Munich Personal RePEc Archive

Predictable markets? A news-driven

model of the stock market

Gusev, Maxim and Kroujiline, Dimitri and Govorkov, Boris

and Sharov, Sergey V. and Ushanov, Dmitry and Zhilyaev,

Maxim

24 April 2014

Online at https://mpra.ub.uni-muenchen.de/58831/

MPRA Paper No. 58831, posted 25 Sep 2014 02:48 UTC

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Predictablemarkets?Anews‐drivenmodelofthe

stockmarket

MaximGusev1,DimitriKroujiline*2,BorisGovorkov1,SergeyV.Sharov3,DmitryUshanov4andMaximZhilyaev51IBCQuantitativeStrategies,Tärnaby,Sweden,www.ibctrends.com.2 LGT Capital Partners, Pfäffikon, Switzerland, www.lgt‐capital‐partners.com. Email:dimitri.kroujiline@lgt.com.(*correspondingauthor)3N.I.LobachevskyStateUniversity,AdvancedSchoolofGeneral&AppliedPhysics,NizhnyNovgorod,Russia,www.unn.ru.4 Moscow State University, Department of Mechanics and Mathematics, Moscow, Russia,www.math.msu.su.5MozillaCorporation,MountainView,CA,USA,www.mozilla.org.Abstract:Weattempttoexplainstockmarketdynamicsintermsoftheinteractionamongthree variables: market price, investor opinionand information flow. We propose aframework for such interaction and apply it to build amodel of stockmarket dynamicswhich we study both empirically and theoretically. We demonstrate that this modelreplicates observed market behavior on all relevant timescales (from days to years)reasonably well. Using the model, we obtain and discuss a number of results that poseimplicationsforcurrentmarkettheoryandofferpotentialpracticalapplications.Keywords: stockmarket,marketdynamics, returnpredictability, news analysis, languagepatterns, investor behavior, herding, business cycle, sentiment evolution, referencesentiment level, volatility, return distribution, Ising, agent‐basedmodels, price feedback,nonlineardynamicalsystems.

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IntroductionThereisasimplechainofeventsthatleadstopricechanges.Priceschangewheninvestorsbuyorsellsecuritiesanditistheflowofinformationthatinfluencestheopinionsofinvestors,accordingtowhich theymake investmentdecisions.Although thisdescription is too general tobeofpracticaluse,ithighlightsthepointthatwiththecorrectchoiceofhypothesesabouthowprices,opinionsandinformationinteractitcouldbepossibletomodelmarketdynamics.In the first part of this paper (Section 1),we develop amechanism that links information toopinionsandopinionstoprices.First,weselectthemarketforUSstocksastheobjectofstudydueto itsdepth,breadthandhighstanding inthefinancialcommunity,whichplaces it in thefocusofglobalfinancialnewsmediaresultinginthedisseminationoflargeamountsofrelevantinformation.We then collect information using proprietary online news aggregators aswell as news archivesoffered by data providers. Next we analyze collected news with respect to their influence oninvestors’marketviews.Ourapproach is to treateach individualnews itemas if itwere a “salespitch”thatmotivatesinvestorstoeitherenterintoorwithdrawfromthemarketandapplymethodsfrommarketingresearchtoassesseffectiveness.We then proceed tomodel opinion dynamics. It is reasonable to expect that, along with theimpact of information, such a model should also account for interactions among investors andvarious idiosyncratic factors that can be assumed to act as random disturbances. There aresimilarities between this problem and some well‐studied problems in statistical mechanics,allowingustoborrowfromtheexistingtoolkittoderiveanequationfortheevolutionof investoropinion in analytic form. At this juncture, we investigate the connection between opinions andprices.Tounderstandit,wemustanswerquestionspertainingtohowinvestorsmakedecisions;forinstance, is an investor more likely to invest if her market outlook has been stably positive orwhether it has recently improved? We suggest a simple solution based on observed investorbehavior.

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Theproceduredescribedaboveisusedtoconstructatimeseriesofdailypricesfrom1996to2012basedon collected information and compare itwith the time series of actualmarketpricesfrom that sameperiod.Wedemonstrate that themodel replicates the observedmarket behavioroverthestudiedperiodreasonablywell.IntheendofSection1wereportanddiscussresultsthatmayberelevantformarkettheoryandpracticalapplication.Although this (empirical)model enablesus to translate a given information flow intomarketprices, it cannot sufficiently highlight the nature of complex market behaviors precisely for thereasonthatinformationistreatedinthemodelasagiven.Togainfurtherinsightintotheoriginsofmarketdynamicswemustextendthisframeworkbyincludinginformationasavariable,alongwithinvestor opinion andmarket price. In the second part of this paper (Section 2), we incorporateassumptionsonhowinformationcanbegeneratedandchanneledthroughoutthemarkettodevelopaclosed‐form,self‐containedmodelofstockmarketdynamicsfortheoreticalstudy.We find that informationsupplied to themarket canbe representedby twocomponents thatplayimportantbutdifferentrolesinmarketdynamics.Thefirstcomponent,whichconsistsofnewscausedbyprice changes themselves, induces a feedback loopwhereby information impactspriceand price impacts information. The resulting nonlinear dynamic explains the appearance in themodelofessentialelementsofmarketbehaviorsuchastrends,ralliesandcrashesandleadstothefamiliarnon‐normalshapeof thereturndistribution.Additionally, thisdynamicentailsacomplexcausalrelationbetweeninformationandprice,ascauseandeffectbecome,inasense,intertwined.The second informational component contains any other relevant news. It acts as a randomexternalforcethatdrivesmarketdynamicsawayfromequilibriumand,fromtimetotime,triggerschangesofmarketregimes.

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We conclude Section 2 by comparing the characteristic behaviors of theoretically‐modeled,empirically‐modeled and observed market prices and by discussing the possibility of marketforecasts.Finally,weprovideanoverallsummaryofconclusionsinSection3.Thepresentstudyhasbeencarriedoutwithaviewtowardpotentiallypredictingstockmarketreturns.Thisviewissupportedbytheempiricalresearchoverthelast30years,whichsuggeststhatreturnsarepredictable,especiallyoverlonghorizons(seeFamaandFrench(1988,1989),CampbellandShiller(1988a,b),Cochrane(1999,2008),BakerandWurgler(2000),CampbellandThompson(2008)). This empirical research has primarily focused on identifying potentially predictivevariables,suchasthedividendyield,earnings‐priceratio,creditspreadandothers,andverifyingorrefutingtheircorrelationwithsubsequentreturns,typicallyapplyingregressionmethods.Ourobjectiveistocapturethebasicmechanismsunderlyingthispredictability.1Weshowthatthe theoretical model (Section 2), which treats information, opinion and price as endogenousvariables,canreproduceobservedmarketfeaturesreasonablywell,includingthepricepathandthereturn distribution, under the realistic choice of parameter values consistent with the valuesobtainedusingtheempiricaldata(Section1).Mostimportantly,thismodelpermitsmarketregimeswheredeterministicdynamicsdominaterandombehaviors,implyingthatreturnsare,inprinciple,predictable.Becausethismodelisfundamentallynonlinear,thecausalrelationamongthevariables                                                            

1Theexistingliteratureexaminesthelong‐termpredictability(e.g.monthlyandannualreturns)andlinksit to economic fluctuations and changes in riskperceptionof investors.Our findings support the view thatlong‐termpredictabilityexistsand,furthermore,indicatethatreturnsmayalreadybepredictableonintervalsofseveraldays.Wereachthisconclusionusingaframeworkwhichattributespricechangestothedynamicsofinvestoropinion.Interestingly,wefindthat,overlonghorizons,thereisaconnectionbetweentheevolutionofinvestoropinionandeconomicfluctuations(Fig.9,Section1.4.1).

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is substantiallymore complex than regressiondependence. It therefore follows that the standardapproach to return prediction, based on regressionmethods,may need to be redefined. Such anapproachwouldcombine,analogoustoweatherforecasting,theoreticalmodelswithempiricaldatatopredictreturns(seeSection2.2.3).ThestockmarketmodeldevelopedinthispaperbelongstothefamilyofIsing‐typemodelsfromstatistical mechanics (see Section 1.2.1.). The Ising model is a subset of agent‐based models –theoretical models that attempt to explain macroscopic phenomena based on the behaviors ofindividual agents. When applied to economics, the Ising model simulates dynamics amonginteracting agents capable ofmaking discrete decisions (e.g. buy, sell or hold) subject to randomfluctuations(duetoidiosyncraticdisturbances)and,ifany,externalinfluence(e.g.publiclyavailableinformation)andvariousconstraints(e.g.wealthoptimization).Astherearenumerouschoicesforfactors affecting the dynamics of agents, the main goal is the selection of a reasonably simplecombinationthatallowsthereplicationofdistinctivefeaturesofactualmarketbehaviorinamodel.Anumberofagent‐basedmodelshavebeenproposedinthecontextoffinancialmarkets,e.g.Levy,Levy and Solomon (1994, 1995), Caldarelli, Marsili and Zhang (1997), Lux and Marchesi (1999,2000), Cont and Bouchaud (2000), Sornette and Zhou (2006), Zhou and Sornette (2007). Thesemodelsdifferby choicesmade in the selectionof factors, suchas, for example, the formof agentheterogeneity,thetopologyof interaction,thenumberofdecision‐makingchoices,theformoftheagents’utilityfunctionsandthenatureofexternalinfluenceonagents.Inthepresentmodelwechosetoapplytheminimalnumberofpossibleopinions(buyorsell)and the simplest interaction pattern (all‐to‐all) to facilitate its study. We also made no a prioriassumptionsoninvestors’behaviors,preferencesortradingstrategies.Ourcontributionisfocusedon other areas. First, we identify informational patterns that can effectively influence investors’opinions and measure the corresponding information flow (Section 1.1). Second, we analyzeopinion dynamics in the presence of this information flow using a classic (homogeneous) Ising

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modelandconstructanempiricaltimeseriesdescribingtheevolutionofinvestoropinion(Section1.2).Third,wededucetherelationbetweeninvestoropinionandmarketprice.Whileintheexistingliteratureopinionsaretypicallyequatedwithinvestmentdecisions,resultinginpricechangesbeingproportional to the difference between the number of positive opinions (buyers) and negativeopinions (sellers) in a model2, we derive an equation for price formation by proposing thatinvestors tend to act on their opinions differently over short and long horizons and use thisequation to obtainanempirical time series ofmodelprices (Section1.3). Fourth,we formulate atheoreticalmodelofmarketdynamicsasaheterogeneousIsingmodel(Section2)whichconsistsoftwo types of agents: investors (whose function is, naturally, to invest and divest) and “analysts”(whose function is to interpret news, form opinions and channel them to investors). Thismodelyields a closed‐form, nonlinear dynamical system shown to generate behaviors that are inagreementwithboththeempiricalmodelandtheactualmarket.Thispaper,whichisprimarilyintendedforeconomistsandinvestmentprofessionals, isbasedon ideas and methods from various scientific and practical disciplines, including statisticalmechanics and dynamical systems. To preserve its readability, we have endeavored to strike abalancebetween thedepthof thematerial and theease of its presentation.To this end,wehaveplaced technical discussions and derivations in the appendices and provided the first principles‐basedexplanationsofutilizedconceptstomaketheworkself‐contained.

1.PartI–EmpiricalstudyofstockmarketdynamicsIntheempiricalstudywedevelopamodel thattranslates information intoopinionsandopinionsintoprices.Section1.1examineswhichinformationcanberelevantinthefinancialmarketscontext                                                            

2Sometimesprice changes are assumed tobeproportional to a function of thedifference between thenumberofpositiveandnegativeopinions,e.g.Zhang(1999)appliedasquarerootofthisdifference.

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andoutlinesourapproach tomeasuring itandconstructing theempirical timeseries.Section1.2derives amodel of opinion dynamics and applies it to the time series from the previous section.Section 1.3 develops a model of price formation that, based on the modeled investor opinion,produces the empirical prices for comparison with the actual stock market prices. Section 1.4discussesresultsandapplications.TherelevanttechnicaldetailsareinAppendicesAandB.1.1.InformationInformation comes in many forms. It differs across various dimensions such as content, source,pattern and distribution channel, and there are myriad possibilities for “slicing and dicing” it.Choices made on how information is handled may lead to different theories and practicalapplications.

Different methods of handling information form the foundations of quintessentially differenttrading strategies in the field. For example, systematic hedge fund managers apply quantitativetechniques to analyze price data to detect trends; global macro managers base their bets onmacroeconomic factors; and fundamental long/short funds employ financial analysis to identifymispricedassets.3                                                            

3Wenotethatincreasesinthequantityofinformationontheinternetanditsaccessibility,thepopularityofsocialnetworksandimprovementsinnaturallanguageprocessingandstatisticalmachinelearningduringthe last decade have prompted the development of trading strategies where news analytics complementtraditional sources of information. This has also motivated recent empirical research on the correlationbetweendisseminatedinformation,includingsocialmediacontent,andpricemovements(e.g.SchumakerandChen(2009);Lietal.(2011);Rechenthin,StreetandSrinivasan(2013);Preis,MoatandStanley(2013)).Thispaper broadly shares the motivation and aims to advance research in this area by employing a novelapproach.

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In the theory of finance, a seminal hypothesis on information’s influence on investors – theefficient market hypothesis – was developed by Fama (1965, 1970) and Samuelson (1965). Theefficient market hypothesis essentially considers that, as a result of competition among rationalinvestors,informationrelatedtopastevents,aswellastoanticipatedfutureevents, isreflectedinspotprices.Thistheoryimpliesthatpriceschangeonlyasunexpectednewinformationisreceivedand,becausesuchinformationisrandom,pricechangesmustalsoberandomvariablesthatcannotbepredicted.Webaseourapproachontheobservationthatinthecontextoffinancialmarketsinformationisimportantonlyduetoitsimpactoninvestors.Therefore,itwouldbesensibletoconsideronlythoseinformational patterns that can effectively impact investors’ decision making.We put forward ahypothesis that the effectiveness of information is determined by the degree of directness of itsinterpretation in relation to the expectations of future market performance, and assume thatinformationwhichismoredirectwillbemoreeffectivelyperceivedbyinvestors.Wewilltestthishypothesisonempiricaldatainthenextsections.Herewebrieflydiscusstherationalebehindit.Thebasicideaisintuitivelysimple:wevieweachnewsitemasifitwerea“salespitch” to buy or sell the market aimed at investors. Sales professionals will confirm that astraightforward,unambiguousmessage is imperativefor ittobeeffective,and inthepresentcasetheleastambiguousmessageistheonethattellsinvestorsdirectlywhetherthemarketisexpectedtogoupordown.Information that does not succinctly spell this message out requires individually subjectiveinterpretationleadingtoconflictingviewsastoitsimplicationsforanticipatedmarketperformance.Wemay suppose that newswhich cannot be easily interpreted in terms ofmarket reactionwillresult in a roughly equal number of positive and negative views or in a lack of strong opinionsaltogether and so in average are unlikely to impactmarket prices. Conversely, news that can be

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easily interpretedwill be quickly interpreted (orwill have alreadybeen interpretedprior to anynews release as amarket scenario) andwill thenbe followedby informationabout the expectedmarket reaction. It therefore seems reasonable to conclude that information suggesting thedirectionoftheexpectedmarketmovementisimportantforpricedevelopment.Thequantifyingofsuchpatterns–henceforthreferredtoasdirectinformation–inthegeneralnewsflowisanareaofourresearchfocus.Thus, in the general daily news flow,wewish tomeasure the number of news releases thatcontainthisdirectinformation.WelimitourstudytotheUSstockmarketandselecttheS&P500Index as its proxy. In particular, we consider only the English language media and search forpatterns that indicate future returns, e.g. “the S&P 500will increase / decrease”, or thosewhichimplya trend,e.g. “theS&P500hasgrown/ fallen”,assuming that investorswouldreact tosuchinformationbyextrapolatingthetrendintothefuture.Wehavechosennottoassignweightstonewsitemsasameasureoftheirrelativeimportance.Thisfactorshouldemergenaturallybycapturingtherepetitionsorechoingoftheoriginalreleaseby other news outlets. Thus, tomeasure the direct information correctly, we collect all relevantnewsreleases4includingtheduplicates.Basedontheaboveconsiderations,wehavedevisedanumberofrulesandappliedthemtodailynewsdatafortheperiodfrom1996to2012retrievedfromDJ/Factivadatabase.Thesesameruleshavealsobeenappliedtothenewsthatwehavedirectlycollectedonadailybasisfromonlinenewssourcessince2010(oneoftheadvantagesofassemblingaproprietarynewsarchiveistheabilitytocontroltheselectionofnewssourcestoensuredataconsistencyovertime).                                                            

4Wealsotakeintoaccountnewsreach,whichisamarketingresearchterm,referringheretothenumberofpeopleexposedtoanewsoutletduringagivenperiod,akintoanewspaper’scirculation.

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WemeasurethenumberofnewsreleasescontainingpositiveornegativedirectinformationforeachdaywheretheNYSEwasopenforbusinessduring1996‐2012toobtainatimeseriesofdailydirectinformation overthatperiod:,

where isthenumberofnewsitemscontainingpositivedirect information, isthenumberofnewsitemscontainingnegativedirectinformationand isthenumberofallrelevantnewsitems(e.g.wherethephrase“S&P500”ismentioned);ordinarily, since containsneutralinformation along with and . For illustration, the time series of daily and aredisplayedonasampleinterval2010‐2012(Fig.1).5                                                            

5Technicalremarks:1. fromDJ/Factivafor1996‐2012isusedasaninputintothemarketmodelthatwedevelopoverthenext sections. In this context it is important to note that the volume gradually rose during thatperiod,fluctuatingonaverageintherangeof25‐50perdayin1996‐2001,50‐100perdayin2002‐2009and100‐200perdayin2010‐2012.Accordingly,wehavetoexercisecautionwithregardstointerpretingmodelingresultsfortheperiod1996‐2001duetolowerdataquality.2. frombothsourceshavebeencalibratedtoreduce(butnoteliminate)thepositivebiasoftheoriginalseries.Thishasbeenachievedbyincreasingtheweightof _ by30%inthecaseofDJ/Factivaand45%inthecaseoftheproprietarydata.Thecalibrationisappliedtoachievemorerealisticmodelbehavior;wenote,however,thattheresultsobtainedwithandwithoutthecalibrationarenotsubstantiallydifferent.Theexcesspositivebiasmaybepartiallyrelatedtogeneralasymmetryintheperceptionofnegativeandpositiveevents(Baumeisteretal.,2001),which,forexample,impliesanaversiontolossesinthecontextoffinancialmarkets(KahnemanandTversky,1979),howeverthisdiscussionfallsoutsidethescopeofthepresentwork.

 

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whetherformedthroughrationalanalysisor irrationalbeliefs,basedonallpossible informationalinputs.6In the previous section we have proposed that direct information can effectively influenceinvestors’views.Inthissectionweuseananalogyfromphysicstoexplorehowinvestorsentimentcanevolveinresponsetodirectinformationflow.

1.2.1.ModelofsentimentdynamicsInthissectionwemakeafirstprinciples‐basedintroductiontotheIsingmodel(Ising,1925)fromstatisticalmechanics, as it applies to the problempresented here, and followwith a preliminarydiscussion of the relevant effects. Although the Ising model has been broadly applied to study                                                            

6 Wenote that the term sentiment appears in the finance literature usually in the sense of an opinionpromptedbyfeelingsorbeliefs,asopposedtoanopinionreachedthroughrationalanalysis. Inthiscontext,the notion of sentiment has been applied to describe the driving forces behind price deviations fromfundamental values that cannot be explained within the classic framework of rational decision making.Various empiricalmeasures of sentiment utilized in the literature (e.g. Brown and Cliff (2004), Baker andWurgler (2007), Lux (2011)) include indices based on periodic surveys of investor opinion aswell as theproxiessuchasclosed‐endfunddiscounts,advancingvs.decliningissues,callvs.putcontractsandothers.Ourapproachreliesonthe fact that it is the investoropinionpersethat leadsto investmentdecisions,irrespectiveofwhetherithasbeenformedrationallyorirrationally.Theunderstandingofhowthissummaryopinion – which we have herein defined as sentiment – evolves is one of the goals of the present work.Accordingly, inthissectionwemodelthesentimentdynamicswithout invokingexplicitassumptionsontherationality of agents and apply this model to obtain sentiment empirically from the time series of directinformationasmeasuredinSection1.1. 

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problems in social and economic dynamics, we have not come across its explanation from firstprinciplesinthesocioeconomiccontext.Weconsideramodelwithalargenumberofinvestors( ≫ 1),identicalinallrespectsexcepttheabilityto formdifferingbinaryopinions( 1)as towhetherthemarketwillriseor fall.Letussay that the ‐th investor has the sentiment 1if she opines that the market will rise and1ifsheopinesthatthemarketwillfall.Weintroduceafunction,calledenergyinphysics,thatdescribes themacroscopic statesof such a system.This function should contain those factors forwhichwewishtoaccountinthemodel.Basically,therearetwosuchfactors.The first factorcapturesthe impactdue to the flowofdirect information.Earlierweassumedthat direct information, either positive (the market will go up) or negative (the market will godown),canbeeffectiveinforcinginvestorstochangetheiropinions,i.e.direct informationactstoorienttheinvestors’sentimentsalongits(positiveornegative)direction.Weexpresstheenergyofimpact on the ‐th investor as , where is direct information received by the investor.Energyisnegativewheretheinvestor’sopinionanddirectinformationarecoalignedbecause andareofthesamesignandispositiveotherwise.Itmeansthat,accordingtothefamiliarprincipleofminimumenergyfromphysics,directinformationactsto(re)alignsentimentalongitsdirectiontoreducetheenergy.Theoverallenergyduetotheimpactofdirectinformationflowontheinvestorsis ∑ , obtained by summing for all investors and where is a positive coefficientdeterminingthestrengthoftheimpact.The second factor is the interaction among the individual investors (agents). When peopleexchangeopinions,theymayinfluencetheopinionsofothersandinturnbeinfluencedaswell.Sowecangenerallysay thatpeople tendto “coalign” theiropinions. Inourcontext,where investorsexchangeviewsaboutmarketperformance(i.e.sentiments),wecanwritetheenergyofinteractionbetween the ‐th and ‐th investors as , where is a positive coefficient determining the

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strengthof interaction.Energywillbeminimal (negative)whenthesentimentsarecoalignedandmaximal (positive) otherwise, which means that, in line with the principle of minimum energy,interaction tends to make investors co‐orient their views. The overall energy of interaction∑ isobtainedbysumming acrossallpairsof investors( )andmultiplyingthesumby½toeliminatedoublecounting.Consequently,thetotalenergyofthesystemtakestheform:

12 . 1 Provided that the same information is available to all investors, it follows that the energy (1) isminimized if investors’ sentiments are coaligned. We should therefore expect to find that allinvestors in themodelshare thesamepositiveornegativemarketoutlook.Yetweknowthat thereality is different: there will always be investors whose market sentiments are contrary to thepopular opinion. This discrepancy arises for the reason that our model does not integrate theinfinitenumberof different specific influences experiencedby investors that lead to randomnessanddisorderpresentinrealworld.

Toincorporatetheeffectofthisrandomnessweagainborrowfromphysicswheretemperatureservesasameasureofdisorder–thehigherthetemperature, themorestochastic‐likeasystem’sbehavior. We apply this same methodology and introduce an economic analog to temperature,whichwehenceforthsimplycalltemperatureor ,thatwillindicatethedegreeofdisorderinourmodel.Thismeansthateachinvestorwillbesubjecttorandomdisturbancesthatmaycausehertooccasionally change sentiment irrespective of other investors in the model; in other words,investors’sentimentswillbesubjecttorandomfluctuations.Asaresult,thestudyofsystem(1)nowrequiresstatisticalmethodsofanalysisthatwewillapplyinthenextsection.

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Let us briefly discuss the effects that suchmicroscopic random fluctuationsmay have on themacroscopiccharacteristicsof thesystem.At lowtemperatures,randomfluctuationsareweak,sothat the interactions among investors lead the system toward one of two ordered macroscopic,polarizedstateswherethetotalsentiment iseitherpositiveornegative. In thiscase,accordingtothe model, investors’ behavior is characterized by a high degree of herding that continues toincreaseasthetemperaturefalls.Athightemperatures,randominfluencesprevailandthesystemas awhole appears disordered,with total sentiment fluctuating around zero, i.e. investors fail toestablish a consensus opinion and proceed to act randomly. In physics, the above‐describedmacroscopic states are called phases and when a system changes state it is known as phasetransition.In reality it is probably seldom, if at all, that investors behave either randomly or in perfectsynchronicity. Itwould be reasonable to suppose that the actualmarket ismostly confined to atransitionalregimebetweendisorderedandorderedphases,whererandombehaviorandherdingorcrowdbehaviorcanbeofroughlyequalimportance.InSection1.3.2wewillprovideevidenceinsupportofthisconjecture.Thisfamilyofmodels,originallydevelopedtodescribeferromagnetisminstatisticalmechanics,is broadly called the Isingmodel. The Isingmodel has been applied tomany problems in socialdynamicsover the last30years (seeCastellano,FortunatoandLoreto (2009)). Ineconomics, theIsingmodelwasutilized for the first timebyVaga (1990),whoadapted it to financialmarkets toinfer the existence of certain characteristicmarket regimes. Since themid‐1990s there has beenextensiveeconomicresearch in thisareaandtherelated fieldofagent‐basedmodeling(seeLevy,LevyandSolomon(2000),Samanidouetal.(2007),Lux(2009)andSornette(2014)).OurapplicationoftheIsingmodeldiffersfromotherresearchinthisareaintwomainaspects:(i) in the empirical part of the paper (Section 1), we use direct information flow, whichwe can

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measure, as an external force in the (homogeneous)model and arrive at a closed‐form dynamicequationthatgovernstheevolutionofthesystem’ssentiment;and(ii)inthetheoreticalpartofthepaper (Section 2),wedevelop a heterogeneous, two‐component extensionof the abovemodel togaininsightintothedynamicsoftheinteractionbetweensentimentanddirectinformation.1.2.2.EquationforsentimentevolutionAs noted earlier, system (1), in which investors’ sentiments are subject to random fluctuation,necessitates statisticalmethodsof analysis. In this sectionwe study theevolutionof the system’stotalsentimentasastatisticalaverage7: ⟨ ⟩/ ,where ∑ and⟨⟩denotesthestatisticalaverage.Wenotethat 1 1because⟨ ⟩canvarybetween and .

Toderive theequation for inanalytic form,wemake two simplifying assumptions: (i) allinvestorsreceivethesameinformationand(ii)eachinvestorinteractswithallotherinvestorswiththe same strength, which yields an all‐to‐all interaction pattern8. As a result, the system’s totalenergy(1)canbewrittenas12 , 2

                                                            

7Alsocalledtheensembleaverage.Todistinguishbetweenthestatisticalorensembleaverage,ontheonehand,andtheaveragewithrespecttotime,ontheotherhand,wedenotetheformerusinganglebracketsandthelatterusinghorizontalbars.8Theuseoftheall‐to‐allinteractionpatternisasensiblefirststepforstudyingthisproblemasitisalsothe leading‐order approximation for a general interaction topology in the Isingmodel.Wewill discuss theimplicationsofthisapproximationinSection2.2.3.

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where is the uniform information flow ( ) and is the constant strength ofinteraction( ).Aconsiderationofthekineticsofsystem(2)leadstothefollowingequationfor (AppendixA,eq.A16):9

tanh , 3 where the dot denotes the derivative with respect to time, is the rescaled strength ofinteraction10, hasbeenintroducedintheprevioussectionastheeconomicanalogtotemperature,i.e.theparameterwhichindicatesthelevelofdisorderinthesystem,andw 1/ with definedas the characteristic time overwhich randomdisturbanceswillmake individual sentiment flipandsoindicatestheinvestor’saveragememorytime‐span.

Letusconsiderequation(3).Tobeginwith,itwouldbemoreconvenienttorewriteitas, tanh , 4

where has themeaning of the total force acting on the sentiment , / is adimensionlessparameterwhich,being inverselyproportional to temperature,determines thedegreeoforder inthesystemand / .                                                            

9Equation(3)isobtainedinthispaperasaspecialcaseofthesystemofdynamicequationsderivedinthestatistical limit → ∞for the all‐to‐all interaction pattern (see Appendix A) that we study in Section 2.Equation(3)wasoriginallyobtainedbySuzukiandKubo(1968),whousedthemean‐fieldapproximationtoderiveit.10Note that ∼ 1/ ensures that coupling energy is finite in the all‐to‐all interaction case, therefore1 .

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Forillustrationpurposesletusassumethattanh σ ≪ 1,whereσ isthestandarddeviationof the time series .11This assumption enables us to represent by a sum of the time‐independentandtime‐dependentcomponents,whichtakesthefollowingleadingorderform:, tanh sech tanh .

Equation(4)canthenbewrittenassech tanh , 5

where thecoefficient is zero in accordancewith (4) (wewillneed for interpreting (5) in thenext paragraph) and the time‐independent component of has been expressed via the function,calledpotential,givenwiththeprecisionuptoaconstantby 12 1 ln cosh . 6

And so,we have arrived at the equation for an overdamped, forced nonlinear oscillator. Theoverdampingmeansthat inertia( ) is smallandcanbeneglectedrelativetodamping( ).Thusequation(5)canbe interpretedasgoverning themotionofa zero‐mass( 0)dampedparticledriven by the force applied, which is dependent on , inside the potential well, the shape ofwhich is determined by . Accordingly, takes on the meaning of the particle’s coordinate.Thereforethemotionoftheparticlewilltracetheevolutionpathofthesentiment.Inotherwords,equation(5)describesasituationwherethetime‐dependentforce(informationflow)actstodisplacetheparticle(sentiment)insidethepotentialwellfromitsat‐restequilibrium                                                            

11As we will see later, σ 1(Table I), so that this assumption is within the relevant range ofparametervalues.

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(where the consensus of opinion is reached) while the restoring force (the interactions amonginvestors subject to random influences) counteracts it by compelling sentiment back towardequilibrium.12Theexpression for thepotential (eq.6) reveals theexistenceofdisorderedandorderedstateswithin the system as a function of temperature.13Thepotential is symmetric. It has theU‐shapewith one stable equilibriumpoint 0for 1(the high temperature phase: ) andthe W‐shape with one unstable ( 0) and two stable ( ) equilibrium points that aresymmetricwithrespecttotheoriginfor 1(thelowtemperaturephase: )(Fig.2).TheU‐shapecorrespondstothedisorderedstatesincesentimentiszeroatequilibriumreflectingthefactthatinvestorstendtobehaverandomlyinthisphase.TheW‐shapecorrespondstotheorderedstatewheresentimentsettlesateitherthenegative( )orpositive( )valueatequilibrium,asherdingbehaviorprevailsinthisphase.Thus,themodelcontainsthephaseregimesthatwerediscussedinSection1.2.1.Figure2showsthat inthedisorderedstate( 1)decreasing causesthepotentialwelltocontract,sothatsentimentbecomesentrenchedaroundzero.Similarly, intheorderedstate(1)when increases the negativewell ( 0) and the positivewell ( 0) quickly deepen andsimultaneously shift toward the boundaries ( 1), so that sentiment becomes trapped at the                                                            

12Thismotion is finiteas 1 1, 0at 1and 0at 1.However, the systemdoesnotpermit freeoscillations around equilibrium: the absenceof inertiamakes theparticle fall directlytowardtheequilibriumpoints,whicharethestablenodesindynamicalsystemsterminology.13Itmaybeeasiertounderstandthebehaviorof ifitisexpandedintoatruncatedTaylorseriesin

thepowersof as ,whichholdsreasonablywellforany when ~1.Notethatthephasetransitionat 1correspondstothechangeofsignofthequadraticterm.

 

extreme1isunrealis

Figure2the stabpointstoandoneFinahas nosentimensolelybythechan

1.2.3.MIntheprinvestordailyme

enegative ors likely toticallyrestri

2:(a)Thepoble equilibriothemaximunstableanlly,itisworintrinsic mntforanarbythestatistngeofsentimModeledrevioustwors’marketseeasured

r positive vbe prevalenicted.

otentialum pointsmaof .Tndtwostablerthwhilenotemory andbitraryticalpropertmentif dsentimeosectionswentiment (,butfortha

values. It folnt, otherwis

for 0correspondThereisoneeequilibriumtingthattheso it woul.Thepossibtiesof .isautocorreentehavedeveeq.4).Nowatwemustf

lows that, ase the latit

0.3and0.9.s to the miestableequimpointsintabsenceofld generallybilityofmakForexamplelated.Wewelopedthemwecanconfirstestimat

aswe havetude of the

(b) foinima oflibriumpointheorderedinertiaineqy be imposskingameanile, itcouldiwillreturntomodelthatcnstructtheeethevalues

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r 1.1anand thentinthedisostate(Fig.2quation(5)isible to preingfulforecanprincipleothispointinconvertsdirempiricaltimoftheparam

Pa

he transitionsentiment

nd1.3.Theunstable eorderedstat2b).impliesthatedict the evastisthendbepossiblenSection1.4ectinformatmeseriesmeters ,

age20of97

nal regimewould be

locationofquilibriumte(Fig.2a)themodelvolution ofdeterminedtopredict4.2.tion intofromtheand .

 

Firstbehaviorthe initidominatinvestorseems rroughlyWeobtain tlengthbthebehacorrespotrendsosentimenweeklyb

t, as notedrandherdinial estimatetingnornegr’smemoryheasonable t1month.Acsubstitutethe daily serbetween199aviorsof theondencebetonvarioustint values anbasisand75

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would be reareequallyt may provethereforetakssomewheres the initialweestimateseries igure 3 show2.OurfirstosentimentaurningpointThisobservag returns anthlybasisa

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to considerhichcorrespassign ~1stheinitialendowfromsorrespondin0.04.on (4) witthe S&P 50isthatthereual index.Foentevolutioportedbythrrelated atlag.

a situationpondsto ~1so that thestimate.Thseveraldaysng to 25 buth 1,00 Index foearecertainor instance,onandthetuestatisticst55% on a d

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where bot~1.Sowesete termhird,intuitivestoseveralusiness days1andr periods onsimilaritiethereappeaurningpointhattheincrdaily basis,

age21of97

th randomt 1asis neitherely,typicalmonths.Its, which is0.04toof differentesbetweenars tobeatsofindexrementsof69% on a

 

Figure3500 IndsimilaritOurthepositand dowcorrespocompatibmotion ipositivepotentiaWecstretcheobservatexperienthirdsofterritoryskewed,

3:Thesentimdex duringtybetweensecondobsetiveregionbwnward) anond tomarkblewiththeinside thepwells.WewalwellinSeccanalsosees of marketion issurprncedthelargf this timeby. It hints atwhichwou

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Pa

d 1.0an2011. Noteetarydata(Foundedmotindextrende negative rh behaviorsrmercanbeross thenentconfiguratduringrelatd bear marwhichtheUSockmarketinbearmariment to behaviorobser

age22of97

ndtheS&Pthe closeFig.3b).tionwithinds(upwardregion thats aremoreerelatedtogative andtionof thetively longrkets. ThisSeconomyspenttworket returnpositivelyrvedinthe

Page23of97 

stock market (perhaps best exemplified by the fact that the DJ Industrial Average Index hasreturnedonaveragearound7.5%p.a.overthelast100years).Accordingtoourmodel,itrequiresapositivemeanof ( 0)tokeepsentimentpositiveon average, i.e. the volume of positive direct information must, in the long run, exceed that ofnegativedirectinformation.Indeed,thedailymean measuredfortheperiodcoveredbyourdataispositive(TableI).Thispositiveinformationbiasispossiblyrelatedtotheeconomicgrowthandinflation.Theasymmetryofsentiment’sbehaviorinducedby 0canalsobeexplainedintermsoftheperturbationofpotentialwell. IfwedecomposeH into twopartsas ,where isthe constant mean of and is its time‐dependent component, such that 0, thenequations(5)and(6)canbewrittenas

sech tanh , 7a with

12 1 ln cosh , 7b where 0.

Figure4showsthatapositivecbreaks thesymmetryof thepotential.For 1thepositivewell( 0)deepensandthenegativewell( 0)flattens,sothattheprobabilityof crossinginto

 

the negabeyondconfigurequilibripositiveshows th≳ 0.01consisteperiod.

Figure 4correspo0.02         

14 Ththirdterm

ative territowhich therationtransfium.Theovwellinterruhat even a1for 1nt with the

4: The poteonds to the2.                      

he potentiamasitislinea

ory decreasenegative sforms intoaveralleffectuptedbyrelasmall can.Usingthee markedly

ential minimum

                      

l canarin .

es.14For larstable equilasingle‐welofthisdistoativelyshorhave a prorelevantvaasymmetric

for 1.1of at

       

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rger the shlibrium poilconfiguratortionisthartexcursionsonounced efluesfor ac behavior o

1and 0.0, surv

ded into athe asymmet

hallow wellint ( )tion inwhichatsentimentsintonegatiffect on theand (Tableof the senti

01, 0.02. Onvives the dis

a truncatedtry relative to

is further) vanishesh thesentimttendstospivesentimenshape of theI),weobtaiment obser

nly one equstortion of t

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age24of97

o the levelouble‐wellitiveat thetimeintheThefigurel well (e.g.17whichishe studied

oint, whichal well for

written asrges via the

Page25of97 

Lastly,wewouldliketomentionthatdespitesimilaritiesinthebehaviorbetweensentimentandthe S&P 500 Index, the differences are nevertheless substantial. Perhaps most importantly, theevolutionofsentimentisboundedwhereasthestockmarketis,onaverage,experiencinggrowth.Onthe other hand, it seems reasonable to suppose that allmarket developments, including its long‐termgrowth,occurthroughdevelopmentsinsentiment.Tofindwhetherornotthisistrue,wemustunderstandtherelationbetween investorsentimentandmarketprice,which is thesubjectof thenextsection.1.3.PriceInthissectionwewillbeconcernedwithinvestigatinghowsentiment canimpactmarketprice .Weproposeamodelofpriceformationandapplyittoconstructanempiricaltimeseriesofmodelpricesthatwewillcomparewithobservedmarketprices.1.3.1.ModelofpriceformationPositiveornegativesentimentmeansthatonaverageinvestorsbelievethatthemarketwillriseorfall,respectively.Butdoesitalsoimplythatinvestorswouldnecessarilyactontheirsentimentandproceedtobuyorsellassets,asthecasemaybe?

Letusconsideraninvestorwhohasjustallocatedcapitaltoastockmarket.Nextday,allotherthingsbeingequal,theinvestorisunlikelytoincreaseordecreasehermarketexposureunlesshersentimenthaschangedbecauseshehadalreadydeployedcapitalintheamountreflectingthatsamelevelofsentiment.Itisthereforesensibletoassumethat,ignoringexternalconstraints(e.g.capital,risk, diversification), investment decisions are mainly driven by the change in sentiment ontimescaleswhere the investor’smemoryofpast sentiment levelspersists, determinedby∆ ≪ (seeSection1.2.2).

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Reasoningsimilarly,weconcludethatonlongertimescales(∆ ≫ )investorswouldinvestordivestbasedprimarilyon the levelof sentiment itselfbecause theirpreviousallocationdecisionswouldnolongerbelinkedintheirmemorytothepreviouslevelsofsentiment(recallthatourinitialguessfor isapproximately1month).Asthechangeofmarketpriceisdeterminedbythenetflowofcapitalinoroutofthemarket,weconclude thatprice changeswoulddependprimarilyon sentiment changeson timescales shorterthan (i.e. probably days to weeks) and on sentiment itself on timescales longer than (i.e.probablymonthstoyears),thatis

∼ ,i. e. ∼ for∆ ≪ , 8a ∼ ,i. e. ∼ for∆ ≫ , 8b

where isthelogarithmofprice ,i.e. ln .15We reiterate that equations (8a) and (8b) provide only the asymptotic relations betweensentimentandpriceatrespectivelyshortand longtimescales.Aswedonotknowthe formoftheactualequationgoverningtheevolutionofprice,wesimplysuperposebothasymptoticviewsandseekpricepatanytimetas

,orequivalently , 9                                                             

15Bytakingthelogarithmof ,wenormalizethepricesothat or resultsintherelativepricechangeln ,i.e.thereturn,whichisastandardprocedure.Ifthepricehadnotbeennormalized, or wouldhavecauseddifferentpercentagechangesinthepricedependingatwhichpricelevelitoccurs.

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in the hope that solutions given by it approximate true solutions reasonably well. Note thatconstants and arepositive,whereasconstants and cantakeanysign.Equation(9)canbewrittenas

∗ , 10 where ∗ .

Equation (10) implies that the change in price at∆t ≫ τ is proportional to the deviation ofsentimentfromacertainvaluegivenbys∗.Thuss∗in(10)servesasayardstick,averagedacrosstheinvestment community, relative to which investors appraise sentiment: if sentiment is above orbelow it, they may invest or divest, respectively. A nonzeros∗can be interpreted as an impliedreferencesentimentlevelthatinvestorsareaccustomedtoandconsidernormal.16Wedonotwishto impose any a priori constraints ons∗. Instead we will determine its value by fitting priceobservationstothemodelinthenextsection.1.3.2.ModeledpriceIn this sectionweapplyequation(10) toconstruct themodelprice fromthe timeseries reportedinFigure3.

We search for the coefficients in equation (10) that minimize the mean‐square deviationbetween andthelogpricesoftheS&P500Indexovertheperiodcoveredbyourdata.Figure5a,whichdepictstheevolutionofmodelpricesvs.indexprices,showsthat approximatedtheindexbehavioroverthisperiodwell.Wenote,first,thatthedailymodelpricesandthedailyindex                                                            

16Anonzero ∗wouldbecompatiblewiththeadaptationleveltheoryputforwardbyHelson(1964)andthereferencepointconceptintroducedbyKahnemannandTversky(1979)intheirprospecttheory.

 

prices acointegrexpectedwhileth

Figureobtained3. The eandslowWenthat ∗stconsidersentimenIt issentimenequilibri

are 83% corated. Additid,the leadineleadingco

5: (a) Thedfromeq.(1estimated cowcomponennotethatthetays on aver normal annt(seeSectishown inAnt(as isexpiumvalueis

orrelated aionally, Figungcomponeomponentat

model pric10)withcoeoefficients antsof giveconstant ∗erage positind so is likeion1.2.3).AppendixBpectedgivendetermined

nd, secondure 5b depintat∆ ≪t∆ ≫ (eq

ce vs. thefficientsesare 0.32venbyeq.(8∗ispositiveive as it mely to be rethat in then itscontextdsolelybyo

, that stanicts the beh(eq.8a)eq.8b)exhibi

he S&P 500stimatedusin22;. 0.08a)andeq.(overthispemanifests aelated to thleading ordtas thebacorderparam

dard testshavior of thexhibitsrapitsslow,larg

0 log pricengtheleast003; 6.(8b),respeceriod.Indeedbackgroundhe positive,der ∗coincidkgroundrefeter ,whi

show thathe componenid,small‐to‐ge‐scalevari

s (1996‐20tsquaresfitt564; and ∗tively.ditwouldbed referencelong‐term bdeswith theferencevaluchistheinv

Pa

t these varnts of (mid‐scaleflationwithti

12). The prtingand0.076. (bereasonablevalue thatbias exhibite equilibriumue).Weknoverselyprop

age28of97

iables are(eq. 8). Asuctuationsime.

rice isfromFig.b) The fastetoexpectt investorsted by themvalue ofwthat theortionalto

Page29of97 

temperature .17This means that temperature has a direct impact on price dynamics. Astemperatureindicatesthebalancebetweenrandombehaviorandherdingbehaviorinthemarket,itwouldbesensibletopresumethatsuchbalancemaygraduallyshiftovertimeandconsider tobeslowly varying with time. Consequently, we can expect the reference sentiment level ∗to be aslowlyvaryingfunctionoftimeaswell.Assumingthat wecalculate fromequations(4)and(10)usinganiterativeprocessforminimizingthemean‐squaredeviationbetweenthemodelandtheindexfor and

∗ ∗ ,whilekeepingotherparametersconstant.18Asaresult,weobtainabetterfitbetweenthemodelandthe index:now matchesthe indexbehavioratvarioustimescalesmoreclosely(Fig.6)andthecorrelationbetweenthedailymodelpricesandthedailyindexvalueshasimprovedto97%.

                                                            

17The equilibrium points are determined by the condition that 0and 0in equation (4) (orequivalently by the extrema of (eq. 6)), given by the equation tanh , so that the equilibriumvaluedependsonlyon .18As follows from equation (10), the reference sentiment level ∗is an important factor for pricedynamics. In the leading order ∗is exclusively a function of : ∗ ∗ (Appendix B). Aswe areinterestedintheeffectofaslowlyvarying on ∗,weconsider tobeafunctionoftimebutforsimplicityholdotherparametersconstant.

 

FigureconstrucshownAvaluesoof and

Letuin FigurThese varound tsentimenprevalenWeo2002 an

6: The mocted througandAppendixB.oftheconstad areshousnowconse 7.We obsalues of the averagent evolved,ntthroughouobservethatnd 2008) an

odel pricegh an itera∗ ∗TheestimatantcoefficienowninFig.3siderserve thatcorresponde value of 1as expectedutthestudiettheperiodnd that the

vs. thetive least sand the retedmeanpants(eq.10).obtainedexhibiteto a slight.1,maintaind, inside thedperiod,al

dsofhigherperiods of

S&P 500squares fittelation betwarametervalare 0.usingtheaed small vartly supercritning a valuee double‐welthoughitsintemperaturlower temp

log pricesting involviween ∗andlues: 1376; 0bove‐outlinriations, alwtical (i.e. above unitell potentianfluencetenecorresponperature cor

(1996‐201ing eq. (4)is determi.12and ∗0.003; andediterativeways remain≳ 1), whty at all timl and that hndedtoebbandtobearmrrespond to

Pa

12). The prand (10),ined from e0.173.The6.293.Tprocessandning lower thich indeedes. It followherding behandflowwit

marketregimo bull marke

age30of97

rice isin whicheq. (B5) aseestimatedThevaluesdreportedthan unity.fluctuatedws that thehavior wasthtime.mes(2001‐et regimes

 

(2003‐2for2009

FigureTemperathecriticAspbe logicasentimeneasierfobetweenandmar         

19Figtemperatdecline involumes

007)witht9‐2012,that

7: Temperaaturefluctuacalvalueofuperiodsofhial to suppont (expressorinformationthesetworketvolatility                      

gure 7 also sturepatternpntemperaturforthatperio

henotableetookplacei

ature (meaatedlessthaunity,whichighertempese that thesed differentonflowtomvariables,wy,whichisa                      

shows elevatepriorto2000re in1999‐20od.

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asured in uan10%arouhmarkstheperature indicse periods ctly: rising temovesentimwhileFigurealsotobeex       

ed temperatu0maybeunr000suggests

fthepost‐2lyhightemp

units of (eundthemeaphasetransicate increascoincidewitemperaturement).Figure8bshowsspected.

ures during teliableduetothat the leve

008‐crisismperatureenv

eq. 3)) as aanvalueof0itionbetweeed importanth periods oflattens the8ademonssimilaritiesi

themarket raolowdatavoelsof1996‐1

marketrallyvironment19

a function o0.9,alwayssenorderedanceofrandoof increasede potentialstratesanevinbehavior

ally of the laolumes.Furth1998couldbePa

,capturedi9.

of time (19stayingnearanddisorderombehaviord volatility iwell, whichvidentcorresbetweentem

ate 1990s. Hohermore,theeanartifactage31of97

nourdata

996‐2012).butbelowredstates.r, itwouldin investorh makes itspondencemperature

owever, thesubsequentof lowdata

 

Note2010‐20followin2007, thconditiorisk‐on,

Figure8calculateperiodaIndex vodeviatioIn sureproduthatthe

ethatthepo012,whichisg theCredithe bull marnsof subdurisk‐offenvi

8: (a)Temped as the stanchoredtoolatility ovenof21‐dayummary, wecemarketpbalancebet

ost‐2008‐crisscompatibltCrisis.A rrket duringuedeconomironment).

perature antandarddevthedate foer that samlogreturnse have dempricebehavitweenherdin

sisbullmarkewiththeaeasonablee2010‐2012ic recovery

nd the volatviation ofmorwhichtheme period. Susingthesamonstrated torasafuncngbehavior

ketexhibitedaboveobservexplanationwas basicaandhighun

tility of sentonthly (21 devolatility isS&P 500 Inamemethodthat the moctionofinverandrandom

danunexpevationthattto it is thatally a liquidncertainty i

timent (19days) incremscomputed.dex volatiliasin(a).odel can, wiestorsentimmbehavior,

ectedlyhighthetemperat, unlike thedity‐driven rnoverall in

996‐2012). Sments in o. (b)Tempety is calculithin a relatment.Additioindicatedby

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sentimentvaturedidnobullmarkerally occurrnvestor senti

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age32of97

volatilityinotdecreaseet in2003‐ring in theiment (the

volatility isng 300‐daydS&P500e standardtolerance,aveshownmicanalog

Page33of97 

oftemperature ,cangraduallychangeovertime.AswewillseeinSection2, playsanimportantroleininfluencingmarketdynamics.20Forconvenience,theestimatesofparametersarereportedinthetablebelow.TableIParametersappliedformodeling and

Parameter Value Source 0.017 DJ/Factiva(Section1.1)σ 0.408 DJ/Factiva(Section1.1) 25(businessdays) Initialguess(Section1.2.3)1/ 0.040 1.100 Average (Fig.7) 1.000 Initialguess(Section1.2.3) 0.374 LeastsquaresfittingEq.10 0.002 LeastsquaresfittingEq.10 6.500 LeastsquaresfittingEq.10∗ 0.131 LeastsquaresfittingEq.10 0.017

                                                            

20Acommentrelevanttothefollowingsections:Weapplynon‐constant and ∗ ∗ for two purposes, to analyze the variation of with time and to demonstrate that by accounting for thisvariationweachieveacloserfitbetweenthemodelandthemarket.Sincethevariationof issmallandforthe sake of simplicity, we will henceforth use the time series and as calculated with constantparameters. The values of parameters are shown in Figures 3 and 5, except that we apply 1.1(theaveragevalueof overthestudiedperiod)insteadof 1.0.Theparametervaluesarealsosummarizedin Table I, however the graphs and are not shown for the economy of space and due to theirsimilaritytothosedepictedinFigures3and5,respectively.

Page34of97 

1.4.DiscussionIn this section we summarize the findings of the empirical study and discuss their theoreticalimplicationsandpracticalapplications.1.4.1.ResultsWehave shown that themodelprice replicated the S&P500 Index valuesover theperiod1996‐2012withinreasonable tolerance, lendingcredenceto theassumptionsmadeearlyontodevelopthemodel.Thusweconclude:

First,investors’sentimentisinfluencedbydirectinformationflow.Second,inadditiontodirectinformation,sentimentisalsoinfluencedbytheinteractionamonginvestors and idiosyncratic influences that can be assumed random for our purposes. The Isingmodel, in which temperature serves as a measure of random influences, provides the relevantframeworkforstudyingsentimentdynamicsand, inparticular,enablesustoobtainaclosed‐formequationfortheevolutionofsentiment.Third, the mechanism of price formation as a function of sentiment works differently ondifferent timescales. On timescales shorter than the investors’ average memory horizon, marketprice changes proportionally to the change in sentiment. On longer timescales, price changesproportionallyto thedeviationofsentimentfromareference level that isgenerallynonzero.Asaresult,pricedevelopmentisnaturallydecomposedintoaslow,large‐scalevariationwithtimeandfast,small‐to‐mid‐scalefluctuations.We have seen that for the studied period the effects in connectionwith herding behavior ofinvestors were more prevalent than those related to random behavior. Consequently, sentimentevolved insideaW‐shapedpotential,withanegativeequilibriumvalue inonewellandapositivevalue in the other. Direct information flow,which acts as a forcemoving sentiment inside these

Page35of97 

wells, was on average positive during the studied period. Two results became evident: first,sentimentspentmostofthetimeinthepositivewellandcrossedintothenegativewellonlyduringthe time of crises and, second, the reference sentiment level stayed mostly positive during thisperiod.Althoughourdata coverage is limited to 16years,we tend to think that the above‐describedasymmetry of sentiment dynamics, caused by the positive bias in direct information flow, maygenerallypersist.This isbecausedirect informationmustbeonaveragepositive tobeconsistentwith the long‐term growth of the stock market. The asymmetric behavior of sentiment can bealternativelyexplained in termsof theperturbationofpotential: the constantproportional to the(positive) mean of direct information flow is a parameter that distorts the shape of potential,makingthepositivewelldeeperthanthenegativewell.Asaresult,sentimenthasapropensitytobeonaveragepositiveasittakesmoresignificantnewstoturnitfrompositivetonegativethanfromnegativetopositive.Wewouldlikenowtocommentoncertaineffectspertainingtothelong‐term(monthstoyears)behavior of sentiment and price. We begin with the economic analog of temperature, whichdetermines the relative importance of random behavior vs. herding behavior. Temperature cansubstantiallyimpactmarketdynamicsintwoways:first,byalteringtheshapeofpotentialtherebyaffectingsentimentevolution(e.g.theprobabilityofcrossingthewells)and,second,byinfluencingthereferencesentimentlevel(eq.10)whichissensitivetotemperaturechanges.Wehaveobservedthattemperatureandthevolatilityofsentimentvariedovertimeinasimilarpattern. Itseemsreasonabletoassumethatperiodsof increasedsentimentvolatilityimplyahigh

 

uncertaiperiodsliquidityasitexh8a).Wenvaried idowntursentimenthatthemarket.

         

21“Rihighvoladuring20

intyofinvesof lowsenty‐drivenmarhibitstypicanowbrieflyn a patternrn and decrnt also fluctmodelcaptu

                      

isk‐on,risk‐oatilityof the r010‐2012the

storopinionimentvolatirketrecoverlrisk‐onrisconsideran resemblingreasing durituated appruressomee

                      

off”isalooselriskappetiteemarketwas

nsymptomatilitycorrespryduring20sk‐offfeaturconnectiong economicng the upturoximately seffectsofthe

       

lydefinedterof investorsinarisk‐offr

ticofbearmpondtothe010‐2012apres,exemplifwith theec fluctuationurn. Figuressynchronouseinteraction

rmthatpract. It is awiderisk‐onregim

marketsandpositive,treppearstobefiedbytheeconomy.Figns, typically9c,d showslywith econbetweenth

titionersusetelyheldviewme.

drisk‐on,risendingmarkeanexceptioelevatedsengure9a,bshincreasingthat directonomy. Theshegenerale

todescribemacross the inPa

sk‐offregimkets.Wenotontothisobntimentvolaows that temduring theinformationse observatieconomyand

marketscharanvestment inage36of97

es21,whiletethatthebservation,atility(Fig.mperatureeconomicn flow andions implydthestock

acterizedbyndustry that

 

Figure9mirror iinformatthan 85similarlyFinawillbedWewilltheoryu

1.4.2.PWehaveinsideththemid‐declinescorrespoA coevolutiotermin

9: (a) Tempmageoftion flow0 days, andysmoothedlly,wehavedone,brieflyrevisit thisunderlyingthPracticaleobservedthepositivew‐termmarke and recoondencebetonsiderationn( 0)an(10)iszero

peratureand the s, smoothed the US reas abovenotyettouy,inthenextissue in thehismodel,emlapplicattwotypesofwellandtheet trends,woveries. Ittweenthetun of equationdthe turnio( ∗).It

and the dameGDP gredusingaFeal GDP perve,andthesucheduponttsection,inwe secondpamphasizingtionsfbehaviorchinfrequentwhile the lattwould beurningpointsn (10) leadingpointsofollowstha

detrended Uraph as shoFourier filter capita growsameGDPgrtheissueofwhichwehiart of thispathepotentiaharacteristiccrossingsbeterappearsinterestingsofsentimeds to a concofprice trenatincertain

S real GDPown in (a) foer that remowth (YoY)raphasshowwhetherorighlighttwoaper (Sectioalformarkectosentimeetweenthewtodrive thg to investenttrendsanclusion thatnds( 0)casesthetu

per capitaor ease of covedharmon(1996‐2012wnin(c).rnotthismoorelevantpron2),whichetforecastingnt:themid‐wells.Thefoe large‐scaltigate whendmarketprthe turningnevercoincurningpoint Pa

(1996‐2012comparison.nicswithpe2). (d) Sentiodelisprediropertiesofh furtherdeg. ‐termboundormerseemsedynamicsether therericetrends.g points ofcideunless ttsofsentimage37of97

2). (b) The(c)Directeriods lessiment ,ictive.Thisthemodel.velops the

dedmotionstoinduceofmarketis somesentimentthesecondmenttrends

 

can occutrendinto be poreversesreversedsentimenTovmultivarIndexanFigure 1secondttrailsthehasoverindex onmoment

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age39of97

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consistentwiththeresultsproducedbytheextended,self‐containedmodelwedevelopinthenextsection.2.PartII–TheoreticalstudyofstockmarketdynamicsIn the empirical study we have assumed that information that speculates about future marketreturns (direct information)maybe effective in influencing investors’ opinions about themarket(investors’ sentiment). We have established a mechanism that translates direct information viainvestors’sentimentintomarketprice.Thetranslationmechanismitselfismathematicallysimpleinthe sense that the equation for sentiment dynamics (eq. 4), which defines how investors formopinions based on received information, and the equation for price formation (eq. 10), whichdefineshowinvestorsmakeinvestmentdecisionsbasedontheiropinions,constituteanuncoupledandintegrablesystemofequations.

It is therefore not surprising that the source of complex behavior, characteristic to actualmarkets, in the model is contained within the direct information flow that we have treated asexogenous.Thismeans thatwehave investigated justonepartof theproblemand to learnmoreabout the origins ofmarket behaviorwe have to extend the framework by incorporating into itassumptionsonhowdirectinformationflowisgeneratedandchanneledinthemarket.Inthispartofthestudywedevelopatheoretical(self‐contained)modelofthestockmarketthatincludes direct information (which we will call to set it apart from the empirical case) as avariable,alongsidesentiment andprice .Thismodelissimplifiedbyconstructiontofacilitateitsstudy, so its solutions cannot reproduce the full range and the precise detail of actual marketregimes. Itspurpose is to capture theessential elementsofmarketbehavior andexplain them intermsoftheinteractionamongdirectinformation,investorsentimentandmarketprice.

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This part proceeds as follows: Section 2.1 develops a self‐contained model of stock marketdynamics and Section 2.2 provides the analysis, results and discussion. The relevant technicaldetailsareinAppendicesAandC.2.1.News‐drivenmarketmodelDirectinformation,whichisbasedontheinterpretationofgeneralnews,consistsofopinionsaboutfuturemarket performance that can reach a large number of investors in a short period of time.Such opinions can come from financial analysts, economists, investment professionals, marketcommentators,businessnewscolumnists,financialbloggers–basically,anyonewithaviewaboutthemarket,themeanstodeliverittoalargeinvestoraudienceandthecredibilitytoinstilltrust.Forconvenience,letusrefertothemcollectivelyasmarketanalysts.Inthecontextofourframework,analystsperformavitalfunction:theyinterpretnewsfromvarioussources,opinehowthemarketmight react and transmit their views through mass media. That is, they convert any type ofinformationintodirectinformationandmakeitavailabletomarketparticipants.22

                                                            

22We consider only publicly available direct information simply becausewe know how tomeasure it.Usually,priortoanexpertopinionbeingexpressedpublicly,itisdiscussedinprofessionalcirclesanddiffusesacrosstheinvestmentcommunity.Thus,itwouldbenormalthatprivatelyexpressedviewswouldhavealeadover those which are publicly expressed.We tend to think, however, that on the timescale of investmentdecision‐making by institutional investors, this informational lag is negligible due to the speedwithwhichinformation becomespublic through online publication, email andblogs. The topic of howopinions transitfromprivatetopublicisdeservingofitsownresearch,butisoutsidethescopeofthiswork.Hereweassumethat,forourpurposes,thetimelagbetweenthepublicandprivatedirectinformationcanbeneglectedwhiletheirmagnitudes(asmeasuredby )canbeconsideredproportional.

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LetusconstructanIsing‐typemodelwithtwotypesof interactingagents: investorswhohavecapital to investbut cannot interpretnewsandanalystswhocan interpretnewsbutdonothavecapitaltoinvest(theironyofwhichisnotlostonus).Atanypointintime,eachinvestorandeachanalysthaseitherapositive(+1)ornegative(‐1)opinionabout futuremarketperformance–thebinarymarketsentiment inthecaseofinvestorsandthebinarymarketsentiment inthecaseofanalysts. Investors can interact with each other and with analysts and, similarly, analysts caninteractwitheachotheraswellaswith investors.They interactby literally imposingopinionsoneach other as every sentiment tries to align other sentiments along its directional view. It isimportanttonotethatalthoughthenatureofinteractionisthesame(theexchangeofopinions),themeansofinteractioninthemodelareverydifferent:analysts,whoareoutnumberedbyinvestors,areassumedtoexertadisproportionallystrongforceoninvestorsduetoaccesstomassmedia.Wecannowreplacethehomogeneous,single‐componentIsingsystem(1),wheretheexternalflowofdirect information acted on investors, by its heterogeneous, two‐component extension, inwhichinvestorsandanalysts(insteadofinformationflow)interactwitheachother.Wedefine and as individual sentiments of investors and analysts, respectively, and , as theirnumbers( ≫ ≫ 1)and,asbefore,writethesystem’stotalenergy(AppendixA,eq.(A1))andapplytheall‐to‐allinteractionpatterntoobtaininthelimit → ∞, → ∞thedynamicequationsthat describe the evolution of the variables as statistical averages, namely: ⟨ ⟩/ where∑ and ⟨ ⟩/ where ∑ .Thegeneralformofthedynamicequationsisasfollows(AppendixA,eq.A14):

tanh , 11 tanh , 11

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where,analogous toequation (3), thecoefficient defines thestrengthof interactionamongtheinvestors; – the strengthwithwhich the analysts act on the investors; – the strengthwithwhich the investorsactontheanalysts; – thestrengthof interactionamongtheanalysts; and areanygeneralexternalforces,sofarundefined,actingrespectivelyontheinvestorsandthe analysts, where and determine their impacts; 1/ with being the characteristichorizon of the investors’ memory and 1/ with being the characteristic horizon of theanalysts’ memory; and has been identified in equation (3) to be the economic analog oftemperatureortheparameterthatdeterminesthe impactofrandominfluencesandconsequentlythelevelofdisorderinthesystem.Notethatparameters , , , , , , , and arenon‐negativeconstants.Inour framework, analystsmust react tonew information faster than investors, so that theirresistancetoachangeinsentimentisweakerthanthatof investors.Accordingly,weassume tobe an order of magnitude smaller than , resulting in 2.5business days, This value isconsistentwith thebehaviorof theautocorrelationof (Fig.11a) that showsa rapiddecayof“memory”effectsontheorderof1‐3businessdays.Henceweset 0.4.Letusconsidertheexternalforces and ,whichhavethemeaningofexternalsourcesof information tapped by investors and analysts, respectively. We set 0by assuming thatinvestorsinthemodelreceivedirectinformationonlythroughanalysts;inotherwords,investors’sentiment maychangeonlybyinteractingwithanalysts’sentiment .Thecaseof ismoreinterestingbecauseoftheanalysts’functiontotranslatenewsintodirectinformationandsoprovideachannelthroughwhichexogenousinformationentersthemodel.Sinceanynewsmaybeofrelevanceinthiscontext–e.g.economicdatarelease,corporatenews,centralbankannouncements,politicaleventsor,say,changeofweatherconditions–wecanmaketheusualassumption that new information arrives randomly and therefore represent news flowvia noise.

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Thereis,however,aparticularsourceofinformationthatshouldbeconsideredseparately.Itisthechangeofmarketpriceitself.23Itsimportanceissupportedbyempiricalevidence:today’svaluesofare correlated with yesterday’s and today’s S&P 500 log returns at over 30% and 50%,respectively.24Thus,wedivide informationavailable toanalysts in themodel into informationrelatedtothechangeinmarketpriceandallother(external)information.Wewrite ρ ρ ,where ismarket price change (eq. 10), is randomnews flow (noise) andρ (ρ 0 alongwithρ arescaling constants. Note that by writing in the expression for , we have assumed that thefeedbackintheformofpricechangeiscontinuousandcontemporaneous,whichisconsistentwiththefactthatpricescanbeobserved–directlyasdata–atanytimeandinreal‐time.                                                            

23Feedbackinconnectionwithpriceobservationshasbeenimplementedinseveralagent‐basedmodels.Forexample,Caldarelli,MarsiliandZhang(1997)usedtherateofchangeinpricealongwithitshigher‐orderderivatives, Bouchaud and Cont (1998) applied price deviations from fundamental values, and Lux andMarchesi (1999, 2000) considered both the rate of change inprice andpricedeviations from fundamentalvalues. The idea that price observations create a feedback loopwhich can produce complicated dynamics,leading tomarket rallies and crashes, has roots in the 19th century (see a review by Shiller (2003)).Withregard to early feedbackmodels,we takeparticularnoteof Shiller’s (1990)modelwith lagged, cumulativefeedbackoperatingoverlongtimeintervals,implyingthatinformationrelatedtopastpricechangeshaslong‐lastingeffects.24Interestingly, the cross‐correlation function is approximately zero at positive time lags. We couldplausiblyexplainthisbystatingthatonshorttimescales(e.g.intraday)thereisanefficientmarketresponsetonews, whereby new information is almost immediately reflected in prices as obvious price anomalies areswiftly arbitraged away by investors (e.g. the high frequency traders). The response to news on longertimescaleshasadifferentnature,anditsmechanicsarethesubjectofthepresentwork.

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Wecannowsubstitutetheexternalforces 0and ρ ρ intoequations(11)andwritethemodelintermsof , and asfollows:∗ , 12

tanh , 12 tanh , 12

whereforconveniencewehaveregroupedtheconstantstohave , , , ,and .Wewishtomaketheinitialinvestigationeasierbyfurthersimplifyingequation(12c).First,letusassumethattheinfluenceofinvestorsonanalysts(~ )andtheinteractionsamongtheanalysts(~ )arelessimportantinrespectofopinionmakingthantheimpactduetomarketperformance(~ )andexogenousnewsflow(~ ),andthereforeweneglectthecorrespondingtermsin(12c).Second,letusapproximate in(12c)as ,where isapositiveconstantthatcarriesthemeaningofthelong‐termgrowthrateofthestockmarket.Thatis,wereplace ∗ by whenwesubstitute from(12a) into(12c).Aswehaveseen inSection1.3.2, ∗ changesslowlywith time as compared with , therefore the approximation is valid for sufficiently short timeintervals.Inotherwords,bymakingthisapproximationweneglectthelong‐termfeedbackeffectson relativetotheanalogousshort‐termeffects.Asaresult,system(12)canbesimplifiedas

∗ , 13 tanh , 13 tanh , 13

 

whereThenonlineafor a gisentimenmodel(e13),asssimple,ptheemp

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age46of97

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2.2.ModelanalysisandresultsInthissectionwereportanddiscusstheresultsofthestudyofsystem(13).Wenotethatequations(13b,c)constituteaclosedsystemthatwecansolve for and andsubsequentlysubstituteinto equation (13a) to obtain .We first consider the autonomous case 0and thenstudythetime‐dependentsituation 0.2.2.1.Autonomouscase: The two‐dimensionalnonlineardynamical system(13b,c) is studied indetail inAppendixC.Herewereportonlythemainresultsfortherelevantrangeofparametervalues.

At 1system (13b,c) experiencesphase transition between the disordered statewith oneequilibriumpoint( 1)andtheorderedstatewiththreeequilibriumpoints( 1).Asbefore,weareinterestedintheorderedstateinthevicinityofthephasetransition( ≳ 1).If 0,thereisoneunstableequilibriumpointattheoriginandtwostableequilibriumpointsat locatedsymmetricallywithrespecttotheoriginontheaxis 0.If 0,theequilibriumpoints shift tonew, asymmetricpositionsalong theaxis for ≪ 1.Once exceedsa criticalvaluethedistortionoftheinitiallysymmetricconfigurationbecomessostrongthatonlythepositiveequilibriumpointat survives.Inthisrespectthebehaviorofsystem(13b,c)issimilartothatoftheone‐componentsentimentmodel(4).Tounderstandwhereanewbehaviormayoriginate,letustakeacloserlookatequations(13b,c).Directinformationflow,describedbyanalysts’sentiment ,causesinvestors’sentiment tochange(eq.13b).Inturn,anychangein alsoforcesachangein asduetothefeedbackterm (eq.13c).Asaresult,investors’sentimentisnowcoupledtodirectinformationandviceversa,anditisthiscouplingthatleadstotheemergenceofinertiaandnewbehaviorsrelatedtoit.Indeed,asisshowninAppendixC,system(13b,c)canbeapproximatedby

 

,where t(C16), rparticlewhich acquiredcompon

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focusinthenegativewell(redasterisk),astablefocusinthepositivewell(greenasterisk)andthetrajectories passing near the saddle point located between the wells. (d) The potential corresponding to (a) and (c). Parameters corresponding to Fig. 13(a‐d): 1.1; 0.55;0.03; 0.04; 0.4.Most importantly, inertia enables free oscillations of and , provided that the feedbackcoefficient isnottoosmall.25Figures13a,bshowtwotypesoffreeoscillationsthatemergeinthesystem.Thefirsttypereflectsdecayingoscillationswithacharacteristicperiodofroughly6monthsaroundtheequilibriumpointinsidethedeepwell.Thesecondtypereflectsalarge‐scalestablelimitcyclethatcarriessentimentfromthedeepwellintotheshallowwellandbackinapproximately14months. Such large‐scale motion is self‐sustaining, as it is fuelled by the coupling of directinformation and sentiment, such that large swings in cause large swings in that cause largeswingsin thatcauselargeswingsin andsoforth.Imaginethattheflowofexogenousnewsweretobeinterrupted.Then,accordingtothemodel,two scenarios are possible. First, themarketmay converge to a steady state inwhich sentimentlikely settles at thepositive equilibriumvalue inside thedeepwell26,while simultaneously thelevel of direct information approaches . Alternatively, themarketmay go into a steady state in                                                            

25Whenγis increased from zero, it induces the following bifurcations of the equilibrium points at thebottomofeachpotentialwell (s s ): stablenode ‐>stable focus ‐>unstable focus ‐>unstablenode.Freeoscillations become possible starting from the first bifurcation in the above sequence. Note that theequilibriumpointatthecuspofthepotentialalwaysremainsanunstablesaddle.26Itwouldberatherunusualifsweretocometorestats because,first,sentimentdoesnotoftencrossinto the shallowwell and, second, suchequilibriumstatemaybeunstableornot exist at all, as the criticalvaluesofδatwhichs vanishesarewithintheadmissiblerangeofparametervalues.

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which sentiment anddirect information are attracted to the limit cyclewhere they exhibit large‐scaleself‐perpetuatingoscillationsbetweenextremenegativeandextremepositivevalues.Figure13cshowsaregimeimmediatelyprecedingtheformationofthelimitcycle.Thisregimeisaprimecandidateforgeneratingarealisticsentimentbehavior.Itincludesmid‐termoscillationsaroundthefocusinthedeepwell,consistentwiththeboundedmotionthatwehaveobservedintheempiricalmodelandconnectedwithmid‐termmarket trends inSection1.2.3. Italsocontainsthelarge‐scale trajectories, located in the vicinity of the (about‐to‐be‐formed) limit cycle, that leadsentiment via quasi‐self‐sustaining motion into and out of the negative sentiment territory. Theevolutionofsentimentalongthesetrajectoriesisconsistentwithitsobservedbehavioratthetimeofmarketcrashesandrallies(Section1.2.3).Lastly,wenotethatthisregime,whichweexpecttoberelevanttothereal‐worldstockmarkets,existsundertherealisticchoiceofparametervaluesconsistentwiththevaluesobtainedusingtheempiricaldata.27

                                                            

27Theparameters in the theoreticalmodel (eq.13andFig.13c)havebeenchosen to coincidewith theparametersoftheempiricalmodel(TableI),exceptthat 0.55(theoretical)whereas 1.0(empirical).Additionally: 1. 56therefore 2.2, so that the feedback term participates in the leading orderdynamics in equation (13c) while not dominating other terms. 2. 0.03so that the constant0.017,whichdeterminesthedistortionofthepotential,isclosetothevalueintheempiricalmodel(TableI).3.It follows from thedefinitionsof and inequation (13) that / .Using and fromTable I andthe daily average logarithmic growth rate 2.24 10 of the DJ Industrial Average Index since 1914(interpolated using annual data), we estimate / ~0.034which is close to 0.030in the theoreticalmodel.

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2.2.2.Non‐autonomouscase: The news flow acts as a random force in system (13). To study its influence, we make theassumptionthat isnormally‐distributedwhitenoisewithzeromeanandunitvariance.28Thesystem’sbehaviorwilldependsignificantlyontherelativemagnitudeof theterms andin equation (13c). Indeed, if | | ≪ | |, the dominant noise will drive and randomlyaround the potential well, leading to nearly stochastic behavior in price . Conversely, if| | ≫ | |, the system’s dynamics will primarily consist of a piecewise motion along thesegmentsofphasetrajectoriesintheautonomouscase,asonaverage and willtravelfaralongatrajectoryby the time thenoisewill havebeenable todisplace them.29In this case, the resultingbehaviorofmarketpricewillhaveadiscernibledeterministiccomponenttoit.Weset 1asthispermitsbothtermstoparticipateintheleadingorderdynamics.30However,as | |isafunctionof and (eq.13b),therelativeimportanceofthesetermsactuallydependsonthelocationofanygiventrajectory,whichmeansthattheabove‐discussedcharacteristicdynamics                                                            

28Notethatinsimulationsthenoise is,strictlyspeaking,whiteonlyonadaily(weekly,monthly,etc.)basis,i.e.when canbeexpressedinmultiplesofbusinessday.Thisisbecause canhavenonzerointradayautocorrelationsduetointradayinterpolationinthenumericalscheme.29This statement is only valid if the system is structurally stable (i.e. non‐chaotic)when perturbed bynoise,which is likely the case aswehavenot been able to find the tracesof chaos (e.g. positiveLyapunovexponents)initfornonzero intherelevantrangeofparametervalues.(Asasidecomment,weobservedpositive Lyapunov exponents in situations where system (13) was forced by a periodic function of time,insteadofnoise).30 1,so | |~ 1,where istheunitstandarddeviationof ,therefore | |hasthesameorderofmagnitudeas | |~ ≲ 2.2for| | ≲ 1.

 

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Fig.16andpatternofatwehavereplaysanlopmentofof returnscsofactualeexplaineduse strongtionacrossmetryofthen leadtoamodel, thecalandtheluding the

 

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of1.27and.

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Figure 18: Distributions of 21‐day log returns. (a,b) Theoretical model (60 years, i.e. 720 datapoints)andtheS&P500Index(1950‐2012). (c,d)Empiricalmodel(1996‐2012)andtheS&P500Index(1996‐2012).Tomakecomparisoneasier,thedistributionsarenormalizedtoyieldzeromeanandunitvariance.Thecorrespondingnormaldistributionsareplottedaswell.Thehorizontalaxis,denominated in the units of standard deviation, indicates the normal distribution 1st and 5thpercentiles.The results presented in this section enable us to conclude that, first, there exist substantialsimilaritiesbetweenthebehaviorofinvestorsentimentproducedbythetheoreticalmodelandtheempiricalmodel;second,thereexistsubstantialsimilaritiesinthepricebehaviorgeneratedbythetheoreticalmodelandthatobservedinthestockmarket;and,third,thetheoreticalmodelreplicatescharacteristicfeaturesofthestockmarket,suchasnon‐normallydistributedreturns.

2.2.3.DiscussionResultsproducedbymodel(13)areinagreementwiththeempiricalstudy.Inparticular,themodelexhibitstwodistinctbehaviorsrepresentedbydifferentcharacteristicfrequenciesatwhichinvestorsentiment evolves: mid‐term oscillations in the positive well consistent with mid‐term markettrendsandlarge‐scale,quasi‐self‐sustainingmotionconsistentwithmarketralliesandcrashes.Exogenousnewsflowisarandomforcethatthrustssentimentinsidethepotentialwell.Itcanoccasionallyforcesentimentacrosstheregionsofdifferentbehavior,resultinginaswitchofmarketregimes.Wehaveseenthattheinfluenceofnewsisnotuniform.Thereisawiderangeofregimesbetweentwoextremes:thefirstwherethenews(noise)dominateleadingtoanearlyrandompricewalkandthesecondwherethenews(noise)havelowimpactresultinginanalmostdeterministicpricedevelopment.Thisimpliesthat,accordingtothemodel,themarket’sevolutionpathcantakeitthroughavarietyofregimes,someofwhichmayhaveelementsofdeterministicdynamics.

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Wehavedemonstratedthatastochasticallyforced,deterministicnonlinearmodeldescribedbyequations(13)iscapableofgeneratingcomplexbehaviorsrepresentativeofactualmarketregimes.For example,we have been able to approximately reproduce the behavior of the S&P 500 Indexfrom2004to2010bysubstitutingtheempiricaltemperatureprofileforthatsameperiodintothemodel (which implies that temperature plays a significant role in the development of marketregimes). Themodel has been shown to replicate essential features of the actual distributions ofstockmarketreturns,suchasnonzeroskewness,positivekurtosisandfat(albeittruncated)tails.Inaddition, the model has provided an explanation of market trends and crashes in terms ofcharacteristicfrequenciesofsentimentdevelopment.Theappearanceofcharacteristicfrequenciesinthemodelisintriguinganddeservesconjectureon its causesand implications.Model (13) isaheavilyapproximatedversionof themoregeneralIsingmodeldefinedbyequation(A1)inAppendixA,thedynamicsofwhicharelargelydeterminedbythetopologyofinteractionamongtheagentsthatconstituteit.Inparticular,model(13)hasbeenderivedundertheassumptionthatthepatternofinteractioninequation(A1)isalltoall.Itisknownthat the all‐to‐all pattern prevents Ising spins (agents) from assembling into clusters, whereasinteractionsthatarenotasradicallysimplifiedcanproduceheterogeneousstructures(e.g.ContandBouchaud (2000)) – which, in our case, represent the clusters of investors and analystscharacterizedbycoalignedsentiments.33                                                            

33Asmentionedearlier,theall‐to‐alltopologyistheleading‐orderapproximationforgeneralinteractiontopologyintheIsingmodelandso isaconvenientstartingpoint forstudyingitskeyfeatures.Wenotethatpricesgeneratedbytheempiricalmodel(eq.4and10),whichalsoemploystheall‐to‐allinteractionpattern,cancontainindirectevidenceofclusterdynamicsbecausetheinputvariable isbasedonempiricaldata.

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Thepresenceofclusterscanleadtodiversedynamicsofinteractionsonvarioustimescales.Thisis because a cluster’s size determines its reaction time to various disturbances, such as adjacentclusters,randominfluencesorexternalforces(thenews),suchthatthelargerthesize,theslowerthe reaction.As different reaction times of differently‐sized clusters to incoming information canrepresentinvestmenthorizonsorallocationprocessesofvariousinvestorgroups,thegeneralmodel(eq.A1)cansimulatethedynamicsofmarketparticipantsrangingfromprivateinvestorstopensionplans.Consequently, the generalmodel (eq. A1) can exhibitmultiple frequencies of interaction.Wemayexpect that theactualstockmarketalsocontainsdifferentcharacteristic frequencies,eachofwhichsignifiesadistinctbehaviorthattogether,throughinteractions,produceaconstantlyshiftingpatternofregimesdrivingthedevelopmentofmarketprice.Andindeed,thefactthatadrasticallysimplifiedmarketmodelcapturedtwocharacteristicfrequenciesthatcorrespondtoactualmarketregimessupportsthisexpectation.34Wewouldliketoconcludebyreturningtothediscussionofthepossiblepredictabilityofmarketreturns thatwehave touchedupon in thebeginningof this section.Recall that the characteristicfrequenciesemergeinthemodelasaninertialeffectcausedbythefeedbackrelationbetweendirectinformation and price. The existence of inertia has another consequence: the model acquires aresistance to change in direction and, as a result, its behavior becomes predictable in situationswhere inertial effects outweigh noise. In other words, based on the theoretical results and thesupport of theempirical evidence,we can expect the stockmarket to include regimeswith someelementsofdeterministicdynamics,whichcaninprinciplebepredicted.                                                            

34Thearguablyquasiperiodicpatternoftheautocorrelationsof (Fig.11a)canbeafurthersignofthepresenceofcharacteristicfrequenciesinthemarketpriceseries.

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Asanillustration,letusconsiderhowthemodelsdevelopedherecouldbeappliedformakingamarketforecast.Apossibleprocedureisasfollows:measuredaily ,obtain fromequation(4), findthecorresponding location , onthephaseportrait (e.g.Fig.13c)andforecastthefollowingday’sreturnexpectationwhich,beingafunctionofthecurrentday’slocationonthephaseportrait,maybenonzero.Ofcourse,model(13)islikelytoocrudetoproduceameaningfulforecast.Moresophisticatedpatternsofinteractionmaybeneededtoconstructmorerealisticmodels.Inthelight of the above discussion, such randomly‐driven deterministic models are likely to generatecomplex dynamics, resulting in the presence of random, deterministic chaotic and deterministicnon‐chaotic behaviors.However, unlike the case of the randomwalk, forecasting the behavior ofcomplexdeterministicsystemscanbepossiblebyblendingmodelswithobservations,asisdone,forexample, inmeteorology and oceanography and can probably be done aswell in the case of thefinancialmarkets.353.ConclusionThis paper introduced a framework for understanding stock market behavior, upon which themodel of stockmarketdynamicswasdevelopedand studied.Wehavedemonstrated, empiricallyand/ortheoretically,thataccordingtothismodel:1. Thestockmarketdynamicscanbeexplainedintermsoftheinteractionamongthreevariables:direct information (defined in Section 1.1), investor sentiment (defined in Section 1.2) andmarketprice–asthetheoreticalandempiricalresultsareinagreementandshowareasonablygoodfitwiththeobservations.

                                                            

35Thisisthedirectionofourfutureresearch.Asapreliminaryfinding,wewouldliketoreportthatsimpleprototypes of algorithmic trading strategies built upon the framework developed here have producedpromisingbacktestedresults,supportiveoftheabove‐describedapproachtomarketforecasting.

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2. Theeffectiveness of the influence of informationon investors is determinedby the degreeofdirectness of its interpretation in relation to the expectations of futuremarket performance.Direct information – information that explicitlymentions the direction of anticipatedmarketmovement–impactsinvestorsmost.3. In addition to being forced by direct information, the evolution of investor sentiment is alsosignificantly influenced, first, by the interaction among investors and, second, by the level ofdisorder in the market given by the vacillating balance between the herding and randombehavior of investors. The influence due to the second factor is determined by the economicanalogof temperature(introduced inSection1.2.1), theprofileofwhichcanbededucedfromtheempiricalmodel.4. Marketpricedevelopsdifferentlyondifferent timescales:onshorter timescales(e.g.daysandweeks),pricechangesproportionallytothechangeinsentiment,whileonlongertimescales(e.g.monthsandyears),pricechangesproportionallytothedeviationofsentimentfromareferencelevelthatisgenerallynonzero.Asaresult,priceevolutionisnaturallydecomposedintoalong‐term,large‐scalevariationandshort‐term,small‐to‐mid‐scalefluctuations.5. Inthemarket,herdingbehaviorprevailed(toasmalldegree)overrandombehaviorduringthestudiedperiod(1996‐2012).However,thelevelatwhichthesebehaviorswerebalancedvariedgraduallyovertimesuchthat,generally,herdingincreasedduringbullmarketsanddecreasedduringbearmarkets.Thelevelofbalanceaffectsmarketvolatilityandtheprobabilityofmarketcrashes. There is a connection between changes in this balance and observed economicfluctuations(thebusinesscycle).6. Long‐term sentiment trends show a substantial temporal lead over long‐term market pricetrends. This result may be of practical importance for the development of trend‐followingstrategies,asachangeinsentimenttrendcouldbeaprecursortoachangeinpricetrend.7. Informationrelatedtopricechangesplaysanimportantroleinmarketdynamicsbyinducingafeedback loop: information ‐> sentiment ‐> price ‐> information. This leads to a nonlinear

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dynamicofinteraction,whichexplainstheexistenceofcertainmarketbehaviors,suchastrends,rallies and crashes. It is responsible for the familiar non‐normal shape of the stock marketreturndistribution.8. Exogenousnewsactasarandomforcethatdisplaces investorsentimentfromtheequilibriumand occasionally causes market dynamics to switch from one regime to another. Twoequilibriumstatesarepossible.Inthefirstregime,sentimentandpriceremainconstant.Inthesecond regime, sentiment oscillates between extreme negative and positive values, drivingalternatingbearandbullmarkets.9. The coupling of price and information creates feedback, whereby information causes pricechanges and price changes generate information (paragraph 7). Additionally, the resultingdynamicfurthercomplicatesthepicture,leadingtoadelayedreactionbetweenthesevariables,such that, forexample, today’sprice changemaybe influencedby informationrelated topastpricechanges,alongwithothernews,overpreviousdays,weeksandmonths.Thisimpliesthatthenotionof causeandeffectdoesnot simplistically apply tomarketdynamics, as causeandeffectbecome,inasense,intertwined.10. Thenonlineardynamicunderlyingsentimentevolutioncontainsbothdeterministicandrandomcomponents. The deterministic component in some market regimes is stronger, while otherregimesmaybedominatedbytherandomcomponent.Developinganunderstandingofwhichmarketregimetakesplaceatwhichtimecanbepossiblebycombiningmodelsandobservations,whichcouldpotentiallyfacilitatemarketreturnforecasts.AcknowledgementsWe are grateful to Dow Jones & Company for providing access to Factiva.com news archive.Wewould also like to thank John Orthwein for editing this paper and contributing ideas on itsreadability.

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AppendixA:EquationforsentimentdynamicsWe base the derivation of a dynamic equation for sentiment evolution in the model studied inSection2onaphysicalanalogyinwhichtwosetsofinteractingIsingspins 1, 1… and1, 1… are acted upon by external magnetic fields and , respectively.Additionally, this analysisyields, as aparticular case, adynamicequation for a single setof Isingspins 1, 1… in the presence of an external magnetic field , which is a physicalanalogyofthemodelstudiedinSection1.Thisparticularcaseissimilartothestatisticalmechanicsproblem treated by Glauber (1963), Suzuki and Kubo (1968), and Ovchinnikov and Onishchuk(1988).A1.Propertiesofthetwo‐componentIsingsystemTheHamiltonian36oftheIsingsystemhastheform:

12 12 , ,

where , and arecoefficientsthatdeterminethestrengthofinteractionamongspins; and aremagneticmomentsperspin;and and areexternalmagneticfields.Ifweassumethat

,,, ,

                                                            

36Inthisappendixwereverttothenotationandterminologycommonlyusedinphysics.

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,,

thentheHamiltonianbecomes, , 2 2 , 1

where∑ , ∑ .Inwhat followswe assume that in the thermodynamic limit ( → ∞, → ∞) the values of, , and arefiniteanddenotethemas

lim→ , lim→ , lim→→ , lim→→ . 2 In situations where the external fields are stationary, and , the Isingsystemcanbeinthestateofthermodynamicequilibrium.Inthiscase,theprobabilityoffindingitinthestatewiththevaluesoftotalspinsequalto and ,respectively,isgivenby, e , , 3

where , , istheenergyofthesystemand istemperaturemeasuredintheunitsofenergy.Thefunctions and ,givenby!2 ! 2 ! and !2 ! 2 !,

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arethedegreesofdegeneracyofthestatesforwhichtotalspinsareequalto and ,respectively. isanormalizationfactordeterminedfromthecondition, 1,

whichleadsto, , e , ,

calledthepartitionfunction.In the limit of large and ≫ 1, ≫ 1 , it is convenient to change from discretevariables , 2, … 2, and , 2, … 2, to quasi‐continuousvariables

, 1 1 ,, 1 1 .

Wecanrewritetheaboveequationsinnewvariables and toobtain, 2 2 2 2 ,, ln

2 ln 11 ln 1 4 1 ln 212 ln 11 ln 1 4 1 ln 21 .

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Thentheequilibriumdistributionfunction , takestheform:, exp , , exp , ,

where, , ,

2 2 ln 11 ln 1 4 12 2 ln 11 ln 1 4 1 .

Let us find the values of and for which the distribution function , has a maximum(minimum). Neglecting terms and in , , we find from the extremum condition, 0and , 0that

2 2 ln 11 ,2 2 ln 11 .Accountingfor ,wecanrewritethissystemofequationsas

2 ln 11 ,2 ln 11 ,

orequivalently:tanh ,tanh , 4

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where 1/ .Asasimpleexample,letusconsideracaseinwhich 0, .Then4 isreducedtotanh ,tanh ,

where . It follows that and tanh 2 . If2 1, there is only one solution:0(zeromagnetization).If2 1,thereexistthreesolutionswith 0, ,twoofwhich,namely (nonzero magnetization), correspond to the maxima of the distribution function, .Thus,thecriticaltemperatureofthephasetransitionisinthiscase 2 .WenotethatamoregeneralcaseistreatedinAppendixC.Itisworthwhilenotingthatphasetransitionispossibleeveninthecaseof 0.Then:tanh ,tanh ,

where , ,whichyieldsnonzerosolutionsfor 1.A2.Master(kinetic)equationTodescribenon‐equilibriumdynamics,weintroducedistributionfunction , , thatdefinestheprobability that the values of the total spins of configurations and at time t are equal toand ,respectively.If , , isknown,wecancalculateanystatisticalquantityatanymoment.Forinstance,theexpectationvaluesoftotalspins and aregivenby

⟨ ⟩ , , , 5

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⟨ ⟩ , , . 5

Notethatsummationin(A5)iscarriedoutacrosstheintegervalues: , 2, …2, and , 2, … 2, . We can obtain , , by deriving and solving thecorrespondingmasterequationfortheevolutionof , , .Toformulatethemasterequation,wedefinetheprobabilityofthesystemtransitingfromoneconfigurationofspinsintoanotherperunittimeas , ; , ,whereit isassumedthattransitionsoccurwhenspinsspontaneouslyflipas a result of uncontrolled energy exchangewith a heat bath. Considering that , , evolvesonlyduetothesetransitions,weconcludethatitmustsatisfythefollowingmasterequation: , , , ; , /// , ,

, /, , /; ,// . 6 Notethatthe firstsumin 6 representsthetotalprobabilityperunit timethat thesystem‘flipsout’of thestate , ,whereasthesecondsumcorrespondstothetotalprobabilityperunit timethatthesystem‘flipsin’tothestate , .

Let us assume that transitions occur only due to single spin flips, i.e. and change at eachtransitionby 2.Thenequation 6 becomes , , , ; 2, , ; 2, , ; , 2

, ; , 2 , , 2, ; , 2, ,2, ; , 2, , , 2; , , 2,, 2; , , 2, . 7

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Therefore,theproblemisnowreducedto, first,deducingtheformof andthensolving 7 .Todetermine wecanusethedetailedbalancecondition:, ; , , , ; , , .Here , istheequilibriumdistributionfunctiongivenby 3 .Whenthedetailedbalancecondition is satisfied, the number of transitions per unit time in the thermodynamic equilibriumstatefromanyconfiguration , intoanyotherconfiguration , exactlyequalsthenumberof transitions in the opposite direction. This condition imposes certain constraints on theprobability ,asitmustsatisfy, ; ,, ; , ,, / / exp , , . 8

Accordingto 8 ,theprobabilities in 7 satisfythefollowingequations:, ; 2,2, ; , 2,, 2 exp 2, , ,, ; , 2, 2; , , 2, 2 exp , 2 , .

Usingexpressionsfor , and , ,weobtain, ; 2,2, ; , ∓22 1 exp 2 1 ,, ; , 2, 2; , ∓22 1 exp 2 1 . 9

Weseek , ; , thatsatisfy 9 intheform:, ; 2, ∓2 1 exp ∓2 1 ,

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2, ; , 2 1 1 exp 2 1 ,, ; , 2 ∓2 1 exp ∓2 1 ,, 2; , 2 1 1 exp 2 1 ,

where and arethecorrespondingrelaxationratesand 1/ ,asdefinedbefore.Letusdefineforconvenience, ; 2, ∓2 , ; 2, ,2, ; , 2 1 2, ; , ,

, ; , 2 ∓2 , ; , 2 ,, 2; , 2 1 , 2; , ,

where, ; 2, 1 exp ∓2 1 ,2, ; , 1 exp 2 1 ,

, ; , 2 1 exp ∓2 1 ,, 2; , 1 exp 2 1 .

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Thensystem(A8)takesthefinalform: , , 2 , ; 2, 2 , ; 2,

2 , ; , 2 2 , ; , 2 , ,2 1 2, ; , 2, ,2 1 2, ; , 2, ,2 1 , 2; , , 2,2 1 , 2; , , 2, . 10

Forsolving 10 itisnecessarytodefinetheinitialdistribution , , 0 , ,aswellastheboundary conditions,which in this case are , , ≡ 0for| | and| | .We also notethatequations 10 areapplicableinthecaseofnonstationaryexternalfields and .A3.DynamicequationsforaveragespinsWe can derive the equations of motion for average spins ⟨ ⟩ and ⟨ ⟩ bydifferentiatingequation 5 withrespecttotimeandusingequations 10 : , , , , , ,

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wherefunctions and representtheresultsofsummationofthe ‐thterminther.h.s.of10 multipliedby and ,respectively.Weobtainthefollowingexpressionsfor and :

2 , ; 2, , ,12 ⟨ , ; 2, ⟩ ,

2 , ; 2, , ,12 ⟨ , ; 2, ⟩ ,

2 , ; , 2 , ,12 ⟨ , ; , 2 ⟩ ,

2 , ; , 2 , ,12 ⟨ , ; , 2 ⟩ ,

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22 2, ; , 2, ,12 / 2 / /, ; / 2, /, ,/ 12 / 2 / /, ; / 2, /, ,/ 12 ⟨ 2 , ; 2, ⟩ .

To derive this last expression, we, having changed variables / 2, omitted the term with/ 2as 2, , ≡ 0and added the term with / as the summated functioncontains the multiplier / that vanishes at such value of /, which has enabled us torepresentthisexpressionincanonicalformabove.Similarly,

22 2, ; , 2, ,12 / 2 / /, ; / 2, /, ,/ 12 / 2 / /, ; / 2, /, ,/ 12 ⟨ 2 , ; 2, ⟩ ,

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22 , 2; , , 2,12 / 2 / , /; , / 2 , /,/12 / 2 / , /; , / 2 , /,/12 ⟨ 2 , ; , 2 ⟩ ,

22 , 2; , , 2,12 / 2 / , /; , / 2 , /,/12 / 2 / , /; , / 2 , /,/12 ⟨ 2 , ; , 2 ⟩ .

Wecannowsumtheaboveexpressionstoobtainthefollowingequationsfor and :⟨ , ⟩ ,⟨ , ⟩ , 11

where, , ; 2, , ; 2, ,, , ; , 2 , ; , 2 .

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Wenotethatsystem 11 isnotclosedbecause⟨ , ⟩ and⟨ , ⟩ cannotingeneralbereduced tobe functionsof and .However, itwouldbe reasonable toassume that in thethermodynamic limit ( ≫ 1, ≫ 1 ) fluctuations of total spins and around theirinstantaneous average values and are small in comparison with these same averagevalues.Thissituationisanalogoustothatwhereasystemisinthermodynamicequilibrium.Wecanthenwrite⟨ , ⟩ ⟨ ⟩ , ⟨ ⟩ , and⟨ , ⟩ ⟨ ⟩ , ⟨ ⟩ , .

Weintroducenewvariables,⟨ ⟩ ,⟨ ⟩ ,

andexpress , and , viathesevariablestoobtaininthelimit → ∞, →∞thefollowingequations:, 1 1 exp 2

1 1 exp 2tanh and

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, 1 1 exp 21 1 exp 2tanh .

Substituting theseexpressions into 11 ,weobtain the following closedsystemofequations forand :tanh ,tanh . 12

The following condition determine the stationary points of 12 for const , const:tanh ,tanh , 13

which coincideswith condition 4 that determines the extrema of the equilibrium distributionfunction , .Revertingtonotation 1/ ,weexpress 12 as

tanh , 14 tanh . 14

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A4.Dynamicequationintheone‐componentIsingsystemFinally, we consider a simpler Ising system that consists of spins 1, 1… and theexternalmagneticfield .TheHamiltonianofthissystemisequalto2 . 15

AstheHamiltonian 15 isaparticularcaseoftheHamiltonian 1 ,equations 14 containthe dynamic equation for average spin in the one‐component system: ⟨ ⟩ . Thisequationcanbederivedfrom 14 bysetting 0andrenaming and :tanh , 16

wherelim → .AppendixB:ThereferencesentimentlevelHerewe derive an approximate expression of the reference sentiment level ∗, relative towhichdeviations in sentiment result in a change of market price, as described by the second term inequation(10).B1.Equationfor ∗Asa firststep,wetakea long‐termaverageofbothsidesofequation(10)withrespecttotimetoobtainitsasymptoticforminwhichtheaveragedvariablesaretime‐independent: ∗ , 1 where denotestheconstantrateofchangeinmarketpriceand 0because isbounded.Thus,wecanrewriteequation(B1)as

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∗ , 2 where satisfiesthesimilarlyaveragedequation(4)(with 0): tanh . 3 Now,letusconsiderasituationwheretheflowofdirectinformation isonaverageneutral,i.e. 0. Sinceaccording toourmarketmodel, it is the flowofdirect information thatdrivesmarketpricedevelopment,thecaseof 0impliesthattherateofchangeinpriceisonaveragezero,i.e. 0,andthusequation(B2)becomes

∗ . 4 Therefore ∗equals intheregimewherethedirectinformationflowisonaveragezero,whichis the regime described by the potential for which 0(eq. 6).37 There is but onecaveattothisderivation.Namelythatbecausewehaveusedequation(10)which–beinga linearcombinationoftwoasymptoticregimesoftheactualequationgoverningpriceevolution(thetrueformofwhichwedonotknow)–isanapproximation,theaboverelationbetween ∗and maynotbethetruerelation.However,weexpectthatthisrelationholdsapproximately,asitleadstoresultsthatareconsistentwithobservations,asisshowninSection1.3.2.Wecanobtain anapproximate solution for fromequation (B3)byexpanding thehyperbolictangentinpowersof intheneighborhoodof andaveragingtheresultingseriestruncatedabovethequadraticterms.Equation(B3)thentakestheapproximateform:                                                            

37In theferromagneticcase( 1), canbenonzero,givennon‐symmetric initialconditions( 0 0),finiteaveragingperiodandsufficientlysmall .

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tanh 1 tanh 1 , 5 where isthestandarddeviationof .Usingthetimeseries (Section1.1)and (Section 1.2.3),we estimate ~0.5, so can be seen to remain reasonablyclosetothecenterofexpansion ,whichjustifiestheaboveapproximationfortherelevantrangeofparametervalues.

Equation(B5)isappliedtodetermine from ∗intheiterativeleastsquaresfittingprocessforthetimeseries inSection1.3.2.Forthispurpose,weapply 0.3insteadof0.5.Thischoiceisdictated by the following considerations. Unlike the empiricalmodel (Section 1), the behavior ofwhichisdrivenprimarilybythemeasured ,thetheoreticalmodel(Section2)issensitiveto .Itwouldthereforebesensibletoselect basedonthetheoreticalmodelbehavior.Theestimateof0.3results in a more realistic behavior of the theoretical model than0.5which leads to highervaluesof thanthosefollowingfromthetheoreticalmodel.Thechoiceof0.3insteadof0.5seemsacceptable, given the approximate nature of equation (B5) and the inevitable imprecision inmeasuringandcalibrating .B2.Perturbativesolutionfor ∗Treating asasmallparameter(asfollowsfromtheestimates intheprecedingsection, ~0.25or0.09),wecanconstructaperturbativesolutiontoequation(B5).Wefindthatintheleadingorderthesolutionis tanh , 6 i.e.itcoincideswiththestableequilibriumpointsofsystem(4).Consequently, ∗becomes∗ for 1, 7 ∗ 0for 1. 7

 

Soluparticlecoincidethat forsolutionorigin.Acco

∗ ∗ TheAppendiandnegaoriginth(Fig.B1)totheeq

tion(B7a)ccannot leaves intheleadvery small(B7b)meanountingfort

10forsecond terixC, 1ativefor .hanthatof).For 1quilibriump

canbeundeve the welldingorderwamplitudesnsthatthepthenextorde

111.m in (B8a)1isnTherefore,,whichisa1,theterm(ointattheo

rstoodasfoin which itwiththestaofmotion tparticle’saveercorrection1 for

corrects fonegativeforowingtotharesultofth(B8b)iszeroorigin.

ollows.Ifthet is initiallyableequilibrthe shapeoerageposition, ∗canbes

1,or the asymany andheasymmetrheouterwalobecausein

eamplitudey found, theriumpoint,f thewell isoncoincidesshowntoequ

mmetry of t,thustheryofthewellofthewellnthiscaseth

ofvariationn the particor , ints symmetricswiththeequalthe well arocorrectionll,thelocatilbeingsteephewellissy

Pa

nin issosmcle’s averagthatwell.Thcaround .quilibriump

ound . Astermisposionof iscloperthantheymmetricwi

age80of97

mallthatage positionhis implies. Similarly,pointatthe

8 8shown initivefor osertotheinnerwallithrespect

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FigureB1:Graphicrepresentationofthereferencelevel ∗for 1.1and 0.3: 0.503(eq.B6)and ∗ 0.313(eq.B8a).AppendixC:AnalysisofdynamicalsystemHere we investigate certain properties of the system of equations (13). As equation (13a) isdecoupledfrom(13b,c),thestudyofsystem(13)isreducedtotheanalysisofthetwo‐dimensionalnonlineardynamicalsystem(13b,c),thesolutionstowhichmustbesubstitutedinto(13a)tosolvesystem(13)completely.C1.EquilibriumpointsIfwe considera situationwhere the time‐dependent force is absent, equations (13b,c) taketheform:

tanh , 1 tanh , 1

where (as a reminder)| | 1, | | 1and the coefficients , , , , and are non‐negativeconstants. Solutions to equations (C1) , for the initial conditions 0 , 0 determinetrajectoriesofmotioncorrespondingtothevelocityfield( , )givenbyther.h.s.of(C1).Themotionisboundedbecausethevelocityattheboundaries 1and 1isdirectedintotheregionofmotion.Theequilibriumpointsofsystem(C1)(where 0)aregivenby

∗ tanh ∗ ∗ , 2 ∗ tanh . 2

 

Asconvenietanh

andsolvof ,, ∗∗

providesone solufunctionterms,a

FigureCcase(If(Fig.C1ban

isgiven, ∗enttowrite, ∗

veitgraphic∗ isnon‐n∗

∗ 1sfortheexisution ∗ nof ,suchtparamagne

C1:Graphic1 .1,theneb).Whenthnd , w

isknownathisequatio12 ln 11

cally(Fig.C1egativeat ∗0,

stenceofas . The ethat → 0aticphase–o

solutions toquation(C3hevalueofwhere 0

andcanbe sonas∗∗ ∗

1).Asfollow∗ 0,i.e.

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ontoequatisolution hus, correm.

(C3). (a)Paeitheroneoritical(d

into (C2a) tureC1a,the

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age82of97

for ∗. It is 3 derivative

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solution is negative for nonzero and approaches zero as → 0. The existence of nonzerosolutions in the limit → 0meansthat thesystemmayhaveordered(ferromagnetic)equilibriumstates.When ,equation(C3)hasasinglesolution ,astwoothersolutionsmergeandvanishat.Wenotethat isnotsensitivetothebifurcationat ,thatis, anditsderivativesdonothaveasingularityatthispoint.Itisnotdifficulttofind tanh .AsfollowsfromFigureC1b, isdeterminedbythevalueof , ∗ (eq.C3)atitsmaximumwithrespectto ∗.Fromtheextremumcondition, , ∗∗ 0,itfollowsthat

∗ 1, 4 where the solution corresponding to the maximum of , ∗ has been selected. Substituting∗ ∗ into(C3),weobtain

tanh 1 , 1 1 12 ln 1 11 1 1 . 5

When the system is in thevicinityof thephase transitionat 1from ferromagnetic state intoparamagneticstate,then0 1 ≪ 1,sotheaboveexpressionfor becomes~ 23 1 .To sum up, system (C1) admits two types of bifurcations with respect to the number ofequilibriumpoints.First,thereisaphasetransitionat 1betweenthedisordered,paramagneticstate with one equilibrium point ( 1) and the ordered, ferromagnetic state with three

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equilibriumpoints( 1).Second,whenthesystemisintheferromagneticstate,twoequilibriumpointsmergeandvanishat ,sothatonlyoneequilibriumpointremainsfor .C2.StabilityanalysisWeproceed to study the stability of system (C1) near the equilibriumpoints. It is convenient torescaletimeas andrewriteequations(C1a)and(C1b)as≡ , tanh , 6 ≡ , tanh , 6

where and .Linearization of system (C6) in the neighborhood of the equilibrium points ( ∗, ∗) leads tosolutions and in the form of linear combinations of exp and exp witheigenvalues givenby12 tr tr 4det ,

whereJacobian ,tracetr anddeterminantdet aredefinedas∗, ∗

∗∗,tr ,det .

Substituting , and , from(C6)intotheexpressionsfor ,tr anddet ,weobtain1 ,tr ,det ,

where

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1 1 ∗ , sech sech , 1 ∗ sech 1 ∗ sech .

Asaresult,thecharacteristicequationforeigenvalues becomes12 4 . 7

If thediscriminant 4 isnon‐negative, then arereal.Theeigenvaluesmayhave the sameoropposite signs. In the formercase theequilibriumpoint is calledanode, in thelattercasetheequilibriumpointiscalledasaddle.Ifatleastoneoftheeigenvaluesispositive,theequilibrium is unstable because there is an exponentially growing solution to (C6) near theequilibrium point. If is negative, then are complex conjugates, so that there is an oscillatorycomponenttothemotioninthevicinityoftheequilibriumpoint,whichiscalledafocus,providedtheeigenvalueshavenonzerorealparts.Inthissituation,thesignoftherealpartoftheeigenvaluesdeterminesstabilityattheequilibrium.Notethattheeigenvaluesaredependentonthepositionofthe equilibrium point ∗. Therefore, we can expect that each of the three equilibrium points inferromagneticstatehasuniquestabilityproperties.Letusconsiderthecharacteristicequation(C7)indetail.

1. If the second term in is negative, then 0and 0, so that the equilibriumpoint is a(unstable)saddle.Thisconditionisfulfilledwhen 1 ∗ 1.2. If 1 ∗ 1,i.e.thesecondtermin ispositive,thentheequilibriumisstable(unstable)provided φ ψ isnegative(positive).Thisconditioncanbewrittenas

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≶ 1 1 1 ∗1 ∗ sech . 8 These inequalitiesdeterminethevaluesofparametersforwhichtheequilibriumisstable(thesign above)orunstable(thesign above).

Ifsimultaneouslywith(C8) 4 ,then ispositive,theeigenvaluesarerealandofthesamesign,sothattheequilibriumpointisastable( 0)orunstable(0)node.Itisnotdifficulttoshowthatthecorrespondingrangesofparametervaluesaredefinedby1 1 1 ∗1 ∗ sech 9

forthestablenodeandby1 1 1 ∗1 ∗ sech 10

fortheunstablenode.On the other hand, if simultaneously with (C8) 4 , then is negative, theeigenvaluesarecomplexconjugates,so that theequilibriumpoint isastable( 0)orunstable( 0)focus.Thecorrespondingrangesofparametervaluesaredeterminedby

1 1 1 ∗1 ∗ sech 1 1 1 ∗1 ∗ sech 11 forthestablefocusand

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1 1 1 ∗1 ∗ sech 1 1 1 ∗1 ∗ sech 12 fortheunstablefocus.

Itfollowsfrom(C9‐12)thatthefollowingsequenceofbifurcationsalwaystakesplacewhenincreasesfromzerotoinfinity:stablenode‐>stablefocus‐>unstablefocus‐>unstablenode.Also, equations (C9) and (C10) show that the range of parameter values for the focus‐typeequilibrium points diminishes with the decreasing ratio and vanishes in the limit→ 0.3. In the paramagnetic phase ( 1), the condition 1 ∗ 1is satisfied for any ∗,therefore the analysis in the paragraph 2 above applies. It means that the equilibrium point∗, ∗ , tanh goes through the above‐described series of bifurcations: stable node ‐>stablefocus‐>unstablefocus‐>unstablenode.4. Intheferromagneticphase( 1),thesystemmayhaveeither(i)threeequilibriumpointsat∗ , , and ∗ tanh or (ii) one equilibrium point at ∗ and ∗ tanh ,depending on whether or (eq. C5), respectively. As follows from Figure C1b,| | | ∗ |.Substituting ∗ givenbyequationC4,weobtainthatthecondition 1 ∗

1is satisfied for any . Conversely, | ∗ |, therefore the condition 1 ∗ 1issatisfiedforany and .Thus,theequilibriumpointcorrespondingto isalwaysasaddleinaccordancewithparagraph1,whereastheequilibriumpointscorrespondingto aresubjecttothe sequence of bifurcations stable node ‐> stable focus‐>unstable focus ‐> unstable node inaccordancewithparagraph2.It is not difficult to show that in the ferromagnetic case with three equilibrium points( ), each bifurcation from the sequence stable focus‐> unstable focus ‐> unstable node

 

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C3.InterpretationasanoscillatorEquation(C6a)canbedifferentiatedwithrespecttotime38and,usingequation(C6b),expressedas, 1 tanh arctanh . 13

This equation can be interpreted as an equation of motion for a particle of unit mass with thecoordinate and the velocity , the motion of which is driven by the applied force , . It isinstructive to expand , into a Taylor series and write equation (C13) in the followingapproximateform:, 0, 14

where hasthemeaningofadampingcoefficientandisequalto, 1 2 2 2 15

andthepotential isgivenby12 234 . 16

Toobtainequations(C15)and(C16),theTaylorseriesof , havebeentruncatedattermsabovecubicin ,linearin andlinearin ,sothat,strictlyspeaking,thisapproximationisonlyvalidintheregionwhere| | ≪ 1and| | ≪ 1,whichdefinestheneighborhoodoftheequilibriumpointat∗ (and ∗ ) forsmall .39However, theexpressionfor thepotential ,whichdoesnot                                                            

38 / where .39Recallthatboth and arebounded:| | 1and| | 1,asfollowsfromequation(C6a).

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containthemoreheavilytruncatedterms~ ,isexpectedtoholdreasonablywellalsooutsidethatregionfortherelevantrangeofparametervalues,namely , , ~1and ≪ 1.Equation (C14) is the equation of a damped oscillator that describes themotion of a particlesubjectedtotherestoringforce / andthedampingforce . Ifdamping isnot toostrong,themotion is expected to be oscillatory. The potential is useful for visualizing the system’sbehavior;forinstance,thechangeintheshapeofthepotentialfordifferent and helpstoexplainthe occurrence of the bifurcationswith respect to the number of equilibriumpoints (e.g. Fig. C4below).Thedampingcoefficient determinesregionswhereenergy isabsorbedorsupplied.Thiscanbeseenbyconsideringtherateofchangeofthetotalenergyofsystem(C14):

12 .Thus,regionsforwhich 0aretheregionswhereenergyisextractedfromthesystem(positiveor true damping) and regions forwhich 0are the regionswhere energy is pumped into thesystem(negativedamping).Asfollowsfromequation(C15), ≷ 0if ≶ 1 2 2 21 .This condition,validnear theorigin| | ≪ 1for ≪ 1, shows thatdamping changes sign frompositive to negative with growing , provided 1. In the ferromagnetic symmetric case( 0, 1),dampingbecomesnegativefirstattheorigin 0 ,sothatfortherelevantvaluesofparameters, ≳ 1, ~1and ≫ 1,thecritical ,atwhichdampingchangessign,isgivenby

~1, orequivalently ~ 1 ≫ 1.

 

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