Adaptive beliefs and the volatility of asset pricesAdaptive beliefs and the volatility of asset...

CEJOR manuscript No.(will beinsertedby theeditor)

Adaptivebeliefsand the volatility of assetprices

Andr eaGaunersdorfer

Departmentof BusinessStudies,Universityof ViennaBrunnerStraße72,1210Vienna,Austria(e-mail:[email protected])

Thedateof receiptandacceptancewill beinsertedby theeditor

Abstract I presenta simplemodelof anevolutionaryfinancialmarketwith het-erogeneousagents,basedon theconceptof adaptivebelief systemsintroducedbyBrock andHommes[8]. Agentschoosebetweendifferentforecastrulesbasedonpastperformance,resultingin an evolutionarydynamicsacrosspredictorchoicecoupledto theequilibriumdynamics.Themodelgeneratesendogenouspricefluc-tuationswith similar statisticalpropertiesas thoseobserved in real return data,suchasfat tails andvolatility clustering.Thesesimilaritiesaredemonstratedfordatafrom theBritish, German,andAustrianstockmarket.

Key words heterogeneousexpectations– boundedrationality – evolutionarylearning– adaptivedynamics– endogenouspricefluctuationsin financialmarkets– bifurcationandchaos

1 Intr oduction

In the pastdecades,the degreeto which financialmarketsareefficient hasbeena matterof heavy debateamongacademicsaswell aspractitioners.Theefficientmarket hypothesis(EMH) still seemsto be oneof the dominatingparadigmsinfinance.A secondparadigmin economictheoryandcloselyrelatedto theEMH, istheconceptof rationalexpectationsequilibria(REE),introducedby Muth [46]. Itis assumedthatinvestorshavecompleteknowledgeof thefundamentalstructureoftheeconomy. They take immediatelyall availableinformationinto consideration,optimizeexpectedutilities accordingto amodelwhich is commonknowledgeandare able to make arbitrarily difficult logical inferences.Sinceall tradersare ra-tional andprocessinformationquickly, assetpricesreflectall publically availableinformationandit is not possibleto earn“abnormal” or “excess”profits on av-

2 AndreaGaunersdorfer

erageby tradingon public information.1 Thevalueof a risky assetis completelydeterminedby its fundamental price, equalto thepresentdiscountedvalueof theexpectedstreamof futuredividends.Changesin assetpriceswouldbecompletelyrandom,solely drivenby randomeconomic“news” aboutchangesin fundamen-tals.

This view wasalreadyquestionedby Keynes[35], who took thepositionthatinvestors’sentimentandmasspsychology(“animal spirits”) playasignificantrolein financialmarkets.Keynesarguedthat stockpricesarenot governedby anob-jective view of “fundamentals”,but by “what averageopinion expectsaverageopinionto be.” Arthur [2] writes:“Not knowing how otherinvestorsarriveat theirexpectations,I cannotform mine in a well-definedway. And so I make assump-tions– subjectiveones– abouthow otherinvestorsform expectationsandbehave.”He talksabouta problem complexity boundary beyondwhich therequirementsofperfectrationality fail and the way how humanagentsarrive at decisionsis notwell-defined.He suggeststo usethe term “rational” in a wider sense“that in-vestorsarefreeto usetheultimatein deductiveandinductivemethodsin trying todeterminehow pricedynamicsworks.”

Also a glanceat financialmagazinessuggeststhat real financialmarketsdif-fer from a perfectlyrationalworld. Financialanalystshave differentexpectationsaboutfuturepricesanddividends.They believe thatmarketsare,at leastto somedegree,forecastablefrom pastpricesandthatbubblesandcrashesaretheresultofherdbehavior ratherthanadjustmentto new information.“Traders,in fact,seethemarketalmostasif it hada personalityof its own, a complex psychologythatcanbereadandunderstoodandprofitedfrom acquaintanceshipwith it deepens.” [2]

Theconceptof rationalexpectationshascertainlyimportanceasa normativetool andprovidesa useful referencepoint. It is useful in demonstratinglogicalequilibriumoutcomesandanalyzingtheirconsequences.However, if somepeopledeviate,theworld thatis createdmaychange,sothatothersshouldlogically expectsomethingdifferentanddeviatetoo. Traditionaltheoryseemsnot to beappropri-ateto understandtheactualbehavior andto explain a lot of pervasive patternsinfinancialdata.For example,stockpricesmove too muchrelative to fundamentals(Shiller [52]), the volatility of pricesis temporarycorrelated,andreturnsdistri-butionsexhibit fat tails. In RE modelsevendisagreementamonginvestorsduetoasymmetricinformationdoesnot generatetradingif all investorsarerationalandrationality is commonknowledge.High tradingvolumesandthepursuitof activeinvestmentstrategiesseemto beinconsistentwith this paradigm.

A comprehensivediscussionof stylizedfactsthatarecharacteristicsof finan-cial seriesis givenby Pagan[48] andBrock [7], seealsoHaugen[33], who dis-cusses“sevenmysteriesof thestockmarket” relatedto thenatureof stockvolatil-ity. In thefollowing we list someof thesefacts.

– Stockpricesarearguabletoo volatile, relative to thedividendsandunderlyingcashflows thatthey arebasedupon.

– Stockvolatility, itself, is too unstable(heteroscedasticity).

1 For reviews on theEMH seee.g.[21], [22].

Adaptive beliefsandthevolatility of assetprices 3

– Volatility shows high persistence,that is, high (small) price changesarefol-lowedby high (small)pricechanges(volatility clustering).

– Stockreturnsexhibit excesskurtosis(fat tails).– Whenvolatility goesup, stockpricesgo down, so as to increasethe sizeof

the risk premiumin the market’s aggregatefuture expectedreturns(leverageeffect).

– Most of the largestchangesin price and volatility are unconnectedto real-world events.

– When the numberof trading hours per week gets smaller (exchangeholi-days),stockvolatility goesdown; whenthenumberof tradinghoursgetslarger(cross-listing),stockvolatility, andshort-termreversalpatternsbecomemorepronounced.

– Autocorrelationfunctions(ACF) of returns,volatility of returns,volumemea-sures,andcross-correlationsof thesemeasureshavesimilar shapesacrossdif-ferentsecuritiesanddifferentindices.Assetreturnscontainlittle serialcorrela-tion (whichisconsistentwith theweakform of EMH), but thereissubstantiallymorecorrelationbetweenabsoluteor squaredreturns.

Thesefactssuggestthat price movementsarenot solely driven by news, butthatmarketshave internaldynamicsof their own, which amplify, distort,andcre-ateinformation.Sucha view may alsobe supportedby the resultsof dynamicalsystemstheorysincesimplenonlineardeterministicmodelscangeneratecompli-cateddynamicbehavior. Long-runpredictionof achaoticsystemis impossibledueto sensitive dependenceon initial conditions.Chaosmay not be distinguishablefrom white noiseby linear statisticalmethodsandnonlineardynamicalsystemscangenerateany given autocorrelationstructure.Thoughmostempiricalstudieshaverejectedthehypothesisthatfinancialdataaregeneratedby low dimensional,purelydeterministicchaos,thepresenceof noisy chaos cannotbeexcluded.Thus,at leastpartof thehighly irregularfluctuationsin financialdatamaybeexplainedby a nonlinearlaw of motion.In fact,investorsresponseto “pseudosignals”,fol-low advicesof market gurus,extrapolatepasttime series,or imitate successfultraders.

Haugen[33] statesthehypothesisthatvolatility consistsof threecomponents,the“event-drivenvolatility” (accurateresponsesto real-world events),the“error-drivenvolatility” (time-varyingmistakesin pricing, relative to the receiptof newinformation),andthe“price-drivenvolatility”, which Haugenthinksto bethe“byfar most importantpart”. Investorsfocuson price changeswhich they astradersco-create,causingpricereactionsthatfeedon themselves.Thus,themarket reactsin a complicatedway to its own pricehistory, creatingprice-driven volatility.

FarmerandLo [23] arguethatpatternsin thepricetendto disappearasagentsevolve profitablestrategiesto exploit them,but this occursonly over anextendedperiodof time. They view financialmarketsasevolutionarysystems,“in whichmarkets,instruments,institutions,andinvestorsinteractandevolve dynamicallyaccordingto the ‘law’ of economicselection.Under this view, financial agentscompeteandadapt,but they donot necessarilydo soin optimalfashion.”


It seemsnaturalthat heterogeneityandevolutionaryswitching betweendif-ferent tradingstrategiesplaysan importantrole in financialmarkets.2 Someau-thorstake the view thatasa new paradigmthe heterogeneous market hypothesisemergesasanalternativeto theEMH, which is closelyrelatedto thehypothesis ofbounded rationality (seee.g.[50]). Financialmarketsareconsideredasevolution-ary systems,consistingof heterogeneousagentsusingdifferentboundedrationalforecastingrules.Thedynamicsof thesesystemsarehighly nonlinearandcaneas-ily leadto market instabilityandcomplicatedpricefluctuations,aboveandbeyondtherationalexpectationsequilibrium(REE-)fundamentalprice(cf. [34]).

Typically, heterogeneousagentsmodelsinclude“fundamentalists”or “smartmoney traders”(traderswho believe that the price of an assetis completelyde-terminedby its economicfundamentals)and “technical traders”,“chartists”, or“noise traders”(believing that assetpricescanbe predictedby simple technicaltradingrulesbaseduponpatternsin pastprices).In theclassicaleconomists’view“irrational” tradersloosetheir money to arbitrageursandwill bedrivenout of themarket.However, De Long et al. [20] show thatif arbitrageursarerisk averseandhave a finite time horizon,noisetradersmay on averageearnhigherprofits thansmartmoney tradersandsurvive in themarketwith positiveprobability. A surveyon thenoisetraderapproachis givenby ShleiferandSummers[54]. Frankel andFroot [25] interpretchartistsastraderswho think shorttermandfundamentalistsas thosewho think long term. LeBaron[39] shows for a simpleagentbasedfi-nancialmarket that shorterhorizon agentsincreasevolatility keepingthe largerhorizonagentsfrom evergettinga footholdin themarket.3 BlumeandEasley [5],who look at theevolutionof simplestrategiesfind thatthelink betweenrationalityand(evolutionary)fitness(wherethecriterion is expectedgrowth ratesof wealthshareaccumulation)is weak.

Otherrecentapproachesdeviating from theperfectRE paradigmarepsycho-logical approaches(see[18] for a review on the work in behavioral finance)andmodelsthat relax the assumptionthat investorshave completeknowledgeabouttheeconomicstructurebut maintainthe(perfect)rationalityassumption.4

Agentbasedmodelsarebasedoninteractionsandlearningdynamicsin groupsof traderslearningabout the relationsbetweenpricesand market information.

2 GoetzmannandMassa[31] examineinvestors’tradingbehavior andcharacterizehet-erogeneityin termsof their investmentpatternsin anS&P 500index mutualfund.

3 Furtherexamplesfor heterogeneousagentsmodelsinclude[60], [4], [13], [30], [17],[19], [51], [36], [41], [42].

4 Timmermann[56], [57] shows that excessvolatility in stock returnscan ariseunderlearningprocessesthat converge (slowly) to RE. Routledge[49] investigatesan adaptivelearningmodel in a repeatedversionof the Grossman-Stiglitzmodel wheretraderscanchooseto acquirea costly signalaboutdividends.He derivesconditionsunderwhich thelearningprocessconvergesto a rationalexpectationsequilibrium. In his model the frac-tion of informedtradersis fixed over time. Daniel et al. [16] andOdean[47], for exam-ple, analyzehow investoroverconfidenceinfluencefinancialmarkets.A dynamic,rationalequilibriummodeladdressingover- andunderreactionto news is Veronesi[59]. Brav andHeaton[6] provide acomparative analysisof behavioral modelsandapproachesrelyingonrationalstructuraluncertainty.


Price fluctuationsare generatedby internal dynamicscausedby the interactionof diversetradingstrategies.Examplesof computationalbasedmodelsincludetheSantaFeartificial stockmarket [3], [40] andthestochasticmulti-agentmodelsofLux andMarchesi[43], [44]. LeBaron[38] andTesfatsion[55] provideoverviewsonthiskind of models.Farmer[24] usesanout-of-equilibriummarketmechanismunderwhichdifferenttradingstrategiesevolve.Heshowsthattrendstrategiestendto inducepositiveautocorrelationsin theprice,whereasvalue-investingstrategiesinducenegative autocorrelations.If theusedstrategiesarenonlinearthediversityof viewsresultsin excessvolatility.

BrockandHommes[8], henceforthBH, have introducedtheconceptof Adap-tive Belief Systems (ABS). Agentsadapttheir beliefsby choosingamonga finitenumberof predictor(expectationsor belief) functionswhich arefunctionsof pastobservations.Eachpredictorhasapublicallyavailableperformanceor fitnessmea-sureattached.Accordingto pastperformanceagentsmake a (boundedly)rationalchoicebetweenthepredictors.This leadsto theso-calledAdaptive Rational Equi-librium Dynamics, an evolutionarydynamicsacrosspredictorchoice,coupledtothedynamicsof equilibriumprices.BH show thatsuchABS incorporateageneralmechanismleadingto local instabilityof theequilibriumsteadystateandto chaosasthesensitivity to switchto betterpredictionstrategiesincreases.

BH [9], [10], [11] have appliedthis approachto a simpleassetpricing model(seealso [7] and [34] for comprehensive discussions).Extensionsof the modelhave beenstudiedby Gaunersdorfer[26], Gaunersdorferand Hommes[27] –henceforthGH –, andChiarellaandHe[14], [15]. Undertheassumptionof homo-geneous,rationalexpectationspricesapproachtheEMH fundamentalprice.Thus,themodelneststheusualrationalexpectationstypeof models.Introducinghetero-geneousbeliefs,however, introducesnonlinearityinto themarketandrich dynam-ics emerge, with bifurcationroutesto strangeattractors.The chaoticprice fluc-tuationsarecharacterizedby a switchingbetweenphaseswherepricesarecloseto thefundamentalprice,phaseswith temporaryupwardspeculativebubbles,andphasesof “pessimism”characterizedby decliningprices.BH call this “the marketis drivenby rationalanimalspirits”.

This paperprovides a review of this approach.I describethe model in thenext sectionandpresenta simpleexamplewith two typesof traders(fundamen-talistsand trendfollowers),wherepredictorchoiceis not only baseduponevo-lutionaryfitness,but alsoon market conditions,in section3. Themodelcapturesstylized factslike volatility clustering(with low autocorrelationsin returnsandsignificantly positive autocorrelationsin absolutereturns),excessvolatility, andfat tails asobservedin realfinancialdata.This is dueto thefactthatthedynamicsis characterizedby two phenomena:intermittency (chaoticpricefluctuationswithphasesof almostperiodicfluctuations,irregularly interruptedby suddenburstsoferraticfluctuations)andcoexistenceof attractors(coexistenceof a locally funda-mentalsteadystateanda“larger” attractor, likea limit cycle).Whenthesystemisbuffetedwith dynamicnoise,irregularswitchingoccursbetweenphasesof smallprice changesandlow volatility, whenthe market is dominatedby fundamental-ists,andphasesof largepricefluctuations,whenthemarket is dominatedby trendfollowers.I extendthemodelof GH [27] by introducinginformationcostsin the


performancemeasureandpresentfurther examples,showing that the resultsarealsovalid for moregeneralcases(memoryin theperformancemeasure,variationof theparametersin theexpectationsfunctionof thefundamentalists)thandemon-stratedin GH [27]. Moreover, I comparethe statisticalpropertiesof the returnseriesgeneratedby themodelwith seriesfrom theBritish, German,andAustrianstockmarket, soasto furtherdocumenttheempiricalvalidity of themodel.Sec-tion 4 concludes.

2 Adaptivebelief systemsin a simpleassetpricing model

In this sectionI presenta simple assetpricing model in a basicmeanvarianceframework as describedin [7] and usedin [9], [10]. Considera financial mar-ket whereonerisky assetandonerisk-freeassetaretraded.Therisk-freeassetisperfectlyelasticallysuppliedatgrossreturn

�. Therisky assetpaysuncertaindiv-

idends�� in futureperiods� , andthereforehasanuncertainreturn.Let �� denotethepriceex-dividendof this risky asset.Thedynamicsof wealthof investortype�is givenby �� ! �#"where � denotesthe numbersof sharesof the risky assetpurchasedat time � .Variablescarrying tildes denoterandomvariables.$ � �&%'� ��()� "*� ��(,+ ".-/-.-/"!� ��(0� "� ��(�+ "/-.-/-/1 definesa publically availableinformationsetconsistingof pastpricesanddividends( $ � mayalsocontainothereconomicvariables,suchastradingvol-ume,interestrates,etc.,or othernewsabouteconomicfundamentals).Let 2 � and3, � denotetheexpectationsor beliefs of investortype� aboutexpectation24�6587 $��!�andvarianceVar �6587 $��6� conditionalon this information set.We assumethat in-vestorsaremyopicmean-variancemaximizers,sothatdemandfor sharessolves9;:�< %=2 �#� � ��/�>�@?A 3� �B� � ��.�#1C" i.e. �>� 2 �D� �� /�? 3 � � �� " (1)

where ?FE Gcharacterizesrisk aversionand

�� 4� �� H� � �� is theexcessreturnpershare.

Let JI�� and K � denotethe supplyof sharesper investorandthe fraction ofinvestorsof type � at time � , respectively. Thenequilibriumof supplyanddemandimplies L K � � �M I�� -

Assumingthat thenumberof outstandingsharesper investor, IN� , is constantover time,we canstick to thesimplecase I��O G .5 Thus,themarketequilibriumequationwritesas6 L K � � � G - (2)

5 In thegeneralcaseonecanrewrite themarket equilibriumequationasin (2) by intro-ducingrisk adjusteddividendsPRQS8T�U�V P S8T�UXWZYR[]\ S!^`_ SaT0U,b P S8T�U!c�dDe�S , see[7].

6 In contrastto this Walrasianmarket clearingmechanism,whereeachagentis viewedasa pricetaker, ChiarellaandHe [15] usea market maker scenario.


In orderto obtainabenchmarkREE fundamental solution considerthespecialcasethatall investorsareperfectlyrationalandtheir expectationsof futurepricesanddividendsaregivenby theconditionalexpectations24�6587 $��!� . Thenthemarketequilibriumequation(2) reducesto� � � �M2f� �� 7 $ � �B- (3)

Thatis, today’spriceof therisky assetmustequalthesumof expectedtomorrow’sprice and dividend,discountedby the risk-free rate of return. In sucha worldwhereinvestorsarehomogeneousandperfectlyrationalandwhereit is commonknowledgethatall investorsarerational,theREEor fundamentalpriceis givenbythediscountedsumof expectedfuturedividends,

��g� �ihLjBk � 24�� j 7 $��6�� j -

This solutionis basedon the fact that the fundamental� g� mustbe finite, whichmeansthatthelong rungrowth ratesof dividendsandpricesmustbesmallerthanthe risk-freegrowth rate

� �ml . Note, that alsoso-calledspeculative bubbleso-lutionsgrowing at theconstantrisk-freeratesatisfythemarket equilibriumequa-tion (3) (for moredetailssee[34]). Along sucha rationalbubblesolution,traderswould have perfectforesight.But in a world whereall tradersare rational theyrealizethat sucha rationalbubblecannotlast forever andtherefore,it will nevergetstarted.Thus,all tradersbelieve thatthevalueof therisky assetequalsits fun-damentalpriceforeveranddeviationsaresolelydrivenby unexpectedchangesindividendsandrandomnews abouteconomicfundamentals.However, in a marketwheretradershaveheterogeneousbeliefsthesituationwill bequitedifferent.

We make thefollowing assumptionsaboutthebeliefsof thetraders.

A1. Investorshave homogeneousbeliefsaboutonestepaheadconditionalmeansof earneddividends,2 � � �� n�o2f� �� 7 $ � � , p)�)"q� .

A2. Beliefs about the conditional meansof next period prices are of the form2 �B� ��D�n�srB��t � �� M2f� �� g�� 7 $��N�0�Fu �*v��()�J"qv��(,+ "/-.-/-'� , p0�X"!� , wherev��>��w�x� g� and u is a deterministic,agentspecificfunction of pastdeviationsfrom acommonlysharedview of thefundamentalprice.

Realizedexcessreturnsoverperiod � to period �)�ol , canbecomputedas�� v �� v � � �y �� "where

�y �� g�� z2f�{� g�� 7 $��6� is amartingaledifferencesequencew.r.t. $�� , representingintrinsic uncertaintyabouteconomicfundamentals.Thus,realizedexcessreturnscanbedecomposedin anEMH-term

�y � andanendogenousdynamictermexplainedby thetheorypresentedhere.7 This termis nonzeroif theprice deviatesfrom the fundamentalvalueandthusallows for the possibility ofexcessvolatility.

7 Thisextra termis ratherlike “endogenousuncertainty”asin Kurz [37], whopresentsatheoryof rationalbelief equilibria.


We focuson the simplecaseof an iid dividendprocesswith constantmean24� ��7 $��6�|�~}� and finite variance,so that the fundamentalprice is given by� g ��}��]� � ��l�� . In thatcase�y �� }� . This implies2 �D� �� D�n�x� t � �� g�� {��X��g.�

and �� 2 � � �� n� �� Z� t�� y �� - (4)

AssumingthatCov � �v�� v��#" �y ��/� O G and3 �D� �y ��D��sr � +� is constant,we

obtain3 �D� �� .�� 3 �B� �v��)� � v��6�C�� +� .8 Gaunersdorfer[26] studiesanexample

whereinvestorshave time varying homogeneousbeliefsabouttheseconditionalvariances.Thebeliefsabout

3 � �*v �� v � � aremodeledasexponentiallymov-ing averages.Thequalitative characteristicsof the modelarequite similar to thecasewherethebeliefsaboutconditionalvariancesareassumedto beconstant.Inparticular, the local behavior of thesteadystatescoincidesin thecaseswith timevarying beliefsandwith constantones.Also the bifurcationroutesto chaosaresimilar. ChiarellaandHe [14], [15] introduceheterogeneityin the beliefsaboutvariances.They studyhow the dynamicsof assetpricing is affectedby differentrisk attitudesof variousinvestors,ascharacterizedby the differentrisk aversioncoefficients.They show that different attitudesto risk have an influenceon thepricedynamics.In particular, whenthefundamentalistsaremorerisk aversethanthetrendtradersthemarketbecomesmoreunstable,which is in line with DeLonget al. [20].

In whatfollows,we assumeconstantbeliefsaboutconditionalvariances,3 �D� �� .� O � + - (5)

In orderto completethe modelwe have to specifythe way the fractions K �areformedover time.Thesefractionsaredefinedasdiscretechoice“Gibbs” prob-abilities(see,for example[45] or [1])K � �M��'� �)� choosepredictor � at time �!��M� < �X�*�� ()� �q�.� �� L � < �X�� (0� �D" (6)

where � � ��(0� is a fitnessfunction or performancemeasure.The parameter� iscalled intensity of choice. It measureshow fast the massof investorsswitchesbetweenthedifferentpredictorsandthushow sensitive tradersareto differencesin the fitness.If �� G

traderschoosepredictorsrandomly(and independentlyof their performance)with probability lJ�J� , where � is the numberof differentinvestortypes.In theotherextremecase,�x�� , all traderschoosethepredictorwhichperformedbestin therecentpast.For finite,positive � agentsareboundedlyrationalin thesensethatfractionsof thechosenpredictorsarerankedaccordingtotheir fitness.BH [8], [9] andGaunersdorfer[26] show that this parameterplaysa

8 In theterminologyof Haugen[33] onecouldinterpret [�S ^#�� SaT0U0WZ� � S�c aspricedrivenvolatility.


crucial role in thebifurcationrouteto chaoticpricefluctuations.In particular, forlargevaluesof � thesystemis closeto having a homoclinicorbit, which is a keyfeatureof chaoticsystems.

To definethe fitnessfunction rememberthat investorsaremyopic mean-va-riancemaximizers.We thereforedefinethe performancemeasureasutilities de-rivedfrom realizedprofitsor pastrealizedrisk-adjustedprofits(see[26]). Let� � r{��2 � � � �� D-Utilities of realizedprofitsin period � aregivenby� �nr{� � � � ��" � �6�wr`� � ��B ,� � �6�>�m?A�� + + � � �6�B" (7)

where ,� � � �n� : �'� 9;:J< � � � � �? � + �o � -Note, thatmaximizingexpectedutility of profits is equivalentto maximizingex-pectedutility of wealth,thus ,� � �6� coincideswith (1). Since(6) doesnot changeif thesametermis subtractedoff theexponents(i.e. fractionsareindependentofutility levels),wemayconsider� �� >� � � � ��" � �6�� " � ��/�� lA ? � + ��2 � � �� D� +�� lA ? � + �� t � �� y �� +(seeequation(4)), where � � � � � � ��J" � ��/� areutilities of profits if investorshadperfectforesight.

Sincewe will mainly focuson the deterministicskeleton,that is, the nonlin-earassetprice dynamicswith

y �� O Gandconstantdividends }� , we definethe

performancemeasureas� �>� � � ��(0�� ()��¢¡�� (0�� lA ? � + �{�� t � � + �x¡�� ()�=" (8)

wherethe parameter¡ , G�£ ¡ £ l , represents“memory strength”.The perfor-mancemeasureis thusdeterminedby squaredpredictionerrors.In thespecialcase¡¢� G

, fitnessis given by utilities of realizedprofits in the mostrecentpast,forpositive ¡ it is an exponentiallymoving averageof pastutilities of profits.Note,that the performancemeasure(andhencethe fractions)in period � dependuponobservablepricesat theendof theprevious(beginningof this)period.

Anotherpossibilityfor theperformancemeasureis to takepastrealizedprofits((7)withoutrisk-adjustmentterm)asin BH [9], [10], [11], with � �� D �� N� .For ¡¤�¥l thefitnessfunctioncoincideswith accumulatedwealth.In thisnon-riskadjustedcasethe fitnessfunction is however inconsistentwith the tradersbeingmyopic mean-variancemaximizersof wealth.BH (1999)show for this non-riskadjustedcasethat increasingmemorystrength¡ hasa similar effect concerningthebifurcationrouteto complicatedpricedynamicsasincreasingthe intensityof


choice � . They conjecture,however, that thesituationis differentfor therisk ad-justedcase,wheremorememoryactsasstabilizing force (if costsfor all tradingstrategiesarezero).

3 A simpleexample:fundamentalistsversustr end followers

Considerasimpleexamplewith two typesof traderswherethebeliefsaboutfuturepricesaregivenby linearpredictionrules2z�N�D¦ ��'§ O � t � � �� g �¢¨,��()�w�Z� g �D" G¤£ ¨ £ l "2©+!�D¦ ��'§ O � t+ � �� (0��xª0�{��()��Z��(,+��B" ªf« G -The first trader type are so-calledfundamentalistswho believe that priceswillmove towardthefundamentalvaluein thenext period(if ¨�¬�l ). Thecasef�lcorrespondsto traderstakingtoday’s priceasbestforecastfor tomorrow’s.Thesetradersbelieve that markets are efficient and pricesfollow a randomwalk. Wecall themEMH-believers.Thesecondtradertypearetrendfollowersextrapolatingthe latestobserved price change.Thus, the predictor function for future pricesof this tradertype incorporatestwo time lags, in contrastto BH [9], [10], [11]who only considerpredictionfunctionswith onelag. ChiarellaandHe [14], [15]studytheimpactof learningschemeswith differentlag lengthsin theformationofexpectations.

The fractionsareupdatedaccordingto pastperformanceasdescribedin theprevioussection,conditioneduponthedeviationsof actualpricesfrom thefunda-mentalprice (cf. the SantaFe artificial stockmarket [3], [40]). The fundamentalpredictormay becostly sinceit needssomeeffort to understandhow the marketworksandto believethatit will priceaccordingto thefundamental.Thefirst, evo-lutionarypartof theupdatingof fractionsis determinedby risk adjustedrealizedprofitsasdescribedin theprevioussection,®K��N��o� < �X��n��n� � ��(0�w�x¯��!�'�/��®K)+!��o� < �X��+ � ��()�D�'�/�>�� +L k � � < �X��°� � ��()� ��¯ �!�D" (9)

where � � ��(0� is definedby (8) and ¯��©�m¯« G arethecoststo obtainthefunda-mentalpredictor(representingtraderseffort for informationgathering),w+±� G .9The secondpart of updatingconditionson deviations from the fundamental.Ifpricesaretoo high or too low technicaltradersmight getnervousanddo not be-lieve that a price trendwhich continuedfor a while will go on any longeranda

9 GH [28] usepastrealizedprofitsasperformancemeasure.It turnsoutthatthedynamicsof themodelandthetime seriespropertiesarevery similar to thecasewhererisk adjustedrealizedprofitsareusedasfitnessmeasure.


correctionto the fundamentalis aboutto occur. That is, tradersbelieve that tem-porary speculative bubblesmy arise,but thesebubbleswill not last forever. Thefractionsof thetwo tradertypesarenow modeledasK +q� � ®K +!� � < �)¦²�±�{� ��()� ��g.� + ��³)§�" ³ E G (10)K��6�n�l��´K)+q�#- (11)

Accordingto (10) fractionsarealmostcompletelydeterminedby evolutionaryfit-nessaslong aspricesdo not deviate too muchfrom the fundamentalvalue.Butwhenthecorrectionterm � < �X¦²�±�{� ��(0� �� g � + �J³X§ becomestoo smallmostchartistswitchto thefundamentalpredictor. Thus,technicaltradersareconditioningtheirchartsuponinformationaboutfundamentals.

BH [9], [10], [11] only take theevolutionarypart for theupdatingof thefrac-tions. Gaunersdorfer[26] introducesa so-calledstabilizing force in the perfor-mancemeasureof thefundamentalistswhich hasa similareffectasthecorrectiontermin (10).

Setting®� �nr{�� ()� , weobtainthefollowing dynamicalsystemfor theprice

dynamics, �� l� �{� t � � �� xK)+q�D�{� t+ � �� t� � �� X��}�]�®� ��@� lA ? � + �{��(0�� t � ��(0� � + �¢¡ ®� � ��(0�J"µ�f�@lR" A - (12)

Sincethe fractions K)+q� arenonlinearfunctionsof previous prices,this definesanonlinear six dimensionalsystemin ��="��+ "*��¶ "*��· "!¸)�="q¸,+.� , where ��¹'��¥l=�Z��(�¹ and ¸ ��ol=�©� ®� � ��()�º�� (�+ . Expectationsaboutfutureprices,whicharedeterminedby pastprices,feedbackon theactualprice.

3.1 Dynamics of the Asset Pricing Model

In the following I briefly discussthe dynamicsof system(12). A comprehensiveanalysisfor the casewithout informationcosts,»� G

, is providedby Gauners-dorfer, Hommes,andWagener[29], henceforthGHW.

Lemma1 summarizesthestabilityanalysisof thefundamentalsteadystateforthegeneralcaseof positive informationcosts.

Lemma 1 (Existenceand stability of steadystates)Let ¡¥¬¼l . System (12) has a unique steady state �{� g "�� g "*� g "�� g " G " G � which islocally stable if� lA � � �Ml��M�*¨ � � �!½ (�¾R¿ �q�w¬Fªf¬ � �6l��F½ (�¾R¿ �B-(1) For parameters satisfyingA ª � � �Ml��M�*¨ � � �6½ (�¾R¿ � G " (13)

the steady state undergoes a perioddoubling(flip) bifurcation.


(2) For parameters satisfying ªÀ� � �!l��¢½ (�¾R¿ �� G " (14)

the steady state undergoes a Hopf (Neimark-Sacker)bifurcation.

Proof. Thecharacteristicpolynomialof theJacobianat thesteadystateis givenby

��Á,�n�MÁ + ��¡Â��Á,� +sÃ Á + � l��xª �¢¨C½ (�¾C¿� �6l��F½ (�¾R¿ � Ás� ª� �!l��¢½ (�¾R¿ �JÄ - (15)

Thus,theeigenvaluesof theJacobianare0, ¡ (bothof multiplicity 2), lying insidetheunit circle,andtherootsof thelastbracket.For therestof theproofproceedinanalogyto theproofsof Lemmas1 and2 in [29]. ÅÆ

Lemma1 shows that the local dynamicsaroundthe steadystateis the samewithoutmemory( ¡¤� G ) andwith memory(

G ¬Ç¡Z¬Èl ).

1 2 3 4 5 6 7beta

1.2

1.4

1.6

1.8

2

g

Ps

Pu

H

Ch1

Ch2

S

1.15 1.2 1.25 1.3 1.35g

0.2

0.4

0.6

0.8

1v

Ch

Ps Pu

S

H

(a) (b)

Fig. 1 Bifurcation diagramsfor parametervalues É V�Ê�Ë Ì , � V�Í.Ë Ê Í , Î V�ÍBÊ , Ï V�Ê ,Y�Ð�Ñ V�Í , ÒP V�Í , _�Ó VÔÍ#Ê�Ê in the (a) Õ - Ö -planefor × V¥Ê�Ë Ø , (b) × - Ö -planefor Õ VÚÙ .Û¼Ü Ö V � ^ Í b¢Ý=Þ�ßJà0c is the Hopf bifurcationcurve. Ch arepointswherea Chencinerbifurcationoccurs.Thesepointsareorigin pointsof asaddle-nodebifurcationcurve á (dot-tedcurve) of invariantcircles.In region â e VmãDÊ;ä Ö ä � ^ Í b�Ý Þ�ß=à c6å thesteadystateis locally stable,in region â,æ V¥ã ÖZç � ^ Í b�Ý=ÞCßJà)c6å it is unstable.As É@è Ê , Ch2 indiagram(a) movesto infinity. As É is increased,the two Chencinerpointsapproacheachotheranddisappearaftercollision (i.e. for É largeenoughtheHopf bifurcationis alwayssupercritical).

Figure1 shows bifurcationdiagramsof thesteadystatein the � - ª andin theª - ¨ -plane,respectively. ChdenotepointswheretheHopf bifurcationis degenerateandaso-calledChenciner bifurcation occurs.At thesepointstheHopf bifurcationchangesfrom supercriticalto subcritical(or vice versa).Thesepointsareoriginpointsof acurve é of saddle-nodebifurcationsof invariantcircles.10 In theregionê I|r{�ë%/ªo¬ � �6lº��½ (�¾R¿ �B1 the steadystateis stable.Whencrossingthe Hopfbifurcationcurve � to theregion

ê�ì r`��%/ª E � �6lw��½ (�¾R¿ �#1 at parametervalue

10 In thethreedimensionalparameterspaceã ^ Õ�íNÖ]í°× c6å theHopf bifurcationmanifoldÛ

andthe setof saddle-nodebifurcationsof invariantcircles á definetwo dimensionalsur-faces.


(a) (b)

Fig. 2 (a) Phasespaceprojectionon the _ S Þ U -_ S -planefor parametervalues Õ V&Í.Ë Ì ,× V@Ê Ë Ø , Ö VÍ.Ë î=Ì , É V@Ê Ë Ì , � VÍ�Ë Ê Í , Î VÍ#Ê , Y Ð Ñ VÚÍ , Ï VmÊ , ÒP VÍ , _ Ó VÍBÊ.Ê(“volatility clusteringregion” betweená and

Û): coexisting stablesteadystate(marked

asa square)andstablelimit cycle. (b) One-parameterbifurcationdiagram_ S versusÕ for× VxÊ Ë Ø , Ö VïÍ�Ë î=Ì , É VxÊ Ë Ì , � V�Í�Ë Ê Í , Î VïÍ#Ê , Y Ð Ñ V�Í , Ï VxÊ , ÒP V�Í , _ Ó V�ÍBÊ�Ê .�Ô¬ð��ñ�ò � or � E �,ñ�ò + in Figure1(a), or ¨ E ¨�ñ�ò in Figure1(b), the steadystatebecomesunstableby a supercriticalHopf bifurcationanda stableinvariantcircle is created.Crossing� from region

ê ìinto the region between� and é ,

a subcriticalHopf bifurcationoccursandan unstableinvariantcircle arisesfromthe unstablesteadystate.The steadystatebecomesstable,whereasthe “outer”stableinvariantcircle still exists.Thus,in the region betweenthe two curves �and é two attractorscoexist: the stablesteadystateanda stablelimit cycle (seefigure2(a)).GHW [29] call this the“volatility clusteringregion”. “Between”thesetwo attractorsliesanunstableinvariantcircle.Thestableandtheunstableinvariantcirclescollideanddisappearat thebifurcationcurve é . Theemergenceof thelimitcycle is clearlyseenin thebifurcationdiagramin figure2(b).Notethebig jumpintheamplitudeof theattractorat �ÇóÔl - ô , whenparameterscrossthesaddle-nodecurveof invariantcircles é (cf. alsofigure1(b)).11 In thediagramweseewindowswherethedynamicsarephaselockedandperiodiccyclesexist.

GHW [29] show examplesfor ¯ð� Gwherethe limit circle in the “volatilty

clusteringregion” may undergo homoclinic bifurcationsand turn into a strangeattractor. In suchacaseastablesteadystateandachaoticattractor“around” it co-exist. Figure3 indicatesparameterregionswherethelargestLyapunov exponentsof orbitson theattractorsarepositive.12 In theseregionsthesystemhasa chaoticattractor. In order to accountfor coexisting attractors,for every pair of parame-

11 Theexactvalue,wherethesaddle-nodebifurcationof invariantcirclesoccurs,canbepicked out from figure1(b). Of course,it is not possibleto concludetheexactvaluefromdiagram2(b).Becauseof coexistenceof attractors,dependingoninitial conditions,atrajec-tory mayeitherconvergeto thesteadystateor to thelimit cycle.In diagram2(b)1000pointswereplottedfor every Õ -valueaftercuttingoff thefirst 10000points.For every Õ -valuethesameinitial valuewaschosen.

12 I amindebtedto FlorianWagenerfor providing theseplots.


ter valuesthe Lyapunov exponentshave beencomputedfor ten different initialconditions.

(a) (b)

Fig. 3 Regionsof positive Lyapunov exponents(a) Õ - Ö -planefor × VoÊ Ë Ø , (b) in the × - Ö -planefor Õ VxÙ , with otherparametersasin Figure1 (horizontalaxis:(a) Õ , (b) × ; verticalaxis: Ö )

If system(12) is buffetedwith dynamicnoise,the price dynamicsis charac-terizedby fairly regularphaseswith pricescloseto the fundamentalsteadystate,suddenlyinterruptedby largepricechangescausedby excursionsto thelimit cycletriggeredby technicaltrading.Suchaphenomenonis known asintermittency. Thepricedynamicsof thechaoticattractorin figure4 alsoshows thesecharacteristics:phasesof low price fluctuationscloseto the fundamentalinterchangewith largefluctuationswhenpricesdeviate from the fundamental,suggestingsomeform ofvolatility clustering.

Thegeometricshapeof thechaoticattractorin figure4 givessomeinsightintothe structureof the dynamics.Considera situationwherepricesarecloseto thefundamentalsteadystate� g . For ª E � �6lz��½ (�¾R¿ � the steadystateis a saddlepoint (andhenceunstable),thuspriceswill moveaway. Sincethefundamentalistsexpectthatpricesreturnto thefundamentalvaluetheirperformanceis badandthefractionsof thetrendfollowerswill increase.If pricesdeviatetoo muchfrom thefundamental,thefactor � < �X¦²�±�{��*�!��¤� g � + �J³X§ startsto playa roleandthefractionof thetrendfollowers K + decreasesandthefundamentaliststake over. In thecasewhenthefundamentalistsdominate( K��ó¥l , K)+Hó G ), thepricedynamicsis closeto ��!�>��gH� ¨� �{��>�Çl=��Z�0g/�D"which definesa line in the � ��()� -� � -plane,and priceswill move along this linetowardsthe fundamentalvalue(this line is clearly seenin figure 4(a)).For moredetailssee[29].

Thepresenteddeterministicmodelis in facttoosimpleto capturethedynamicsof a financialmarket. In the following sectiona dynamicnoiseterm is addedto(12),representingmodelapproximationerror. Theanalysisgivenaboveis howeverimportantto understandtimeseriespropertiesof thenoisymodel.


(a) (b)

Fig. 4 (a) Phasespaceprojectionof a chaoticattractoron the _ S Þ U -_ S -planeand(b) cor-respondingpriceseriesshowing intermittentchaos;Õ VmÍ#Ê , × VMÊ�Ë Ì , Ö V�õJË{Í , É VMÊ�Ë õ ,� V�Í�Ë Ê Í , Î V�ÍBÊ , Y�Ð Ñ V�Í , Ï VxÊ�Ë ö , ÒP VoÍ , _ Ó V�ÍBÊ.Ê3.2 Time series properties

Figure5 shows the (continuouslycompounded)returnseries÷ � �ùøú� �� #� ��(0� �for theFTSE,theDAX (Germantradedindex), theATX (Austriantradedindex),and the AustrianstockOMV (Austrian oil company).13 Table 1 reportsthe de-scriptivestatisticsof theseseries.

Fig. 5 Returnsseriesof the FTSE (04/20/94–03/13/98),the DAX (04/06/94–03/31/98),theATX (04/13/94–03/31/98), andtheOMV (03/16/94–03/31/98),1000observationseach

13 I amindebtedto RobertTompkinsfor providing thesedata.Tompkins[58] providesananalysisof returnsandvolatility of theAustriandata.


Themeansandmediansarecloseto zerofor all series.All seriesexhibit ex-cesskurtosisandthe hypothesisof normaldistribution is clearly rejectedby theJarque-Berateststatistics.14 Theskewnessstatisticssignificantlydeviatesfrom anormaldistribution at the 1% level for the DAX andthe ATX, but is not signif-icant for the FTSEandthe OMV. As pointedout for exampleby Tompkins[58]andotherauthors,theskewnessstatisticsis notsignificantnorof samesignfor allmarkets.Nevertheless,someauthorsexaminethe skewnessin additionto excesskurtosis.In a recentpaperHarvey andSiddique[32] arguethatskewnessmaybeimportantin investmentdecisionsbecauseof inducedasymmetriesin realizedre-turns.They presentanassetpricing modelwhereskewnessis pricedandsuggestto analyzeportfolios in a conditionalmean-variance-skewnessframework insteadof a conditionalmean-varianceframework.

Table1 Descriptive statisticsof returns

Series r FTSE r DAX r ATX r OMVMean 0.000656 0.000795 0.000269 0.000527Median 0.000925 0.001238 8.32E-05 Wnû Ë û ö E-05Maximum 0.040191 0.044072 0.069210 0.066413Minimum W Ê�Ë Ê.Ù=õ û Ì�ü W Ê�Ë Ê=Ì.Ø�Ì.Ù.Ø W Ê�Ë Ê�ö�ö�ÍDÌ.õ W Ê Ë Ê û ü�ÍBØ.öStd.Dev. 0.008998 0.010976 0.011182 0.014929Skewness W Ê�Ë Ê�Ì.ö.Ù û Ø W Ê�Ë õ.Ê�Ê�õ�Í#Ø ** W Ê�Ë Ì.ö=ü.õ�ÍBü ** W Ê Ë Ê.î�Ù=ü/î�öKurtosis 4.310033** 4.867463** 12.60597** 6.640624**Jarque-Bera 72.09555 151.9901 3904.221 552.5750

** null hypothesisof normalityrejectedat the1% level

Figure6 shows autocorrelationplots for thefirst 36 lagsof the returnsseriesandthe absolutereturnsseries.For the indicesall low orderautocorrelationsareinsignificant.OMV hassignificant,but smallautocorrelationat thefirst andthirdlag. In contrast,theautocorrelationsof theabsolutereturns(which area measurefor thevolatility) areclearlypositivesignificantat low orderlags.Thatis,all seriesexhibit volatility clustering.

Ouraimis to captureasmany aspossibleof theseobservedfactsby ourmodel.BH [9] calibratetheir modelto tenyearsof monthly IBM pricesandreturns.Theautocorrelationstructureof their noisychaoticpriceandreturnseriesaresimilarto thoseof therealdata.However, squaredreturnsdonothavesignificantautocor-relations.

GH [27] comparethe time seriespropertiesof the returnpathsgeneratedbythe modelwithout costs( ¯ý� G

) with 40 yearsdaily S&P 500 data.They showthat the model is ableto generateclusteredvolatility andto capturesomeof the

14 TheJarque-Berastatisticsmeasuresthedifferenceof theskewnessandkurtosisof theserieswith thosefrom thenormaldistribution. Underthenull hypothesisof normality theJarque-Berastatisticis Chi-squareddistributedwith two degreesof freedom.The criticalvalueat theonepercentlevel is 9.21.


Fig. 6 Autocorrelationsof returnsandabsolutereturnsof the FTSE,the DAX, the ATX,andtheOMV

descriptive statisticalproperties.HereI presentsomefurtherexamples,includinganexamplewith positivecosts for thefundamentalpredictor.

Returnsaredefinedasrelativepricechanges

÷/�� (0��(0� -


Thepriceseriesaregeneratedby system(12)with a dynamicnoisetermaddedtothefirst equationof thesystem,15 i.e.

�� l� �{� t � � �� xK)+q�D�{� t+ � �� t� � �� X��}�]�X�xþJ�®� ��@� lA ? � + �{��(0�� t � ��(0� � + �¢¡ ®� � ��(0�J"µ�f�@lR" A "where þ � are iid normally distributed randomvariables.This noiseterm canbeinterpretedas unexpectednews about fundamentalsor noisecreatedby “noisetraders”,agentswhosetradingis not modeled,but exogenouslygiven.

The return seriesfor the following examplesare shown in Figure 7. I haveonly introducedpositive costsfor the case ¨�¬ÿl . In the case ¨m� l the firsttradertypetakesthelastobservedpriceaspredictorfor tomorrow’sprice.It seemsunreasonableto assumepositive informationcostsfor usingsucha predictor. Inbracketsthedynamicalbehavior of thesystemwithout noiseis indicated.

example1: example2:

example3: example4:

Fig. 7 Returnseriesof examples1–4

15 Wealsoransimulationswith thestochasticterm� S , definedin equation(2), includedin

theperformancemeasure.However, since� S hasto besmallin comparisionto ÒP , thishardly

affectstheresults.


Example1 (coexistingstablesteadystateandstablelimit circle):�ï�� , � �ðlR- G l , ¨´�»l , ª´� A, ¯ � G

, ¡�� G , }�´�»l , ³M�ðl G , ? � + �ýl ,þ��¢� G "!ô��Example2 (stablesteadystate):�F� A , � �ùl - G l , ¨��ùl , ª��Ôl -�� , ¯�� G , ¡ � G , }�|�Ôl , ³��ùl G , ? � + �ùl ,þ��¢� G "C�Example3 (stablelimit circle,unstablesteadystate):�m�� , � �ëlR- G l , ¨F� G - � , ªF�ël -�� , ¯ � G -�� , ¡Ç� G - A , }�¢�ël , ³��ël G ,? � + �¥l , þ ��¢� G "��R�Example4 (stablesteadystate):� �� , � �¥lR- GRG , ¨;�¥l , ª¤��l -�� , ¯¥� G , ¡4� G - A , }�;�¥l , ³´�Úl G , ? � + �Úl ,þ��¢� G "�C�

Table2 shows thedescriptivestatisticsfor theseexamples,theautocorrelationfunctionsof the returnsandabsolutereturnsareplotted in Figure8. The seriesclearly have excesskurtosis,comparablein size to thoseof the FTSE and theDAX. Theautocorrelationsof thereturnsarefor all examplessmallor not signifi-cant,whereasthelow orderautocorrelationsof theabsolutereturnsareall signifi-cantandpositive.Thus,themodelis ableto capturethephenomenonof volatilityclustering.In examples1–3therangeof thereturnsis largerthanfor therealdata.If�

is decreasedthe rangebecomessmallerasdemonstratedby example4. Infact,for daily dataasmallervalueof

�seemseconomicallymorereasonable.16

Table2 Descriptive statisticsfor examples1–4

Series Example1 Example2 Example3 Example4Mean 0.000732 W Ê�Ë Ê�Ê�Ê�ü.Ù.ö 0.001296 0.000206Median 0.000732 W Ê�Ë Ê�Ê ÍBÌ.Ø�ü 0.001972 W Ê Ë Ê�Ê.Ê=Ì�õ.üMaximum 0.217724 0.132335 0.247417 0.078711Minimum W Ê�Ë�Í'î=Ø�õ.Ø�Ø W Ê�Ë{ÍBÙ û Ê�ö�ü W Ê�Ë õ�õ�ü.õ.Ê û W Ê Ë Ê�Ø�õ�Ì û õStd.Dev. 0.038897 0.031364 0.050282 0.015815Skewness 0.258151** W Ê�Ë Ê=õ�õDî=Ê�ü 0.168476* Ê Ë õ/î�Ù.î�Ê�ö **Kurtosis 4.882617** 4.076606** 5.028881** 4.517690**Jarque-Bera 158.7839 48.37866 176.2455 105.8490

* null hypothesisof normalityrejectedat the5%level** null hypothesisof normalityrejectedat the1% level

Our simulationsdemonstratethatvolatility clusteringmayevenoccurfor pa-rameterswherethedeterministicskeletonof themodelhasagloballystablesteadystate.Notethatfor parametervaluescloseto the“volatility clusteringregion”, thesystemis close to having a limit cycle andorbitsspiralslowly towardsthesteadystateandappearalmost(quasi)periodic.In particular, the transientphasesin and

16 I chosemost exampleswith � V Í�Ë Ê Í for numericreasons.Sincepricesincreaseswhen � is decreasedtheexponentsin equation(9) becomeratherlargernumbersresultingsometimesin overflows in thesimulations.


Fig. 8 Autocorrelationsof returnsandabsolutereturnsof examples1–4

outof thevolatility clusteringregionareverysimilar. Whenthesystemis buffetedwith noise,it is not possibleto distinghuishif parametersarechosenin or out ofthe region wherea limit cycle exists. In fact, the bifurcationsof the determinis-tic system(cf. Figure2) cannotbedetectedin thenoisymodel.Nevertheless,theanalysisof the deterministicsystemgivesimportantinsight into the dynamicsofthenoisysystem.

Weobtainthebestresultswhentheparameter¨ is equalor closeto 1, i.e.whenEMH-believersinteractwith trendfollowers.EMH-believersthink thatpricesfol-


low a randomwalk. Therefore,whenthis type of tradersdominatesthe market,pricesarehighly persistentandprice changesaresmall, only driven by randomnews. Thus,returnsarecloseto zeroandvolatility is small. As pricesmove to-wardsthefundamentalvalue,trendfollowersperformbetterthanEMH-believersandtheir fractiongraduallyincreases.Whenthey startdominatingthemarket,bigprice changesoccurandvolatility becomeslarge. However, aspricesmove toofar away from the fundamentalvalue,technicaltradersstop chasingtrends,thefraction of EMH-believersincreaseandthe story repeats.Thoughour systemisstationary(dueto the fact thatwe assumeda simpleiid dividendprocessimply-ing a constantfundamentalvalue),it is close to having a unit root. Note that for¨ � � �ùl and ¯ù� G thecharacteristicpolynomial(15) hasaneigenvalue1.17

This is illustratedby the resultsof unit roots test,seetable3. For the exampleswith ¨��Ôl thenull hypothesisof a unit root (i.e. non-stationarity)is not rejectedfor a samplesizeof 1000observations,whereasit is clearly rejectedif ¨È¬&l .Nevertheless,example3 demonstratesthat the resultsarerobust with respectto(small) changesof theparameter . If thesamplesizeis increasedthe teststatis-tics decreasesandthenull hypothesisis rejectedalsoin thecasef�l for longertime series.Thepriceseriesarehighly persistentfor ¨¤�Úl , theautocorrelationofthefirst is closeto one(seetable3) andis only decreasingvery slowly at higherlags,asit is thecasein realpriceseries.

Table3 Unit root teststatisticsandautocorrelationof first lagonpriceseries

Example1 Example2 Example3 Example4AugmentedDickey-Fuller W õ�Ë Ù û Ø=ÌDî=Ê W õ�Ë Ø û îJÌ/Ø.î W Ø�Ë ö û Ø.Ø ÍBö W Í.Ë û Ø�Ø.Ø�Ù�ÊPhilipsPerron W õ�Ë õ/ö=Ì û Ì û W õ�Ë ü.õ�ÍDÌ/Ê=ü W üJË üJÍBö.Ù û î W Í.Ë û Ø�Ø û�û�ûACF (lag 1) 0.991 0.980 0.895 0.996

MacKinnoncritical valuesfor rejectionof hypothesisof a unit root:1%: W Ù�Ë î�Ù�ö.Ø , 5%: W õ�Ë û Ø/î û , 10%: W õJË Ì.Ø û Ì

Finally, I estimateGARCH(1,1)modelson thereturnseries.TheGARCH(1,1)modelis specifiedas÷.��.�� y �#" y ��¢� G "q��N�� +� ��D+n� � � y +��(0� � � +/� +��(0� -Themodelcanberewritten in termsof errorsy +� �� + �� + � y +��()� �� + � ��()� "where�J�>� y +� �Z� +� is theerrorin squaredreturns.Thus,thesquarederrorsfollowa heteroskedasticARMA(1,1) process.Theautoregressiveroot whichgovernsthepersistenceof volatility shocksis the sumof � � (ARCH term) and � + (GARCH

17 NotethattheJacobianof the lineardifferenceequationP S V Î�� b�� U Î � P S Þ � hasaneigenvalue1 if andonly if thetimeseriesP S V Î�� b�� U Î � P S Þ � b! /S hasaunit rootequalto 1.


term).In realreturndatathis root is usuallyvery closeto unity, which meansthatshocksdie out ratherslowly. This fact canalsobe observed in the seriesof theFTSE,theDAX, andtheOMV (seetable4). TheGARCH termsandtheARCHtermsareclearlysignificantandtheGARCH termis muchlargerthantheARCHterm.This is alsothefact for thecoefficientsof theGARCH andARCH termsofthesimulatedseries,in particular, they resemblethoseof theFTSE,theDAX, andtheOMV.18

Table4 GARCH(1,1)estimationsfor returnseries,with (standarderrors)and[ _ -values]

Series " U " Ñ # U #RÑ # U,b #RÑ0.000657 7.15E-07 0.039852 0.950747 0.990599

FTSE (0.000256) (4.02E-07) (0.010714) (0.014192)[0.0103] [0.0755] [0.0002] [0.0000]0.000801 1.78E-06 0.061592 0.923463 0.985055

DAX (0.000307) (7.17E-07) (0.013120) (0.017482)[0.0090] [0.0131] [0.0000] [0.0000]0.000392 2.09E-05 0.143104 0.682189 0.8252994

ATX (0.000311) (3.97E-06) (0.018830) (0.052898)[0.2083] [0.0000] [0.0000] [0.0000]0.000324 2.40E-06 0.046694 0.944774 0.991468

OMV (0.000421) (8.36E-07) (0.007515) (0.008817)[0.4424] [0.0041] [0.0000] [0.0000]0.000834 3.07E-05 0.080059 0.900093 0.980152

Example1 (0.001028) (1.38E-05) (0.017367) (0.022698)[0.4169] [0.0259] [0.0000] [0.0000]W Ù Ë Ê=õ E-05 1.42E-05 0.099011 0.890440 0.989451

Example2 (0.000832) (8.21E-06) (0.023513) (0.024322)[0.9710] [0.0829] [0.0000] [0.0000]0.000997 3.54E-05 0.061999 0.922654 0.984653

Example3 (0.001327) (1.77E-05) (0.015077) (0.019353)[0.4523] [0.0457] [0.0000] [0.0000]0.000695 1.90E-06 0.039280 0.952265 0.991545

Example4 (0.000453) (1.11E-06) (0.008329) (0.010159)[0.1246] [0.0869] [0.0000] [0.0000]

4 Conclusions

I havepresenteda classof modelswherefinancialmarketsareconsideredasnon-linearadaptiveevolutionarysystems.Endogenouspricefluctuationsarecausedbytheinteractionof differenttypesof traders,whochoosepredictorsfor futureprices

18 A residualtestshowsthatthesquaredresidualsfrom theestimatedGARCH(1,1)modelhave almostnosignificantautocorrelations.


accordingto their performancein thepast,conditioneduponthestateof themar-ket. In ourmodelonly expectationsor forecastingrulesarechangingwhile every-thingelsestaysconstant.Theseexpectationsfeedbackinto themarketequilibriumequation,generatingnew pricesandthusaffect theexpectationsof investors.Theresultingvolatility of realizedmarket pricesis muchhigherthantheRE volatility(“expectationdrivenexcessvolatility”).

I haveshown for a simpleexamplewith only two typesof traders(fundamen-talistsor EMH-believersandtrendfollowers)anda commonlysharedview abouta constantfundamental,that the model is able to generatesomeof the stylizedfactsobservedin realdata,suchasexcesskurtosisandvolatility clustering.Excessvolatility is createdby thetradingprocessitself. Deviationsfrom thefundamentalvaluetriggeredby noisemay be amplifiedby technicaltrading.In real markets,wherethe“true” fundamentalvalueis not exactlyknown, goodor badnewsabouteconomicfundamentalsmaybereinforcedby evolutionaryforces,leadingto over-andundervaluationof risky assets.

Volatility clustering,with significantlow orderautocorrelationsin absolutere-turns,arisesin our modeldueto intermittency andcoexistenceof attractors,a sta-ble steadystateanda “larger” attractor– like a (quasi)periodicorbit or a chaoticattractor– dueto aChencinerbifurcation.We obtainthestrongestform of volatil-ity clusteringwhenthe systemis closeto having a unit root andfundamentalistsonly adaptslowly into thedirectionof thefundamentalvalue.It shouldbepointedout thatthemodelsof BH [9] andGaunersdorfer[26] werenotableto producethephenomenonof volatility clustering.In fact,it is not true– asoftenstatedascriticsagainstboundedrationality models– that one“can get everything” by deviatingfrom theconceptof RE andintroducingnew parametersin amodel.

Comparedto stochastictimeseriesmodelssuchastructuralnonlineareconom-ical modelaspresentedherehasthe advantagethat it givesinsight into the eco-nomic mechanismthat causesthe observedpatternsin financialdata.Our modelis still tractable,generatingfluctuationsin assetreturnssimilar to thoseobservedin real data.It may alsohelp to understandthe morecomplicatedcomputationaloriented,multi-agentsmodelsbasedon computersimulationsof assettrading.19

Intermittency andcoexistenceof attractorsaregenericphenomenaandoccurnat-urally in nonlineardynamicalsystems.Moreover, they arerobustwith respecttoandsometimesreinforcedby dynamicnoise.Suchphenomenamay alsoplay arole in themorecomplicatedsystems.

An interestingquestionis, if statisticaltestsareableto detectthestructureinthe data,which are generatedby a low dimensionaldynamicalsystembuffetedwith dynamicnoise.Thisdynamicnoisedestroyspredictabilityin returns,but pre-servesstructurein volatility, measuredby absolutereturns.In particular, it wouldbeinterestinghow statisticaltestsbehave in thecaseof coexistingattractors.

19 BH [11] and Brock, Hommesand Wagener[12] study so-calledLarge Type Limits(LTL) systems,deterministicapproximationsof marketswith many typesof traders,whereinitial beliefsaredrawn from somerandomdistribution. This typeof modelmayserve asa bridgebetweensimplemodelswith only a few typesof tradersand the computationalorientedmulti-agentmodels.


Acknowledgements I would like to thankCarsHommesandHelmut Elsingerfor manyhelpful andstimulatingdiscussions,EngelbertDocknerand two anonymousrefereesforusefulcomments.I amindebtedto FlorianWagenerfor makingtheplotsof Lyapunov expo-nents,to RobertTompkinsfor providing thereturnsdatato meandto Roy vanderWeideforproviding thesimulationsprogram.Supportfrom theAustrianScienceFoundation(FWF)undergrantSFB# 010 (‘Adaptive InformationSystemsandModelling in EconomicsandManagementScience’)is gratefullyacknowledged.

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