Herd behavior and aggregate fluctuations in financial markets 1

Transcription

1 Herd behavior and aggregate flutuations in finanial markets 1 Rama CONT 1;2;3 and Jean-Philippe BOUCHAUD 1;2 1) Servie de Physique de l Etat Condensé Centre d Etudes de Salay Gif sur Yvette, Frane 2) Siene & Finane Researh Group rue V. Hugo, Levallois, Frane. 3) Department of Eonomis Amerian University 31 avenue Bosquet Paris, Frane Abstrat We present a simple model of a stok market where a random ommuniation struture between agents generially gives rise to a heavy tails in the distribution of stok prie variations in the form of an exponentially trunated power-law, similar to distributions observed in reent empirial studies of high frequeny market data. Our model provides a link between two well-known market phenomena: the heavy tails observed in the distribution of stok market returns on one hand and herding behavior in finanial markets on the other hand. In partiular, our study suggests a relation between the exess kurtosis observed in asset returns, the market order flow and the tendeny of market partiipants to imitate eah other. Keywords: ommuniation, market organization, random graphs. JEL Classifiation number: C0, D49, G19 PACS : Ak, n 1 This work is a ondensed version of Chapter 5 the first author s dotoral dissertation at Université de Paris XI. R. Cont gratefully aknowledges an AMX fellowship from Eole Polytehnique (Frane) and thanks Siene & Finane for their hospitality. We thank Yann Braoueze for helpful remarks on a preliminary draft of this artile and for numerous bibliographial indiations. ont@ens.fr 1

2 Empirial studies of the flutuations in the prie of various finanial assets have shown that distributions of stok returns and stok prie hanges have fat tails that deviate from the Gaussian distribution [38, 39, 12, 27, 42, 14, 43] espeially for intraday time sales [14]. These fat tails, haraterized by a signifiant exess kurtosis, persist even after aounting for heteroskedastiity in the data [9]. The heavy tails observed in these distributions orrespond to large flutuations in pries, bursts of volatility whih are diffiult to explain only in terms of variations in fundamental eonomi variables [47]. The fat that signifiative flutuations in pries are not neesarily related to the arrival of information [16] or to variations in fundamental eonomi variables [47] leads to think the high variability present in stok market returns may orrespond to olletive phenomena suh as rowd effets or herd behavior. Although herding in finanial markets is by now relatively well doumented empirially, there have been few theoretial studies on the impliations of herding and imitation for the statistial properties of market demand and prie flutuations. In partiular some questions one would like to answer is: how does the presene of herding modify the distribution of returns? What are the impliations of herding for relations between market variables suh as order flow and prie variability? These are some of the questions whih have motivated our study. The aim of the present study is to examine, in the framework of a simple model, how the existene of herd behavior among market partiipants may generially lead to large flutuations in the aggregate exess demand, desribed by a heavy-tailed non-gaussian distribution. Furthermore we explore how empirially measurable quantities suh as the exess kurtosis of returns and the average order flow may be related to eah other in the ontext of our model. Our approah provides a quantitative link between the two issues disussed above: the heavy tails observed in the distribution of stok market returns on one hand and the herd behavior observed in finanial markets on the other hand. The artile is divided into four setions. Setion 1 reviews well known empirial fats about the heavy-tailed nature of the distribution of stok returns and various models proposed to aount for it. Setion 2 presents previous empirial and theoretial work on herding and imitation in finanial markets in relation to the present study. Setion 3 disusses the statistial properties of exess demand resulting from the aggregation of of a large number of random individual demands in a market. Setion 4 defines our model and presents analytial results. Setion 5 interprets the results in eonomi terms, ompares them to empirial data and disusses possible extensions. Details of alulations are given in the appendies. 2

3 1 The heavy-tailed nature of asset return distributions It is by now a well known fat that the distribution of returns of almost all finanial assets -stoks, indexes and futures- exhibit a slow asymptoti deay that deviates from a normal distribution. This is quantitatively refleted in the exess kurtosis, defined as: κ µ 4 = σ 4 3 (1) where µ 4 is the fourth entral moment and σ the standard deviation of the returns. κ should be zero for a normal distribution but ranges between 2 and 50 for daily returns [12, 43] and is even higher for intraday data. Careful study of the tails of the distribution shows an exponential deay for most assets [14, 40]. Many statistial models have been put forth to aount for the heavy tails observed in the distribution of asset returns. Well known examples are Mandelbrot s stable paretian hypothesis [38] the mixture of distributions hypothesis [13], and models based on onditional heteroskedastiity [18]. It is well known that in the presene of heteroskedastiity, the unonditional distribution of returns will have heavy tails. In most models based on heteroskedastiity, the proess of return is assumed to be onditionally Gaussian : the shoks are loally Gaussian and the non-gaussian harater of the unonditional distribution is an effet of aggregation. It is obtained by superposing a large number of loal Gaussian shoks. In this desription, sudden movements in pries are interpreted as orresponding to a high value of onditional variane. On one hand, it has been shown that although onditional heteroskedastiity does lead to fat-tails in unonditional distributions, ARCH-type models annot fully aount for the kurtosis of returns [29, 9]. On the other hand, from a theoretial point of view there is no a priori reason to postulate that returns are onditionally normal: although onditional normality is onvenient for parameter estimation of the resulting model, non-normal onditional distributions possess the same qualitative features as for volatility lustering while aounting better for heavy tails. Gallant and Tauhen [23] report signifiant evidene of both onditional heteroskedastiity and onditional non-normality in the daily NYSE value-weighted index. Similarly, Engle and Gonzalez-Rivera [17] show that when a GARCH(1,1) model is used for the onditional variane of stok returns the onditional distribution has onsiderable kurtosis, espeially for small firm stoks. Indeed several 3

4 authors have proposed GARCH-type models with non-normal onditional distributions [9]. Stable distributions [38] offer an elegant alternative to heteroskedastiity for generating fat tails, with the advantage that they have a natural interpretation in terms of aggregation of a large number of individual ontributions of agents to market flutuations: indeed, stable distribution may be obtained as limit distributions of sums of independent or weakly dependent random variables, a property whih is not shared by alternative models. Unfortunately, the infinite variane property of these distributions is not observed in empirial data: sample varianes do not inrease indefinitely with sample size but appear to stabilize at a ertain value for large enough data sets. We will disuss stable distributions in more detail in Setion 3. A third approah, first advoated by Clark [13], is to model stok returns by a subordinated proess, typially subordinated Brownian motion. This amounts to stipulating that through a stohasti time hange one an transform the ompliated dynamis of the prie proess into Brownian motion or some other simple proess. It an be shown that, depending on the hoie of the subordinator, one an obtain a wide range of distributions for the inrements all of whih possess heavy tails i.e. positive exess kurtosis. As a matter of fat, even stable distributions may be obtained as a subordinated Brownian motion. In the original approah of Clark [13], the subordinator was taken to be trading volume. Other hoies whih have been proposed are the number of trades [24] or other loal measures of market ativity. However, none of these hoies for the subordinator lead to a normal distribution for the inrements of the time-hanged proess, indiating that large flutuations in prie may not be ompletely explained by large flutuations in trading volume or number of trades. In short, although heteroskedastiity and time deformation partly explain the kurtosis of asset returns, they do not explain it quantitatively: even after aounting for these effets, one is left with an important residual kurtosis in the resulting transformed time series. Moreover, these approahes are not based on any partiular model of the market phenomenon generating the data that they attempt to desribe. Reent works by Bak, Pazuski and Shubik [3] and Caldarelli, Marsili and Zhang [11] have tried to explain the heavy tailed nature of return distributions as an emergent property in a market where fundamentalist traders interat with noise traders. Bak, Pazuski and Shubik onsider several types of trading rules and study the resulting statistial properties for the time series of asset pries in eah ase. Computer simulations of their model do seem to yield fat-tailed 4

5 distributions for asset returns whih at least qualitatively resemble empirial distributions of stok returns, showing that the appearane of fat-tailed distributions an be regarded as an emergent property in large markets. However, the model has two drawbaks: first, it is a fairly ompliated model with many ingredients and parameters and it is diffiult to see how eah ingredient of the model affets the results obtained, whih in turn diminishes its explanatory power. Seond, the omplexity of the model does not allow expliit alulations to be performed, preventing the model parameters to be ompared with empirial values. We present here an alternative approah whih, by modeling the ommuniation struture between market agents as a random graph, proposes a simple mehanism aounting for some non-trivial statistial properties of stok prie flutuations. Although muh more rudimentary and ontaining fewer ingredients than the model proposed by Bak, Pazuski and Shubik, our model allows for analyti alulations to be performed, thus enabling us to interpret in eonomi terms the role of eah of the parameters introdued. The basi intuition behind our approah is simple: interation of market partiipants through imitation an lead to large flutuations in aggregate demand, leading to heavy tails in the distribution of returns. 2 Herd behavior in finanial markets A number of reent studies have onsidered mimeti behavior as a possible explanation for the exessive volatility observed in finanial markets [5, 47, 49]. The existene of herd behavior in speulative markets has been doumented by a ertain number of studies: Sharfstein and Stein [44] disuss evidene of herding in the behavior of fund managers, Grinblatt et al. [26] report herding in mutual fund behavior while Trueman [50] and Welh [51] show evidene for herding in the foreasts made by finanial analysts. On the theoretial side several studies have shown that, in a market with noise traders, herd behavior is not neessarily irrational in the sense that it may be ompatible with optimizing behavior of the agents [48]. Other motivations may be invoked for explaining imitation in markets, suh as group pressure [6, 44]. Various models of herd behavior have been onsidered in the literature, the most well known approah being that of Bannerjee [4, 5] and Bikhhandani et al. [6]. In these models, individuals attempt to infer a parameter from noisy observations and deisions of other agents, typially through a Bayesian proedure, giving rise to information asades [6]. An important feature of these models is 5

6 the sequential harater of the dynamis: individuals make their deisions one at a time, taking into aount the deisions of the individuals preeding them. The model therefore assumes a natural way of ordering the agents. This assumption seems unrealisti in the ase of finanial markets: orders from various market partiipants enter the market simultaneously and it is the interplay between different orders that determines aggregate market variables 2. Non-sequential herding has been studied in a Bayesian setting by Orléan [41] in a framework inspired by the Ising model. Orléan onsiders imitation between agents in whih any two agents have the same tendeny to imitate eah other. In terms of aggregate variables, this model leads either to a Gaussian distribution when the imitation is weak, or to a bimodal distribution with non-zero modes, whih Orléan interprets as orresponding to olletive market phenomena suh as rashes or panis. In neither ase does one obtain a heavy-tailed unimodal distribution entered at zero suh as those observed for stok returns. The approah proposed in this paper is different from both approahes desribed above. Our model is different from that in [4, 6]. in that herding is not sequential. The unrealisti nature of the results in [41] result from the fat that all agents are assumed to imitate eah other to the same degree. We avoid this problem by onsidering the random formation of groups of agent who imitate eah other but suh that different groups of agents make independent deisions, whih allows for a heterogeneous market struture. More speifially, our approah onsiders the interations between agents as resulting from a random ommuniation struture, as explained below. 3 Aggregation of random individual demands Consider a stok market with N agents, labeled by an integer 1 i N, trading in a single asset, whose prie at time t will be denoted x(t). During eah time period, an agent may hoose either to buy the stok, sell it or not to trade. The demand for stok of agent i is represented by a random variable φ i, whih an take the values 0, -1 or +1: a positive value of φ i represents a bull - an agent willing to buy stok-, a negative value a bear, eager to sell stok while φ i = 0 means that agent i does not trade during a given period. The random harater of 2 Bikhhandani et al [6] do not onsider their model as appliable to finanial markets but for another reason: they remark that as the herd grows, the ost of joining it will also grow, disouraging new agents to join. This aspet, whih is not taken into aount by their model, is again unavoidable to the sequential harater of herd formation. 6

7 individual demands may be due either to heterogeneous preferenes or to random resoures of the agents, or both. Alternatively, it may result from the appliation by the agents of simple deision rules, eah group of agents using a ertain rule. However, in order to fous on the effet of herding, we do not expliitly model the deision proess leading to the individual demands and model the result of the deision proess as a random variable φ i. In ontrast with many binary hoie models in the miroeonomis literature, we allow for an agent to be inative i.e. not to trade during a given time period t. This, as we shall see below, is important for deriving our results. Let us onsider for simplifiation that, during eah time period, an agent may either trade one unit of the asset or remain inative. The demand of the agent i is then represented by φ i 2f 1;0;+1g, φ i = 1 representing a sell order. The aggregate exess demand for the asset at time t is therefore simply D(t) = N i=1 φ i (t) (2) given the algebrai nature of the φ i. The marginal distribution of agent is individual demand will is assumed to be symmetri : P(φ i =+1)=P(φ i = 1)=a P(φ i = 0)=1 2a suh that the average aggregate exess demand is zero i.e. the market is onsidered to flutuate around equilibrium. A value of a < 1=2 allows for a finite fration of agents not to trade during a given period. We are onerned here with obtaining a result whih ould then be ompared with atual market data and the short term exess demand is not an easily observable quantity. Also, most of the studies on the statistial properties of finanial time series have been done on returns, log returns or prie hanges. We therefore need to relate the aggregate exess demand in a given period to the return or prie hange during that period. The aggregate exess demand has an impat on the prie of the stok, ausing it to rise if the exess demand is positive, to fall if it is negative. A ommon speifiation, whih is ompatible with standard tatonnement ideas, is to assume a proportionality between prie hange (or return) and exess demand: x = x(t + 1) x(t) = 1 λ 7 N i=1 φ i (t) (3)

8 where λ is a fator measuring the liquidity or, more preisely, the market depth [32]: it is the exess demand needed to move the prie by one unit: it measure the sensitivity of prie to flutuations in demand. Eq. (3), emphasizes the prie impat of the order flow as opposed to other possible auses for prie flutuations. Eq.(3) may be onsidered either in absolue terms with x(t) being the prie, or as representing relative variations of the prie, x(t) then being onsidered as the log of the prie and its inrement as the instantaneous return. The latter has the advantage of guaranteeing the positivity of the prie but for short-run dynamis the two speifiations do not differ substantially sine the two quantities have the sameempirialproperties. Asimilarmodelfortheprieimpatoftrades has been onsidered by Hausman, Lo and MaKinlay [28]. Although in the long run eonomi fators other than short term exess demand may influene the evolution of the asset prie, resulting in mean-reversion or more omplex types of behavior, we fous here on the short-run behavior of pries, for example on intra-day time sales in the ase of stok markets, so this approximation seems reasonable. The linear nature of this relation may also be questioned: indeed, some empirial studied seem to indiate that the prie impat of trades may be non-linear [12, 32]. First, note that these studies deal with the prie impat of trades and not of order flow (exess demand), whih is muh harder to measure. Results reported by Farmer and o-workers [21] based on the study of the prie impat of bloks of orders of different sizes sent to the market seem to indiate a linear relationship for small prie hanges with nonlinearity arising when the size of bloks is inreased. Moreover, if the one-period return x is a non-linear but smooth funtion h(d) 3 of the exess demand, then a linearization of the inverse demand funtion h (a first order Taylor expansion in D) shows that Eq.(3) still holds for small flutuations of the aggregate exess demand with h 0 (0) =1=λ. In order to evaluate the distribution of stok returns from Eq.(3), we need to know the joint distribution of the individual demands (φ i (t)) 1iN. Let us begin by onsidering the simplest ase where individual demands φ i of different agents are independent identially distributed random variables. We shall refer to this hypothesis as the independent agents hypothesis. In this ase the joint distribution of the individual demands is simply the produt of individual distributions and the prie variation x is a sum of N iid random variables with finite variane. When the number of terms in Eq.(3) is large the entral limit theorem applied to 3 It is interesting to note that if x = h(d), wherehis an inreasing funtion of D and if the individual demands (φ i (t)) are sequenes of independent random variables (a somewhat extreme assumption), then it is easy to show that the overall wealth of all traders inreases on average with time. 8

9 the sum in Eq.(3) tells us that the distribution of x is well approximated by a Gaussian distribution. Of ourse, this result still holds as long as the distribution of individual demands has finite variane. This an be seen as a rationale for the frequent use of the normal distribution as a model for the distribution of stok returns: indeed, if the variation of market prie is seen as the sum of a large number of independent or weakly dependent random effets, it is plausible that a Gaussian desription should be a good one. Unfortunately, empirial evidene tells us otherwise: the distributions both of asset returns [43, 12] and of asset prie hanges [38, 39, 14, 15] have been repeatedly shown to deviate signifiantly from the Gaussian distribution, exhibiting fat tails and exess kurtosis. But the independent agent model is also apable of generating aggregate distributions with heavy tails: indeed, if one relaxes the assumption that the individual demands φ i have a finite variane then under the hypothesis of independene (or weak dependene) of individual demands, the aggregate demand -and therefore the prie hange if we assume Eq.(3)- will have a stable (Pareto-Lévy) distribution. This is a possible interpretation for the stable-paretian model proposed by Mandelbrot [38] for the heavy tails observed in the distribution of the inrements of various market pries. The infinite variane of the φ i then reflets the heterogeneity of the market, for example in terms of broad distribution of wealth of the partiipants as proposed by Levy & Solomon [37]. Mandelbrot s stable-paretian hypothesis has been ritiized for several reasons, one of them being that it predits an infinite variane for stok returns whih implies in pratie that the sample variane will indefinitely inrease with sample size, a property whih is not observed in empirial data. More preisely, a areful study of the tails of the distribution of inrements for various finanial assets shows [10, 14] that they have heavy tails with a finite variane. Many distributions verify these onditions [12]; a partiular example proposed by the authors and others [14] is an exponentially trunated stable distribution the tails of the density then have the asymptoti form of an exponentially trunated power law: p( x) j xj! C j xj 1+µexp ( x x 0 ) (4) The exponent µ is found to be lose to 1.5 (µ ' 1:4 1:6) for a wide variety of stoks and market indexes [10]. This asymptoti form allows for heavy tails 9

10 (exess kurtosis) without implying infinite variane. However, it is known the entral limit theorem also holds for ertain sequenes of dependent variables: under various types of mixing onditions [7], whih are mathematial formulations of the notion of weak dependene, aggregate variables will still be normally distributed. Therefore the non-gaussian and more generally non-stable harater of empirial distributions, be it exess demand or the stok returns, not only demonstrates the failure of the independent agent approah, but also shows that suh an approah is not anywhere lose to being a good approximation: the dependene between individual demands is an essential harater of the market struture and may not be left out in the aggregation proedure, they annot be assumed to be weak (in the sense of a mixing ondition [7]) and do hange the distribution of the resulting aggregate variable, Indeed, the assumption that the outomes of deisions of individual agents may be represented as independent random variables is highly unrealisti: suh an assumption ignores an essential ingredient of market organization, namely the interation and ommuniation among agents. In real markets, agents may form groups of various sizes whih then may share information and at in oordination. In the ontext of a finanial market, groups of traders may align their deisions and at in unison to buy or sell; a different interpretation of a group may be an investment fund orresponding to the wealth of several investors but managed by a single fund manager. In order to apture suh effets we need to introdue an additional ingredient, namely the ommuniation struture between agents. One solution would be to speify a fixed trading group struture and then proeed to study the resulting aggregate flutuations. Suh an approah has two major drawbaks. First, a realisti market struture may require speifying a ompliated struture of lusters and rendering the resulting model analytially intratable. More importantly, the resulting pattern of aggregate flutuations will ruially depend on the speifiation of the market struture. An alternative approah, suggested by Kirman [33], is to onsider the market ommuniation struture itself as stohasti. One way of generating a random market struture is to assume that market partiipants meet randomly and trades take plae when an agent willing to buy meets and agent willing to sell. This proedure, alled random mathing by some authors [30], has been previously onsidered in the ontext of formation of trading groups by Ioannides [30] and in the ontext of a stok market model by Bak, Pazuski and Shubik [3]. Another way is to onsider that market partiipants form groups or lusters through a 10

11 random mathing proess but that no trading takes plaes inside a given group: instead, members of a given group adopt a ommon market strategy (for example, they deide to buy or sell or not to trade) and different groups may trade with eah other through a entralized market proess. In the ontext of a finanial market, lusters may represent for example a group of investors partiipating in a mutual fund. This is the line we will follow in this paper. 4 Presentation of the model More preisely, let us suppose that agents group together in oalitions or lusters and that, one a oalition has formed, all it s members oordinate their individual demands so that all individuals in a given luster have the same belief regarding future movements of the asset prie. In the framework desribed in the preeding setion, we will onsider that all agents belonging to a given luster will have the same demand φ i for the stok. In the ontext of a stok market, these lusters may orrespond for example to mutual funds e.g. portfolios managed by the same fund manager or to herding among seurity analysts as in [50, 51]. The right hand side of the equation (3) may therefore be rewritten as a sum over lusters : k x 1 = α φ α (t) = λ α=1w 1 λ n α=1 X α (5) where W α is the size of luster α, φ α (t) the (ommon) individual demand of agents belonging to the luster α, n the number of lusters (oalitions) and X α = φ α W α. One may onsider that oalitions are formed through binary links between agents, a link between two agents meaning that they undertake the same ation on the market i.e. they both buy or sell stok. For any pair of agents i and j, let p ij be the probability that i and j are linked together. Again, in order to simplify, we assume that p ij = p is independent of i and j: all links are equally probable. (N 1)p then denotes the average number of agents a given agent is linked to. Sine we are interested in studying the N! limit, p should therefore be hosen in suh a way that (N 1)p has a finite limit. A natural hoie is p ij = =N,any other hoie verifying the above ondition being asymptotially equivalent to this one. The distribution of oalition sizes in the market is thus ompletely speified by a single parameter,, whih represents the willingness of agents to align their 11

12 ations: it an be interpreted as a oordination number, measuring the degree of lustering among agents. Suh a struture is known as a random graph in the mathematial literature [19, 8]: in terms of random graph theory, we onsider agents as verties of a random graph of size N, and the oalitions as onneted omponents of the graph. Suh an approah to ommuniation in markets using random graphs was first suggested in the eonomis literature by Kirman [33] to study the properties of the ore of a large eonomy. Random graphs have also been used in the ontext of multilateral mathing in searh problems by Ioannides [30]. A good review of the appliations of random graph theory in eonomi modeling is given in [31]. The properties of large random graphs in the N! limit were first studied by Erdös and Renyi [19]. An extensive review of mathematial results on random graphs is given in [8]. The main resullts of the ombinatorial approah are given in Appendix 1. One an show [8] that for = 1 the probability density for the luster size distribution dereases asymptotially as a power law: P(W ) A W! W 5=2 while for values of lose to and smaller than 1 ( 0 < 1 size distribution is ut off by an exponential tail: << 1), the luster A ( 1)W P(W ) W! W 5=2 exp( ) (6) W 0 For =1, the distribution has an infinite variane while for < 1 the variane beomes finite beause of the exponential tail. In this ase the average size of a oalition is of order 1=(1 ) and the average number of lusters is then of order N(1 =2). Setting the oordination parameter lose to 1 means that eah agent tends to establish a link with one other agent, whih an be regarded as a reasonable assumption. This does not rule out the formation of large oalitions through suessive binary links between agents but prevents a single agent from forming multiple links, as would be the ase in a entralized ommuniation struture where one agent (the autioneer ) is linked to all the others. As argued by Kirman [33] the presene of a Walrasian autioneer orresponds to suh star-like, entralized, ommuniation strutures. We are thus exluding suh a situation by onstrution: we are interested in a market where information is distributed and not entralized, whih orresponds more losely to the situations enountered in 12

13 real markets. More preisely, the loal struture of the market may be haraterized by the following result [8]: in the limit N!, the number ν i of neighbors of a given agent i is a Poisson random variable with parameter : P(ν i = ν) = e ν ν! (7) A Walrasian autioneer w would be onneted to every other agent: ν w = N 1. The probability for having a Walrasian autioneer is therefore given by NP(ν w = N 1) whih goes to zero when N!. A given market luster is haraterized by its size W α and its nature i.e. whether the members are buyers or sellers. This is speified by a variable φ α 2 f 1;0;1g. It is reasonable to assume that W α and φ α are independent random variables: the size of a group does not influene its deision whether to buy or sell. The variable X α = φ α W α is then symmetrially distributed with a mass of 1 2a at the origin. Let Then the distribution of X α is given by F(x) = P(X α xjx α 6= 0) (8) G(x) = P(X α x) =(1 2a)H(x)+2aF(x) (9) where H is a unit step funtion at 0 (Heaviside funtion). We shall assume that F has a ontinuous density, f. f then deays asymptotially as in (6): f (x) jxj! A jxj 5=2 e ( 1)jxj W 0 (10) The expression for the prie variation x therefore redues to a sum of n iid random variables X α ;α = 1::n with heavy-tailed distributions as in (6): x = 1 λ n X α α=1 Sine the probability density of X α has a finite mass 1 2a at zero, only a fration 2a of the terms in the sum (5) are non-zero; the number of non-zero terms in the sum is of order 2an 2aN(1 =2)=N order (1 =2) where N order = 2aN is the average number of market partiipants who atively trade in the market 13

14 during a given period. For example, N order an be thought of as the number of orders reeived during the time period [t;t + 1] if we assume that different orders orrespond to net demands, as defined above, of different lusters of agents. For a time period of, say, 15 minutes on a liquid market suh as NYSE, N order = is a typial order of magnitude. The distribution of the prie variation δx is then given by P( x = x) = N P(n = k) k=1 k j=0 k j (2a) j (1 2a) k j f j (λx) (11) where denotes a onvolution produt, n being the number of lusters. The above equation enables us to alulate the moment generating funtions F of the aggregate exess demand D in terms of f (see Appendix 3 for details): F (z) exp[n order (1 N! 2 )( f (z) 1)] (12) The moments of D (and those of x) may now be obtained through a Taylor expansion of Eq.(12) (see Appendix 4 for details). The alulation of the variane and the fourth moment yields: µ 2 (D) = N order (1 2 )µ 2(X α ) (13) µ 4 (D) = N order (1 2 )µ 4(X α )+3Norder 2 (1 2 )2 µ 2 (X α ) 2 (14) An interesting quantity is the kurtosis of the asset returns whih, in our model, is equal to the kurtosis of exess demand κ(d): κ(d) = µ 4 (X α ) N order (1 2 )µ 2 (X α ) The moments µ j (X α ) may be obtained may be obtained by an expansion in 1/N where N is the number of agents in the market (see Appendix 2). Substituting their expression on the above formula yields the kurtosis κ(d) as a funtion of and the order flow: 2+1 κ(d) = N order (1 2 )A()(1 ) 3 (15) where A() is a normalization onstant with a value lose to 1 defined in Appendix 2, tending to a finite limit as! 1. This relation may be interpreted as 14

15 follows: a redution in the volume of the order flow results in larger prie flutuations, haraterized by a larger exess kurtosis. This result orresponds to the well known fat that large prie flutuations are more likely to our in less ative markets, haraterized by a smaller order flow. It is also onsistent with results from various market mirostruture models where a larger order flow enables easier regulation of supply and demand by the market maker. It is interesting that we find the same qualitative feature here although we have not expliitly integrated a market maker in our model. This result should be ompared to the observation in [17] that, even after aounting for heteroskedastiity, the onditional distribution of stok returns for small firms is higher than that of large firms. Small firm stoks being haraterized by a smaller order flow N order, this observation is ompatible with our results. More importantly, Eq. (15) shows that the kurtosis an be very large even if the number of orders is itself large, provided is lose to 1. Sine A(1) is lose to 1/2, one finds that even for = 0:9andN order = 1000, the kurtosis κ is still of order 10, as observed on very ative markets on time intervals of tens of minutes. Atually, one an show that provided 2aN is not too large, the asymptoti behaviour of P( x) is still of form given by Eq. (6). This model thus leads naturally to the value of µ = 3=2, lose to the value observed on real markets. Of ourse, the value of ould itself be time dependent. For example, herding tendeny tends to be stronger during periods of unertainty, leading to an inrease in the kurtosis. When reahes one, a finite fration of the market shares simultaneously the same opinion and this leads to a rash. An interesting extension of the model would be one in whih the time evolution of the market struture is expliitly modeled, and the possible feedbak effet of the prie moves on the behavior of market partiipants. 5 Disussion We have exhibited a model of a stok market whih, albeit its simpliity, gives rise to a non-trivial probability distribution for aggregate exess demand and stok prie variations, similar to empirial distributions of asset returns. Our model illustrates the fat that while a naive market model in whih agents do not interat with eah other would tend to give rise to normally distributed aggregate flutuations, taking into aount interation between market partiipants through a rudimentary herding mehanism gives a result whih is quantitatively omparable to empirial findings on the distribution of stok market returns. 15

16 One of the interesting results of our model is that it predits a relation between the fatness of the tails of asset returns as measured by their exess kurtosis and the degree of herding among market partiipants as measured by the parameter. This relation is given by Eq.(15). Although we impliitly assumed that t represents hronologial time, one ould formulate the model by onsidering t as market time, leading to a subordinated proess in real time as in [13], with the differene that the underlying proess will not be a Gaussian random walk. Our model raises several interesting questions. As remarked above, the value of is speified as being less than, and lose to 1. Fine-tuning a parameter to a ertain value may seem arbitrary unless one an justify suh an assumption. An interesting extension of the model would be one in whih the time evolution of the market struture is expliitly modeled in suh a way that the parameter remains in the ritial region (lose to 1). One approah to this problem is via the onept of self-organized ritiality, introdued by Bak et al [2]: ertain dynamial systems generially evolve to a state where the parameters onverge to the ritial values leading to saling laws and heavy-tailed distributions for the quantities modeled. This state is reahed asymptotially and is an attrator for the dynamis of the system. Bak, Chen, Sheinkman and Woodford [1] present a simple model of an eonomi system presenting self-organized ritiality. Note however that, for the above results to hold, one does not need to adjust to a ritial value : it is suffiient for to be within a ertain range of values. As noted above, when approahes 1 the lusters beome larger and larger and a giant oalition appears when 1. In our model the ativation of suh a luster would orrespond to a market rash (or boom). In order to be realisti, the dynamis of should be suh that the rash (or boom) is not a stable state and the giant luster disaggregates shortly after it is formed: after a short period of pani, the market resumes normal ativity. In mathematial terms, one should speify the dynamis of (t) suh that the value = 1 is repulsive. This an be ahieved by introduing a feedbak effet of pries on the behavior of market partiipants: a nonlinear oupling between an lead to a ontrol mehanism maintaining in the ritial region. Yet another interesting dynamial speifiation ompatible with our model is obtained by onsidering agents with threshold response. Threshold models have been previously onsidered as possible origins for olletive phenomena in eonomi systems [25]. One an introdue heterogeneity by allowing the individual threshold θ i to be random variables: for example one may assume the θ i stobe 16

17 iid with a standard deviation σ(θ). A simple way to introdue interations among agents is through an aggregate variable: eah agent observes the aggregate exess demand D(t) given by Eq.(2) or eventually D(t) +E(t),where Eis an exogeneous variable. Agents then evolve as follows: at eah time step, an agents hanges its market position φ(t) ( flips from long to short or vie versa) if the observed signal D(t) rosses his/her threshold θ i. Aggregate flutuations an then our through asades or avalanhes orresponding to the flipping of market positions of groups of agents. This model has been studied in the ontext of physial systems by Sethna et al [46] who have shown that for a fairly wide range of values of σ(θ) one observes aggregate flutuations whose distribution has power-law behavior with exponential tails, as in Eq.(6). These issues will be adressed in a forthoming work. Referenes [1] Bak P., Chen, Sheinkman J. & Woodford M. (1993) Aggregate flutuations from independent setorial shoks: self-organized ritiality in a model of prodution and inventory dynamis Rierhe Eonomihi, 47, [2] Bak P., Tang C. & Wiesenfeld, K. (1987) Self-organized ritiality Physial Review Letters, 59, 381. [3] Bak P., M. Pazuski & M. Shubik (1996) Prie variations in a stok market with many agents Santa Fe Institute Working Paper. [4] Bannerjee, A. (1992) A simple model of herd behavior Quarterly Journal of Eonomis, 107, [5] Bannerjee, A. (1993) The eonomis of rumours Review of Eonomi Studies, 60, [6] Bikhhandani S., Hirshleifer D. & Welh I. (1992) A theory of fads, fashion, ustom and ultural hanges as informational asades Journal of Politial Eonomy, 100, [7] Billingsley, P. (1975) Convergene of probability measures, NewYork:Wiley. [8] Bollobas, B. (1985) Random graphs New York: Aademi Press. 17

18 [9] Bollerslev T., Chou R.C. & Kroner K. (1992) ARCH modeling in finane Journal of Eonometris, 52, [10] Bouhaud J.P. & Potters M. (1997) Théorie des risques finaniers, Paris: Ala Salay. [11] Caldarelli G., Marsili M. & Zhang Y.C. (1997) A prototype model of stok exhange Université de Fribourg preprint. [12] Campbell J., Lo A.H. & MKinlay C. (1997) The eonometris of finanial markets, Prineton University Press. [13] Clark, P.K. (1973) A subordinated stohasti proess model with finite variane for speulative pries Eonometria, 41, [14] Cont R., Potters M. & Bouhaud J.P. (1997) Saling in stok market data: stable laws and beyond to appear in Sale invariane and beyond, Proeedings of the CNRS Workshop on Sale Invariane (Les Houhes, Frane), Marh [15] Cont, R. (1997) Saling properties of intraday prie hanges Siene & Finane Working Paper (ond-mat/ ). [16] Cutler D.M., Poterba J.M., & Summers, L. (1989) What moves stok pries? Journal of Portfolio Management, Spring, [17] Engle, R. & Gonzalez-Rivera (1991) Semiparametri ARCH models Journal of Business and Statistis, 9, [18] Engle, R. (1995) ARCH: seleted readings, Oxford: Oxford University Press. [19] Erdös, P. & Renyi, A. (1960) On the evolution of random graphs Publiations of the Mathematial Institute of the Hungarian Aademy of Sienes, 5, [20] Fama, E.F. (1965) The Behavior of Stok market Pries Journal of Business, 38, [21] Farmer, D. (1997), Seminar presented at UniversitédeParisVI. 18

19 [22] Feller, W. (1950) Introdution to Probability theory and its appliations, II, 3rd ed., New York: John Wiley & Sons. [23] Gallant, A.R. & Tauhen, G. (1989) Semi non-parametri estimation of onditional onstrained heterogeneous proesses, Eonometria, [24] Geman, H. & Ané, T. (1996) Stohasti subordination RISK, September. [25] Granovetter, M. & Soong, R. (1983 ) Threshold models of diffusion and olletive behavior Journal of Mathematial Soiology, 9, [26] Grinblatt M., Titman S. & Wermers R. (1995) Momentum investment strategies, portfolio performane and herding: a study of mutual fund behavior Amerian Eonomi Review, 85 (5), [27] Guillaume D.M., Daorogna M.M., Davé R.R., Müller U.A., Olsen R.B., Pitet O.V. (1997) From the birds eye to the mirosope: a survey of new stylized fats of the intra-day foreign exhange markets Finane and Stohastis, 1, [28] Hausman J., Lo A.W. & MaKinlay C. (1992) An ordered probit analysis of transation stok pries Journal of finanial eonomis,31, [29] Hsieh, D.A. (1991) Chaos and non-linear dynamis: appliation to finanial markets, Journal of Finane, 46, [30] Ioannides, Y.M. (1990) Trading unertainty and market form International Eonomi review, 31, 3, [31] Ioannides, Y.M. (1996) Evolution of trading strutures in Arthur,B.W., Lane,D. & Durlauf, S.N. (eds.) The Eonomy as an evolving omplex system (Santa Fe Institute), Addison Wesley. [32] Kempf, A. & Korn, O. (1997) Market depth and order size, Lehrstuhl für Finanzierung Working Paper 97-05, Universität Mannheim. [33] Kirman, A. (1983) Communiation in markets: a suggested approah Eonomis Letters, 12, 1-5. [34] Kirman A., Oddou C. & Weber S. (1986) Stohasti ommuniation and oalition formation Eonometria, 54,

20 [35] Kirman, A. (1996) Interation and markets, GREQAM Working Paper. [36] Lakonishok J., Shleifer A. & Vishny R. (1992) Impat of institutional investors on stok pries, Journal of finanial Eonomis, 32, [37] Levy, M. & Solomon, S. (1997) New Evidene for the Power Law Distribution of Wealth Physia A, 242, [38] Mandelbrot, B. (1963) The variation of ertain speulative pries Journal of Business, XXXVI, [39] Mandelbrot, B. (1997) Fratals and saling in finane, Berlin: Springer. [40] R. Mantegna & H. E. Stanley Nature, 376, (1995). [41] Orléan, A. (1995) Bayesian interations and olletive dynamis of opinion Journal of Eonomi Behavior and Organisation, 28, [42] Pitet O.V., Daorogna M., Muller U.A., Olsen R.B. & Ward J.R. (1997) Statistial study of foreign exhange rates, empirial evidene of a prie hange saling law and intraday analysis Journal of Banking and Finane, 14, [43] Pagan, A. (1996) The eonometris of finanial markets Journal of Empirial Finane,3, [44] Sharfstein D.S. & Stein J.C. (1990) Herd behavior and investment Amerian Eonomi Review, 80, [45] Sethna J.P., Dahmen K., Kartha S., Krumhans J.A., Roberts B.W. & Shore J. (1993) Hysteresis and hierarhies: dynamis of disorder-driven first-order phase transformations Phys. Rev. Lett., 70, [46] Sethna J.P., Perkovi O. & Dahmen, K. (1997) Hysteresis, avalanhes and Barkhausen noise to appear in Sale invariane and beyond, Proeedings of the CNRS Workshop on Sale Invariane (Les Houhes, Frane), Marh [47] Shiller R. (1989) Market volatility Cambridge, MA: MIT Press. [48] Shleifer, A. & Summers, L.H. Crowds and pries: towards a theory of ineffiient markets, University of Chiago Center for Researh in Seurity Pries Working paper

21 [49] Topol, R. (1991) Bubbles and volatility of stok pries: effet of mimeti ontagion Eonomi Journal, 101, [50] Trueman, B. (1994) Analysts foreasts and herding behavior Review of finanial studies, 7 (1), [51] Welh, Ivo (1996) Herding among seurity analysts UCLA Working Paper Appendies: Unless speified otherwise, f (N; ) g(n; ) means f (N;) g(n;) = 1 + o N! (1) uniformly in on all ompat subsets of ]0,1[. Appendix 1: some results from random graph theory In this appendix we will review some results on asymptoti properties of large random graphs. Proofs for most of the results may be found in [19] or [8]. Consider N labeled points V 1 ;V 2 ; :::V N, alled verties. A link (or edge) is defined as an unordered pair fi,j g. A graph is defined by a set V of verties and a set E of edges. Any two verties may either be linked by one edge or not be linked at all. In the language of graph theory, we will onsider non-oriented graphs without parallel edges. We shall always denote the number of verties by N. Apath is defined as a finite sequene of links suh that every two onseutive edges and only these have a ommon vertex. Verties along a path may be labeled in two ways, thus enabling to define the extremities of the path. A graph is said to be onneted if any two verties V i ;V j are linked by a path i.e. there exists a path with V i and V j as extremities. A yle (or loop) is defined as a path suh that the extremities oinide. A graph is alled a tree if it is onneted and if none of its subgraphs is a yle. A graph is alled ayli if all its subgraphs are trees. Consider now a graph built by hoosing, for eah pair of verties V i ;V j whether to link them or not through a random proess, the probability for seleting any given edge being p > 0, the deisions for different edges being independent. A graph obtained by suh a proedure is termed a random graph of type G(N; p) in the notations of [8] 4. 4 This definition orresponds to random graphs of type Γ n;n in [19] (see [19],p. 20). 21

22 In the following, we will be speifially interested in the ase p = =N. Various graph-theoretial parameters of suh graphs are random variables whose distributions only depends on N and. We shall be partiularly interested in the properties of large random graphs of this type i.e. G(N;=N) in the limit N!. The following results have been shown by Erdös and Renyi [19] and Bollobas [8]: If < 1 then in the limit N! all point of the random graphs belong to trees exept for a finite number U of verties whih belong to uniyli omponents. Moreover, the probability of a vertex belonging to a yli omponent tends to zero as N 1=3. For desribing the struture of large random graphs for < 1itis therefore suffiient to aount for verties belonging to trees; yli omponents do not essentially modify the results, exept when = 1. More preisely ( [8], Theorem V.22) U 1 2 σ 2 (U ) 1 2 (e ) k k 3 k=3 j=0 k=3 k j j! k(e ) k k 3 k j j=0 j! The above expressions are valid for 6= 1. Appendix 2: Distribution of luster sizes in a large random graph Let p 1 (s) be the probability for a given vertex to belong to a luster of size s in the N! limit. The moment generating funtion Φ 1 of the p 1 is defined by : Φ 1 (z) = e sz p 1 (s) s=1 We shall now proeed to derive a funtional equation verified by Φ 1 in the large N limit when the effet of loops (yles) are negleted. Let p 1 N (s) be the orresponding probability in a random graph with N verties. Adding a new vertex to the graph will modify the pattern of links, the probability of k new links from the new vertex to the old ones being (=N) k (1 =N) N k N. As shown above (Appendix 1) the probability of reating a yle tends to zero for large N. The onstraint that no new yles are reated by the k new links imposes that the k links are made to verties in k different lusters of sizes. If s 1 ;s 2 ; :::s k are the sizes of these lusters, the new links will reate a new luster of size s 1 + s 2 + ::: + s k

23 p 1 N+1 (s)= N k=1 N s 1 ;:::;s k =1 N k ( N )k (1 N )N k δ(s 1 +s 2 +:::+s k +1 s)p 1 N (s 1)p 1 N (s 2):::p 1 N (s k) Multiplying both sides by e sz and summing over s gives: whih gives in the large N limit: Φ 1 (z;n + 1) =e z [1+ N +Φ 1(z;N) N ]N Φ 1 (z) =e z+(φ 1(z) 1) from whih various moments and umulants may be alulated reursively. The distribution of lusters sizes p(s) is then given by p(s) =A() p 1(s) s where A() is a normalizing onstant defined suh that R p(s)ds = 1. Appendix 3: Number of lusters in a large random graph Let n (N) be the number of lusters ( onneted omponents) in a random graph of size N defined as above. n is a random variable whose harateristis depend on N and the parameter. In this setion we will show that n has an asymptoti normal distribution when N! and that for large N, the j-th umulant C j of of n is given by: ( 1) j N C j N! 2 From a well known generalization of Euler s theorem in graph theory l(n) N + n (N) =χ(n) where χ(n) is the number of independent yles and l(n) the number of links. This implies in turn n (N) =N(1 2 )+O(1) 23