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Predictablemarkets?Anews‐drivenmodelofthe

stockmarket

MaximGusev1,DimitriKroujiline*2,BorisGovorkov1,SergeyV.Sharov3,DmitryUshanov4

andMaximZhilyaev5

1IBCQuantitativeStrategies,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,Nizhny

Novgorod,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:Weattempttoexplainstockmarketdynamicsintermsoftheinteractionamong

three variables: market price, investor opinionand information flow. We propose a

framework for such interaction and apply it to build amodel of stockmarket dynamics

which we study both empirically and theoretically. We demonstrate that this model

replicates 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 pose

implicationsforcurrentmarkettheoryandofferpotentialpracticalapplications.

Keywords: stockmarket,marketdynamics, returnpredictability, news analysis, language

patterns, investor behavior, herding, business cycle, sentiment evolution, reference

sentiment level, volatility, return distribution, Ising, agent‐basedmodels, price feedback,

nonlineardynamicalsystems.

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Introduction

Thereisasimplechainofeventsthatleadstopricechanges.Priceschangewheninvestorsbuyor

sellsecuritiesanditistheflowofinformationthatinfluencestheopinionsofinvestors,accordingto

which theymake investmentdecisions.Although thisdescription is too general tobeofpractical

use,ithighlightsthepointthatwiththecorrectchoiceofhypothesesabouthowprices,opinionsand

informationinteractitcouldbepossibletomodelmarketdynamics.

In the first part of this paper (Section 1),we develop amechanism that links information to

opinionsandopinionstoprices.First,weselectthemarketforUSstocksastheobjectofstudydue

to itsdepth,breadthandhighstanding inthefinancialcommunity,whichplaces it in thefocusof

globalfinancialnewsmediaresultinginthedisseminationoflargeamountsofrelevantinformation.

We then collect information using proprietary online news aggregators aswell as news archives

offered by data providers. Next we analyze collected news with respect to their influence on

investors’marketviews.Ourapproach is to treateach individualnews itemas if itwere a “sales

pitch”thatmotivatesinvestorstoeitherenterintoorwithdrawfromthemarketandapplymethods

frommarketingresearchtoassesseffectiveness.

We then proceed tomodel opinion dynamics. It is reasonable to expect that, along with the

impact of information, such a model should also account for interactions among investors and

various idiosyncratic factors that can be assumed to act as random disturbances. There are

similarities between this problem and some well‐studied problems in statistical mechanics,

allowingustoborrowfromtheexistingtoolkittoderiveanequationfortheevolutionof investor

opinion in analytic form. At this juncture, we investigate the connection between opinions and

prices.Tounderstandit,wemustanswerquestionspertainingtohowinvestorsmakedecisions;for

instance, is an investor more likely to invest if her market outlook has been stably positive or

whether it has recently improved? We suggest a simple solution based on observed investor

behavior.

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Theproceduredescribedaboveisusedtoconstructatimeseriesofdailypricesfrom1996to

2012basedon collected information and compare itwith the time series of actualmarketprices

from that sameperiod.Wedemonstrate that themodel replicates the observedmarket behavior

overthestudiedperiodreasonablywell.IntheendofSection1wereportanddiscussresultsthat

mayberelevantformarkettheoryandpracticalapplication.

Although this (empirical)model enablesus to translate a given information flow intomarket

prices, it cannot sufficiently highlight the nature of complex market behaviors precisely for the

reasonthatinformationistreatedinthemodelasagiven.Togainfurtherinsightintotheoriginsof

marketdynamicswemustextendthisframeworkbyincludinginformationasavariable,alongwith

investor opinion andmarket price. In the second part of this paper (Section 2), we incorporate

assumptionsonhowinformationcanbegeneratedandchanneledthroughoutthemarkettodevelop

aclosed‐form,self‐containedmodelofstockmarketdynamicsfortheoreticalstudy.

We find that informationsupplied to themarket canbe representedby twocomponents that

playimportantbutdifferentrolesinmarketdynamics.Thefirstcomponent,whichconsistsofnews

causedbyprice changes themselves, induces a feedback loopwhereby information impactsprice

and price impacts information. The resulting nonlinear dynamic explains the appearance in the

modelofessentialelementsofmarketbehaviorsuchastrends,ralliesandcrashesandleadstothe

familiarnon‐normalshapeof thereturndistribution.Additionally, thisdynamicentailsacomplex

causalrelationbetweeninformationandprice,ascauseandeffectbecome,inasense,intertwined.

The second informational component contains any other relevant news. It acts as a random

externalforcethatdrivesmarketdynamicsawayfromequilibriumand,fromtimetotime,triggers

changesofmarketregimes.

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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 market

forecasts.Finally,weprovideanoverallsummaryofconclusionsinSection3.

Thepresentstudyhasbeencarriedoutwithaviewtowardpotentiallypredictingstockmarket

returns.Thisviewissupportedbytheempiricalresearchoverthelast30years,whichsuggeststhat

returnsarepredictable,especiallyoverlonghorizons(seeFamaandFrench(1988,1989),Campbell

andShiller(1988a,b),Cochrane(1999,2008),BakerandWurgler(2000),CampbellandThompson

(2008)). This empirical research has primarily focused on identifying potentially predictive

variables,suchasthedividendyield,earnings‐priceratio,creditspreadandothers,andverifyingor

refutingtheircorrelationwithsubsequentreturns,typicallyapplyingregressionmethods.

Ourobjectiveistocapturethebasicmechanismsunderlyingthispredictability.1Weshowthat

the theoretical model (Section 2), which treats information, opinion and price as endogenous

variables,canreproduceobservedmarketfeaturesreasonablywell,includingthepricepathandthe

return distribution, under the realistic choice of parameter values consistent with the values

obtainedusingtheempiricaldata(Section1).Mostimportantly,thismodelpermitsmarketregimes

wheredeterministicdynamicsdominaterandombehaviors,implyingthatreturnsare,inprinciple,

predictable.Becausethismodelisfundamentallynonlinear,thecausalrelationamongthevariables

                                                            

1Theexistingliteratureexaminesthelong‐termpredictability(e.g.monthlyandannualreturns)andlinks

it to economic fluctuations and changes in riskperceptionof investors.Our findings support the view that

long‐termpredictabilityexistsand,furthermore,indicatethatreturnsmayalreadybepredictableonintervals

ofseveraldays.Wereachthisconclusionusingaframeworkwhichattributespricechangestothedynamicsof

investoropinion.Interestingly,wefindthat,overlonghorizons,thereisaconnectionbetweentheevolutionof

investoropinionandeconomicfluctuations(Fig.9,Section1.4.1).

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is substantiallymore complex than regressiondependence. It therefore follows that the standard

approach to return prediction, based on regressionmethods,may need to be redefined. Such an

approachwouldcombine,analogoustoweatherforecasting,theoreticalmodelswithempiricaldata

topredictreturns(seeSection2.2.3).

ThestockmarketmodeldevelopedinthispaperbelongstothefamilyofIsing‐typemodelsfrom

statistical 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 of

individual agents. When applied to economics, the Ising model simulates dynamics among

interacting agents capable ofmaking discrete decisions (e.g. buy, sell or hold) subject to random

fluctuations(duetoidiosyncraticdisturbances)and,ifany,externalinfluence(e.g.publiclyavailable

information)andvariousconstraints(e.g.wealthoptimization).Astherearenumerouschoicesfor

factors affecting the dynamics of agents, the main goal is the selection of a reasonably simple

combinationthatallowsthereplicationofdistinctivefeaturesofactualmarketbehaviorinamodel.

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). These

modelsdifferby choicesmade in the selectionof factors, suchas, for example, the formof agent

heterogeneity,thetopologyof interaction,thenumberofdecision‐makingchoices,theformofthe

agents’utilityfunctionsandthenatureofexternalinfluenceonagents.

Inthepresentmodelwechosetoapplytheminimalnumberofpossibleopinions(buyorsell)

and the simplest interaction pattern (all‐to‐all) to facilitate its study. We also made no a priori

assumptionsoninvestors’behaviors,preferencesortradingstrategies.Ourcontributionisfocused

on other areas. First, we identify informational patterns that can effectively influence investors’

opinions and measure the corresponding information flow (Section 1.1). Second, we analyze

opinion dynamics in the presence of this information flow using a classic (homogeneous) Ising

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modelandconstructanempiricaltimeseriesdescribingtheevolutionofinvestoropinion(Section

1.2).Third,wededucetherelationbetweeninvestoropinionandmarketprice.Whileintheexisting

literatureopinionsaretypicallyequatedwithinvestmentdecisions,resultinginpricechangesbeing

proportional to the difference between the number of positive opinions (buyers) and negative

opinions (sellers) in a model2, we derive an equation for price formation by proposing that

investors tend to act on their opinions differently over short and long horizons and use this

equation to obtainanempirical time series ofmodelprices (Section1.3). Fourth,we formulate a

theoreticalmodelofmarketdynamicsasaheterogeneousIsingmodel(Section2)whichconsistsof

two 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). Thismodel

yields a closed‐form, nonlinear dynamical system shown to generate behaviors that are in

agreementwithboththeempiricalmodelandtheactualmarket.

Thispaper,whichisprimarilyintendedforeconomistsandinvestmentprofessionals, isbased

on ideas and methods from various scientific and practical disciplines, including statistical

mechanics and dynamical systems. To preserve its readability, we have endeavored to strike a

balancebetween thedepthof thematerial and theease of its presentation.To this end,wehave

placed technical discussions and derivations in the appendices and provided the first principles‐

basedexplanationsofutilizedconceptstomaketheworkself‐contained.

1.PartI–Empiricalstudyofstockmarketdynamics

Intheempiricalstudywedevelopamodel thattranslates information intoopinionsandopinions

intoprices.Section1.1examineswhichinformationcanberelevantinthefinancialmarketscontext

                                                            

2Sometimesprice changes are assumed tobeproportional to a function of thedifference between the

numberofpositiveandnegativeopinions,e.g.Zhang(1999)appliedasquarerootofthisdifference.

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andoutlinesourapproach tomeasuring itandconstructing theempirical timeseries.Section1.2

derives 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 1.4

discussesresultsandapplications.TherelevanttechnicaldetailsareinAppendicesAandB.

1.1.Information

Information 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 practical

applications.

Different methods of handling information form the foundations of quintessentially different

trading strategies in the field. For example, systematic hedge fund managers apply quantitative

techniques to analyze price data to detect trends; global macro managers base their bets on

macroeconomic factors; and fundamental long/short funds employ financial analysis to identify

mispricedassets.3

                                                            

3Wenotethatincreasesinthequantityofinformationontheinternetanditsaccessibility,thepopularity

ofsocialnetworksandimprovementsinnaturallanguageprocessingandstatisticalmachinelearningduring

the last decade have prompted the development of trading strategies where news analytics complement

traditional sources of information. This has also motivated recent empirical research on the correlation

betweendisseminatedinformation,includingsocialmediacontent,andpricemovements(e.g.Schumakerand

Chen(2009);Lietal.(2011);Rechenthin,StreetandSrinivasan(2013);Preis,MoatandStanley(2013)).This

paper broadly shares the motivation and aims to advance research in this area by employing a novel

approach.

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In the theory of finance, a seminal hypothesis on information’s influence on investors – the

efficient market hypothesis – was developed by Fama (1965, 1970) and Samuelson (1965). The

efficient market hypothesis essentially considers that, as a result of competition among rational

investors,informationrelatedtopastevents,aswellastoanticipatedfutureevents, isreflectedin

spotprices.Thistheoryimpliesthatpriceschangeonlyasunexpectednewinformationisreceived

and,becausesuchinformationisrandom,pricechangesmustalsoberandomvariablesthatcannot

bepredicted.

Webaseourapproachontheobservationthatinthecontextoffinancialmarketsinformationis

importantonlyduetoitsimpactoninvestors.Therefore,itwouldbesensibletoconsideronlythose

informational patterns that can effectively impact investors’ decision making.We put forward a

hypothesis that the effectiveness of information is determined by the degree of directness of its

interpretation in relation to the expectations of future market performance, and assume that

informationwhichismoredirectwillbemoreeffectivelyperceivedbyinvestors.

Wewilltestthishypothesisonempiricaldatainthenextsections.Herewebrieflydiscussthe

rationalebehindit.Thebasicideaisintuitivelysimple:wevieweachnewsitemasifitwerea“sales

pitch” to buy or sell the market aimed at investors. Sales professionals will confirm that a

straightforward,unambiguousmessage is imperativefor ittobeeffective,and inthepresentcase

theleastambiguousmessageistheonethattellsinvestorsdirectlywhetherthemarketisexpected

togoupordown.

Information that does not succinctly spell this message out requires individually subjective

interpretationleadingtoconflictingviewsastoitsimplicationsforanticipatedmarketperformance.

Wemay suppose that newswhich cannot be easily interpreted in terms ofmarket reactionwill

result in a roughly equal number of positive and negative views or in a lack of strong opinions

altogether 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 any

news release as amarket scenario) andwill thenbe followedby informationabout the expected

market reaction. It therefore seems reasonable to conclude that information suggesting the

directionoftheexpectedmarketmovementisimportantforpricedevelopment.Thequantifyingof

suchpatterns–henceforthreferredtoasdirectinformation–inthegeneralnewsflowisanareaof

ourresearchfocus.

Thus, in the general daily news flow,wewish tomeasure the number of news releases that

containthisdirectinformation.WelimitourstudytotheUSstockmarketandselecttheS&P500

Index as its proxy. In particular, we consider only the English language media and search for

patterns that indicate future returns, e.g. “the S&P 500will increase / decrease”, or thosewhich

implya trend,e.g. “theS&P500hasgrown/ fallen”,assuming that investorswouldreact tosuch

informationbyextrapolatingthetrendintothefuture.

Wehavechosennottoassignweightstonewsitemsasameasureoftheirrelativeimportance.

Thisfactorshouldemergenaturallybycapturingtherepetitionsorechoingoftheoriginalrelease

by other news outlets. Thus, tomeasure the direct information correctly, we collect all relevant

newsreleases4includingtheduplicates.

Basedontheaboveconsiderations,wehavedevisedanumberofrulesandappliedthemtodaily

newsdatafortheperiodfrom1996to2012retrievedfromDJ/Factivadatabase.Thesesamerules

havealsobeenappliedtothenewsthatwehavedirectlycollectedonadailybasisfromonlinenews

sourcessince2010(oneoftheadvantagesofassemblingaproprietarynewsarchiveistheabilityto

controltheselectionofnewssourcestoensuredataconsistencyovertime).

                                                            

4Wealsotakeintoaccountnewsreach,whichisamarketingresearchterm,referringheretothenumber

ofpeopleexposedtoanewsoutletduringagivenperiod,akintoanewspaper’scirculation.

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Wemeasurethenumberofnewsreleasescontainingpositiveornegativedirectinformationfor

eachdaywheretheNYSEwasopenforbusinessduring1996‐2012toobtainatimeseriesofdaily

directinformation overthatperiod:

,

where isthenumberofnewsitemscontainingpositivedirect information, isthenumberof

newsitemscontainingnegativedirectinformationand isthenumberofallrelevantnewsitems

(e.g.wherethephrase“S&P500”ismentioned);ordinarily, since containsneutral

information along with and . For illustration, the time series of daily and are

displayedonasampleinterval2010‐2012(Fig.1).5

                                                            

5Technicalremarks:

1. fromDJ/Factivafor1996‐2012isusedasaninputintothemarketmodelthatwedevelopoverthe

next sections. In this context it is important to note that the volume gradually rose during that

period,fluctuatingonaverageintherangeof25‐50perdayin1996‐2001,50‐100perdayin2002‐2009

and100‐200perdayin2010‐2012.Accordingly,wehavetoexercisecautionwithregardstointerpreting

modelingresultsfortheperiod1996‐2001duetolowerdataquality.

2. frombothsourceshavebeencalibratedtoreduce(butnoteliminate)thepositivebiasoftheoriginal

series.Thishasbeenachievedbyincreasingtheweightof _ by30%inthecaseofDJ/Factivaand45%

inthecaseoftheproprietarydata.Thecalibrationisappliedtoachievemorerealisticmodelbehavior;we

note,however,thattheresultsobtainedwithandwithoutthecalibrationarenotsubstantiallydifferent.

Theexcesspositivebiasmaybepartiallyrelatedtogeneralasymmetryintheperceptionofnegativeand

positiveevents(Baumeisteretal.,2001),which,forexample,impliesanaversiontolossesinthecontext

offinancialmarkets(KahnemanandTversky,1979),howeverthisdiscussionfallsoutsidethescopeofthe

presentwork.

 

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whetherformedthroughrationalanalysisor irrationalbeliefs,basedonallpossible informational

inputs.6

In the previous section we have proposed that direct information can effectively influence

investors’views.Inthissectionweuseananalogyfromphysicstoexplorehowinvestorsentiment

canevolveinresponsetodirectinformationflow.

1.2.1.Modelofsentimentdynamics

Inthissectionwemakeafirstprinciples‐basedintroductiontotheIsingmodel(Ising,1925)from

statisticalmechanics, as it applies to the problempresented here, and followwith a preliminary

discussion 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 opinion

promptedbyfeelingsorbeliefs,asopposedtoanopinionreachedthroughrationalanalysis. Inthiscontext,

the notion of sentiment has been applied to describe the driving forces behind price deviations from

fundamental 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 and

Wurgler (2007), Lux (2011)) include indices based on periodic surveys of investor opinion aswell as the

proxiessuchasclosed‐endfunddiscounts,advancingvs.decliningissues,callvs.putcontractsandothers.

Ourapproachreliesonthe fact that it is the investoropinionpersethat leadsto investmentdecisions,

irrespectiveofwhetherithasbeenformedrationallyorirrationally.Theunderstandingofhowthissummary

opinion – which we have herein defined as sentiment – evolves is one of the goals of the present work.

Accordingly, inthissectionwemodelthesentimentdynamicswithout invokingexplicitassumptionsonthe

rationality of agents and apply this model to obtain sentiment empirically from the time series of direct

informationasmeasuredinSection1.1. 

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problems in social and economic dynamics, we have not come across its explanation from first

principlesinthesocioeconomiccontext.

Weconsideramodelwithalargenumberofinvestors( ≫ 1),identicalinallrespectsexcept

theabilityto formdifferingbinaryopinions( 1)as towhetherthemarketwillriseor fall.Letus

say that the ‐th investor has the sentiment 1if she opines that the market will rise and

1ifsheopinesthatthemarketwillfall.Weintroduceafunction,calledenergyinphysics,that

describes themacroscopic statesof such a system.This function should contain those factors for

whichwewishtoaccountinthemodel.Basically,therearetwosuchfactors.

The first factorcapturesthe impactdue to the flowofdirect information.Earlierweassumed

that direct information, either positive (the market will go up) or negative (the market will go

down),canbeeffectiveinforcinginvestorstochangetheiropinions,i.e.direct informationactsto

orienttheinvestors’sentimentsalongits(positiveornegative)direction.Weexpresstheenergyof

impact on the ‐th investor as , where is direct information received by the investor.

Energyisnegativewheretheinvestor’sopinionanddirectinformationarecoalignedbecause and

areofthesamesignandispositiveotherwise.Itmeansthat,accordingtothefamiliarprincipleof

minimumenergyfromphysics,directinformationactsto(re)alignsentimentalongitsdirectionto

reducetheenergy.Theoverallenergyduetotheimpactofdirectinformationflowontheinvestors

is ∑ , obtained by summing for all investors and where is a positive coefficient

determiningthestrengthoftheimpact.

The second factor is the interaction among the individual investors (agents). When people

exchangeopinions,theymayinfluencetheopinionsofothersandinturnbeinfluencedaswell.So

wecangenerallysay thatpeople tendto “coalign” theiropinions. Inourcontext,where investors

exchangeviewsaboutmarketperformance(i.e.sentiments),wecanwritetheenergyofinteraction

between the ‐th and ‐th investors as , where is a positive coefficient determining the

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strengthof interaction.Energywillbeminimal (negative)whenthesentimentsarecoalignedand

maximal (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( )andmultiplying

thesumby½toeliminatedoublecounting.

Consequently,thetotalenergyofthesystemtakestheform:

12

. 1

Provided that the same information is available to all investors, it follows that the energy (1) is

minimized if investors’ sentiments are coaligned. We should therefore expect to find that all

investors in themodelshare thesamepositiveornegativemarketoutlook.Yetweknowthat the

reality is different: there will always be investors whose market sentiments are contrary to the

popular opinion. This discrepancy arises for the reason that our model does not integrate the

infinitenumberof different specific influences experiencedby investors that lead to randomness

anddisorderpresentinrealworld.

Toincorporatetheeffectofthisrandomnessweagainborrowfromphysicswheretemperature

servesasameasureofdisorder–thehigherthetemperature, themorestochastic‐likeasystem’s

behavior. We apply this same methodology and introduce an economic analog to temperature,

whichwehenceforthsimplycalltemperatureor ,thatwillindicatethedegreeofdisorderinour

model.Thismeansthateachinvestorwillbesubjecttorandomdisturbancesthatmaycauseherto

occasionally change sentiment irrespective of other investors in the model; in other words,

investors’sentimentswillbesubjecttorandomfluctuations.Asaresult,thestudyofsystem(1)now

requiresstatisticalmethodsofanalysisthatwewillapplyinthenextsection.

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Let us briefly discuss the effects that suchmicroscopic random fluctuationsmay have on the

macroscopiccharacteristicsof thesystem.At lowtemperatures,randomfluctuationsareweak,so

that the interactions among investors lead the system toward one of two ordered macroscopic,

polarizedstateswherethetotalsentiment iseitherpositiveornegative. In thiscase,accordingto

the model, investors’ behavior is characterized by a high degree of herding that continues to

increaseasthetemperaturefalls.Athightemperatures,randominfluencesprevailandthesystem

as awhole appears disordered,with total sentiment fluctuating around zero, i.e. investors fail to

establish a consensus opinion and proceed to act randomly. In physics, the above‐described

macroscopic states are called phases and when a system changes state it is known as phase

transition.

In reality it is probably seldom, if at all, that investors behave either randomly or in perfect

synchronicity. Itwould be reasonable to suppose that the actualmarket ismostly confined to a

transitionalregimebetweendisorderedandorderedphases,whererandombehaviorandherding

orcrowdbehaviorcanbeofroughlyequalimportance.InSection1.3.2wewillprovideevidencein

supportofthisconjecture.

Thisfamilyofmodels,originallydevelopedtodescribeferromagnetisminstatisticalmechanics,

is broadly called the Isingmodel. The Isingmodel has been applied tomany problems in social

dynamicsover the last30years (seeCastellano,FortunatoandLoreto (2009)). Ineconomics, the

Isingmodelwasutilized for the first timebyVaga (1990),whoadapted it to financialmarkets to

infer the existence of certain characteristicmarket regimes. Since themid‐1990s there has been

extensiveeconomicresearch 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 dynamic

equationthatgovernstheevolutionofthesystem’ssentiment;and(ii)inthetheoreticalpartofthe

paper (Section 2),wedevelop a heterogeneous, two‐component extensionof the abovemodel to

gaininsightintothedynamicsoftheinteractionbetweensentimentanddirectinformation.

1.2.2.Equationforsentimentevolution

As noted earlier, system (1), in which investors’ sentiments are subject to random fluctuation,

necessitates statisticalmethodsof analysis. In this sectionwe study theevolutionof the system’s

totalsentimentasastatisticalaverage7: ⟨ ⟩/ ,where ∑ and⟨⟩denotesthestatistical

average.Wenotethat 1 1because⟨ ⟩canvarybetween and .

Toderive theequation for inanalytic form,wemake two simplifying assumptions: (i) all

investorsreceivethesameinformationand(ii)eachinvestorinteractswithallotherinvestorswith

the same strength, which yields an all‐to‐all interaction pattern8. As a result, the system’s total

energy(1)canbewrittenas

12

, 2

                                                            

7Alsocalledtheensembleaverage.Todistinguishbetweenthestatisticalorensembleaverage,ontheone

hand,andtheaveragewithrespecttotime,ontheotherhand,wedenotetheformerusinganglebracketsand

thelatterusinghorizontalbars.

8Theuseoftheall‐to‐allinteractionpatternisasensiblefirststepforstudyingthisproblemasitisalso

the leading‐order approximation for a general interaction topology in the Isingmodel.Wewill discuss the

implicationsofthisapproximationinSection2.2.3.

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where is the uniform information flow ( ) and is the constant strength of

interaction( ).

Aconsiderationofthekineticsofsystem(2)leadstothefollowingequationfor (Appendix

A,eq.A16):9

tanh , 3

where the dot denotes the derivative with respect to time, is the rescaled strength of

interaction10, hasbeenintroducedintheprevioussectionastheeconomicanalogtotemperature,

i.e.theparameterwhichindicatesthelevelofdisorderinthesystem,andw 1/ with defined

as the characteristic time overwhich randomdisturbanceswillmake individual sentiment flip

andsoindicatestheinvestor’saveragememorytime‐span.

Letusconsiderequation(3).Tobeginwith,itwouldbemoreconvenienttorewriteitas

, tanh , 4

where has themeaning of the total force acting on the sentiment , / is adimensionless

parameterwhich,being inverselyproportional to temperature,determines thedegreeoforder in

thesystemand / .                                                            

9Equation(3)isobtainedinthispaperasaspecialcaseofthesystemofdynamicequationsderivedinthe

statistical limit → ∞for the all‐to‐all interaction pattern (see Appendix A) that we study in Section 2.

Equation(3)wasoriginallyobtainedbySuzukiandKubo(1968),whousedthemean‐fieldapproximationto

deriveit.

10Note that ∼ 1/ ensures that coupling energy is finite in the all‐to‐all interaction case, therefore

1 .

Page18of97

 

Forillustrationpurposesletusassumethattanh σ ≪ 1,whereσ isthestandarddeviation

of the time series .11This assumption enables us to represent by a sum of the time‐

independentandtime‐dependentcomponents,whichtakesthefollowingleadingorderform:

, tanh sech tanh .

Equation(4)canthenbewrittenas

sech tanh , 5

where thecoefficient is zero in accordancewith (4) (wewillneed for interpreting (5) in the

next paragraph) and the time‐independent component of has been expressed via the function

,calledpotential,givenwiththeprecisionuptoaconstantby

12

1ln cosh . 6

And so,we have arrived at the equation for an overdamped, forced nonlinear oscillator. The

overdampingmeansthat inertia( ) is smallandcanbeneglectedrelativetodamping( ).Thus

equation(5)canbe interpretedasgoverning themotionofa zero‐mass( 0)dampedparticle

driven by the force applied, which is dependent on , inside the potential well, the shape of

which is determined by . Accordingly, takes on the meaning of the particle’s coordinate.

Thereforethemotionoftheparticlewilltracetheevolutionpathofthesentiment.

Inotherwords,equation(5)describesasituationwherethetime‐dependentforce(information

flow)actstodisplacetheparticle(sentiment)insidethepotentialwellfromitsat‐restequilibrium

                                                            

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

parametervalues.

Page19of97

 

(where the consensus of opinion is reached) while the restoring force (the interactions among

investors subject to random influences) counteracts it by compelling sentiment back toward

equilibrium.12

Theexpression for thepotential (eq.6) reveals theexistenceofdisorderedandordered

stateswithin the system as a function of temperature.13Thepotential is symmetric. It has theU‐

shapewith one stable equilibriumpoint 0for 1(the high temperature phase: ) and

the W‐shape with one unstable ( 0) and two stable ( ) equilibrium points that are

symmetricwithrespecttotheoriginfor 1(thelowtemperaturephase: )(Fig.2).TheU‐

shapecorrespondstothedisorderedstatesincesentimentiszeroatequilibriumreflectingthefact

thatinvestorstendtobehaverandomlyinthisphase.TheW‐shapecorrespondstotheorderedstate

wheresentimentsettlesateitherthenegative( )orpositive( )valueatequilibrium,asherding

behaviorprevailsinthisphase.Thus,themodelcontainsthephaseregimesthatwerediscussedin

Section1.2.1.

Figure2showsthat inthedisorderedstate( 1)decreasing causesthepotentialwellto

contract,sothatsentimentbecomesentrenchedaroundzero.Similarly, intheorderedstate(

1)when increases the negativewell ( 0) and the positivewell ( 0) quickly deepen and

simultaneously shift toward the boundaries ( 1), so that sentiment becomes trapped at the

                                                            

12Thismotion is finiteas 1 1, 0at 1and 0at 1.However, the system

doesnotpermit freeoscillations around equilibrium: the absenceof inertiamakes theparticle fall directly

towardtheequilibriumpoints,whicharethestablenodesindynamicalsystemsterminology.

13Itmaybeeasiertounderstandthebehaviorof ifitisexpandedintoatruncatedTaylorseriesin

thepowersof as ,whichholdsreasonablywellforany when ~1.Notethat

thephasetransitionat 1correspondstothechangeofsignofthequadraticterm.

 

extreme

1is

unrealis

Figure2

the stab

pointsto

andone

Fina

has no

sentimen

solelyby

thechan

1.2.3.M

Inthepr

investor

dailyme

enegative or

s likely to

ticallyrestri

2:(a)Thepo

ble equilibri

othemaxim

unstablean

lly,itiswor

intrinsic m

ntforanarb

ythestatist

ngeofsentim

Modeled

revioustwo

rs’marketse

easured

r positive v

be prevalen

icted.

otential

um points

maof .T

ndtwostable

rthwhilenot

emory and

bitrary

ticalpropert

mentif

dsentime

osectionsw

entiment (

,butfortha

values. It fol

nt, otherwis

for 0

correspond

Thereisone

eequilibrium

tingthatthe

so it woul

.Thepossib

tiesof .

isautocorre

ent

ehavedeve

eq.4).Now

atwemustf

lows that, a

se the latit

0.3and0.9.

s to the mi

estableequi

mpointsint

absenceof

ld generally

bilityofmak

Forexampl

elated.Wew

elopedthem

wecancon

firstestimat

aswe have

tude of the

(b) fo

inima of

libriumpoin

theordered

inertiaineq

y be imposs

kingameani

le, itcouldi

willreturnto

modelthatc

nstructthee

ethevalues

expected, th

change of

r 1.1an

and the

ntinthediso

state(Fig.2

quation(5)i

sible to pre

ingfulforeca

nprinciple

othispointin

convertsdire

mpiricaltim

oftheparam

Pa

he transition

sentiment

nd1.3.The

unstable e

orderedstat

2b).

impliesthat

edict the ev

astisthend

bepossible

nSection1.4

ectinformat

meseries

meters ,

age20of97

nal regime

would be

locationof

quilibrium

te(Fig.2a)

themodel

volution of

determined

topredict

4.2.

tion into

fromthe

and .

 

First

behavior

the initi

dominat

investor

seems r

roughly

We

obtain t

lengthb

thebeha

correspo

trendso

sentimen

weeklyb

t, as noted

randherdin

ial estimate

tingnorneg

r’smemoryh

easonable t

1month.Ac

substitute

the daily ser

between199

aviorsof the

ondencebet

onvariousti

nt values an

basisand75

earlier, it w

ngbehavior

. Second, it

ligible.Wet

horizonlies

to choose as

ccordinglyw

the daily s

ries . Fig

96and2012

emodelled

tweenthetu

imescales.T

nd index lo

5%onamon

would be re

areequally

t may prove

thereforetak

ssomewhere

s the initial

weestimate

series i

gure 3 show

2.Ourfirsto

sentimenta

urningpoint

Thisobserva

g returns a

nthlybasisa

easonable t

present,wh

e useful to

ke 1as

einthewin

l guess co

1⁄

into equati

ws and

observation

andtheactu

tsofsentime

ationissupp

re cross‐cor

atzerotimel

to consider

hichcorresp

assign ~1

stheinitiale

ndowfroms

orrespondin

0.04.

on (4) wit

the S&P 50

isthatthere

ual index.Fo

entevolutio

portedbyth

rrelated at

lag.

a situation

pondsto ~

1so that th

estimate.Th

severaldays

ng to 25 bu

th 1,

00 Index fo

earecertain

or instance,

onandthetu

estatisticst

55% on a d

Pa

where bot

~1.Soweset

e term

hird,intuitive

stoseveral

usiness days

1and

r periods o

nsimilaritie

thereappea

urningpoin

thattheincr

daily basis,

age21of97

th random

t 1as

is neither

ely,typical

months.It

s, which is

0.04to

of different

esbetween

ars tobea

tsofindex

rementsof

69% on a

 

Figure3

500 Ind

similarit

Our

theposit

and dow

correspo

compatib

motion i

positive

potentia

Wec

stretche

observat

experien

thirdsof

territory

skewed,

3:Thesentim

dex during

tybetween

secondobse

tiveregionb

wnward) an

ond tomark

blewiththe

inside thep

wells.Wew

alwellinSec

canalsosee

s of marke

tion issurpr

ncedthelarg

f this timeb

y. It hints at

whichwou

ment co

(a) 1996‐2

basedo

ervationcon

between

nd infrequen

ket crashes

eW‐shapeof

positivewell

willdemons

ction1.3.2.

e that sentim

t losses, an

risinggiven

gestfinancia

below itshi

t thepossib

uldbeconsis

onstructedfr

2012; (b) 2

nDJ/Factiva

ncernstwod

0.2and

nt large‐scal

and rallies.

fthepotenti

lwhile the

strate that th

mentremain

nd becomes

nthatweha

alcrisissinc

igh‐waterm

ility that th

stentwithth

from fo

010‐2012;

adataand

distinctbeha

0.4thatco

le movemen

Althoughw

ial for

latter canb

heW‐shape

nspositivem

negative o

aveconsider

cetheGreat

markandon

ere is anat

heasymmet

r 0.04,

and (c) 07

basedon

aviorsinsen

orrespondst

nts into and

we have set

r 1(eq.

be related to

e is indeedt

mostof the

only during

redtheperio

tDepression

equarterof

tural tenden

tryofthelon

, 1, and

7/2011‐11/2

ntheproprie

ntiment:abo

tomid‐term

d out of the

1, such

6)asthefo

omotion ac

theprevalen

time,even

pronounce

odduringw

nandthest

f this time i

ncy for senti

ng‐termbeh

Pa

d 1.0an

2011. Note

etarydata(F

oundedmot

indextrend

e negative r

h behaviors

rmercanbe

ross thene

ntconfigurat

duringrelat

d bear mar

whichtheUS

ockmarket

inbearmar

iment to be

haviorobser

age22of97

ndtheS&P

the close

Fig.3b).

tionwithin

ds(upward

region that

s aremore

erelatedto

gative and

tionof the

tively long

rkets. This

Seconomy

spenttwo

rket return

positively

rvedinthe

Page23of97

 

stock market (perhaps best exemplified by the fact that the DJ Industrial Average Index has

returnedonaveragearound7.5%p.a.overthelast100years).

Accordingtoourmodel,itrequiresapositivemeanof ( 0)tokeepsentimentpositive

on average, i.e. the volume of positive direct information must, in the long run, exceed that of

negativedirectinformation.Indeed,thedailymean measuredfortheperiodcoveredbyourdata

ispositive(TableI).Thispositiveinformationbiasispossiblyrelatedtotheeconomicgrowthand

inflation.

Theasymmetryofsentiment’sbehaviorinducedby 0canalsobeexplainedintermsofthe

perturbationofpotentialwell. IfwedecomposeH into twopartsas ,where is

the constant mean of and is its time‐dependent component, such that 0, then

equations(5)and(6)canbewrittenas

sech tanh , 7a

with

12

1ln cosh , 7b

where 0.

Figure4showsthatapositivecbreaks thesymmetryof thepotential.For 1thepositive

well( 0)deepensandthenegativewell( 0)flattens,sothattheprobabilityof crossinginto

 

the nega

beyond

configur

equilibri

positive

shows th

≳ 0.01

consiste

period.

Figure 4

correspo

0.02

         

14 Th

thirdterm

ative territo

which the

rationtransf

ium.Theov

wellinterru

hat even a

1for 1

nt with the

4: The pote

onds to the

2.

                      

he potentia

masitislinea

ory decrease

negative s

forms intoa

veralleffect

uptedbyrela

small can

.Usingthe

e markedly

ential

minimum

                      

l can

arin .

es.14For lar

stable equil

asingle‐wel

ofthisdisto

ativelyshor

have a pro

relevantva

asymmetric

for 1.1

of at

       

be expand

.Note that t

rger the sh

librium poi

lconfigurat

ortionistha

rtexcursions

onounced ef

luesfor a

c behavior o

1and 0.0

, surv

ded into a

the asymmet

hallow well

int ( )

tion inwhich

atsentiment

sintonegati

ffect on the

and (Table

of the senti

01, 0.02. On

vives the dis

a truncated

try relative to

is further

) vanishes

h thesentim

ttendstosp

ivesentimen

shape of th

eI),weobta

iment obser

nly one equ

stortion of t

Taylor se

o theorigin

Pa

distorted to

and the d

ment isposi

pendmoret

ntterritory.

he potential

ain 0.01

rved over th

uilibrium po

the potentia

eries and w

( 0) emer

age24of97

o the level

ouble‐well

itiveat the

timeinthe

Thefigure

l well (e.g.

17whichis

he studied

oint, which

al well for

written as

rges via the

Page25of97

 

Lastly,wewouldliketomentionthatdespitesimilaritiesinthebehaviorbetweensentimentand

the S&P 500 Index, the differences are nevertheless substantial. Perhaps most importantly, the

evolutionofsentimentisboundedwhereasthestockmarketis,onaverage,experiencinggrowth.On

the other hand, it seems reasonable to suppose that allmarket developments, including its long‐

termgrowth,occurthroughdevelopmentsinsentiment.Tofindwhetherornotthisistrue,wemust

understandtherelationbetween investorsentimentandmarketprice,which is thesubjectof the

nextsection.

1.3.Price

Inthissectionwewillbeconcernedwithinvestigatinghowsentiment canimpactmarketprice .

Weproposeamodelofpriceformationandapplyittoconstructanempiricaltimeseriesofmodel

pricesthatwewillcomparewithobservedmarketprices.

1.3.1.Modelofpriceformation

Positiveornegativesentimentmeansthatonaverageinvestorsbelievethatthemarketwillriseor

fall,respectively.Butdoesitalsoimplythatinvestorswouldnecessarilyactontheirsentimentand

proceedtobuyorsellassets,asthecasemaybe?

Letusconsideraninvestorwhohasjustallocatedcapitaltoastockmarket.Nextday,allother

thingsbeingequal,theinvestorisunlikelytoincreaseordecreasehermarketexposureunlessher

sentimenthaschangedbecauseshehadalreadydeployedcapitalintheamountreflectingthatsame

levelofsentiment.Itisthereforesensibletoassumethat,ignoringexternalconstraints(e.g.capital,

risk, diversification), investment decisions are mainly driven by the change in sentiment on

timescaleswhere the investor’smemoryofpast sentiment levelspersists, determinedby∆ ≪

(seeSection1.2.2).

Page26of97

 

Reasoningsimilarly,weconcludethatonlongertimescales(∆ ≫ )investorswouldinvestor

divestbasedprimarilyon the levelof sentiment itselfbecause theirpreviousallocationdecisions

wouldnolongerbelinkedintheirmemorytothepreviouslevelsofsentiment(recallthatourinitial

guessfor isapproximately1month).

Asthechangeofmarketpriceisdeterminedbythenetflowofcapitalinoroutofthemarket,we

conclude thatprice changeswoulddependprimarilyon sentiment changeson timescales shorter

than (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 .15

We reiterate that equations (8a) and (8b) provide only the asymptotic relations between

sentimentandpriceatrespectivelyshortand longtimescales.Aswedonotknowthe formofthe

actualequationgoverningtheevolutionofprice,wesimplysuperposebothasymptoticviewsand

seekpricepatanytimetas

,orequivalently , 9

                                                            

15Bytakingthelogarithmof ,wenormalizethepricesothat or resultsintherelativepricechange

ln ,i.e.thereturn,whichisastandardprocedure.Ifthepricehadnotbeennormalized, or

wouldhavecauseddifferentpercentagechangesinthepricedependingatwhichpricelevelitoccurs.

Page27of97

 

in the hope that solutions given by it approximate true solutions reasonably well. Note that

constants 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 of

sentimentfromacertainvaluegivenbys∗.Thuss∗in(10)servesasayardstick,averagedacrossthe

investment community, relative to which investors appraise sentiment: if sentiment is above or

below it, they may invest or divest, respectively. A nonzeros∗can be interpreted as an implied

referencesentimentlevelthatinvestorsareaccustomedtoandconsidernormal.16Wedonotwish

to impose any a priori constraints ons∗. Instead we will determine its value by fitting price

observationstothemodelinthenextsection.

1.3.2.Modeledprice

In this sectionweapplyequation(10) toconstruct themodelprice fromthe timeseries

reportedinFigure3.

We search for the coefficients in equation (10) that minimize the mean‐square deviation

between andthelogpricesoftheS&P500Indexovertheperiodcoveredbyourdata.Figure

5a,whichdepictstheevolutionofmodelpricesvs.indexprices,showsthat approximatedthe

indexbehavioroverthisperiodwell.Wenote,first,thatthedailymodelpricesandthedailyindex

                                                            

16Anonzero ∗wouldbecompatiblewiththeadaptationleveltheoryputforwardbyHelson(1964)and

thereferencepointconceptintroducedbyKahnemannandTversky(1979)intheirprospecttheory.

 

prices a

cointegr

expected

whileth

Figure

obtained

3. The e

andslow

Wen

that ∗st

consider

sentimen

It is

sentimen

equilibri

are 83% co

rated. Additi

d,the leadin

eleadingco

5: (a) The

dfromeq.(1

estimated co

wcomponen

notethatthe

tays on ave

r normal an

nt(seeSecti

shown inA

nt(as isexp

iumvalueis

orrelated a

ionally, Figu

ngcompone

omponentat

model pric

10)withcoe

oefficients a

ntsof giv

econstant ∗

erage positi

nd so is like

ion1.2.3).

AppendixB

pectedgiven

determined

nd, second

ure 5b depi

ntat∆ ≪

t∆ ≫ (eq

ce vs. th

efficientses

are 0.32

venbyeq.(8

∗ispositive

ive as it m

ely to be re

that in the

n itscontext

dsolelybyo

, that stan

icts the beh

(eq.8a)e

q.8b)exhibi

he S&P 500

stimatedusin

22;. 0.0

8a)andeq.(

overthispe

manifests a

elated to th

leading ord

tas thebac

orderparam

dard tests

havior of th

exhibitsrapi

tsslow,larg

0 log price

ngtheleast

003; 6.

(8b),respec

eriod.Indeed

background

he positive,

der ∗coincid

kgroundref

eter ,whi

show that

he componen

id,small‐to‐

ge‐scalevari

s (1996‐20

tsquaresfitt

564; and ∗

tively.

ditwouldbe

d reference

long‐term b

deswith the

ferencevalu

chistheinv

Pa

t these var

nts of (

mid‐scalefl

ationwithti

12). The pr

tingand

0.076. (b

ereasonable

value that

bias exhibit

e equilibrium

ue).Wekno

verselyprop

age28of97

iables are

(eq. 8). As

uctuations

ime.

rice is

fromFig.

b) The fast

etoexpect

t investors

ted by the

mvalue of

wthat the

ortionalto

Page29of97

 

temperature .17This means that temperature has a direct impact on price dynamics. As

temperatureindicatesthebalancebetweenrandombehaviorandherdingbehaviorinthemarket,it

wouldbesensibletopresumethatsuchbalancemaygraduallyshiftovertimeandconsider tobe

slowly varying with time. Consequently, we can expect the reference sentiment level ∗to be a

slowlyvaryingfunctionoftimeaswell.

Assumingthat wecalculate fromequations(4)and(10)usinganiterativeprocess

forminimizingthemean‐squaredeviationbetweenthemodelandtheindexfor and

∗ ∗ ,whilekeepingotherparametersconstant.18Asaresult,weobtainabetterfitbetween

themodelandthe index:now matchesthe indexbehavioratvarioustimescalesmoreclosely

(Fig.6)andthecorrelationbetweenthedailymodelpricesandthedailyindexvalueshasimproved

to97%.

                                                            

17The equilibrium points are determined by the condition that 0and 0in equation (4) (or

equivalently by the extrema of (eq. 6)), given by the equation tanh , so that the equilibrium

valuedependsonlyon .

18As follows from equation (10), the reference sentiment level ∗is an important factor for price

dynamics. In the leading order ∗is exclusively a function of : ∗ ∗ (Appendix B). Aswe are

interestedintheeffectofaslowlyvarying on ∗,weconsider tobeafunctionoftimebutforsimplicity

holdotherparametersconstant.

 

Figure

construc

shownA

valueso

of and

Letu

in Figur

These v

around t

sentimen

prevalen

Weo

2002 an

6: The mo

cted throug

and

AppendixB.

oftheconsta

d aresho

usnowcons

e 7.We obs

alues of

the average

nt evolved,

ntthroughou

observethat

nd 2008) an

odel price

gh an itera

∗ ∗

Theestimat

antcoefficien

owninFig.3

sider

serve that

correspond

e value of 1

as expected

utthestudie

ttheperiod

nd that the

vs. the

tive least s

and the re

tedmeanpa

nts(eq.10)

.

obtained

exhibite

to a slight

.1,maintain

d, inside th

edperiod,al

dsofhigher

periods of

S&P 500

squares fitt

elation betw

arameterval

are 0.

usingthea

ed small var

tly supercrit

ning a value

e double‐we

lthoughitsin

temperatur

lower temp

log prices

ting involvi

ween ∗and

lues: 1

376; 0

bove‐outlin

riations, alw

tical (i.e.

above unit

ell potentia

nfluenceten

ecorrespon

perature cor

(1996‐201

ing eq. (4)

is determi

.12and ∗

0.003; and

editerative

ways remain

≳ 1), wh

ty at all tim

l and that h

ndedtoebba

ndtobearm

rrespond to

Pa

12). The pr

and (10),

ined from e

0.173.The

6.293.T

processand

ning lower t

hich indeed

es. It follow

herding beh

andflowwit

marketregim

o bull marke

age30of97

rice is

in which

eq. (B5) as

eestimated

Thevalues

dreported

than unity.

fluctuated

ws that the

havior was

thtime.

mes(2001‐

et regimes

 

(2003‐2

for2009

Figure

Tempera

thecritic

Asp

be logica

sentimen

easierfo

between

andmar

         

19Fig

temperat

decline in

volumes

007)witht

9‐2012,that

7: Tempera

aturefluctua

calvalueofu

periodsofhi

al to suppo

nt (express

orinformatio

nthesetwo

rketvolatility

                      

gure 7 also s

turepatternp

ntemperatur

forthatperio

henotablee

tookplacei

ature (mea

atedlesstha

unity,which

ighertempe

se that thes

ed different

onflowtom

variables,w

y,whichisa

                      

shows elevate

priorto2000

re in1999‐20

od.

exceptionof

inarelativel

asured in u

an10%arou

hmarksthep

erature indic

se periods c

tly: rising te

movesentim

whileFigure

alsotobeex

       

ed temperatu

0maybeunr

000suggests

fthepost‐2

lyhightemp

units of (e

undthemea

phasetransi

cate increas

coincidewit

emperature

ment).Figure

8bshowss

pected.

ures during t

eliabledueto

that the leve

008‐crisism

peratureenv

eq. 3)) as a

anvalueof0

itionbetwee

ed importan

th periods o

flattens th

e8ademons

similaritiesi

themarket ra

olowdatavo

elsof1996‐1

marketrally

vironment19

a function o

0.9,alwayss

enordereda

nceofrando

of increased

e potential

stratesanev

inbehavior

ally of the la

olumes.Furth

1998couldbe

Pa

,capturedi

9.

of time (19

stayingnear

anddisorder

ombehavior

d volatility i

well, which

videntcorres

betweentem

ate 1990s. Ho

hermore,the

eanartifact

age31of97

nourdata

996‐2012).

butbelow

redstates.

r, itwould

in investor

h makes it

spondence

mperature

owever, the

subsequent

of lowdata

 

Note

2010‐20

followin

2007, th

conditio

risk‐on,

Figure8

calculate

perioda

Index vo

deviatio

In su

reprodu

thatthe

ethatthepo

012,whichis

g theCredit

he bull mar

nsof subdu

risk‐offenvi

8: (a)Temp

ed as the st

anchoredto

olatility ove

nof21‐day

ummary, we

cemarketp

balancebet

ost‐2008‐cris

scompatibl

tCrisis.A r

rket during

uedeconom

ironment).

perature an

tandarddev

thedate fo

er that sam

logreturns

e have dem

pricebehavi

tweenherdin

sisbullmark

ewiththea

easonablee

2010‐2012

ic recovery

nd the volat

viation ofmo

rwhichthe

me period. S

usingthesa

monstrated t

orasafunc

ngbehavior

ketexhibited

aboveobserv

explanation

was basica

andhighun

tility of sent

onthly (21 d

evolatility is

S&P 500 In

amemethod

that the mo

ctionofinve

randrandom

danunexpe

vationthatt

to it is that

ally a liquid

ncertainty i

timent (19

days) increm

scomputed.

dex volatili

asin(a).

odel can, wi

estorsentim

mbehavior,

ectedlyhigh

thetempera

t, unlike the

dity‐driven r

noverall in

996‐2012). S

ments in o

. (b)Tempe

ty is calcul

ithin a relat

ment.Additio

indicatedby

Pa

sentimentv

aturedidno

bullmarke

rally occurr

nvestor senti

Sentiment v

over a rollin

erature and

ated as the

tively close

onally,weha

ytheeconom

age32of97

volatilityin

otdecrease

et in2003‐

ring in the

iment (the

volatility is

ng 300‐day

dS&P500

e standard

tolerance,

aveshown

micanalog

Page33of97

 

oftemperature ,cangraduallychangeovertime.AswewillseeinSection2, playsanimportant

roleininfluencingmarketdynamics.20

Forconvenience,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 this

variationweachieveacloserfitbetweenthemodelandthemarket.Sincethevariationof issmallandfor

the sake of simplicity, we will henceforth use the time series and as calculated with constant

parameters. The values of parameters are shown in Figures 3 and 5, except that we apply 1.1(the

averagevalueof overthestudiedperiod)insteadof 1.0.Theparametervaluesarealsosummarized

in Table I, however the graphs and are not shown for the economy of space and due to their

similaritytothosedepictedinFigures3and5,respectively.

Page34of97

 

1.4.Discussion

In this section we summarize the findings of the empirical study and discuss their theoretical

implicationsandpracticalapplications.

1.4.1.Results

Wehave shown that themodelprice replicated the S&P500 Index valuesover theperiod1996‐

2012withinreasonable tolerance, lendingcredenceto theassumptionsmadeearlyontodevelop

themodel.Thusweconclude:

First,investors’sentimentisinfluencedbydirectinformationflow.

Second,inadditiontodirectinformation,sentimentisalsoinfluencedbytheinteractionamong

investors and idiosyncratic influences that can be assumed random for our purposes. The Ising

model, in which temperature serves as a measure of random influences, provides the relevant

frameworkforstudyingsentimentdynamicsand, inparticular,enablesustoobtainaclosed‐form

equationfortheevolutionofsentiment.

Third, the mechanism of price formation as a function of sentiment works differently on

different timescales. On timescales shorter than the investors’ average memory horizon, market

price changes proportionally to the change in sentiment. On longer timescales, price changes

proportionallyto thedeviationofsentimentfromareference level that isgenerallynonzero.Asa

result,pricedevelopmentisnaturallydecomposedintoaslow,large‐scalevariationwithtimeand

fast,small‐to‐mid‐scalefluctuations.

We have seen that for the studied period the effects in connectionwith herding behavior of

investors were more prevalent than those related to random behavior. Consequently, sentiment

evolved insideaW‐shapedpotential,withanegativeequilibriumvalue inonewellandapositive

value 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,

sentimentspentmostofthetimeinthepositivewellandcrossedintothenegativewellonlyduring

the time of crises and, second, the reference sentiment level stayed mostly positive during this

period.

Althoughourdata coverage is limited to 16years,we tend to think that the above‐described

asymmetry of sentiment dynamics, caused by the positive bias in direct information flow, may

generallypersist.This isbecausedirect informationmustbeonaveragepositive tobeconsistent

with the long‐term growth of the stock market. The asymmetric behavior of sentiment can be

alternativelyexplained in termsof theperturbationofpotential: the constantproportional to the

(positive) mean of direct information flow is a parameter that distorts the shape of potential,

makingthepositivewelldeeperthanthenegativewell.Asaresult,sentimenthasapropensitytobe

onaveragepositiveasittakesmoresignificantnewstoturnitfrompositivetonegativethanfrom

negativetopositive.

Wewouldlikenowtocommentoncertaineffectspertainingtothelong‐term(monthstoyears)

behavior of sentiment and price. We begin with the economic analog of temperature, which

determines the relative importance of random behavior vs. herding behavior. Temperature can

substantiallyimpactmarketdynamicsintwoways:first,byalteringtheshapeofpotentialthereby

affectingsentimentevolution(e.g.theprobabilityofcrossingthewells)and,second,byinfluencing

thereferencesentimentlevel(eq.10)whichissensitivetotemperaturechanges.

Wehaveobservedthattemperatureandthevolatilityofsentimentvariedovertimeinasimilar

pattern. Itseemsreasonabletoassumethatperiodsof increasedsentimentvolatilityimplyahigh

 

uncertai

periods

liquidity

asitexh

8a).

Wen

varied i

downtur

sentimen

thatthe

market.

         

21“Ri

highvola

during20

intyofinves

of lowsent

y‐drivenmar

hibitstypica

nowbriefly

n a pattern

rn and decr

nt also fluct

modelcaptu

                      

isk‐on,risk‐o

atilityof the r

010‐2012the

storopinion

imentvolati

rketrecover

lrisk‐onris

considera

n resembling

reasing duri

tuated appr

uressomee

                      

off”isaloosel

riskappetite

emarketwas

nsymptomat

ilitycorresp

ryduring20

sk‐offfeatur

connection

g economic

ng the uptu

roximately s

effectsofthe

       

lydefinedter

of investors

inarisk‐offr

ticofbearm

pondtothe

010‐2012ap

res,exemplif

with theec

fluctuation

urn. Figures

synchronous

einteraction

rmthatpract

. It is awide

risk‐onregim

marketsand

positive,tre

ppearstobe

fiedbythee

conomy.Fig

ns, typically

9c,d show

slywith eco

nbetweenth

titionersuset

elyheldview

me.

drisk‐on,ris

endingmark

eanexceptio

elevatedsen

gure9a,bsh

increasing

that direct

onomy. Thes

hegenerale

todescribem

across the in

Pa

sk‐offregim

kets.Wenot

ontothisob

ntimentvola

ows that tem

during the

information

se observati

economyand

marketschara

nvestment in

age36of97

es21,while

tethatthe

bservation,

atility(Fig.

mperature

economic

n flow and

ions imply

dthestock

acterizedby

ndustry that

 

Figure9

mirror i

informat

than 85

similarly

Fina

willbed

Wewill

theoryu

1.4.2.P

Wehave

insideth

themid‐

declines

correspo

A co

evolutio

termin

9: (a) Temp

mageof

tion flow

0 days, and

ysmoothed

lly,wehave

done,briefly

revisit this

underlyingth

Practical

eobservedt

hepositivew

‐termmarke

and reco

ondencebet

onsideration

n( 0)an

(10)iszero

perature

and the s

, smoothe

d the US re

as abov

enotyettou

y,inthenext

issue in the

hismodel,em

lapplicat

twotypesof

wellandthe

et trends,w

overies. It

tweenthetu

n of equatio

ndthe turni

o( ∗).It

and the d

ameGDP gr

edusingaF

eal GDP per

ve,andthes

uchedupont

tsection,inw

e secondpa

mphasizing

tions

fbehaviorch

infrequent

while the latt

would be

urningpoints

n (10) lead

ingpointso

followstha

detrended U

raph as sho

Fourier filte

r capita grow

sameGDPgr

theissueof

whichwehi

art of thispa

thepotentia

haracteristic

crossingsbe

terappears

interesting

sofsentime

ds to a conc

ofprice tren

atincertain

S real GDP

own in (a) fo

er that remo

wth (YoY)

raphasshow

whetheror

ighlighttwo

aper (Sectio

alformarke

ctosentime

etweenthew

todrive th

g to invest

enttrendsan

clusion that

nds( 0)

casesthetu

per capita

or ease of c

ovedharmon

(1996‐2012

wnin(c).

rnotthismo

orelevantpr

on2),which

etforecasting

nt:themid‐

wells.Thefo

e large‐scal

tigate whe

ndmarketpr

the turning

nevercoinc

urningpoint

Pa

(1996‐2012

comparison.

nicswithpe

2). (d) Senti

odelispredi

ropertiesof

h furtherde

g.

‐termbound

ormerseems

edynamics

ether there

ricetrends.

g points of

cideunless t

tsofsentim

age37of97

2). (b) The

(c)Direct

eriods less

iment ,

ictive.This

themodel.

velops the

dedmotion

stoinduce

ofmarket

is some

sentiment

thesecond

menttrends

 

can occu

trendin

to be po

reverses

reversed

sentimen

Tov

multivar

Indexan

Figure 1

secondt

trailsthe

hasover

index on

moment

Figure1

pricecom

theS&P

ur before th

sentimenti

ositive, as c

s the trend

d at ∗,

ntagainbre

verifytheabo

riate singula

ndplotthem

10b shows t

terminequ

e index.Thi

rtheindex,a

n long time

tumstrategi

10:MSSAex

mponent(eq

500logpric

he turning p

isreverseda

oefficients

in advance

changes si

eaksthetren

oveconclusi

ar spectrum

minFigure1

the sameM

ation(10),a

iscompensa

and,asares

escales. Thes

es.

xpansion (1

q.8a),andt

ce.

points of pri

at ∗, c

and in

e of the ma

ign from ne

ndbeforethe

ionempirica

m analysis (M

10a.Theind

MSSA expans

andtheS&P

ates for the

sult,theover

se observat

996‐2012)

theS&P500

ice trends. F

changessign

equation (1

arket’s reve

egative to p

emarket.

ally,weextr

MSSA) expan

excanbese

sion for the

P500Index

leadthatse

rallmodelp

tions may b

(a) the sent

logprice;an

For example

nfromposit

10) are pos

ersal. Simila

positive, wh

racttwofirst

nsion of the

eentotrails

slow comp

x.It iseviden

entiment(i.e

priceisnots

be helpful fo

timent ,

nd(b)thesl

e, in situatio

tivetonegat

sitive. This m

arly, when a

hile remain

tquasiperio

e sentiment

sentimentw

ponent of pr

ntthatthes

e. thefirst te

ubstantially

or the devel

which ispr

lowpriceco

Pa

onswhere a

tive,while

means that

a downward

ns negative,

diccompon

and the

withaconsid

rice , giv

slowpricec

erm inequa

yshiftedrela

lopment of

roportional

omponent(e

age38of97

an upward

continues

sentiment

d trend is

such that

entsofthe

e S&P500

derablelag.

ven by the

component

ation(10))

ativetothe

long‐term

to the fast

eq.8b)and

 

Next

wehave

is n

autocorr

different

which in

thatitco

.Fu

discussio

Figure1

theprop

dailydat

dailylog

intervals

Itisw

aproces

pattern

tweconside

epointedou

not white n

relation of t

tfromthea

ndeedmake

ouldinprinc

urtherresear

oninthesec

11:(a)Auto

prietarynew

tavolumes

greturnsoft

sarealsoind

worthwhile

ssthatconta

at time lags

erthepossib

utthatsuch

noise, i.e. its

the dailyme

autocorrelati

es a compell

ciplebeposs

rchinthisa

condpartof

correlations

wsarchiveco

than ba

theS&P500

dicatedinth

notingthat

ainsdecayin

sgreater tha

bilityofpre

possibilitye

s autocorrel

easured

ionoftheS&

lingargume

sibletopred

areamaypro

fthispaper(

sofdaily

overingthe

asedonthe

0Indexacros

hegraphs.

theautocor

ngquasiperio

an2days is

edictingthe

existsinpri

ation is not

does not b

&P500logr

ent for ther

dictthebeha

oveusefulto

(Section2).

acrossdif

period2010

DJ/Factiva

ssdifferent

rrelationpat

odicoscillat

smostlywit

evolutionof

nciple,prov

t a delta fun

behave as a

returnsover

randomwalk

aviorof an

omarketpr

fferenttime

0‐2012asit

archive(see

timelagsfo

tternof

tions,withth

thin thema

fsentiment

videdthatdi

nction. Figu

a delta funct

rthatsame

khypothesi

ndtherefore

ractitioners.

lags.Wetak

hasacleare

eFig.1b).(b

rthesamep

resembles

hecaveatth

arginof erro

Pa

itself.InSe

irectinform

ure 11 show

tion and is

period(the

s).Thus,we

fromthet

Wewillret

ke obta

ersignaldue

b)Autocorr

period.Thec

theautocor

hattheautoc

or.Thisobse

age39of97

ction1.2.3

mationflow

ws that the

noticeably

patternof

e conclude

timeseries

urntothis

ainedfrom

etohigher

elationsof

confidence

rrelationof

correlation

ervation is

Page40of97

 

consistentwiththeresultsproducedbytheextended,self‐containedmodelwedevelopinthenext

section.

2.PartII–Theoreticalstudyofstockmarketdynamics

In the empirical study we have assumed that information that speculates about future market

returns (direct information)maybe effective in influencing investors’ opinions about themarket

(investors’ sentiment). We have established a mechanism that translates direct information via

investors’sentimentintomarketprice.Thetranslationmechanismitselfismathematicallysimplein

the sense that the equation for sentiment dynamics (eq. 4), which defines how investors form

opinions based on received information, and the equation for price formation (eq. 10), which

defineshowinvestorsmakeinvestmentdecisionsbasedontheiropinions,constituteanuncoupled

andintegrablesystemofequations.

It is therefore not surprising that the source of complex behavior, characteristic to actual

markets, in the model is contained within the direct information flow that we have treated as

exogenous.Thismeans thatwehave investigated justonepartof theproblemand to learnmore

about the origins ofmarket behaviorwe have to extend the framework by incorporating into it

assumptionsonhowdirectinformationflowisgeneratedandchanneledinthemarket.

Inthispartofthestudywedevelopatheoretical(self‐contained)modelofthestockmarketthat

includes direct information (which we will call to set it apart from the empirical case) as a

variable,alongsidesentiment andprice .Thismodelissimplifiedbyconstructiontofacilitateits

study, so its solutions cannot reproduce the full range and the precise detail of actual market

regimes. Itspurpose is to capture theessential elementsofmarketbehavior andexplain them in

termsoftheinteractionamongdirectinformation,investorsentimentandmarketprice.

Page41of97

 

This part proceeds as follows: Section 2.1 develops a self‐contained model of stock market

dynamics and Section 2.2 provides the analysis, results and discussion. The relevant technical

detailsareinAppendicesAandC.

2.1.News‐drivenmarketmodel

Directinformation,whichisbasedontheinterpretationofgeneralnews,consistsofopinionsabout

futuremarket 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, market

commentators,businessnewscolumnists,financialbloggers–basically,anyonewithaviewabout

themarket,themeanstodeliverittoalargeinvestoraudienceandthecredibilitytoinstilltrust.For

convenience,letusrefertothemcollectivelyasmarketanalysts.Inthecontextofourframework,

analystsperformavitalfunction:theyinterpretnewsfromvarioussources,opinehowthemarket

might react and transmit their views through mass media. That is, they convert any type of

informationintodirectinformationandmakeitavailabletomarketparticipants.22

                                                            

22We consider only publicly available direct information simply becausewe know how tomeasure it.

Usually,priortoanexpertopinionbeingexpressedpublicly,itisdiscussedinprofessionalcirclesanddiffuses

acrosstheinvestmentcommunity.Thus,itwouldbenormalthatprivatelyexpressedviewswouldhavealead

over those which are publicly expressed.We tend to think, however, that on the timescale of investment

decision‐making by institutional investors, this informational lag is negligible due to the speedwithwhich

information becomespublic through online publication, email andblogs. The topic of howopinions transit

fromprivatetopublicisdeservingofitsownresearch,butisoutsidethescopeofthiswork.Hereweassume

that,forourpurposes,thetimelagbetweenthepublicandprivatedirectinformationcanbeneglectedwhile

theirmagnitudes(asmeasuredby )canbeconsideredproportional.

Page42of97

 

LetusconstructanIsing‐typemodelwithtwotypesof interactingagents: investorswhohave

capital to investbut cannot interpretnewsandanalystswhocan interpretnewsbutdonothave

capitaltoinvest(theironyofwhichisnotlostonus).Atanypointintime,eachinvestorandeach

analysthaseitherapositive(+1)ornegative(‐1)opinionabout futuremarketperformance–the

binarymarketsentiment inthecaseofinvestorsandthebinarymarketsentiment inthecaseof

analysts. Investors can interact with each other and with analysts and, similarly, analysts can

interactwitheachotheraswellaswith investors.They interactby literally imposingopinionson

each other as every sentiment tries to align other sentiments along its directional view. It is

importanttonotethatalthoughthenatureofinteractionisthesame(theexchangeofopinions),the

meansofinteractioninthemodelareverydifferent:analysts,whoareoutnumberedbyinvestors,

areassumedtoexertadisproportionallystrongforceoninvestorsduetoaccesstomassmedia.

Wecannowreplacethehomogeneous,single‐componentIsingsystem(1),wheretheexternal

flowofdirect information acted on investors, by its heterogeneous, two‐component extension, in

whichinvestorsandanalysts(insteadofinformationflow)interactwitheachother.Wedefine

and as individual sentiments of investors and analysts, respectively, and , as their

numbers( ≫ ≫ 1)and,asbefore,writethesystem’stotalenergy(AppendixA,eq.(A1))and

applytheall‐to‐allinteractionpatterntoobtaininthelimit → ∞, → ∞thedynamicequations

that describe the evolution of the variables as statistical averages, namely: ⟨ ⟩/ where

∑ and ⟨ ⟩/ where ∑ .

Thegeneralformofthedynamicequationsisasfollows(AppendixA,eq.A14):

tanh , 11

tanh , 11

Page43of97

 

where,analogous toequation (3), thecoefficient defines thestrengthof interactionamongthe

investors; – the strengthwithwhich the analysts act on the investors; – the strengthwith

which the investorsactontheanalysts; – thestrengthof interactionamongtheanalysts;

and areanygeneralexternalforces,sofarundefined,actingrespectivelyontheinvestorsand

the analysts, where and determine their impacts; 1/ with being the characteristic

horizon of the investors’ memory and 1/ with being the characteristic horizon of the

analysts’ memory; and has been identified in equation (3) to be the economic analog of

temperatureortheparameterthatdeterminesthe impactofrandominfluencesandconsequently

thelevelofdisorderinthesystem.Notethatparameters , , , , , , , and arenon‐

negativeconstants.

Inour framework, analystsmust react tonew information faster than investors, so that their

resistancetoachangeinsentimentisweakerthanthatof investors.Accordingly,weassume to

be an order of magnitude smaller than , resulting in 2.5business days, This value is

consistentwith thebehaviorof theautocorrelationof (Fig.11a) that showsa rapiddecayof

“memory”effectsontheorderof1‐3businessdays.Henceweset 0.4.

Letusconsidertheexternalforces and ,whichhavethemeaningofexternalsources

of information tapped by investors and analysts, respectively. We set 0by assuming that

investorsinthemodelreceivedirectinformationonlythroughanalysts;inotherwords,investors’

sentiment maychangeonlybyinteractingwithanalysts’sentiment .

Thecaseof ismoreinterestingbecauseoftheanalysts’functiontotranslatenewsintodirect

informationandsoprovideachannelthroughwhichexogenousinformationentersthemodel.Since

anynewsmaybeofrelevanceinthiscontext–e.g.economicdatarelease,corporatenews,central

bankannouncements,politicaleventsor,say,changeofweatherconditions–wecanmaketheusual

assumption that new information arrives randomly and therefore represent news flowvia noise.

Page44of97

 

Thereis,however,aparticularsourceofinformationthatshouldbeconsideredseparately.Itisthe

changeofmarketpriceitself.23Itsimportanceissupportedbyempiricalevidence:today’svaluesof

are correlated with yesterday’s and today’s S&P 500 log returns at over 30% and 50%,

respectively.24

Thus,wedivide informationavailable toanalysts in themodel into informationrelatedtothe

changeinmarketpriceandallother(external)information.Wewrite ρ ρ ,where is

market price change (eq. 10), is randomnews flow (noise) andρ (ρ 0 alongwithρ are

scaling constants. Note that by writing in the expression for , we have assumed that the

feedbackintheformofpricechangeiscontinuousandcontemporaneous,whichisconsistentwith

thefactthatpricescanbeobserved–directlyasdata–atanytimeandinreal‐time.

                                                            

23Feedbackinconnectionwithpriceobservationshasbeenimplementedinseveralagent‐basedmodels.

Forexample,Caldarelli,MarsiliandZhang(1997)usedtherateofchangeinpricealongwithitshigher‐order

derivatives, Bouchaud and Cont (1998) applied price deviations from fundamental values, and Lux and

Marchesi (1999, 2000) considered both the rate of change inprice andpricedeviations from fundamental

values. 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)).With

regard to early feedbackmodels,we takeparticularnoteof Shiller’s (1990)modelwith lagged, cumulative

feedbackoperatingoverlongtimeintervals,implyingthatinformationrelatedtopastpricechangeshaslong‐

lastingeffects.

24Interestingly, the cross‐correlation function is approximately zero at positive time lags. We could

plausiblyexplainthisbystatingthatonshorttimescales(e.g.intraday)thereisanefficientmarketresponseto

news, whereby new information is almost immediately reflected in prices as obvious price anomalies are

swiftly arbitraged away by investors (e.g. the high frequency traders). The response to news on longer

timescaleshasadifferentnature,anditsmechanicsarethesubjectofthepresentwork.

Page45of97

 

Wecannowsubstitutetheexternalforces 0and ρ ρ intoequations(11)and

writethemodelintermsof , and asfollows:

∗ , 12

tanh , 12

tanh , 12

whereforconveniencewehaveregroupedtheconstantstohave , , , ,

and .

Wewishtomaketheinitialinvestigationeasierbyfurthersimplifyingequation(12c).First,let

usassumethattheinfluenceofinvestorsonanalysts(~ )andtheinteractionsamongtheanalysts

(~ )arelessimportantinrespectofopinionmakingthantheimpactduetomarketperformance

(~ )andexogenousnewsflow(~ ),andthereforeweneglectthecorrespondingtermsin(12c).

Second,letusapproximate in(12c)as ,where isapositiveconstantthatcarriesthe

meaningofthelong‐termgrowthrateofthestockmarket.Thatis,wereplace ∗ by when

wesubstitute from(12a) into(12c).Aswehaveseen inSection1.3.2, ∗ changesslowly

with time as compared with , therefore the approximation is valid for sufficiently short time

intervals.Inotherwords,bymakingthisapproximationweneglectthelong‐termfeedbackeffects

on relativetotheanalogousshort‐termeffects.

Asaresult,system(12)canbesimplifiedas

∗ , 13

tanh , 13

tanh , 13

 

where

The

nonlinea

for a gi

sentimen

model(e

13),ass

simple,p

theemp

Figure1

accordin

Sentime

investor

newsflo

thehype

Directin

toobtain

(

stockmark

ardynamica

iven equ

ntis,asexpe

eq.4and10

shown inFi

possessesso

iricalstudy

12: (a)A sc

ng to(13a),

nt evolv

rsandanaly

ow and

erbolic tang

nformation

n ,which

0 ,

ketmodel is

alsystemdri

uation 13(b

ected,synon

0),studiedin

gure(12). I

omesurprisi

aswellasw

chematic vie

describing

ves accordi

sts.Directin

bymarketp

gent in(13c)

ismeas

hissubsequ

( 0 an

s now fully

ivenbyther

b) becomes

nymouswith

nSection1,

n thenexts

inglyinteres

withtheactu

ewof the (s

how investo

ng to (13b

nformation

performance

).(b)Asche

sureddailyi

uentlysubsti

nd issimpl

defined. Sy

randomflow

equation (

hdirectinfo

isaparticu

sectionwew

stingdynam

albehavior

simplified) t

orsmake in

b) through

evolves

eapproxima

ematicview

inthegener

itutedintoe

ly renam

ystem (13) r

wofexogeno

(4) where

rmationflow

ularcaseoft

will see that

micsconsiste

ofthestock

theoreticalm

nvestmentd

interactions

saccording

atedbythef

woftheemp

alnewsflow

q.(10)toyi

ed.

represents t

ousnews

is replaced

wandhence

thetheoreti

tmodel (13

entwiththe

market.

model (eq. 1

ecisionsbas

s among in

to(13c)dri

firsttwoter

piricalmode

wandthens

eld .

Pa

the stockm

(Fig.12a)

d by . Thu

etheempiri

calmarketm

3),although

behaviorso

13). Price

sedonsenti

vestors and

ivenbythe

msinthear

lstudied in

substitutedi

age46of97

market as a

).Notethat

us, analyst

calmarket

model(eq.

extremely

bservedin

evolves

iment .

d between

exogenous

rgumentof

Section1.

intoeq.(4)

Page47of97

 

2.2.Modelanalysisandresults

Inthissectionwereportanddiscusstheresultsofthestudyofsystem(13).Wenotethatequations

(13b,c)constituteaclosedsystemthatwecansolve for and andsubsequentlysubstitute

into equation (13a) to obtain .We first consider the autonomous case 0and then

studythetime‐dependentsituation 0.

2.2.1.Autonomouscase:

The two‐dimensionalnonlineardynamical system(13b,c) is studied indetail inAppendixC.Here

wereportonlythemainresultsfortherelevantrangeofparametervalues.

At 1system (13b,c) experiencesphase transition between the disordered statewith one

equilibriumpoint( 1)andtheorderedstatewiththreeequilibriumpoints( 1).Asbefore,

weareinterestedintheorderedstateinthevicinityofthephasetransition( ≳ 1).

If 0,thereisoneunstableequilibriumpointattheoriginandtwostableequilibriumpoints

at locatedsymmetricallywithrespecttotheoriginontheaxis 0.If 0,theequilibrium

points shift tonew, asymmetricpositionsalong theaxis for ≪ 1.Once exceedsa critical

valuethedistortionoftheinitiallysymmetricconfigurationbecomessostrongthatonlythepositive

equilibriumpointat survives.

Inthisrespectthebehaviorofsystem(13b,c)issimilartothatoftheone‐componentsentiment

model(4).Tounderstandwhereanewbehaviormayoriginate,letustakeacloserlookatequations

(13b,c).Directinformationflow,describedbyanalysts’sentiment ,causesinvestors’sentiment to

change(eq.13b).Inturn,anychangein alsoforcesachangein asduetothefeedbackterm

(eq.13c).Asaresult,investors’sentimentisnowcoupledtodirectinformationandviceversa,andit

isthiscouplingthatleadstotheemergenceofinertiaandnewbehaviorsrelatedtoit.

Indeed,asisshowninAppendixC,system(13b,c)canbeapproximatedby

 

,

where t

(C16), r

particle

which

acquired

compon

Figure1

unstable

( 0)a

(b)Thee

around t

the damping

respectively.

of unit mas

ispositivea

damass,an

ent,inertiale

13:(a)The

e limit cycle

andtheequ

evolutionof

the stable f

0,

g coefficien

Equation (

ss inside an

andashallow

ndtherefore

essmodel(4

phaseportr

e (dark red)

uilibriumpo

fsentiment

focus. (c) Th

t , and

(14) can be

n asymmetri

wwellinwh

inertia,resu

4).

rait( 67.

) around a

oint inthesh

inFig.1

he phase po

d the poten

e interprete

ically shape

hich isneg

ultsinaver

7)showing

stable focus

hallow,nega

13(a)alongt

ortrait of the

ntial are

d as the eq

ed potential

gative(Fig.1

rydifferent

alargestab

s (green ast

ativewell(

twotypicalt

emain case

e given by

quation of m

characteriz

13d).Thefa

dynamicas

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terisk) in th

0)which

trajectories,

e ( 56.0)

Pa

equations

motion for

zed by a de

actthatthes

comparedt

le(darkblu

he deep, po

h isanunst

,onthelimit

showing an

age48of97

14

(C15) and

a damped

eep well in

systemhas

totheone‐

ue),asmall

sitive well

tablenode.

tcycleand

n unstable

Page49of97

 

focusinthenegativewell(redasterisk),astablefocusinthepositivewell(greenasterisk)andthe

trajectories 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 feedback

coefficient isnottoosmall.25Figures13a,bshowtwotypesoffreeoscillationsthatemergeinthe

system.Thefirsttypereflectsdecayingoscillationswithacharacteristicperiodofroughly6months

aroundtheequilibriumpointinsidethedeepwell.Thesecondtypereflectsalarge‐scalestablelimit

cyclethatcarriessentimentfromthedeepwellintotheshallowwellandbackinapproximately14

months. Such large‐scale motion is self‐sustaining, as it is fuelled by the coupling of direct

information and sentiment, such that large swings in cause large swings in that cause large

swingsin thatcauselargeswingsin andsoforth.

Imaginethattheflowofexogenousnewsweretobeinterrupted.Then,accordingtothemodel,

two scenarios are possible. First, themarketmay converge to a steady state inwhich sentiment

likely settles at thepositive equilibriumvalue inside thedeepwell26,while simultaneously the

level 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 the

bottomofeachpotentialwell (s s ): stablenode ‐>stable focus ‐>unstable focus ‐>unstablenode.Free

oscillations become possible starting from the first bifurcation in the above sequence. Note that the

equilibriumpointatthecuspofthepotentialalwaysremainsanunstablesaddle.

26Itwouldberatherunusualifsweretocometorestats because,first,sentimentdoesnotoftencross

into the shallowwell and, second, suchequilibriumstatemaybeunstableornot exist at all, as the critical

valuesofδatwhichs vanishesarewithintheadmissiblerangeofparametervalues.

Page50of97

 

which sentiment anddirect information are attracted to the limit cyclewhere they exhibit large‐

scaleself‐perpetuatingoscillationsbetweenextremenegativeandextremepositivevalues.

Figure13cshowsaregimeimmediatelyprecedingtheformationofthelimitcycle.Thisregime

isaprimecandidateforgeneratingarealisticsentimentbehavior.Itincludesmid‐termoscillations

aroundthefocusinthedeepwell,consistentwiththeboundedmotionthatwehaveobservedinthe

empiricalmodelandconnectedwithmid‐termmarket trends inSection1.2.3. Italsocontainsthe

large‐scale trajectories, located in the vicinity of the (about‐to‐be‐formed) limit cycle, that lead

sentiment via quasi‐self‐sustaining motion into and out of the negative sentiment territory. The

evolutionofsentimentalongthesetrajectoriesisconsistentwithitsobservedbehavioratthetime

ofmarketcrashesandrallies(Section1.2.3).

Lastly,wenotethatthisregime,whichweexpecttoberelevanttothereal‐worldstockmarkets,

existsundertherealisticchoiceofparametervaluesconsistentwiththevaluesobtainedusingthe

empiricaldata.27

                                                            

27Theparameters in the theoreticalmodel (eq.13andFig.13c)havebeenchosen to coincidewith the

parametersoftheempiricalmodel(TableI),exceptthat 0.55(theoretical)whereas 1.0(empirical).

Additionally: 1. 56therefore 2.2, so that the feedback term participates in the leading order

dynamics in equation (13c) while not dominating other terms. 2. 0.03so that the constant

0.017,whichdeterminesthedistortionofthepotential,isclosetothevalueintheempiricalmodel(TableI).3.

It follows from thedefinitionsof and inequation (13) that / .Using and fromTable I and

the 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 theoretical

model.

Page51of97

 

2.2.2.Non‐autonomouscase:

The news flow acts as a random force in system (13). To study its influence, we make the

assumptionthat isnormally‐distributedwhitenoisewithzeromeanandunitvariance.28

Thesystem’sbehaviorwilldependsignificantlyontherelativemagnitudeof theterms and

in equation (13c). Indeed, if | | ≪ | |, the dominant noise will drive and randomly

around the potential well, leading to nearly stochastic behavior in price . Conversely, if

| | ≫ | |, the system’s dynamics will primarily consist of a piecewise motion along the

segmentsofphasetrajectoriesintheautonomouscase,asonaverage and willtravelfaralonga

trajectoryby the time thenoisewill havebeenable todisplace them.29In this case, the resulting

behaviorofmarketpricewillhaveadiscernibledeterministiccomponenttoit.

Weset 1asthispermitsbothtermstoparticipateintheleadingorderdynamics.30However,

as | |isafunctionof and (eq.13b),therelativeimportanceofthesetermsactuallydependson

thelocationofanygiventrajectory,whichmeansthattheabove‐discussedcharacteristicdynamics

                                                            

28Notethatinsimulationsthenoise is,strictlyspeaking,whiteonlyonadaily(weekly,monthly,etc.)

basis,i.e.when canbeexpressedinmultiplesofbusinessday.Thisisbecause canhavenonzerointraday

autocorrelationsduetointradayinterpolationinthenumericalscheme.

29This statement is only valid if the system is structurally stable (i.e. non‐chaotic)when perturbed by

noise,which is likely the case aswehavenot been able to find the tracesof chaos (e.g. positiveLyapunov

exponents)initfornonzero intherelevantrangeofparametervalues.(Asasidecomment,weobserved

positive Lyapunov exponents in situations where system (13) was forced by a periodic function of time,

insteadofnoise).

30 1,so | |~ 1,where istheunitstandarddeviationof ,therefore | |hasthesame

orderofmagnitudeas | |~ ≲ 2.2for| | ≲ 1.

 

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portrait

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age52of97

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age53of97

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arketprice.

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age54of97

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age55of97

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Page57of97

 

Figure 18: Distributions of 21‐day log returns. (a,b) Theoretical model (60 years, i.e. 720 data

points)andtheS&P500Index(1950‐2012). (c,d)Empiricalmodel(1996‐2012)andtheS&P500

Index(1996‐2012).Tomakecomparisoneasier,thedistributionsarenormalizedtoyieldzeromean

andunitvariance.Thecorrespondingnormaldistributionsareplottedaswell.Thehorizontalaxis,

denominated in the units of standard deviation, indicates the normal distribution 1st and 5th

percentiles.

The results presented in this section enable us to conclude that, first, there exist substantial

similaritiesbetweenthebehaviorofinvestorsentimentproducedbythetheoreticalmodelandthe

empiricalmodel;second,thereexistsubstantialsimilaritiesinthepricebehaviorgeneratedbythe

theoreticalmodelandthatobservedinthestockmarket;and,third,thetheoreticalmodelreplicates

characteristicfeaturesofthestockmarket,suchasnon‐normallydistributedreturns.

2.2.3.Discussion

Resultsproducedbymodel(13)areinagreementwiththeempiricalstudy.Inparticular,themodel

exhibitstwodistinctbehaviorsrepresentedbydifferentcharacteristicfrequenciesatwhichinvestor

sentiment evolves: mid‐term oscillations in the positive well consistent with mid‐term market

trendsandlarge‐scale,quasi‐self‐sustainingmotionconsistentwithmarketralliesandcrashes.

Exogenousnewsflowisarandomforcethatthrustssentimentinsidethepotentialwell.Itcan

occasionallyforcesentimentacrosstheregionsofdifferentbehavior,resultinginaswitchofmarket

regimes.Wehaveseenthattheinfluenceofnewsisnotuniform.Thereisawiderangeofregimes

betweentwoextremes:thefirstwherethenews(noise)dominateleadingtoanearlyrandomprice

walkandthesecondwherethenews(noise)havelowimpactresultinginanalmostdeterministic

pricedevelopment.Thisimpliesthat,accordingtothemodel,themarket’sevolutionpathcantakeit

throughavarietyofregimes,someofwhichmayhaveelementsofdeterministicdynamics.

Page58of97

 

Wehavedemonstratedthatastochasticallyforced,deterministicnonlinearmodeldescribedby

equations(13)iscapableofgeneratingcomplexbehaviorsrepresentativeofactualmarketregimes.

For example,we have been able to approximately reproduce the behavior of the S&P 500 Index

from2004to2010bysubstitutingtheempiricaltemperatureprofileforthatsameperiodintothe

model (which implies that temperature plays a significant role in the development of market

regimes). Themodel has been shown to replicate essential features of the actual distributions of

stockmarketreturns,suchasnonzeroskewness,positivekurtosisandfat(albeittruncated)tails.In

addition, the model has provided an explanation of market trends and crashes in terms of

characteristicfrequenciesofsentimentdevelopment.

Theappearanceofcharacteristicfrequenciesinthemodelisintriguinganddeservesconjecture

on its causesand implications.Model (13) isaheavilyapproximatedversionof themoregeneral

Isingmodeldefinedbyequation(A1)inAppendixA,thedynamicsofwhicharelargelydetermined

bythetopologyofinteractionamongtheagentsthatconstituteit.Inparticular,model(13)hasbeen

derivedundertheassumptionthatthepatternofinteractioninequation(A1)isalltoall.Itisknown

that the all‐to‐all pattern prevents Ising spins (agents) from assembling into clusters, whereas

interactionsthatarenotasradicallysimplifiedcanproduceheterogeneousstructures(e.g.Contand

Bouchaud (2000)) – which, in our case, represent the clusters of investors and analysts

characterizedbycoalignedsentiments.33

                                                            

33Asmentionedearlier,theall‐to‐alltopologyistheleading‐orderapproximationforgeneralinteraction

topologyintheIsingmodelandso isaconvenientstartingpoint forstudyingitskeyfeatures.Wenotethat

pricesgeneratedbytheempiricalmodel(eq.4and10),whichalsoemploystheall‐to‐allinteractionpattern,

cancontainindirectevidenceofclusterdynamicsbecausetheinputvariable isbasedonempiricaldata.

Page59of97

 

Thepresenceofclusterscanleadtodiversedynamicsofinteractionsonvarioustimescales.This

is because a cluster’s size determines its reaction time to various disturbances, such as adjacent

clusters,randominfluencesorexternalforces(thenews),suchthatthelargerthesize,theslower

the reaction.As different reaction times of differently‐sized clusters to incoming information can

representinvestmenthorizonsorallocationprocessesofvariousinvestorgroups,thegeneralmodel

(eq.A1)cansimulatethedynamicsofmarketparticipantsrangingfromprivateinvestorstopension

plans.

Consequently, the generalmodel (eq. A1) can exhibitmultiple frequencies of interaction.We

mayexpect that theactualstockmarketalsocontainsdifferentcharacteristic frequencies,eachof

whichsignifiesadistinctbehaviorthattogether,throughinteractions,produceaconstantlyshifting

patternofregimesdrivingthedevelopmentofmarketprice.Andindeed,thefactthatadrastically

simplifiedmarketmodelcapturedtwocharacteristicfrequenciesthatcorrespondtoactualmarket

regimessupportsthisexpectation.34

Wewouldliketoconcludebyreturningtothediscussionofthepossiblepredictabilityofmarket

returns thatwehave touchedupon in thebeginningof this section.Recall that the characteristic

frequenciesemergeinthemodelasaninertialeffectcausedbythefeedbackrelationbetweendirect

information and price. The existence of inertia has another consequence: the model acquires a

resistance to change in direction and, as a result, its behavior becomes predictable in situations

where inertial effects outweigh noise. In other words, based on the theoretical results and the

support of theempirical evidence,we can expect the stockmarket to include regimeswith some

elementsofdeterministicdynamics,whichcaninprinciplebepredicted.

                                                            

34Thearguablyquasiperiodicpatternoftheautocorrelationsof (Fig.11a)canbeafurthersignofthe

presenceofcharacteristicfrequenciesinthemarketpriceseries.

Page60of97

 

Asanillustration,letusconsiderhowthemodelsdevelopedherecouldbeappliedformakinga

marketforecast.Apossibleprocedureisasfollows:measuredaily ,obtain fromequation

(4), findthecorresponding location , onthephaseportrait (e.g.Fig.13c)andforecastthe

followingday’sreturnexpectationwhich,beingafunctionofthecurrentday’slocationonthephase

portrait,maybenonzero.Ofcourse,model(13)islikelytoocrudetoproduceameaningfulforecast.

Moresophisticatedpatternsofinteractionmaybeneededtoconstructmorerealisticmodels.Inthe

light of the above discussion, such randomly‐driven deterministic models are likely to generate

complex dynamics, resulting in the presence of random, deterministic chaotic and deterministic

non‐chaotic behaviors.However, unlike the case of the randomwalk, forecasting the behavior of

complexdeterministicsystemscanbepossiblebyblendingmodelswithobservations,asisdone,for

example, inmeteorology and oceanography and can probably be done aswell in the case of the

financialmarkets.35

3.Conclusion

This paper introduced a framework for understanding stock market behavior, upon which the

model of stockmarketdynamicswasdevelopedand studied.Wehavedemonstrated, empirically

and/ortheoretically,thataccordingtothismodel:

1. Thestockmarketdynamicscanbeexplainedintermsoftheinteractionamongthreevariables:

direct information (defined in Section 1.1), investor sentiment (defined in Section 1.2) and

marketprice–asthetheoreticalandempiricalresultsareinagreementandshowareasonably

goodfitwiththeobservations.

                                                            

35Thisisthedirectionofourfutureresearch.Asapreliminaryfinding,wewouldliketoreportthatsimple

prototypes of algorithmic trading strategies built upon the framework developed here have produced

promisingbacktestedresults,supportiveoftheabove‐describedapproachtomarketforecasting.

Page61of97

 

2. Theeffectiveness of the influence of informationon investors is determinedby the degreeof

directness of its interpretation in relation to the expectations of futuremarket performance.

Direct information – information that explicitlymentions the direction of anticipatedmarket

movement–impactsinvestorsmost.

3. In addition to being forced by direct information, the evolution of investor sentiment is also

significantly influenced, first, by the interaction among investors and, second, by the level of

disorder in the market given by the vacillating balance between the herding and random

behavior of investors. The influence due to the second factor is determined by the economic

analogof temperature(introduced inSection1.2.1), theprofileofwhichcanbededucedfrom

theempiricalmodel.

4. Marketpricedevelopsdifferentlyondifferent timescales:onshorter timescales(e.g.daysand

weeks),pricechangesproportionallytothechangeinsentiment,whileonlongertimescales(e.g.

monthsandyears),pricechangesproportionallytothedeviationofsentimentfromareference

levelthatisgenerallynonzero.Asaresult,priceevolutionisnaturallydecomposedintoalong‐

term,large‐scalevariationandshort‐term,small‐to‐mid‐scalefluctuations.

5. Inthemarket,herdingbehaviorprevailed(toasmalldegree)overrandombehaviorduringthe

studiedperiod(1996‐2012).However,thelevelatwhichthesebehaviorswerebalancedvaried

graduallyovertimesuchthat,generally,herdingincreasedduringbullmarketsanddecreased

duringbearmarkets.Thelevelofbalanceaffectsmarketvolatilityandtheprobabilityofmarket

crashes. There is a connection between changes in this balance and observed economic

fluctuations(thebusinesscycle).

6. Long‐term sentiment trends show a substantial temporal lead over long‐term market price

trends. This result may be of practical importance for the development of trend‐following

strategies,asachangeinsentimenttrendcouldbeaprecursortoachangeinpricetrend.

7. Informationrelatedtopricechangesplaysanimportantroleinmarketdynamicsbyinducinga

feedback loop: information ‐> sentiment ‐> price ‐> information. This leads to a nonlinear

Page62of97

 

dynamicofinteraction,whichexplainstheexistenceofcertainmarketbehaviors,suchastrends,

rallies and crashes. It is responsible for the familiar non‐normal shape of the stock market

returndistribution.

8. Exogenousnewsactasarandomforcethatdisplaces investorsentimentfromtheequilibrium

and occasionally causes market dynamics to switch from one regime to another. Two

equilibriumstatesarepossible.Inthefirstregime,sentimentandpriceremainconstant.Inthe

second regime, sentiment oscillates between extreme negative and positive values, driving

alternatingbearandbullmarkets.

9. The coupling of price and information creates feedback, whereby information causes price

changes and price changes generate information (paragraph 7). Additionally, the resulting

dynamicfurthercomplicatesthepicture,leadingtoadelayedreactionbetweenthesevariables,

such that, forexample, today’sprice changemaybe influencedby informationrelated topast

pricechanges,alongwithothernews,overpreviousdays,weeksandmonths.Thisimpliesthat

thenotionof causeandeffectdoesnot simplistically apply tomarketdynamics, as causeand

effectbecome,inasense,intertwined.

10. Thenonlineardynamicunderlyingsentimentevolutioncontainsbothdeterministicandrandom

components. The deterministic component in some market regimes is stronger, while other

regimesmaybedominatedbytherandomcomponent.Developinganunderstandingofwhich

marketregimetakesplaceatwhichtimecanbepossiblebycombiningmodelsandobservations,

whichcouldpotentiallyfacilitatemarketreturnforecasts.

Acknowledgements

We are grateful to Dow Jones & Company for providing access to Factiva.com news archive.We

would also like to thank John Orthwein for editing this paper and contributing ideas on its

readability.

Page63of97

 

AppendixA:Equationforsentimentdynamics

We base the derivation of a dynamic equation for sentiment evolution in the model studied in

Section2onaphysicalanalogyinwhichtwosetsofinteractingIsingspins 1, 1… and

1, 1… are acted upon by external magnetic fields and , respectively.

Additionally, this analysisyields, as aparticular case, adynamicequation for a single setof Ising

spins 1, 1… in the presence of an external magnetic field , which is a physical

analogyofthemodelstudiedinSection1.Thisparticularcaseissimilartothestatisticalmechanics

problem treated by Glauber (1963), Suzuki and Kubo (1968), and Ovchinnikov and Onishchuk

(1988).

A1.Propertiesofthetwo‐componentIsingsystem

TheHamiltonian36oftheIsingsystemhastheform:

12

12

,

,

where , and arecoefficientsthatdeterminethestrengthofinteractionamongspins;

and aremagneticmomentsperspin;and and areexternalmagneticfields.

Ifweassumethat

,

,

, ,

                                                            

36Inthisappendixwereverttothenotationandterminologycommonlyusedinphysics.

Page64of97

 

,

,

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 Ising

systemcanbeinthestateofthermodynamicequilibrium.Inthiscase,theprobabilityoffindingitin

thestatewiththevaluesoftotalspinsequalto and ,respectively,isgivenby

,e

,

, 3

where , , istheenergyofthesystemand istemperaturemeasuredintheunitsof

energy.Thefunctions and ,givenby

!

2 ! 2 !and

!

2 ! 2 !,

Page65of97

 

arethedegreesofdegeneracyofthestatesforwhichtotalspinsareequalto and ,respectively.

isanormalizationfactordeterminedfromthecondition

, 1,

whichleadsto

, , e ,,

calledthepartitionfunction.

In the limit of large and ≫ 1, ≫ 1 , it is convenient to change from discrete

variables , 2, … 2, and , 2, … 2, to quasi‐continuous

variables

, 1 1 ,

, 1 1 .

Wecanrewritetheaboveequationsinnewvariables and toobtain

,2

22

2 ,

, ln

2ln

11

ln14

1ln

21

2ln

11

ln14

1ln

21

.

Page66of97

 

Thentheequilibriumdistributionfunction , takestheform:

,exp ,

,exp

,

,

where

, , ,

22 ln

11

ln14

1

22 ln

11

ln14

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 ln11

,

2 2 ln11

.

Accountingfor ,wecanrewritethissystemofequationsas

2ln

11

,

2ln

11

,

orequivalently:

tanh ,

tanh , 4

Page67of97

 

where 1/ .

Asasimpleexample,letusconsideracaseinwhich 0, .Then

4 isreducedto

tanh ,

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 .Wenotethata

moregeneralcaseistreatedinAppendixC.

Itisworthwhilenotingthatphasetransitionispossibleeveninthecaseof 0.Then:

tanh ,tanh ,

where , ,whichyieldsnonzerosolutionsfor 1.

A2.Master(kinetic)equation

Todescribenon‐equilibriumdynamics,weintroducedistributionfunction , , thatdefinesthe

probability that the values of the total spins of configurations and at time t are equal to

and ,respectively.If , , isknown,wecancalculateanystatisticalquantityatanymoment

.Forinstance,theexpectationvaluesoftotalspins and aregivenby

⟨ ⟩ , , , 5

Page68of97

 

⟨ ⟩ , , . 5

Notethatsummationin(A5)iscarriedoutacrosstheintegervalues: , 2, …

2, and , 2, … 2, . We can obtain , , by deriving and solving the

correspondingmasterequationfortheevolutionof , , .Toformulatethemasterequation,we

definetheprobabilityofthesystemtransitingfromoneconfigurationofspinsintoanotherperunit

timeas , ; , ,whereit isassumedthattransitionsoccurwhenspinsspontaneouslyflip

as a result of uncontrolled energy exchangewith a heat bath. Considering that , , evolves

onlyduetothesetransitions,weconcludethatitmustsatisfythefollowingmasterequation:

, ,, ; , /

//

, ,

, /, , /; ,//

. 6

Notethatthe firstsumin 6 representsthetotalprobabilityperunit timethat thesystem‘flips

out’of thestate , ,whereasthesecondsumcorrespondstothetotalprobabilityperunit time

thatthesystem‘flipsin’tothestate , .

Let us assume that transitions occur only due to single spin flips, i.e. and change at each

transitionby 2.Thenequation 6 becomes

, ,, ; 2, , ; 2, , ; , 2

, ; , 2 , , 2, ; , 2, ,

2, ; , 2, , , 2; , , 2,

, 2; , , 2, . 7

Page69of97

 

Therefore,theproblemisnowreducedto, first,deducingtheformof andthensolving 7 .To

determine wecanusethedetailedbalancecondition:

, ; , , , ; , , .

Here , istheequilibriumdistributionfunctiongivenby 3 .Whenthedetailedbalance

condition is satisfied, the number of transitions per unit time in the thermodynamic equilibrium

statefromanyconfiguration , intoanyotherconfiguration , exactlyequalsthenumber

of transitions in the opposite direction. This condition imposes certain constraints on the

probability ,asitmustsatisfy

, ; ,, ; ,

,,

/ /

exp, ,

. 8

Accordingto 8 ,theprobabilities in 7 satisfythefollowingequations:

, ; 2,2, ; ,

2,,

2exp

2, ,,

, ; , 2, 2; ,

, 2,

2exp

, 2 ,.

Usingexpressionsfor , and , ,weobtain

, ; 2,2, ; ,

∓2

2 1exp 2 1 ,

, ; , 2, 2; ,

∓2

2 1exp 2 1 .

9

Weseek , ; , thatsatisfy 9 intheform:

, ; 2,∓2 1 exp ∓2 1

,

Page70of97

 

2, ; ,2

11 exp 2 1

,

, ; , 2∓2 1 exp ∓2 1

,

, 2; ,2

11 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

,

, ; , 21 exp ∓2 1

,

, 2; ,1 exp 2 1

.

Page71of97

 

Thensystem(A8)takesthefinalform:

, ,2

, ; 2,2

, ; 2,

2, ; , 2

2, ; , 2 , ,

21 2, ; , 2, ,

21 2, ; , 2, ,

21 , 2; , , 2,

21 , 2; , , 2, . 10

Forsolving 10 itisnecessarytodefinetheinitialdistribution , , 0 , ,aswellasthe

boundary conditions,which in this case are , , ≡ 0for| | and| | .We also note

thatequations 10 areapplicableinthecaseofnonstationaryexternalfields and .

A3.Dynamicequationsforaveragespins

We can derive the equations of motion for average spins ⟨ ⟩ and ⟨ ⟩ by

differentiatingequation 5 withrespecttotimeandusingequations 10 :

, , ,

, , ,

Page72of97

 

wherefunctions and representtheresultsofsummationofthe ‐thterminther.h.s.of

10 multipliedby and ,respectively.

Weobtainthefollowingexpressionsfor and :

2, ; 2, , ,

12⟨ , ; 2, ⟩ ,

2, ; 2, , ,

12⟨ , ; 2, ⟩ ,

2, ; , 2 , ,

12⟨ , ; , 2 ⟩ ,

2, ; , 2 , ,

12⟨ , ; , 2 ⟩ ,

Page73of97

 

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 function

contains the multiplier / that vanishes at such value of /, which has enabled us to

representthisexpressionincanonicalformabove.Similarly,

22

2, ; , 2, ,

12

/ 2 / /, ; / 2, /, ,/

12

/ 2 / /, ; / 2, /, ,/

12⟨

2, ; 2, ⟩ ,

Page74of97

 

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 .

Page75of97

 

Wenotethatsystem 11 isnotclosedbecause⟨ , ⟩ and⟨ , ⟩ cannotingeneralbe

reduced tobe functionsof and .However, itwouldbe reasonable toassume that in the

thermodynamic limit ( ≫ 1, ≫ 1 ) fluctuations of total spins and around their

instantaneous average values and are small in comparison with these same average

values.Thissituationisanalogoustothatwhereasystemisinthermodynamicequilibrium.Wecan

thenwrite

⟨ , ⟩ ⟨ ⟩ , ⟨ ⟩ ,

and

⟨ , ⟩ ⟨ ⟩ , ⟨ ⟩ , .

Weintroducenewvariables,

⟨ ⟩,

⟨ ⟩,

andexpress , and , viathesevariablestoobtaininthelimit → ∞, →

∞thefollowingequations:

, 11 exp 2

11 exp 2

tanh

and

Page76of97

 

, 11 exp 2

11 exp 2

tanh .

Substituting theseexpressions into 11 ,weobtain the following closedsystemofequations for

and :

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 distribution

function , .

Revertingtonotation 1/ ,weexpress 12 as

tanh , 14

tanh . 14

Page77of97

 

A4.Dynamicequationintheone‐componentIsingsystem

Finally, we consider a simpler Ising system that consists of spins 1, 1… and the

externalmagneticfield .TheHamiltonianofthissystemisequalto

2. 15

AstheHamiltonian 15 isaparticularcaseoftheHamiltonian 1 ,equations 14 contain

the dynamic equation for average spin in the one‐component system:⟨ ⟩

. This

equationcanbederivedfrom 14 bysetting 0andrenaming and :

tanh , 16

wherelim → .

AppendixB:Thereferencesentimentlevel

Herewe derive an approximate expression of the reference sentiment level ∗, relative towhich

deviations in sentiment result in a change of market price, as described by the second term in

equation(10).

B1.Equationfor ∗

Asa firststep,wetakea long‐termaverageofbothsidesofequation(10)withrespecttotimeto

obtainitsasymptoticforminwhichtheaveragedvariablesaretime‐independent:

∗ , 1

where denotestheconstantrateofchangeinmarketpriceand 0because isbounded.Thus,

wecanrewriteequation(B1)as

Page78of97

 

∗ , 2

where satisfiesthesimilarlyaveragedequation(4)(with 0):

tanh . 3

Now,letusconsiderasituationwheretheflowofdirectinformation isonaverageneutral,

i.e. 0. Sinceaccording toourmarketmodel, it is the flowofdirect information thatdrives

marketpricedevelopment,thecaseof 0impliesthattherateofchangeinpriceisonaverage

zero,i.e. 0,andthusequation(B2)becomes

∗ . 4

Therefore ∗equals intheregimewherethedirectinformationflowisonaveragezero,which

is the regime described by the potential for which 0(eq. 6).37 There is but one

caveattothisderivation.Namelythatbecausewehaveusedequation(10)which–beinga linear

combinationoftwoasymptoticregimesoftheactualequationgoverningpriceevolution(thetrue

formofwhichwedonotknow)–isanapproximation,theaboverelationbetween ∗and maynot

bethetruerelation.However,weexpectthatthisrelationholdsapproximately,asitleadstoresults

thatareconsistentwithobservations,asisshowninSection1.3.2.

Wecanobtain anapproximate solution for fromequation (B3)byexpanding thehyperbolic

tangentinpowersof intheneighborhoodof andaveraging

theresultingseriestruncatedabovethequadraticterms.Equation(B3)thentakestheapproximate

form:

                                                            

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

( 0 0),finiteaveragingperiodandsufficientlysmall .

Page79of97

 

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 reasonably

closetothecenterofexpansion ,whichjustifiestheaboveapproximationfortherelevantrange

ofparametervalues.

Equation(B5)isappliedtodetermine from ∗intheiterativeleastsquaresfittingprocessfor

thetimeseries inSection1.3.2.Forthispurpose,weapply 0.3insteadof0.5.Thischoiceis

dictated by the following considerations. Unlike the empiricalmodel (Section 1), the behavior of

whichisdrivenprimarilybythemeasured ,thetheoreticalmodel(Section2)issensitiveto .

Itwouldthereforebesensibletoselect basedonthetheoreticalmodelbehavior.Theestimateof

0.3results in a more realistic behavior of the theoretical model than0.5which leads to higher

valuesof thanthosefollowingfromthetheoreticalmodel.Thechoiceof0.3insteadof0.5seems

acceptable, given the approximate nature of equation (B5) and the inevitable imprecision in

measuringandcalibrating .

B2.Perturbativesolutionfor ∗

Treating asasmallparameter(asfollowsfromtheestimates intheprecedingsection, ~0.25

or0.09),wecanconstructaperturbativesolutiontoequation(B5).Wefindthatintheleadingorder

thesolutionis

tanh , 6

i.e.itcoincideswiththestableequilibriumpointsofsystem(4).Consequently, ∗becomes

∗ for 1, 7

∗ 0for 1. 7

 

Solu

particle

coincide

that for

solution

origin.

Acco

∗

∗

The

Appendi

andnega

originth

(Fig.B1)

totheeq

tion(B7a)c

cannot leav

es inthelead

very small

(B7b)mean

ountingfort

1

0for

second ter

ixC, 1

ativefor .

hanthatof

).For 1

quilibriump

canbeunde

ve the well

dingorderw

amplitudes

nsthatthep

thenextorde

1

1

1.

m in (B8a)

1isn

Therefore,

,whichisa

1,theterm(

ointattheo

rstoodasfo

in which it

withthesta

ofmotion t

particle’save

ercorrection

1for

corrects fo

negativefor

owingtoth

aresultofth

(B8b)iszero

origin.

ollows.Ifthe

t is initially

ableequilibr

the shapeo

eragepositio

n, ∗canbes

1,

or the asym

any and

heasymmetr

heouterwal

obecausein

eamplitude

y found, the

riumpoint,

f thewell is

oncoincides

showntoequ

mmetry of t

,thusthe

ryofthewe

llofthewell

nthiscaseth

ofvariation

n the partic

or , int

s symmetric

swiththeeq

ual

the well aro

correction

ll,thelocati

lbeingsteep

hewellissy

Pa

nin issosm

cle’s averag

thatwell.Th

caround .

quilibriump

ound . As

termisposi

onof isclo

perthanthe

ymmetricwi

age80of97

mallthata

ge position

his implies

. Similarly,

pointatthe

8

8

shown in

itivefor

osertothe

innerwall

ithrespect

Page81of97

 

FigureB1:Graphicrepresentationofthereferencelevel ∗for 1.1and 0.3: 0.503(eq.

B6)and ∗ 0.313(eq.B8a).

AppendixC:Analysisofdynamicalsystem

Here we investigate certain properties of the system of equations (13). As equation (13a) is

decoupledfrom(13b,c),thestudyofsystem(13)isreducedtotheanalysisofthetwo‐dimensional

nonlineardynamicalsystem(13b,c),thesolutionstowhichmustbesubstitutedinto(13a)tosolve

system(13)completely.

C1.Equilibriumpoints

Ifwe considera situationwhere the time‐dependent force is absent, equations (13b,c) take

theform:

tanh , 1

tanh , 1

where (as a reminder)| | 1, | | 1and the coefficients , , , , and are non‐negative

constants. Solutions to equations (C1) , for the initial conditions 0 , 0 determine

trajectoriesofmotioncorrespondingtothevelocityfield( , )givenbyther.h.s.of(C1).Themotion

isboundedbecausethevelocityattheboundaries 1and 1isdirectedintotheregionof

motion.

Theequilibriumpointsofsystem(C1)(where 0)aregivenby

∗ tanh ∗ ∗ , 2

∗ tanh . 2

 

As

convenie

tanh

andsolv

of ,

, ∗

∗

provides

one solu

function

terms,a

FigureC

case(

If

(Fig.C1b

an

isgiven, ∗

enttowrite

, ∗

veitgraphic

∗ isnon‐n

∗

∗

1

sfortheexis

ution ∗

nof ,sucht

paramagne

C1:Graphic

1 .

1,thene

b).Whenth

nd , w

isknowna

thisequatio

12ln

11

cally(Fig.C1

egativeat ∗

0,

stenceofas

. The e

that → 0a

ticphase–o

solutions to

quation(C3

hevalueof

where 0

andcanbe s

onas

∗

∗∗

1).Asfollow

∗ 0,i.e.

singlesolutio

equilibrium

as → 0.Th

ofthesystem

oequation (

)mayhavee

isbelowcr

0, 0and

substituted

∗

wsfromFigu

ontoequati

solution

hus, corre

m.

(C3). (a)Pa

eitheroneo

ritical(

d

into (C2a) t

ureC1a,the

ion(C3).The

is a sin

spondstoa

ramagnetic

orthreesolu

),equatio

0in the

toobtain the

conditionth

ereforeif

ngle‐valued,

disordered

case (

utions,depen

n(C3)hast

limit → 0

Pa

eequation f

hatthefirst

1,there

continuous

state–or,i

1). (b)Ferr

ndingonthe

threesolutio

0, whereas

age82of97

for ∗. It is

3

derivative

existsonly

s and odd

inphysical

romagnetic

evalueof

ons: ,

the third

Page83of97

 

solution is negative for nonzero and approaches zero as → 0. The existence of nonzero

solutions in the limit → 0meansthat thesystemmayhaveordered(ferromagnetic)equilibrium

states.

When ,equation(C3)hasasinglesolution ,astwoothersolutionsmergeandvanishat

.Wenotethat isnotsensitivetothebifurcationat ,thatis, anditsderivatives

donothaveasingularityatthispoint.Itisnotdifficulttofind tanh .AsfollowsfromFigure

C1b, isdeterminedbythevalueof , ∗ (eq.C3)atitsmaximumwithrespectto ∗.Fromthe

extremumcondition,, ∗

∗ 0,itfollowsthat

∗ 1, 4

where the solution corresponding to the maximum of , ∗ has been selected. Substituting

∗ ∗ into(C3),weobtain

tanh1

,1 1 1

2ln

11

11

1 . 5

When the system is in thevicinityof thephase transitionat 1from ferromagnetic state into

paramagneticstate,then0 1 ≪ 1,sotheaboveexpressionfor becomes

~2

31 .

To sum up, system (C1) admits two types of bifurcations with respect to the number of

equilibriumpoints.First,thereisaphasetransitionat 1betweenthedisordered,paramagnetic

state with one equilibrium point ( 1) and the ordered, ferromagnetic state with three

Page84of97

 

equilibriumpoints( 1).Second,whenthesystemisintheferromagneticstate,twoequilibrium

pointsmergeandvanishat ,sothatonlyoneequilibriumpointremainsfor .

C2.Stabilityanalysis

Weproceed to study the stability of system (C1) near the equilibriumpoints. It is convenient to

rescaletimeas andrewriteequations(C1a)and(C1b)as

≡ , tanh , 6

≡ , tanh , 6

where and .

Linearization of system (C6) in the neighborhood of the equilibrium points ( ∗, ∗) leads to

solutions and in the form of linear combinations of exp and exp with

eigenvalues givenby

12tr tr 4det ,

whereJacobian ,tracetr anddeterminantdet aredefinedas

∗, ∗

∗∗

,tr ,det .

Substituting , and , from(C6)intotheexpressionsfor ,tr anddet ,weobtain

1,tr ,det ,

where

Page85of97

 

1 1 ∗ ,

sech sech ,

1 ∗ sech 1 ∗ sech .

Asaresult,thecharacteristicequationforeigenvalues becomes

12

4 . 7

If thediscriminant 4 isnon‐negative, then arereal.Theeigenvaluesmay

have the sameoropposite signs. In the formercase theequilibriumpoint is calledanode, in the

lattercasetheequilibriumpointiscalledasaddle.Ifatleastoneoftheeigenvaluesispositive,the

equilibrium is unstable because there is an exponentially growing solution to (C6) near the

equilibrium point. If is negative, then are complex conjugates, so that there is an oscillatory

componenttothemotioninthevicinityoftheequilibriumpoint,whichiscalledafocus,provided

theeigenvalueshavenonzerorealparts.Inthissituation,thesignoftherealpartoftheeigenvalues

determinesstabilityattheequilibrium.Notethattheeigenvaluesaredependentonthepositionof

the equilibrium point ∗. Therefore, we can expect that each of the three equilibrium points in

ferromagneticstatehasuniquestabilityproperties.

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

Page86of97

 

≶1 1 1 ∗

1 ∗ sech. 8

These inequalitiesdeterminethevaluesofparametersforwhichtheequilibriumisstable(the

sign above)orunstable(thesign above).

Ifsimultaneouslywith(C8) 4 ,then ispositive,theeigenvaluesarerealand

ofthesamesign,sothattheequilibriumpointisastable( 0)orunstable(

0)node.Itisnotdifficulttoshowthatthecorrespondingrangesofparametervaluesaredefined

by

1 1 1 ∗

1 ∗ sech 9

forthestablenodeandby

1 1 1 ∗

1 ∗ sech 10

fortheunstablenode.

On the other hand, if simultaneously with (C8) 4 , then is negative, the

eigenvaluesarecomplexconjugates,so that theequilibriumpoint isastable( 0)or

unstable( 0)focus.Thecorrespondingrangesofparametervaluesaredeterminedby

1 1 1 ∗

1 ∗ sech

1 1 1 ∗

1 ∗ sech 11

forthestablefocusand

Page87of97

 

1 1 1 ∗

1 ∗ sech

1 1 1 ∗

1 ∗ sech 12

fortheunstablefocus.

Itfollowsfrom(C9‐12)thatthefollowingsequenceofbifurcationsalwaystakesplacewhen

increasesfromzerotoinfinity:stablenode‐>stablefocus‐>unstablefocus‐>unstablenode.

Also, equations (C9) and (C10) show that the range of parameter values for the focus‐type

equilibrium 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 ∗ 1is

satisfiedforany and .Thus,theequilibriumpointcorrespondingto isalwaysasaddlein

accordancewithparagraph1,whereastheequilibriumpointscorrespondingto aresubjectto

the sequence of bifurcations stable node ‐> stable focus‐>unstable focus ‐> unstable node in

accordancewithparagraph2.

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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age88of97

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Page89of97

 

C3.Interpretationasanoscillator

Equation(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 the

coordinate and the velocity , the motion of which is driven by the applied force , . It is

instructive to expand , into a Taylor series and write equation (C13) in the following

approximateform:

, 0, 14

where hasthemeaningofadampingcoefficientandisequalto

, 1 2 2 2 15

andthepotential isgivenby

12

23

4. 16

Toobtainequations(C15)and(C16),theTaylorseriesof , havebeentruncatedatterms

abovecubicin ,linearin andlinearin ,sothat,strictlyspeaking,thisapproximationisonlyvalid

intheregionwhere| | ≪ 1and| | ≪ 1,whichdefinestheneighborhoodoftheequilibriumpointat

∗ (and ∗ ) forsmall .39However, theexpressionfor thepotential ,whichdoesnot

                                                            

38 / where .

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

Page90of97

 

containthemoreheavilytruncatedterms~ ,isexpectedtoholdreasonablywellalsooutsidethat

regionfortherelevantrangeofparametervalues,namely , , ~1and ≪ 1.

Equation (C14) is the equation of a damped oscillator that describes themotion of a particle

subjectedtotherestoringforce / andthedampingforce . Ifdamping isnot toostrong,

themotion is expected to be oscillatory. The potential is useful for visualizing the system’s

behavior;forinstance,thechangeintheshapeofthepotentialfordifferent and helpstoexplain

the occurrence of the bifurcationswith respect to the number of equilibriumpoints (e.g. Fig. C4

below).

Thedampingcoefficient determinesregionswhereenergy isabsorbedorsupplied.Thiscan

beseenbyconsideringtherateofchangeofthetotalenergyofsystem(C14):

12

.

Thus,regionsforwhich 0aretheregionswhereenergyisextractedfromthesystem(positive

or true damping) and regions forwhich 0are the regionswhere energy is pumped into the

system(negativedamping).Asfollowsfromequation(C15), ≷ 0if

≶1 2 2 2

1.

This condition,validnear theorigin| | ≪ 1for ≪ 1, shows thatdamping changes sign from

positive to negative with growing , provided 1. In the ferromagnetic symmetric case

( 0, 1),dampingbecomesnegativefirstattheorigin 0 ,sothatfortherelevantvalues

ofparameters, ≳ 1, ~1and ≫ 1,thecritical ,atwhichdampingchangessign,isgivenby

~1, orequivalently ~1≫ 1.

 

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Economic

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