c)and(d),respectively.

1.Weassumethatthefundamentalvalueischaracterizedbya geo-metricBrownianmotionso,dealingwiththelog-fundamental valuewehave:

Ft+1=Ft+F,t, with F,t∼N(F,2F);

2.thesecondnoiseweintroduceinvolveparameterc, character-izedbyarandomwalkdynamics:

ct+1=ct+c,t, withc,t∼N(c,2c).

Alltherealizationsofthenoisesareindependentandidentically distributedandalsoindependenteachother.

Themotionofthefundamentalvalueisquitecommonandit makesthemodelmorerealistic.Thenoiseaddedtothechartists parameterccanbeinterpretedasa rumoronthereactivity(or relativenumerosity)ofchartists.Obviouslyasimilarnoisecould beaddedtotheotherreactivityparametersbutwewanttokeep the model as parsimonious as it possible and, as we will see, thesetwostochasticelementsaresufficienttoobtaininteresting results.

Fromthedynamicmotionofcweexpecttoseeinthetimeseries atypicalelementoffinancial timeseries,thatis volatility clus-tering.Infact,assuggestedbythebifurcationdiagramsinFig.1, whencislowerthanonethedeterministicskeletonofthemodel isstable,andtheonlyvariabilityisduetothestochasticelements. Attheopposite,forhighervaluesofc,andendogenousdynamics isaddedtothestochasticone.Whencalternatesbetweenvalues higherandlowerthanone,weexpecttoseeclustersof volatil-ity.

Summarizing,thestochasticmodelisthefollowing:

˜ T:

pt+1=pt+˛



f(Ft−pt)3+ct(pt−Ft)+d1t+1+d2t+1



, t+1=

min



ˇ1t+(1−ˇ1)rt,¯



ifrt>t t if−w≤rt≤t max



ˇ2t+(1−ˇ2)rt,0



rt<−w









, t+1=

max



ˇ1t+(1−ˇ1)rt,¯



ifrt<t t ift≤rt≤w min



ˇ2t+(1−ˇ2)rt,0



rt>w









, rt+1=˛



f(Ft−pt)3+ct(pt−Ft)+d1t+1+d2t+1



, Ft+1=Ft+F,t, ct+1=ct+c,t. (7)

4.1. Somefeaturesoffinancialtimeseries

Withourdeterministicmodelitisalreadypossible,under suit-ableparametricconditions,toreplicatesomequalitativefeatures offinancialmarketssuchasthefluctuationofpricesandreturns andsomekindofexcessvolatility,aswehaveseeninthe bifurca-tiondiagramsofFig.1.Thesefeaturescanbeobtainedanytime thedeterministicmodel exhibitschaotic dynamicsandit isnot anoveltyintheHAMliterature.Moreover,inourframeworkthe presenceofconfirmationbiasisnotstrictlynecessarytoreplicate thesefeatures.

Inthissectionwewanttoidentifysomeother,more quantita-tive,featuresoffinancialtimeseries,thatcanbebetterexplained ifweassumethatsometradersareaffectedbyconfirmationbias.

Wetakeasreferencethebehavioroftwoimportantindexes:the FTSEMIBindex(Fig.2)andtheStandardandPoorindex(Fig.3).The

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Fig.2. TimeseriesandprobabilityplotoftheFTSEMIBindex.Theunderlyingtimeseriesrunsfrom2004to2018andcontains3890dailyobservations.

Fig.3. TimeseriesandprobabilityplotoftheS&Pindex.Theunderlyingtimeseriesrunsfrom2004to2018andcontains3890dailyobservations.

Table1

FTSEMIBindex.Wereportthevarianceofthereturns(V)theminimumand maxi-mumreturn(rminandrmax),kurtosis(K)andskewness(S)ofthereturns’distribution.

V rmin rmax K S

0.0002 −0.133 +0.11 9.1929 −0.25

lengthconsideredis3890dailyobservations,from2004to2018. InthepanelontheleftonthetopofFig.2,wecanseethatthe returns,thatareaffectedbyhighvolatility(especiallyaroundday 1250,2000and3200),alternatingwithperiodsoflowvolatility (likeforthefirst700days).Thefigureontherightcontainsthe normalprobabilityplotofreturns:itisevidentthattheyaredistant fromwhatweexpectfromanormaldistribution.

Inordertomeasurethesefeatureswehavecalculatedthe vari-anceofthereturns(V)theminimumandmaximumreturn(rmin

andrmax),kurtosis(K)andskewness(S)ofthereturns’distribution.

ThesemeasuresarereportedinTable1.

Thehighvalueofthekurtosisandtheskewnessdifferentfrom0 confirmthatreturnsarenotnormallydistributed,andthenegative valueoftheskewnesssaysthatthedistributionisnotsymmetric butitslefttailextendsontheleft.

SimilarresultscanbefoundbylookingattheS&Pindex(Fig.3). Wehavestillturbulentperiods(liketheonearoundday1250) alternatingwithcalmerones.Thereturndistributionisagainfar fromnormal,aswitnessedbythemeasuresreportedinTable2.

Table2

S&Pindex.Wereportthevarianceofthereturns(V)theminimumandmaximum return(rminandrmax),kurtosis(K)andskewness(S)ofthereturns’distribution.

V rmin rmax K S

0.00012 −0.095 +0.11 15.75 −0.383

Table3

ValuesoftheparametersfortheMonteCarlomethod.

Parameter ˛ f d1 d2 ¯ ¯ ˇ1 ˇ2

Value 1 200 1 1 0.05 −0.05 0.95 0.98

Itiswellknownthatthefeatureswehaveidentifiedforthese twoindexescanbefoundalsoinalmostalltheothers.

Ouraimnowistofindifwithapropercalibrationofthe param-eterswecangetclosetothesemeasureswithourstochasticmodel, withparticularattentionattheroleofconformationbias.Inorder todothatwehaveperformedaMonteCarlomethod.

4.2. SimulationswiththeMonteCarlomethod

Let usdescribe howwe haveimplemented theMonte Carlo method.Firstofall,wehaveidentified,withatrialanderror pro-cedure,thevaluesofalltheparameterswiththeexceptionofw. Wewanttostressherethemostofthem(suchasFandf)arejust

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Table4

Averagemeasuresandstatisticsof1000simulations(length3980iterationseach). Wereportthevarianceofthereturns(V)theminimumandmaximumreturn(rmin andrmax),kurtosis(K)andskewness(S)ofthereturns’distribution.

Scenario V rmin rmax K S

Noc.b.(w=0) 0.00061 −0.21 +0.2 18.56 +0.016 Weakc.b.(w=0.01) 0.00056 −0.2 +0.19 19.93 −0.058 Strongc.b.(w=0.04) 0.0002 −0.147 +0.13 20.6 −0.11

scaleparametersforourmodel.Table3containsalltheparameters’ valueswekeptfixedduringoursimulations.

Wejustnotethatwehaveassignedthesamereactivityvalueto optimistsandpessimistsinordertoavoidthatonegroupoftrader becomesmoreimportantthantheother.

Thenwehaveconsideredthreevaluesofw:0(noconfirmation bias),0.01(weakconfirmationbias)and0.04(strongconfirmation bias).Inotherwords,intheweakconfirmationbiasscenario opti-mists(resp.pessimists)tradersrequireadrop(resp.rise)ofmore than1%ofthereturnstoadjustdownwards(resp.upwards)their returnexpectations.Inthestrongconfirmationbiasscenariothe thresholdis4%.Weavoidhighervaluesofwbecauseweconsider themexcessiveandhardlyrealistic.

Foreachvalueofwweperformed1000runsofsimulations,each onemadeupby3890iterations,similarlytothedailyobservations oftheindexesweshowedbefore.

Forthestochasticelementsweusedthesevaluesofinitialvalue, averageandvariance:

F−→F0=1; F=0; F2=0.005,

c−→c0=1.5; c=0; 2c =0.05.

Whensimulationsproducenegativevaluesofcweconsiderit0. Whenthesimulationproducedanexploding,divergingtrajectory, wedidnottakeitintoconsideration.4

In Table 4 we summarize the averagevalues that we have obtainedinthethreescenarios(no,weakandstrongconfirmation bias).

Wecaneasilyseethatconfirmationbiasisquiterelevantfor almostallthemeasures.Inparticular,itdecreasesthevarianceand therangeofvariationofthereturns.Itsroleisalsoimportantfor thenegativeskewnessofthedistribution.Infact,without confir-mationbiasthedistributionappearsclosetobesymmetric,witha slightlypositiveskewness.Whenweintroduceconfirmationbias skewnessbecomesmoreandmorenegativeasthebiasis ampli-fied.Theseresultswouldbeconfirmedwithevenhighervaluesof w,butaspreviouslysaidweconsiderthemunrealistic.Thevalue ofthekurtosisisalwaysextremelyhighinallthescenarios.5

Eveniftheseresultsshouldbeconfirmedthroughmore sophis-ticatedanalysisthatweplantoperforminfutureworks,wecan alreadysaythatthescenariowithstrongconfirmationbiasappears theone betterreproducing themeasuresof real financial time serieswehavetakenintoconsideration.So,thebiasseemstobe presentandrelevant.

Inordertogivea possibleinterpretationoftheseresultswe showin Fig.4 a simulation runthat can beobtainedwithour parameters’configuration.

Firstofall,wecannotethatthevalueofcoscillatesalternating valueshigherandlowerthanone(panel(a)).Thetwoseriesof log-returnsandagentsexpectationsareobtainedbyusingthesame

4 Ithappensinparticularwhencbecomeslargerthan1.7.

5 DespitewhatTable4maysuggest,thereisnomonotonic(increasing)relation

betweenthekurtosisofthereturns’distributionandthevalueoftheskewness.If weconsideralsointermediatevaluesofwwewouldseethatkurtosisoscillatesin arangeboundedby17and21.

Fig.4.In(a)itisrepresentedarealizationofthemotionofc.Thecorresponding log-returnswhenw=0andwhenw=0.04areshowninpanels(b)and(c).Finally, panels(d)and(e)showhowtheexpectationsofoptimistsandpessimistsevolvein thetwoscenarios.

realizationofcbutdifferentvaluesofw.In particular,wehave consideredascenariowithnoconfirmationbias(w=0inpanels (b)and(d))andwithstrongconfirmationbias(w=0.04inpanels (c)and(e)).Asexpectedtheerraticmotionofchartists’reactivity isabletoreplicatethefeatureofvolatilityclusteringofthereturns (panels(b)and(c)).Infactthemostturbulentperiodsarethose correspondingwithhighervaluesofc.Lessturbulenceispresent inthescenarioofstrongconfirmationbiasevenifthepeaksofthe returnsarealwaysincorrespondenceofthehighestvaluesofc.Our interpretationisthatwhenreturnsareextremelyhighandthere isnoconfirmationbias,optimistsbecomemoreandmoreactive, whilepessimistsarelessrelevantbecausetheirexpectationsget closetozero.Theoppositewhen returnsareextremelylow,by exchangingtheroleofoptimistsandpessimists.Thecombinationof thesebehaviorsexacerbatestheturbulenceandpermitstoincrease thevolatilityofthereturns.Attheopposite,whenthereisastrong

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confirmationbias,despitethehighvalueofthereturns,pessimists keeppersistintheirideathatreturnsaregoingtodecreaseand theiractionatleastpartiallycompensatethebehaviorofoptimists, reducingthevolatilityofpriceandreturns.

Thisis confirmed bythe time evolution of theexpectations inthetwoscenarios.Inpanel(d),withoutconfirmationbias,we canseethattheexpectationsperiodicallyseemstoconvergence towardszero(whencislow,thatisforthedeterministicskeleton anequilibriumisstable),whiletheyaremoreseparatedinperiods ofturbulence.Whenconfirmationbiasisstronglypresent(panel (e))theaveragedistancebetweentheexpectation islargerand whenoptimistsbelieveinalargepositivereturn,pessimistsare stillconvincedthatreturnsmustdrop.Inouropinionthisis coher-entwithourinterpretationoftheresultsobtainedwiththeMonte Carlomethod.

5. Conclusions

Inthispaperwehavestudiedasimplefinancialmarketmodelin whichfourdifferentgroupsoftraderscoexist.Thesegroupsdiffer inthewaytheyformtheirexpectationsaboutthepriceoftheasset. Themaporiginatedfromthedifferentexpectationsis piecewise-definedandischaracterizedbyamultiplicityofequilibriawhere thepricestaysconstantandreturnsarenull.Wehavedeepenthe roleplayedbyoptimistsandpessimists,whoseexpectationsdiffer accordingtothewaytheyinterpretthenewrealizationofprices andreturnstoadjusttheirexpectationforthefuture.Inparticular, wehaveconsideredtradersaffectbyconfirmationbias,thatleads to(atleastpartially)ignoreevidenceagainsttheircurrentbeliefs. ThispsychologicalphenomenoncausesafailureintheBayesian updating.Ourexpectationshavebeenconfirmedbythesimulations ofourmodel,sinceourresultswerethatconfirmationbiasmakes thedifferentbeliefsmoredistantfromeachother.Despitethefact thatourmodelissimple,wefoundoutthatitisabletoexplain thedynamicspresentinfinancialmarkets,suchastheamountof volatility,volatilityclustering,skewnessandkurtosisinthe distri-butionofreturns.Wehavealsocheckedifthedynamicssimulated byourmodelisrepresentativeofthebehaviorofdata–in particu-laroftheFTSEMIBindexandoftheStandardandPoorindex–and wenoticedthatitprovidesagoodrepresentationofreality espe-ciallyinthescenariowhereconfirmationbiasisstronglypresent. Tobringourmodeltoreality,wehaveintroducedastochastic com-ponentwhichdescribessomerandomeventsthattakeplacein financialmarkets.Ourmodelmaybeextendedandimprovedin variousways:wecanintroduceswitchingstrategiesbetween opti-mistsandpessimists,wecanaddotherstochasticcomponentsalso tothetworeactionparametersoffundamentalistsandchartistsor wecanassumetouseamorecomplexdemandfunction.Another possibleextensionistothinkaboutpossiblepolicyimplicationsto thedynamicswegetfromthemodel.Aboutthis,itisfundamental todesignmorepreciselyhowfinancialmarketswork. Neverthe-lesswhatwehavefoundconfirmsthatpsychologicalbiasaffecting investorsmayhaveimportanteffectsonfinancialtimeseriesand contributeinexplainingsomeoftheirmostimportantandrelevant features.

Acknowledgement

Workdevelopedintheframeworkoftheresearchprojecton “Models of behavioral economics for sustainable development” financedbyDESP-UniversityofUrbino.

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