Endogenous uctuations in a simple asset pricing model …cial "nancial market are Arthur et al. (1997a,b) and LeBaron (1995). A. Gaunersdorfer / Journal of Economic Dynamics & Control

*Tel.: #43-1-29128-466; fax: #43-1-29128-464. I would like to thank Cars Hommes for manyuseful discussions and helpful hints. I also thank Herbert Dawid, Thomas Dangl, and EngelbertDockner for useful comments. Support from the Austrian Science Foundation (FWF) under grantSFB d 010 (&Adaptive Information Systems and Modelling in Economics and ManagementScience') is gratefully acknowledged.

E-mail address: gauner@"nance2.bwl.univie.ac.at (A. Gaunersdorfer)

Journal of Economic Dynamics & Control24 (2000) 799}831

Endogenous #uctuations in a simple assetpricing model with heterogeneous agents

Andrea Gaunersdorfer*

Department of Business Studies, University of Vienna, Bru( nner Stra}e 72, 1210 Vienna, Austria

Accepted 30 April 1999

Abstract

In this paper we study the adaptive rational equilibrium dynamics in a simple assetpricing model introduced by Brock and Hommes (System Dynamics in Economic andFinancial Models, Wiley, Chichester, 1997, pp. 3}44; Journal of Economic Dynamics andControl, 22, 1998, 1235}1274). Traders have heterogeneous expectations concerningfuture prices and update their beliefs according to a risk adjusted performance measureand to market conditions. Further, also their expectations about conditional variances ofreturns vary over time. We show that even for the simple case where agents can onlychoose between two di!erent predictors complicated dynamics arise and we analyse thebifurcation routes to chaos. ( 2000 Elsevier Science B.V. All rights reserved.

JEL classixcation: C61; G14; D84

Keywords: Heterogeneous expectations; Endogenous price #uctuations in "nancial mar-kets; Adaptive dynamics; Bifurcation; Chaos

0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 2 6 - 3

1. Introduction

For the last 40 years economic and "nance theory have been based on theassumption of rational behavior. It is assumed that agents have homogeneousexpectations and are fully rational in the sense that they immediately take allavailable information into consideration, optimize according to a model whichis common knowledge and are able to make arbitrarily di$cult logical inferen-ces. One paradigm of modern "nance is the e$cient market hypothesis (EMH).EMH postulates that the current price contains all available information andpast prices cannot help in predicting future prices. Sources of risk and economic#uctuations are exogenous. Therefore, in the absence of external shocks priceswould converge to a steady-state path which is completely determined byfundamentals and there are no opportunities for consistent speculative pro"ts.

However, there is evidence that markets are not always e$cient and a lot ofphenomena observed in real data cannot be explained by the EMH. Forexample, trading volume and volatility of returns in real markets are large andshow signi"cant autocorrelation, higher returns than expected and calendare!ects are observed. It seems that psychology and heterogeneous expectationsplay an important role in real markets.

Indeed, news on "nancial markets is always accompanied by pictures ofpeople engaged in wild activity which gives the impression that "nancialmarkets are dominated by actions of people. Traders often see markets aso!ering speculative opportunities and believe that technical trading is pro"t-able. In their opinion, something like a market psychology exists and herde!ects unrelated to market news can cause bubbles and crashes. Markets areseen as possessing their own moods and personalities.

In the 1930s, Keynes already argued that agents do not have su$cientknowledge of the structure of the economy to form correct mathematicalexpectations. Under this assumption, it is impossible for any formal theory topostulate unique expectations that would be held by all agents. In the Keynesianview, prices are not only determined by fundamentals but part of observed#uctuations is endogenously caused by nonlinear economic forces and marketpsychology. This implies that technical trading rules need not be systematicallywrong and may help in predicting future price changes. Empirical work whichhas shown that such trading rules may indeed outperform traditional stochastic"nance models includes, for example, Brock et al. (1992), and Genc7 ay andStengos (1997).

By now, there is enough statistical evidence both, to question the EMH and toconsider traders' viewpoints when modelling the price behavior on "nancialmarkets (see e.g. Shiller, 1991). Haugen (1998a,b,c) gives &a collection of theevidence and arguments, which constitute a strong and persuasive case fora noisy stock market that over-reacts to past records of success and failure onthe part of business "rms, and prices with great imprecision'.

800 A. Gaunersdorfer / Journal of Economic Dynamics & Control 24 (2000) 799}831

1Data quality and weakness of statistical tests makes it di$cult to detect chaos in real data.A presentation of techniques to detect chaos in empirical data can be found, for example, in Brocket al. (1991).

Also developments in the theory of nonlinear dynamical systems have con-tributed to new approaches in economics and "nance. Simple deterministicmodels may generate complex (chaotic) dynamics, similar to a random walk.Though empirical evidence for the existence of chaotic dynamics seems to berather weak, chaos o!ers a pragmatic shift in thinking about methods to studyeconomic activity.1 Introducing nonlinearities in the models may improve ourunderstanding of how economic processes work. Chaos is suggestive of path-ways to complex dynamics, it stimulates the search for a mechanism thatgenerates the observed movements in real "nancial data and that minimizes therole of exogenous shocks.

Recently, "nance literature has been searching for alternative theories thatcan explain observed patterns in "nancial data. A number of models weredeveloped which build on boundedly rational, non-identical agents. Financialmarkets are considered as systems of interacting agents which continually adaptto new information. Heterogeneity in expectations can lead to market instabilityand complicated dynamics.

One approach to explain movements in "nancial returns are models withinformed and uninformed traders or with &irrational noise traders' (see forexample Grossman, 1989; Black, 1986; Shleifer and Summers, 1990; De Longet al., 1990a,b; Allen and Gorton, 1993). Brunnermeier (1998) provides a surveyof the literature on the informational aspects of price processes. The literature onbehavioral "nance (see Thaler, 1991, part V, and the papers in Thaler, 1993, partIII) emphasizes the role of quasi-rational, overreacting, and biased traders.

Another view is that agents are intelligent, but since agents do not havecomplete knowledge about the underlying model and do not have the computa-tional abilities assumed in rational expectations models, equally informedtraders may interpret the same information di!erently. This results in heterogen-eous beliefs about the market, traders evaluate their forecasts and trade on thosepredictors which perform best. Prices are therefore driven endogenously andagents' expectations co-evolve in a world they co-create.

A number of structural models with a few strategy types have been introducedemphazising this heterogeneity in expectations formations. Typically, they in-clude fundamental and technical traders (chartists). Examples are Beja andGoldman (1980), Chiarella (1992), Frankel and Froot (1988), Ghezzi (1992), Dayand Huang (1990), De Grauwe et al. (1993), Sethi (1996), Franke and Sethi(1993), Lux (1994, 1995), Lux and Marchesi (1998). Computational modelsrelying on arti"cial intelligence methods where many traders interact in anarti"cial "nancial market are Arthur et al. (1997a,b) and LeBaron (1995).

A. Gaunersdorfer / Journal of Economic Dynamics & Control 24 (2000) 799}831 801

Kurz (1997) presents an equilibrium concept, which he calls rational beliefequilibrium. He introduces heterogenous beliefs about probability distributionswhich give rise to endogenous uncertainty. Agents concentrate on the empiricalconsistency of their forecasting model. Grandmont (1998) and Hommes andSorger (1998) develop models where expectations may become self-ful"lling.Related approaches are also evolutionary models such as Blume and Easley(1992) and learning models (see for example Sargent, 1993). Barucci and Posch(1996) show the emergence of complex beliefs dynamics in linear stochasticmodels as the outcome of boundedly rational learning.

Brock and Hommes (1997a), hereafter BH, consider Adaptive Belief Systems tostudy heterogeneous expectations formations. Agents adapt their predictions bychoosing among a "nite number of predictor or expectations functions whichare functions of past information. Each predictor has a performance measureattached which is publically available to all agents. Based on this performancemeasure agents make a (boundedly) rational choice between the predictors. Thisresults in the Adaptive Rational Equilibrium Dynamics (ARED), an evolutionarydynamics across predictor choice which is coupled to the dynamics of theendogenous variables. BH show that the ARED incorporates a general mecha-nism which may generate local instability of the equilibrium steady state andcomplicated global equilibrium dynamics.

Brock and Hommes (1997b, 1998) apply this concept to a simple asset pricingmodel, building on a theoretical framework formulated by Brock and LeBaron(1996), where traders in a "nancial market use di!erent types of predictors fortheir price forecasts of a risky asset (see also Brock (1997), for a generalformulation of the model and possible generalizations). In this model, underhomogeneous, rational expectations and the assumption of an independentlyidentically distributed dividend process, prices are constant over time. However,introducing heterogeneous price expectations changes the situation substan-tially. Market dynamics may then be characterized by an irregular switching ofperiods where prices are close to the (EMH) fundamental, phases of &optimism'where prices rise far from the fundamental * traders become excited andextrapolate trends by simple technical analysis * and &pessimistic' periodscharacterized by a sharp decline in prices. This irregular switching is caused bythe rational choice between predictors as described above. BH call it &the marketis driven by rational animal spirits'. They show that increasing the &intensity ofchoice' to switch between predictors may result in a bifurcation route tocomplicated price #uctuations where price dynamics takes place on a chaotic(strange) attractor. The model nests the usual rational expectations type ofmodel, e.g. a class of models that are versions of the EMH. But these rationalexpectations beliefs are costly and &compete' with other types of beliefs ingenerating net trading pro"ts as the system evolves over time.

The main purpose of this paper is to extend this model in two ways. We adjustthe performance measure of the predictors by a term capturing risk aversion. It


2BH (1998) give a brief introduction in numerical analysis in nonlinear dynamics. We will notrepeat it in this paper. General references on the theory of nonlinear dynamics and bifurcationtheory are, for example, Guckenheimer and Holmes (1986), Arrowsmith and Place (1990), andKuznetsov (1995).

turns out that predictor choice is then determined by squared prediction errorsof price forecasts. Further, agents do not only update their conditional expecta-tions of prices in every period but also their beliefs about conditional variancesof returns. In every period traders estimate variances as exponential movingaverages of past returns. We focus on a simple version of the model with twotypes of traders, fundamentalists and trend extrapolators (trend chasers). Toprevent prices to diverge to in"nity we introduce a &stabilizing force'. Thisstabilizing force may be interpreted as technical traders do not use the trendchasing predictor if prices are too far away from the fundamental value, even ifthat predictor performed best in the recent past. Predictor choice and thereforethe fractions of the two types of traders are thus not only determinded by pastperformance of the predictors but also conditioned on market conditions. Thequestion is whether the &rational route to randomness' is similar to that in BH(1997a,b, 1998). We give a detailed bifurcation analysis2 applying local bifurca-tion theory to detect primary and secondary bifurcations of the steady statesand using the LOCBIF bifurcation package (Khibik et al., 1992). Further, we usenumerical tools such as phase portrait analysis, bifurcation diagrams, Lyapunovcharacteristic exponents, and compute invariant manifolds to demonstrate theemergence of strange attractors.

This paper is organized as follows. In Section 2 we brie#y recall the assetpricing model of BH and explain our extensions to the model. We study the localbehavior of the system near steady states in Section 3 and the global dynamics inSection 4. By numerical analysis we show the existence of horseshoes andstrange attractors for a wide range of parameter values. Section 5 concludes.Proofs and details of derivations are given in an appendix.

2. The model

We brie#y recall the model used in BH (1997b, 1998). They consider an assetpricing model with one risky asset and one risk-free asset available with grossreturn R. p

tdenotes the price (ex-dividend) of the risky asset and My

tN the

dividend process which is assumed to be independently and identically distrib-uted (IID). The dynamics of wealth is described by

Wt`1

"R=t#R

t`1zt,


where ztis the number of purchased shares of the risky asset at time t and

Rt`1

"pt`1

#yt`1

!Rptis the excess return per share. Bold face type denotes

random variables. We write Et, <

tfor the conditional expectation and condi-

tional variance operators at time t, based on a publically available informationset of past prices and dividends, F

t"Mp

t, p

t~1,2, y

t, y

t~1,2N. The &beliefs' of

investor type h about these conditional expectation and variance are denoted byEht

and <ht.

Assuming that investors are myopic mean variance maximizers the demandfor shares z

htby type h solves

maxGEhtW

t`1!

a

2<

htW

t`1H , i.e. zht"

Eht

Rt`1

a<ht

Rt`1

, (1)

where a50 characterizes risk aversion. Let zst

and nht

denote the supply ofshares per investor and the fraction of investors of type h at time t, respectively.Equilibrium of supply and demand implies

+h

nhtzht"z

st. (2)

Assuming constant supply of outside shares over time we may, without loss ofgenerality, stick to the (equivalent) special case z

st,0 (for the general case see

Brock (1997)).To obtain a benchmark notion of &fundamental solution' consider the case

where there is only one type. Then Eq. (2) becomes Rpt"E

tpt`1

#y6 ;Etyt`1

,y6 is constant since MytN is assumed to be IID. The fundamental solution

pHt,p6 is the only solution which satis"es the &no bubbles' condition

limt?=

(EpHt/Rt)"0, i.e. p6 "y6 /(R!1). Note that p6 is the expectation of the

discounted sum of future dividends. In what follows we express prices indeviations from the benchmark fundamental, x

t"p

t!pH

t"p

t!p6 .

Beliefs are assumed to be of the form

Eht( p

t`1#y

t`1)"E

t( pH

t`1#y

t`1)#f

h(x

t~1,2,x

t~L)"RpH

t#f

ht

for some deterministic function fh. Note that f

ht:"f

h(x

t~1,2,x

t~L)"E

htxt`1

isthe conditional expectation of type h for the price deviation from a commonlyshared fundamental. As in BH (1997b, 1998) we only consider beliefs of thesimple form f

ht"g

hxt~1

#bh. The equilibrium equation (2) now becomes

Rxt"+

h

nht

fht. (3)

Fractions are determined as discrete choice probabilities (cf. Anderson et al.,1993)

nht"exp(b;

h,t~1)/Z

t, Z

t"+

h

exp(b;h,t~1

),


3 It can be shown that under the given assumptions xt`1

is a deterministic function of (xt,x

t~1,2)

(see Brock, 1997).

4Notice that this maximum problem is equivalent to (1), up to a constant, so the optimal choice ofshares of the risky asset is the same.

where;ht

is some &"tness function' or &performance measure'. The parameter b iscalled the intensity of choice. It measures how fast agents switch betweendi!erent predictors, i.e. it is a measure of traders' rationality. For b"0 fractionsare "xed over time and are equal to 1/N, where N is the number of di!erenttypes of traders. If b"R all traders choose immediately the predictor with thebest performance in the recent past. Thus, for "nite, positive b agents areboundedly rational in the sense that fractions of the predictors are rankedaccording to their "tness. The parameter b plays a crucial role in the bifurcationroute to chaos.

Let

ot:"E

tR

t`1"E

txt`1

!Rxt"x

t`1!Rx

t

denote the rational expectations of excess returns3 and let

oht

:"EhtR

t`1"E

htxt`1

!Rxt"f

ht!x

t`1#o

t

be the conditional expectations of type h. De"ne risk adjusted realized pro"ts

nht

:"n(ot, o

ht) :"o

tz(o

ht)!

a

2z (o

ht)2<

htR

t`1, (4)

where the demand for shares

z(oht)"

oht

a<htR

t`1

is the solution of the maximum problem4 maxzMo

htz!(a/2)z2<

htR

t`1N and

<htR

t`t"<

ht(x

t`1!Rx

t#d

t`1)

"<ht

(xt`1

!Rxt)#<

htdt`1

is the belief of type h about the the conditional variance. dt`1

is a martingale dif-ference sequence (see BH, 1998, Eq. (2.11)). We assume Cov(x

t`1!Rx

t,

dt`1

),0,<htdt`1

":p2d to be constant, and<ht(x

t`1!Rx

t)"<

t(x

t`1!Rx

t)":

p8 2t. Thus, we assume that agents have homogeneous expectations on conditional

variances of returns. Nelson (1992) (see also Bollerslev et al., 1994) providessome justi"cation for this assumption. He shows that conditional variances aremuch easier to estimate than conditional means, hence there should be moredisagreement about the mean than about the variance among the traders. BH


(1997b, 1998) study the case where the belief about conditional variances ofreturns is constant over time, <

ht,p2. Here we allow for time varying beliefs

about variances. This is an important generalization of the model. Agentsobserve price behavior to update their beliefs about prices. It seems naturalthat they use the observed data also to update their estimations of the variancesof returns. We assume that traders estimate p8 2

tas exponential moving

averages

p8 2t"wpp8 2t~1

#(1!wp)(xt~1!Rx

t~2!k8

t~1)2,

k8t"wkk8 t~1

#(1!wk)(xt~1!Rx

t~2),

where wp,wk3[0,1]. k8tde"nes the exponential moving averages of returns.

Let us return to the discussion of the performance measure;ht. The "rst term

of nht

in (4) denotes realized pro"ts for type h, the second term captures riskaversion. n

tis de"ned by analogy, dropping index h. BH (1997b, 1998) concen-

trate on the case without risk adjustment, i.e. they drop the second term in (4). Inthis paper we study the model with risk adjustment. The fractions of the di!erenttypes at the end of period t will be determined by the &"tness function' n

h,t~1.

Subtracting o! the same term nt~1

for all types does not change the discretechoice fractions. For updating the fractions we therefore use the di!erence in(risk adjusted) pro"ts of type h beliefs and rational expectations beliefs:

;ht"dn (o

t~1,o

h,t~1) :"n

h,t~1!n

t~1"!

1

2a(p8 2t~1

#p2d )(x

t!f

h,t~1)2.

Thus, in contrary to BH (1997b, 1998) where the "tness measure of each type isdetermined by past realized pro"ts, using risk adjusted pro"ts, squared predic-tion errors determine the predictor choice. Notice that also Arthur et al.(1997a,b) look at squared forecast errors to form expectations about futureprices in their arti"cial market.

The timing of expectations formations is important. In updating fractions ofbeliefs in each period, the most recently observed prices and returns are used.Agents take positions in the market in period t based on forecasts they make forperiod t#1. Which predictors they use depends on the performance of thispredictor in period t!1. That is, at the end of period t!1 (beginning of periodt), after having observed price x

t~1, the fractions n

htand the expectations

oh,t~1

and p8 2t~1

are formed. Note that fh,t~1

and p8 2t~1

depend on xt~2

,xt~3

,2.In period t price x

tis determined, which de"nes o

t~1, etc.

For the class of fundamentalists we will make two further adaptions of the"tness measure, by introducing costs and a &stabilizing force'. We de"ne;I

ht:";

ht!C#ax2

tfor fundamentalists (traders who believe that prices will

return to their fundamentals) and ;Iht

:";ht

for all other types of traders.Fundamentalists have all past prices and dividends in their information set,however, they do not know the fractions of the other belief types. They act as if


all agents were fundamentalists which means that they are not perfectly rational.Costs of the fundamental predictor, C, might be positive since it takes somee!ort to understand how the market works and to believe that it will priceaccording to the fundamental. Thus, the performance function (realized pro"ts)of the fundamental predictor has to be reduced by costs C50.

a50 de"nes an exogenous stabilizing force which, when the derivation fromthe fundamental price becomes too large, should drive prices back to thefundamental. As we will see, the evolutionary dynamics gets easily dominated bytrend followers and prices will grow exponentially without bound. Introducingthis stabilizing parameter means that fundamentalists get more weight as pricesmove further away from the fundamental and the evolutionary dynamics willremain bounded if a is large enough. Thus, fractions are not only determined bythe predictor performance but also by market conditions. If prices are too faraway from the fundamental price, technical traders might not use the trendchasing trading rule. Even when its prediction in the recent period was betterthan the fundamental trading rule, they do not believe that this predictor willalso perform better in future periods. Arthur et al. (1997a,b) also introducecondition/forecast rules for predictors that contain both, a market conditionwhich determines if a certain predictor is used and a forecasting formula for nextperiod's price and dividend. At this point our stabilizing force is exogenouslyspeci"ed and ad hoc. De Grauwe et al. (1993) formulate weights for chartists inan analogous way. In future work one might incorporate &far from equilibrium'forces driving prices back to the fundamental, such as e.g. futures markets orlong-term traders on fundamental. Adding those more realistic economic forcesmight give similar dynamics as the stylized model above. It seems useful to getmore insight into the dynamics of this stylized model before adding anotherlayer of complexity.

The evolution of equilibrium prices, fractions, and beliefs about conditionalvariances is summarized by the Adaptive Rational Equilibrium Equation:

Rxt"

H+h/1

nht

fht, (5)

nht"exp (b;I

h,t~1)/Z

t, (6)

p8 2t"wpp8 2t~1

#(1!wp) (xt~1!Rx

t~2!k8

t~1)2, (7)

k8t"wkk8 t~1

#(1!wk) (xt~1!Rx

t~2). (8)

Considering (5) and (6), prices in period t are determined by ;Ih,t~1

, i.e. bydn(o

t~2,o

h,t~2)"!(x

t~1!f

h,t~2)2/(2ap8 2

t~2). Introducing p

t:"p8

t~1and k

t:"

k8t~1

and replacing (7) and (8) by the corresponding equations for p2t

andktreduces the dimension of the system by one (see below).We restrict to a simple but typical case with only two types of traders,

fundamentalists (type 1) and trend chasers (type 2), i.e. f1t,0 and f

2t"gx

t~1


(g'0). This gives the following dynamical system:

Rxt"n

2tgx

t~1,

n1t"expCbA!

1

2a(p2t~1

#p2d )x2t~1

#ax2t~1

!CBD/Zt,

n2t"expC!

b2a (p2

t~1#p2d )

(xt~1

!gxt~3

)2D/Zt, (9)

p2t"wpp2

t~1#(1!wp)(xt~2

!Rxt~3

!kt~1

)2,

kt"wkkt~1

#(1!wk)(xt~2!Rx

t~3),

which is a third-order di!erence equation. Setting yt:"x

t~1and z

t:"x

t~2and

de"ning

mt:"1!2n

2,t`1"n

1,t`1!n

2,t`1

"tanhCb2A

g

2a(p2t~1

#p2d )xt~2

( gxt~2

!2xt)#ax2

t!CBD,

(9) is equivalent to

xt"

g

2Rxt~1A1!tanhC

b2 A

g

2a (p2t~1

#p2d )zt~1

( gzt~1

!2xt~1

)

#ax2t~1

!C)DB,

yt"x

t~1,

zt"y

t~1,

(10)

p2t"wpp2

t~1#(1!wp) (yt~1

!Rzt~1

!kt~1

)2,

kt"wk k

t~1#(1!wk) (yt~1

!Rzt~1

).

This system is "ve dimensional, in contrast to BH (1997b, 1998) where thecorresponding case with two types of traders is three dimensional. Allowingtime-dependent beliefs about conditional variances of returns introduces twoadditional dynamical variables, the average return k

tand the estimate of the

conditional variance p2t. The model di!ers also in the way fractions are deter-

mined. We adjust the performance measure by a term of risk aversion andcondition the fractions of the di!erent types on market conditions by introduc-ing the parameter a. This allows for a two-parameter bifurcation analysis withrespect to the intensity of choice b and the strength of the stabilizing force a.


5At bM "minMbH, b

FN, de"ned in Proposition 2, a Hopf or #ip bifurcation occurs and the two

non-fundamental steady states become unstable.

3. Steady states and local bifurcations

In this section we study the local behavior near the steady states of the system.Our bifurcation analysis is supported by the program package LOCBIF(Khibik et al., 1992). The LOCBIF program is based on numerical methods tocompute steady states, periodic orbits and bifurcation curves in the parameterspace which are described in Kuznetsov (1995, Chapter 10).

We "rst restrict to the case with constant beliefs about variances,wp"wk"1, and denote p2

t,p2. In this case we can restrict to the three-

dimensional system de"ned by the "rst three equations of (10).

Proposition 1 (Existence and stability of steady states of (10)). Let C'0.

1. For 0(g(R, E1"(0,0,0) is the unique, globally stable steady state ( funda-

mental steady state).2. For R(g(2R a pitchfork bifurcation occurs at bH:"(1/C) log[g/(g!R)].

E1

is stable for b(bH and unstable for b'bH.

(a) For a(aH :"!(1/2ap2) g (g!2) the pitchfork bifurcation is subcritical, i.e.there exist two unstable non-fundamental steady states for b(bH.

(b) For a'aH the pitchfork bifurcation is supercritical, i.e. there exist two stablenon-fundamental steady states for bH(b(bM .5

3. For g*2R there exist three steady states. The fundamental steady state E1

isunstable for all b'0. The non-fundamental steady states are stable for b(bM(see footnote 5).

Remark. Result 3 also holds for the case C"0.

Proof. See the appendix.

Results 1, 2(b), 3 coincide with the results in Proposition 2 in BH (1998). Thus,di!erent performance measures may yield similar primary bifurcations towardsinstability of the evolutionary dynamics. Result 2(a) shows that when there arecosts for the fundamental predictor prices might explode if the stabilizing forcea is small. Near the fundamental steady state both predictors yield smallforecasting errors. Since the fundamental predictor is costly, agents choose thetrend chasing forecasting rule which drives prices away from the fundamental.Then the forecasting error of the fundamental predictor becomes large, so there


6We use these parameter values for all "gures in this paper (in examples where the variance is timedependent, p2 has to be replaced by p2d ).

is no force which brings this upward (or downward) trend to an end. This willnot happen in real markets. When prices are too far from &fundamentals' theremust be some forces which drive them back, though it is not really clear whatkind of forces these are. If there are more types of traders in the market thismight also prevent prices to explode (cf. our discussion in the previous section).

The next proposition shows that the non-fundamental steady states becomeunstable either by a Hopf or a #ip bifurcation when the intensity of choice b isincreased.

Proposition 2 (Secondary bifurcations). Let g'R and b'bH. When b is in-creased, the non-fundamental steady states become unstable either by a Hopf or bya yip bifurcation.

1. For small a, more precisely, for aH(a(aHH:"g2/2ap2, the non-fundamentalsteady states undergo a Hopf bifurcation at a certain value b

H(a) (dexned by Eq.

(20) in the appendix).2. For a'aHH the non-fundamental steady states undergo a yip bifurcation at

bF:"

1

CAg

g!R#log

R

g!RB.


Fig. 1 shows a detailed bifurcation diagram on the ( b, a)-plane (for a"10,p2"0.1, C"1, R"1.01, g"1.2)6 which we generated by the use of theLOCBIF program.

Depending on a, there are di!erent bifurcation routes possible as b increases.If the non-fundamental steady states are destabilized by a Hopf bifurcation (i.e.bH(a)(b

F) a closed invariant curve "lled by quasiperiodic orbits is created. For

certain b-values phase locking phenomena occur that create and destroy stableand unstable periodic orbits. These periodic orbits are located on the closedinvariant curve. Actually, an in"nite number of phase locking windows havingthe form of small tongues* so-called Arnold tongues* are rooted in the Hopfbifurcation curve (see Kuznetsov (1995, Chapter 7.3) for a mathematical treat-ment of this phenomenon). Their origin points correspond to eigenvalues j ofthe Jacobian of the non-fundamental steady states with arg j"arg (A#Bi)"2pp/q for rational p/q, reduced, lying on the unit circle. Arnold tongues de"neparameter regions where points of period q occur. They are delimited by


Fig. 1. Two parameter bifurcation diagram w.r.t. b and a. 2-Hopf denotes a Hopf bifurcation curvefor a 2-cycle, 2-#ip and 4-#ip denote #ip bifurcation curves of 2- and 4-cycles, respectively; SNndenote saddle-node bifurcation curves of period n points. The dots on the Hopf bifurcation curvede"ne origin points of p:q Arnold tongues. Note, that B"(bH, aH) and C"(b

F, aHH).

saddle-node bifurcation curves of period q points, corresponding to a collisionbetween stable and unstable periodic orbits of F (the map which de"nes thedynamical system (10)). In Fig. 1, a 1:10 Arnold tongue is drawn and originpoints of some further Arnold tongues are plotted. As b increases, p/q increases.

B"(bH,aH) is the intersection point of the pitchfork bifurcation curve, theHopf bifurcation curve (20), and the line g (g!2)#2ap2a"0 (see Fig. 1). AtB the pitchfork bifurcation changes from sub- to supercritical and expression(13), which determines the non-fundamental steady states, is not de"ned. Whenapproaching B along the Hopf bifurcation curve one eigenvalue of the Jacobianof the non-fundamental steady states converges to !1

2and the two complex

conjugate eigenvalues on the unit circle converge to 1, i.e. arg j2,3

"0 (fordetails see the appendix). A situation where a double eigenvalue 1 occurs wouldcorrespond to a 1:1 resonance (see for example Kuznetsov, 1995, Chapter 9.5.2).However, in our model B is a singularity on the (b, a)-plane, the dynamicalsystem only consists of "xed points for a"aH. In the appendix we show thatwhen approaching B from di!erent directions the eigenvalues have di!erentlimits.

At point C"( bF, aHH) a codimension two bifurcation occurs. The Jacobian of

the non-fundamental steady states has one eigenvalue j1"!1 and two complex


7 In the class of all dynamical systems C would be a point of codimension three. In our specialfamily of systems, however, the 1:6 resonance follows from the simultaneous occurence of the Hopfand #ip bifurcations.

conjugate eigenvalues j2,3

"A$Bi lying on the unit circle. Sincej1"!j

2j3/(j

2#j

3)"!(A2#B2)/2A (c.f. appendix, proof of Proposition 2),

A"12

and B"J32

, thus arg j2,3

"arctanJ3"2p16. Point C is the origin of

a 1:6 Arnold tongue.7Between the points C and B any resonance p : q with 0/1(p/q(1/6 can be

found, which implies in particular all 1:q (q'6) resonances. p/q increasesmonotonically as b is increased. The computation of these points is described inthe appendix.

For values a'aHH we either have bF(b

H(a) or b

H(a) NR and the secondary

bifurcation of the non-fundamental steady states is a #ip bifurcation. In Fig. 1,besides the Hopf bifurcation curve of the non-fundamental steady state de-scribed in Proposition 2, a Hopf bifurcation curve of the 2-cycle (which wascreated by the #ip bifurcation) starts in point C. Thus, for the chosen parametervalues, the 2-cycle becomes unstable by a Hopf bifurcation which results in twoattracting invariant cycles. After a saddle-node bifurcation of the sixth iterate ofF a stable (and an unstable) 6-cycle is created, lying in an Arnold tongue whichoriginates in C. From the bifurcation diagram in Fig. 2(a) we conclude that this6-cycle is destabilized by a period doubling bifurcation.

Let us now return to the general case of time varying beliefs on conditionalvariances of returns, i.e. wp,wk3 (0, 1). The equilibria of the "ve-dimensionalsystem (10) are the fundamental steady state E

1"(0,0,0,0,0) and the non-

fundamental steady states E2"(xH, xH, xH, 0,( 1!R)xH) and E

3"!E

2, where

xH is given by (13), replacing p2 by p2d .

Proposition 3. In the case of time varying beliefs about conditional variances ofreturns (0(wp, wk(1) the primary bifurcation of the fundamental steady state andthe secondary bifurcations of the non-fundamental steady states are identical as inthe case with constant beliefs about variances (wp"wk"1).


This means that the results of Propositions 1 and 2 also hold for wp,wk3(0,1).That is, the primary and secondary bifurcations of the case with constant beliefsabout variances are exactly the same as in the case with time varying beliefsabout variances. Higher-order bifurcations might be di!erent as can be seenfrom Fig. 2(a) and (b). For example, for the chosen parameter values the 2-cyclewhich is generated by the #ip bifurcation undergoes a further period doublingbefore it is destabilized by a Hopf bifurcation.


Fig. 2. Bifurcation diagrams plotting long-run behavior versus intensity of choice b for (a) constantbeliefs about conditional variances of returns, (b) time varying beliefs about conditional variances ofreturns (wvar"wp, wret"wk ).

Numerical simulations show that increasing b further leads to the occurrenceof strange attractors. We will study the global dynamics and the occurrence ofstrange attractors in the next section.

4. Global dynamics

Proof of Proposition 1 shows that for large b-values the fundamental steadystate is unstable with one unstable and two stable (four in the case of time


8The stable and unstable manifolds of a steady state x0

of a dynamical system corresponding tothe map F are de"ned as =s(x

0)"Mx: lim

n?=Fn(x)"x

0N and =u(x

0)"Mx: lim

n?~=Fn(x)"x

0N,

resp., where Fn denotes the nth iterate of F.

9For a brief introduction to the concept of homoclinic orbits see BH (1997a) and BH (1997b,1998), Goeree and Hommes (1999) in this issue also recall this notion. An extensive mathematicaltreatment of homoclinic bifurcation theory can be found in Palis and Takens (1993).

10For the notion of a strange attractor see for example Palis and Takens (1993, Chapter 7.2).A map F has a horseshoe if there exist rectangular regions R such that Fn(R) is folded over R in theform of a horseshoe, for some n3N.

varying variances) directions. If the unstable and the stable manifolds8 intersect(in a so-called homoclinic point) this gives rise to very complicated behavior (aswas already noticed by PoincareH at the end of last century). To understand theglobal dynamics of the system it is thus useful to study the geometric shapes ofthese manifolds. It will also give insight into the economic mechanism generat-ing complicated #uctuations.

In order to get insight into the dynamics we "rst consider the limiting caseb"R. In this case the stable and unstable manifolds can be analyzed analyti-cally. In what follows we suppress the index 1 for the fundamental steadystate E

1.

Proposition 4. Let C'0 and b"R. If

g'R and a'R2

2ap2dg2(2g!R2)

the unstable manifold of the fundamental steady state E is bounded and all orbitsconverge to the saddle point E.


This proposition shows that for b"R orbits remain bounded for a widerange of parameter values. Since the fundamental steady state is unstable pricesmove away, above or below the fundamental, but will return close to it. Thus,the unstable and stable manifolds have to intersect in a so-called homoclinicpoint. This suggests that for large but "nite b the system is close to havinga homoclinic orbit.9 Homoclinic points imply very complicated behavior andpossibly the existence of strange attractors. Smale (1965) showed that a homo-clinic point implies the existence of (in"nitely) many horseshoes10 which leadsto a situation called topological chaos (i.e. there is a closed invariant set thatcontains a countable set of periodic orbits, an uncountable set of non-periodicorbits, among which there are orbits passing arbitrarily close to any point of theinvariant set, and exhibits sensitive dependence on initial conditions).


11Note, that when the system is studied in the (x,y,z)-plane, (0,1,0) is a generalized eigenvector.

For further analysis of the global geometric properties of the unstable mani-fold we restrict to the case with constant beliefs about variances (wp"wk"1).Recall that the dynamical system is three dimensional in that case. We rewritesystem (10) in the (x, y, m)-space, which yields

xt"

g

2Rxt~1

(1!mt~1

),

yt"x

t~1,

mt"tanhC

b2A

g2

2ap2yt~1Ayt~1

!

1

Rxt~1

(1!mt~1

)B#a

g2

4R2x2t~1

(1!mt~1

)2!CBD. (11)

We will use the notation (xt`1

, yt`1

, mt`1

)"Fb(xt, y

t,m

t).

Next we consider the stable and unstable manifolds of the fundamental steadystate E. The stable manifold of E is tangent to the plane S"Mx"0N since thestable eigenspace is spanned by the eigenvectors (0,1,0) and (0,0,1).11 For b"R

the stable manifold of E contains the plane S since every point in S is mappedonto E. In this case agents switch in"nitely fast between the predictors, thedi!erence of the fractions is given by (22), resp. by

mt"G

#1 ifg2

2ap2yt~1

(yt~1

!

1

Rxt~1

(1!mt))#a

g2

4R2x2t~1

(1!mt)2'C,

!1 ifg2

2ap2yt~1

(yt~1

!

1

Rxt~1

(1!mt))#a

g2

4R2x2t~1

(1!mt)24C.

(12)

Any point lying on the plane Mm"1N is mapped onto the plane S. Let A0

be thepoint on the unstable manifold where all agents switch from being trend chasersto fundamentalists, i.e. the m-coordinate of A

0is !1, but on the the right-hand

side of (12) equality holds. The line segement EA0

lies on the unstable manifoldof E. De"ne A

1:"F

=(A

0), A@

1:"(x(A

1), y(A

1),1), the point with the same x and

y coordinates as A1

lying on the plane Mm"1N (x(A1) denotes the x-coordinate

of A1, etc.), F

=(A

1)": A

2, F

=(A@

1)":A@

2, F

=(A

2)": A

3. The segment A@

2A

3is

mapped onto (0,0,1)": A4

and A4is mapped to E (see Fig. 3). Thus, the "rst "ve

iterates of EA0

are

F=(EA

0)"EA

1,

F2=(EA

0)"EA

1XA@

1A

2,


Fig. 3. Unstable manifold for case 1 (a"0.65, b"500): "rst 6 iterates of the segment EA0.

F3=(EA

0)"EA

1XA@

1A

2XA@

2A

3,

F4=(EA

0)"EA

1XA@

1A

2XA@

2A

3XA

4,

F5=(EA

0)"EA

1XA@

1A

2XA@

2A

3XA

4XE.

F5=(EA

0) de"nes the unstable manifold for b"R. Our analysis implies that for

large but "nite b the system must be close to having a homoclinic orbit betweenthe stable and the unstable manifolds. But to see whether this implies complic-ated dynamics one needs complicated horseshoe constructions.

A numerical analysis of the unstable manifold for "nite b-values is presentedin the appendix. We observe two cases. For su$ciently large a-values (case 2, seeFig. 7) the unstable segment EA

0is expanded and folded over (close to) itself by

the 6th iterate of Fb, which suggests that F6b has a full horseshoe and thussuggests the occurrence of chaotic dynamics. For smaller a-values (case 1, seeFig. 3) our analysis does not show the existence of horseshoes. However,numerical simulations suggest that chaos arises in that case also.


12BH (1998) brie#y explain the concept of Lyapunov exponents, see also the references there.Medio (1992, Chapter 6), also gives an introduction into that concept.

To show that chaos arises indeed we compute the largest Lyapunov character-istic exponents12 (LCE) for the system. LCE measure the asymptotic exponentialrate of divergence (resp. convergence) of two trajectories starting close to eachother. Hence a positive LCE means that nearby trajectories separate exponenti-ally as time goes by and the system exhibits sensitive dependence on initial statesand therefore chaos. When b becomes su$ciently large we "nd positive LCE forboth cases, 1 and 2, and also for the model with time varying beliefs aboutconditional variances. Thus, a high intensity of choice gives rise to chaoticdynamics. Fig. 4 shows the largest LCE for examples corresponding to case 1.

Figs. 5 and 6 show phase portraits in the (xt,x

t~1)-plane and time series of

strange attractors. We observe that the shape of the attractors may di!er for thecases with constant and time varying beliefs about variances as long as theintensity of choice is not too high. For large b the projections of the attractorslook very similar to the projection of the unstable manifold of E on the (x

t,x

t~1)-

plane (cf. Figs. 3 and 7) for all cases.Our analysis also gives insight into the underlying economic mechanism.

When prices are close to the fundamental both predictors give good forecasts.Since the fundamental predictor is costly most agents choose the trend chasingpredictor which causes prices to move away from the fundamental value. Ata certain point the stabilizing force a gives enough weight to the fundamentaliststo push prices back to the fundamental. This leads to an irregular switchingbetween periods where prices are close to and periods where prices are far aboveor below the fundamental. Fig. 6(a) shows a time series with such an irregularswitching between periods where prices are close to the fundamental steadystate, followed by an upward trend, after some unstable phase away from thefundamental value returning back and getting stuck near the locally unstablefundamental steady state and later moving away again. Fig. 6 also shows timeseries of returns. We observe periods with high and low volatilities.

For prices to return to the fundamental value the intensity of choice has to behigh enough. In our numerical examples we observe that, in the case where thebeliefs about variances vary over time, the intensity of choice b has to be evenlarger, compared to the case with constant beliefs about variances (cf. Fig. 5).This stems from the fact that the same parameter values are used for p2 and p2d ,respectively. Therefore, the total variance in the case of time varying variances,p2t#p2d , is larger and hence the performance ;

htis lower, which has a similar

e!ect as decreasing b.Note that the system exhibits a symmetry with respect to the m-axes and the

positive octant of the state space is invariant. That is, taking initial conditions


Fig. 4. Largest Lyapunov characteristic exponents.

where prices lie below the fundamental value results in a dynamics which issymmetrical, with negative price deviations. However, adding some noise tosystem, as prices come close to the fundamental the market might switchbetween &optimistic' periods where prices are higher than the fundamental and&pessimistic' periods with prices #uctuating below the fundamental value.

5. Conclusions

We have studied a simple asset pricing model following BH (1997b, 1998)where traders may choose between two predictors for future prices, a costly


Fig. 5. Projections of strange attractors on the (xt,x

t~1)-plane.

Fig. 6. Time series of price deviations from the fundamental xt, di!erences in the fractions m

t, and

returns for (a) constant beliefs about variances, (b) time varying beliefs about variances. (b) alsoshows the time series of the variances p2

t.


Fig

.7.

Unst

able

man

ifold

forca

se2

(a"

0.75

,b"

2000

):(a

)the"rs

t4

iter

ates

oft

hese

gmen

tEA

0,(b)t

he4t

hiter

ate

and

the

begi

nni

ng

and

end

(dotted

line)

ofth

e5t

hiter

ate

ofA

0A@ 0A

1.


predictor based on fundamentals and a technical, trend chasing predictor.Choice of predictors is not only determined by their performance in the recentpast but also by market conditions. Though the model is still very simple andstylized it possesses rich dynamics. As the intensity of choice to switch betweenpredictors increases, di!erent bifurcation routes to chaotic price dynamics arepossible. They depend (among other things) upon the strength of the stabilizingforce a. One could interpret a as a parameter that determines the probablilitythat traders do not choose the technical predictor when prices are too far awayfrom the fundamental value, even when its recent performance was good. Wehave observed di!erent periods in the market where prices switch between stablephases close to the fundamental and unstable periods with prices much higheror much lower than the fundamental value.

Introducing the parameter a as a stabilizing force has allowed for a twoparameter bifurcation analysis focusing on codimension one bifurcation curves.Such bifurcations generally occur in higher dimensional systems also. Thoughthe way we introduced this stabilizing force here is rather &ad hoc' our analysis isuseful. Similar codimension bifurcation routes may be expected in extensions ofthe model where the stabilizing force is modelled explicitely.

We further have introduced time varying beliefs about conditional variancesof returns. The bifurcation routes to chaos di!er in some details, however theglobal qualitative features of the price dynamics are similar to the case withconstant beliefs about variances. In real markets traders would not only updatetheir expectations about future returns but use observed data also to estimatevariables such as variances of returns. Our analysis gives a justi"cation toconcentrate on the more tractable model with constant beliefs about variances,when trying to understand the behavior of "nancial markets.

The aim of the model is to understand stochastic properties of stock returnsand trading volume observed in real "nancial data and the forces that accountfor these properties. Though statistical techniques are useful they are no substi-tute for a structural model in giving insight into the economic mechanism thatmay generate nonlinearity and observed #uctuations. Simple stylized versions ofadaptive belief systems as presented in this paper also complement computerexperiments like that of Arthur et al. (1997a, b) since they are able to analyzecertain issues in an analytic framework. Such a scienti"c understanding isessential for a design of intelligent regulatory policy and also has practical valueconcerning risk management.

There are of course many ways to develop the model further, an extensive listis presented in BH (1997b) and Brock (1997). In a next step we will extend themodel in order to catch empirical observed patterns in the dynamics of volatilityand volume, such as autocorrelation functions of volatility of returns andtrading volume and cross-autocorrelations betweeen volatility measures andvolume measures. A "rst attempt to calibrate the model to monthly IBM datawas already done in BH (1997b) for a case with four di!erent types of traders.


We have observed that the model generates time series of returns with periodsof high and low volatilities. Though the time series of the actual model do notpossess GARCH e!ects it seems promising that a further development of themodel may generate such e!ects. Introducing more types may give price serieswith stochastic properties which are closer to real "nancial data. Also animprovement in the conditional rules for predictor choice which does not causesuch an abrupt decline in prices as we have seen in some time series will be a stepinto this direction.

6. For further reading

Brock (1993).

Appendix

Proof of Proposition 1. Steady states x are given by

Rx"n2gx"

1!mH

2gx,

where mH is the value of mtfor x

t"x

t~2"xH. It follows that x"0 or

mH"1!2R

g"tanhC

b2 A

g

2ap2xH(g xH!2xH)#axH2!CBD ,

i.e. xH2"2ap2 (log (g/R!1)#bC)

b(g (g!2)#2ap2a). (13)

Thus, besides the fundamental steady state E1

there exist two non-fundamentalequilibria E

2,3"($xH,$xH,$xH) i!

log(g/R!1)#bC

g(g!2)#2ap2a'0 and g'R.

The Jacobian of the fundamental steady state E1

has the eigenvalue j1"

(g/2R)(1#tanh(bC/2)), which lies in the interval (0,1) i! b(bH"(1/C) log[g/( g!R)], and a double eigenvalue j

2"j

3"0.

For g52R, j1'1 (in that case aH'0 and therefore a'aH, and bH(0).

In Fig. 1, aH and bH are the coordinates of B. h

Proof of Proposition 2. The characteristic polynomial of the Jacobian of thenon-fundamental steady states E

2and E

3is given by

p(j)"j3!A1#g!2ap2a

gZBj2#(g!1)Z, (14)


where

Z :"b

ap2(g!R)xH2"2( g!R)

log(g/R!1)#bC

g (g!2)#2ap2a. (15)

Note that a change in a is locally equivalent to a change in the risk parametera near all steady states.

At the pitchfork bifurcation value b"bH, xH"0 (Z"0) and p(j) hasa double eigenvalue 0 and an eigenvalue 1. For b slightly larger than bH (Zslightly positive), p(j) has three real, one negative and two positive eigenvaluesinside the unit circle and the non-fundamental steady states are stable. Increas-ing b, one of the two positive eigenvalues increases, the other one decreases. Fora certain value of b they coincide resulting in a double real positive eigenvalueinside the unit circle, increasing b further, two complex conjugate eigenvaluesoccur. These eigenvalues might cross the unit circle at a value b

H(a), given by

(19) and (20), respectively. In this case a Hopf bifurcation of the non-funda-mental steady states occurs.

Let j1, j

2, j

3denote the eigenvalues, where j

2,3"A$Bi. Since p(j) has no

linear term, j1"!j

2j3/(j

2#j

3). Therefore,

p(j)"(j!j1) (j2!(j

2#j

3)j#j

2j3)

"Aj#A2#B2

2A B (j2!2Aj#A2#B2)

"j3#AA2#B2

2A!2ABj2#

(A2#B2)2

2A. (16)

Hopf bifurcation of the non-fundamental steady states: A Hopf bifurcation of thenon-fundamental steady states occurs if A2#B2"1. By comparing coe$cientsof (14) and (16) this yields

1#g!2ap2a

gZ"!

1

2A#2A and (g!1)Z"

1

2A. (18)

Eliminating A from these equations, we obtain that on the (b, a)-plane the Hopfbifurcation curve is implicitely de"ned by

(g2!2ap2a)(g!1)

gZ2#(g!1)Z!1"0. (19)

Using (15), (19) can be rewritten as

4(g2!2ap2a)(g!1)(g!R)2AlogAg

R!1B#bCB

2

#2g(g!1)(g!R)(g(g!2)#2ap2a)AlogAg

R!1B#bCB

!g(g(g!2)#2ap2a)2"0. (20)


From (19) we further have

Z"

!g(g!1)$Jg2(g!1)2#4g(g!1)(g2!2ap2a)

2(g2!2ap2a)(g!1),

which is real for

a(1

8ap2g(5g!1)": a6 .

On the (b, a)-plane the Hopf bifurcation curve a(b) obtains its maximum at a6(cf. Fig. 1).

Flip bifurcation of the non-fundamental steady states: A #ip bifurcation of thenon-fundamental steady states occurs for j

1"!(A2#B2)/2A"!1. Com-

paring coe$cients of (14) and (16) gives

1#g!2ap2a

gZ"!1#2A and (g!1)Z"A2#B2.

Eliminating A and B, we obtain

Z"

2g

g(g!2)#2ap2a

and plugging this expression into (15) yields

b"bF"

1

CAg

g!R!logA

g

R!1BB.

Plugging bF

into (20) we obtain two solutions for a,

a1"!

g(5g!6)

2ap2(aH Q g'1,

a2"

g2

2ap2"aHH(a6 Q g'1.

Since g'R ('1) (otherwise non-fundamental steady states do not exist), the#ip bifurcation line b"b

Fintersects the Hopf bifurcation curve in C"(b

F, aHH)

(see Fig. 1, recall that B"(bH, aH)). Therefore, if a3(aH,aHH) the non-funda-mental steady states are destabilized by a Hopf bifurcation at b"b

H(a) de"ned

by (20), bH(a)(b

Fin that case. If a3(aHH, a6 ], b

F(b

H(a), if a'a6 , b

H(a) N R, thus

the non-fundamental steady states are destabilized by a #ip bifurcation atb"b

Fif a'aHH. h


A.1. Limits of the eigenvalues when approaching B

On the Hopf bifurcation curve (19), as aPaH, g2!2ap2aP2g (g!1), there-fore for a+aH (19) can approximately be written as

2(g!1)2Z2#(g!1)Z!1"0,

hence lim(b,a)?B

Z"1/(2(g!1)) along the Hopf bifurcation curve. Thus, in thelimit (14) reduces to

j3!32j2#1

2"0,

which has zeros 1 (double) and !12.

Approaching point B along the Hopf bifurcation curve the two complexconjugate eigenvalues on the unit circle therefore converge to 1 and the thirdeigenvalue to !1

2.

On the pitchfork bifurcation line the fundamental and the non-fundamentalsteady states coincide. The eigenvalues of the steady state are 0 (double) and 1.On the line g(g!2)#2ap2a"0 (on the (b, a)-plane this is a horizontal linethrough point B"(bH, aH), see Fig. 1) the characteristic Eq. (14) reduces toa quadratic equation with zeros $1. As a approaches aH (from above) oneeigenvalue tends to in"nity.

A.2. Computations of the origin points of Arnold tongues

On the Hopf bifurcation curve A2#B2"1, using (18) we have

arg j"arctanB

A"arctan

J1!A2

A

"arctan C2g(g!1)ZS1!1

4(g!1)2Z2 D.Hence plugging expression (15) in for Z we obtain

tanA2pp

qB"J16(g!1)2(g!R)2(log(g/R!1)#bC)2!(g (g!2)#2ap2a)2

g (g!2)#2ap2a,

resp.

16(g!1)2(g!R)2AlogAg

R!1B#bCB

2!(g(g!2)#2ap2a)2

"(g (g!2)#2ap2a)tan2A2pp

qB. (21)


For any p/q we obtain the origin points of the Arnold tongues on the (a, b)-plane,as the solutions of (20) and (21). Note that four angles arg j give the same valuefor tan2(arg j). From the second equation of (18) we conclude that the eigen-values corresponding to the Hopf bifurcation curve have non-negative real part(when moving away from B along the Hopf bifurcation curve (20) the twocomplex conjugate eigenvalues move on the unit circle from #1 to $i asbPR). Therefore, p and q have to be chosen such that 2p (p/q)3[0,p/2] (p/qreduced) to get a p : q resonance. From (18) it also follows that p/q increasesmonotonically when b is increased.

Proof of Proposition 3. It is easy to check that the eigenvalues of the funda-mental steady state E

1are (g/2R)(1#tanh (bC/2)) and 0(2) (which are the same

as in the case of constant beliefs about variances) and the two stable eigenvalueswp,wk3(0, 1).

The characteristic polynomial of the Jacobian J of the non-fundamentalsteady states is given by

det (J!jI)"(wp!j)(wk!j) det (J#!jI),

where J#is the Jacobian for the case with constant beliefs about variances when

p2 is replaced by p2d . Therefore, in addition to the stable eigenvalues wp and wk,we have the same eigenvalues as in the case of constant beliefs about vari-ances. h

Proof of Proposition 4. We proceed analogous as in the proof of Lemma 4 in BH(1998). We have

mt"G

#1 ifg

2a(p2t~1

#p2d )xt~2

( gxt~2

!2xt)#ax2

t'C,

!1 ifg

2a(p2t~1

#p2d )xt~2

(gxt~2

!2xt)#ax2

t4C.

(22)

The unstable eigenvector of the fundamental steady state is

AAg

RB2,g

R, 1, 0,

(R!(g/R))(1!wk)wk!(g/R) B

@.

Since xt`1

"(g/2R)xt(1!m

t), x

t`1"(g/R)x

tif m

t"!1 and x

t`1"0 if

mt"1.


13The numerical computations of the unstable manifolds were carried out by plotting 6000equally spaced points on a small unstable eigenvector according to the j-lemma (see Guckenheimerand Holmes, 1986, p. 247). (To get accurate pictures, we further iterated segment A

1A@

1and some

small subsegments on A1A@

1taking even more, namely, 30,000 points.)

Take an initial state x0"e(g/R)2, x

~1"eg/R, x

~2"e, e'0. The expres-

sion determining whether mt"!1 or #1 is

Ct:"e2A

g

RB2t`2

Cg

2a(p2t#p2d )

(R2!2g)#aAg

RB2

Dand note that g/(2a(p2

t#p2d ))4g/2ap2d . If g'R and the expression in brackets

is positive there is some smallest ¹'0 such that CT'C, so that

mT"#1, x

t"0 ∀t5¹#1, and m

t"!1 ∀t5¹#3. Hence in this case

the unstable manifold is bounded and all orbits converge to E. h

A.3. Numerical analysis of the unstable manifold of the fundamentalsteady state E for large but xnite b

Numerical computations suggest that the situation is similar as in BH (1997a)and in Goeree and Hommes (1999). However, our system is three dimensional,therefore the analysis of the unstable manifold of E for large but "nite b is muchmore delicate than for the two-dimensional models in those papers. An approxi-mation of the unstable manifold is not obtained by connecting the segments ofF5=(EA

0) just by straight segments. The limiting manifold will contain line

segments lying on the planes Mm"!1N and Mm"1N which are connected byvertical line segments. Of course, A

1and A@

1are connected by a vertical line

segment. Note that for b"R, mtin (11) is not de"ned if equality holds on the

right-hand side of (12). We have de"ned mt"!1 in that case, however, we

could have taken any other value between !1 and 1. One could interpret thevertical line segement A

1A@

1as image of A

0. For "nite but su$ciently large

b there is a segment A0A@

03EA

1whose image is close to the vertical segment

A1A@

1. Further analysis will be carried out numerically.

We "nd that, depending on a, two cases are possible. Figs. 3 and 7 show the"rst six resp. "ve iterates of the unstable manifold for g"1.2,R"1.01, a"10, p2"0.1, C"1.13 We analyze case 1 for the numericalexample a"0.65, case 2 for a"0.75.

Case 1: For b su$ciently large the unstable manifold is arbitrary close to(see Fig. 3)

F(EA0)"EA

1,

F2(EA0)"F(EA

0)A@

1A

2,


F3(EA0)"F2(EA

0) B

1B@1B@

2B2A@

2A

3,

F4(EA0)"F3(EA

0) C

1C

2C

3C

4A

4,

F5(EA0)"F4(EA

0) A

3A

4E,

F6(EA0)"F5(EA

0) A

4E.

Except for points Ai

(the points which de"ne the segments of the unstablemanifold in the case b"R), X@ denotes points lying on the plane Mm"!1Nhaving the same x and y-coordinates as X3Mm"1N. F denotes the limit of Fbas bPR.

Case 2: Fig. 7(a) shows EA0A@

0and the "rst three iterates of the segment

A0A@

0A

1("rst four iterates of EA

0), Fig. 7(b) shows the fourth iterate of the

segment A0A@

0A

1and the "rst and last part (dotted line) of its "fth iterate. The

"rst three iterates of EA0

are as in case 1. For further iterations we obtain

F4(EA0)"F3(EA

0)C

1C

5C@

5C@

6C

6C

2C

7C@

7C@

8C

8C

3C

4A

4,

F5(EA0)"F4(EA

0)D

1D

2D@

2D@

3D

3D

4D

5D@

5D@

6D

6D

7A

3D

7

D8D@

8D@

9D

9D

10A@

1A

1D@

11D@

12E.

For the 6th iterate we only roughly sketch the approximative way of theunstable manifold:

F5(EA0)A

4D

12A

0A@

02A

22A@

2A

3A@

22A

22A@

0A

02D

1

A4

$&&

F4(EA0) EA

4E.

$&&

F4(EA0) denotes the reverse direction of F4(EA

0). When moving between the

points (close to) A2

and A@23Mm"1N parts of the connecting segment lie in the

plane Mm"!1N, similar as for F4(EA0) and F5(EA

0). Though this description

is rather rough it shows that the 6th iterate of EA0

contains segments which areexpanded and folded along (close to) the unstable segment EA

0. This suggests

that for su$ciently large b-values, F6b has as a full horseshoe over a rectangularregion containing a subsegment of EA

0.

Note that in both cases the unstable manifold gets arbitrarily close to theplane S which is contained in the stable manifold. That is, the unstable manifold"rst moves away from the fundamental steady state but returns close to it lateron (cf. Proposition 4).

In case 2 the situation seems to be similar as in BH (1997a) and suggests theoccurrence of homoclinic bifurcations of period N (N56) saddle points and theassociated occurrence of strange attractors (cf. BH, 1997a, Section 3.4; Viana,1993). Computing Lyapunov characterisic exponents (LCE), which are positivefor su$cient high b-values, gives further evidence of chaotic dynamics (seeSection 4).


Case 1 appears to be similar to the case of Fig. 8(e}f) in Goeree and Hommes(1999). The fourth iterate of EA

0does not contain the segment C

7C@

7C@

8C

8as in

case 2. Indeed, when a is decreased, C7

and C8approach each other turning case

2 into case 1 for a certain value of a. It is not clear whether a horseshoeconstruction as in case 2 is possible. Numerically, all points of the segmentA

1A@

1A

2are mapped to the steady state E after four iterates. The 6th iterate

might continue to move along EA0. Computing LCE shows that chaos arises in

that case also (see Section 4).

References

Allen, F., Gorton, G., 1993. Churning bubbles. Review of Economic Studies 60, 813}836.Anderson, S., de Palma, A., Thisse, J., 1993. Discrete Choice Theory of Product Di!erentiation. MIT

Press, Cambridge, MA.Arrowsmith, D.K., Place, C.M., 1990. An Introduction to Dynamical Systems. Cambridge Univer-

sity Press, Cambridge, UK.Arthur, W.B., Holland, J.H., LeBaron, B., Palmer, R., Tayler, P., 1997a. Asset pricing under

endogenous expectations in an arti"cial stock market. In: Arthur, W.B., Durlauf, S.N., Lane,D.A. (Eds.), The Economy as an Evolving Complex System II. Addison-Wesley, Reading, MA,pp. 15}44.

Arthur, W.B., LeBaron, B., Palmer, R., 1997b. Time series properties of an arti"cial stock market.University of Wisconsin, SSRI working paper 9725.

Barucci, E., Posch, M., 1996. The rise of complex beliefs dynamics. IIASA working paper WP-96-46.Beja, A., Goldman, B., 1980. On the dynamic behavior of prices in disequilibrium. Journal of

Finance 35, 235}248.Black, F., 1986. Noise. Journal of Finance 41, 529}543.Blume, L., Easley, D., 1992. Evolution and market behavior. Journal of Economic Theory 58,

9}40.Bollerslev, T., Engle, R.F., Nelson, D.B., 1994. ARCH models, In: Engle, R.F., McFadden, D.L.

(Eds.), Handbook of Econometrics. Elsevier, Amsterdam, pp. 2959}3038.Brock, W.A., 1993. Pathways to randomness in the economy: emergent nonlinearity and chaos in

economics and "nance. Estudios EconoH micos 8, 3}55.Brock, W.A., 1997. Asset price behavior in complex environments. In: Arthur, W.B., Durlauf, S.N.,

Lane, D.A. (Eds.), The Economy as an Evolving Complex System II. Addison-Wesley, Reading,MA, pp. 385}423.

Brock, W.A., Hommes, C.H., 1997a. A rational route to randomness. Econometrica 65, 1059}1095.Brock, W.A., Hommes, C.H., 1997b. Models of complexity in economics and "nance. In: Heij, C.,

Schumacher, H., Hanzon, B., Praagman, K. (Eds.), System Dynamics in Economic and FinancialModels. Wiley, Chichester, pp. 3}44.

Brock, W.A., Hommes, C.H., 1998. Heterogeneous beliefs and bifurcation routes to chaos in a simpleasset pricing model. Journal of Economic Dynamics and Control 22, 1235}1274.

Brock, W.A., Hsieh, D.A., LeBaron, B., 1991. Nonlinear Dynamics, Chaos, and Instability. MITPress, Cambridge, MA.

Brock, W.A., Lakonishok, J., LeBaron, B., 1992. Simple trading rules and the stochastic properties ofstock returns. Journal of Finance 47, 1731}1764.

Brock, W.A., LeBaron, B., 1996. A dynamical structural model for stock return volatility and tradingvolume. Review of Economics and Statistics 78, 94}110.


Brunnermeier, M.K., 1998. Prices, price processes, volume and their information* A survey of themarket microstructure literature. FMG discussion paper 270, The London School of Economics.

Chiarella, C., 1992. The dynamics of speculative behaviour. Annals of Operations Research 37,101}123.

Day, R.H., Huang, W., 1990. Bulls, bears and market sheep. Journal of Economic Behavior andOrganization 14, 299}329.

De Grauwe, P., Dewachter, H., Embrechts, M., 1993. Exchange Rate Theory. Blackwell, Oxford.De Long, J.B., Shleifer, A., Summers, L.H., Waldmann, R.J., 1990a. Noise trader risk in "nancial

markets. Journal of Political Economy 98, 703}738.De Long, J.B., Shleifer, A., Summers, L.H., Waldmann, R.J., 1990b. Positive feedback investment

strategies and destabilizing rational speculation. Journal of Finance 45, 379}395.Franke, R., Sethi, R., 1993. Cautious Trend-Seeking and Complex Dynamics. University of Bielefeld.Frankel, J.A., Froot, K.A., 1988. Chartists, fundamentalists and the demad for dollars. Greek

Economic Review 10, 49}102.Genc7 ay, R., Stengos, T., 1997. Technical trading rules and the size of the risk premium in security

returns. Studies in Nonlinear Dynamics and Econometrics 2, 23}34.Ghezzi, L.L., 1992. Bifurcations in a stock market model. Systems & Control Letters 19, 371}378.Goeree, J.K., Hommes, C.H., 1999. Heterogeneous beliefs and the nonlinear cobweb model. Journal

of Economic Dynamics and Control, this issue.Grandmont, J.-M., 1998. Expectations formations and stability of large socioeconomic systems.

Econometrica 66, 741}781.Grossman, S.J., 1989. The Informational Role of Prices. MIT Press, Cambridge, MA.Guckenheimer, J., Holmes, P., 1986. Nonlinear Oscillations, Dynamical Systems, and Bifurcation

of Vector Fields. Springer, New York.Haugen, R.A., 1998a. Beast on Wall Street* How Stock Volatility Devours Our Wealth. Prentice-

Hall, Englewood Cli!s, NJ.Haugen, R.A., 1998b. The Ine$cient Market*What Pays O! and Why. Prentice-Hall, Englewood

Cli!s, NJ.Haugen, R.A., 1998c. The New Finance* The Case for an Over-reactive Stock Market. Prentice-

Hall, Englewood Cli!s, NJ.Hommes, C.H., Sorger, G., 1998. Consistent expectations equilibria. Macroeconomic Dynamics 2,

287}321.Khibik, A.H., Kuznetsov, Y.A., Levin, V.V., Nikolaev, E.V., 1992. LOCBIF. An interactive LOCal

BIFurcation analyzer.Kurz, M. (Ed.), 1997. Endogenous Economic Fluctuations. Springer, Berlin.Kuznetsov, Y.A., 1995. Elements of Applied Bifurcation Theory. Springer, New York.LeBaron, B., 1995. Experiments in Evolutionary Finance. Department of Economics, University of

Wisconsin.Lux, T., 1994. Complex dynamics in speculative markets: a survey of the evidence and some

implications for theoretical analysis. Volkswirtschaftliche DiskussionsbeitraK ge 67, University ofBamberg.

Lux, T., 1995. Herd behavior, bubbles and crashes.. The Economic Journal 105, 881}896.Lux, T., Marchesi, M., 1998. Volatility in "nancial markets: a micro-simulation of interactive agents.

Working paper presented at the Third Workshop on Economics with Heterogeneous InteractingAgents, University of Ancona.

Medio, A., 1992. Chaotic Dynamics: Theory and Applications to Economics. Cambridge UniversityPress, Cambridge, UK.

Nelson, D.B., 1992. Filtering and forecasting with misspeci"ed ARCH Models I: getting the rightvariance with the wrong model. Journal of Econometrics 52, 61}90.

Palis, J., Takens, F., 1993. Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurca-tions. Cambridge University Press, Cambridge, UK.


Sargent, T.J., 1993. Bounded Rationality in Macroeconomics. Clarendon Press, Oxford.Sethi, R., 1996. Endogenous regime switching in speculative markets. Structural Change and

Economic Dynamics 7, 99}118.Shiller, R.J., 1991. Market Volatility. MIT Press, Cambridge, MA.Shleifer, A., Summers, L.H., 1990. The noise trader approach to "nance. Journal of Economic

Perspectives 4, 19}33.Smale, S., 1965. Di!eomorphisms with many periodic points. In: Cairns, S.S. (Ed.), Di!erential and

Combinatorical Topology. Princeton University Press, Princeton, NJ, pp. 63}80.Thaler, R.H., 1991. Quasi Rational Economics. Russel Sage Foundation, New York.Thaler, R.H. (Ed.), 1993. Advances in Behavioral Finance. Russel Sage Foundation, New York.Viana, M., 1993. Strange attractors in higher dimensions. Boletim da Sociedade Brasileira de

MathemaH tica 24, 13}62.


Documents

Endogenous uctuations in a simple asset pricing model …cial "nancial market are Arthur et al. (1997a,b) and LeBaron (1995). A. Gaunersdorfer / Journal of Economic Dynamics & Control