Eur. Phys. J. B 71, 489–497 (2009)DOI: 10.1140/epjb/e2009-00337-6

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

The role of communication and imitation in limit order marketsG. Tedeschi1,a, G. Iori2, and M. Gallegati1

1 Department of Economics, Universita Politecnica delle Marche, Ancona, Italy2 Department of Economics, City University London, UK

Received 27 November 2008 / Received in final form 10 September 2009Published online 10 October 2009 – c! EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2009

Abstract. In this paper we develop an order driver market model with heterogeneous traders that imitateeach other on di!erent network structures. We assess how imitations among otherway noise traders, cangive rise to well known stylized facts such as fat tails and volatility clustering. We examine the impactof communication and imitation on the statistical properties of prices and order flows when changingthe networks’ structure, and show that the imitation of a given, fixed agent, called “guru”, can generateclustering of volatility in the model. We also find a positive correlation between volatility and bid-askspread, and between fat-tailed fluctuations in asset prices and gap sizes in the order book.

PACS. 02.50.-r Probability theory, stochastic processes, and statistics – 05.40.-a Fluctuation phenomena,random processes, noise, and Brownian motion – 89.65.Gh Economics; econophysics, financial markets,business and management – 89.65.-s Social and economic systems

1 Introduction

The main purpose of this paper is to understand if andhow herding e!ects may be responsible for the persistenceof asset price volatility. While stocks returns themselvesare relatively uncorrelated, the squares or absolute val-ues of returns are autocorrelated, reflecting a tendencyfor markets to move from relative quiet periods to moreturbulent ones. Significant positive autocorrelation ofabsolute stock returns survives for a year or more, anddecays at a rate which is slower than exponential. A num-ber of econometric time series techniques have been in-troduced for the modelling of time varying variances andcovariances1. However, statistical analysis alone is not suf-ficient to understand the presence or absence of long-rangememory in volatility, and economic mechanisms that canexplain the origin of this phenomena are needed. Someinsights into volatility clustering have been provided byagent-based models. Following the pioneering Santa Fe Ar-tificial Stock Market, SF-ASM [1], several artificial finan-cial markets have been developed over the last 15 years.In particular understanding how sophisticated agents maybe able to reproduce volatility clustering has been widelystudied in the economic literature. For instance ([2–6])are examples of agent based models able to explain the

a e-mail: gabriele.tedeschi@gmail.com1 The first instrument developed in 1982 by Engle for ana-

lyzing the variation in higher order moments is the AutoRe-gressive Conditional Heteroskedastic (ARCH) model.

influence of traders’ behaviours on the persistence of as-set price volatility. As argued by [7], some mechanismsof behavioural switching can generate persistent returnvolatility, even though they are not su"cient to createlong range dependence in absolute returns2.

The models cited above have looked at the e!ect of co-ordination of traders strategies via market mediated inter-actions (for example when agents follow common chartisttrading rules). Collective behaviour nonetheless could re-flects the phenomenon known as herding which occurswhen agents take actions on the basis of directly imitatingeach other ([8–14]). Most of the studies on herding e!ectshave focused on how herding can lead to large price fluc-tuation but only a few papers have investigated its role asa source of volatility clustering ([14–16]). As pointed outby the model of [14] of which this is, in a way, an exten-sion, local communication and imitation among traders, ifdynamically evolving, can induce not only volatility clus-tering but multiscaling as well. The originality of this workas regards [14] is the mechanism of price formation. Weintroduce here an order-driven market where limit ordersare stored in the order book and executed using time pri-ority at a given price and price priority across prices. Theadvantage of an auction market respect to a supply de-mand in-balance approach (as in [14]) is that the price is

2 A short memory process has autocorrelations decaying tozero exponentially as the lag increases. In contrast, the au-tocorrelation of the long memory process decays to zero at ahyperbolic rate with an exponent smaller than 1.

490 The European Physical Journal B

determined through the trading mechanism itself withoutad-hoc pricing rules for reaching an equilibrium price.

The mechanism of price formation following the sup-ply demand in-balance requires the presence of a marketmaker. If in the market there is a market maker there is noexplicit mechanism of trading. The market maker is riskneutral and endowed with unbounded liquidity, which isused to absorb excess demand and make trading alwaysviable, regardless of its size. In each period, the marketmaker adjusts the price to reduce the excess demand. In-spired by the metaphor of the Walrasian tatonnement,this price-adjusting rule fails to recognize that, in a realmarket, trade occurs whenever two agents can match theirrequests at a given price. The supply demand in-balanceapproach shortcuts the study of the out-of-the-equilibriumprice dynamics by jumping to the stationary points. Anorder driven, instead, enables us to understand how theeconomy behaves out of equilibrium. Moreover, since inthe order driven the clearing mechanism is totally algorith-mic, it is similar to electronic crossing markets (ECN’s),and several public markets, such as the Paris Bourse (nowEuronext) and the London Stock Exchange. Here the pricedynamic is determined by the structure of the exchangeprocess without adding any ad hoc deus ex machina.

Another important point di!erentiates this modelfrom [14]. While in [14] agents were placed on a regulargrid and could only interact with their nearest neighbours,here we introduce a random communication structure thatcan evolve dynamically. The communication network istaken as exogenous. In this setting, we compare the e!ectsof random interaction versus the imitation of a given, fixedagent, that we name the guru.

Agents in our model imitate the expectations ofthe Guru and not its actions3. Using expectation in-stead of action is a modeling choice. In economic models(see [20–22]), there are many possible mechaninsm of imi-tation through which information could be shared outsideprice system. “Obviously, the correct model for informa-tion sharing is not identifiable, but it is clear that someimitation must take place in financial markets” [16].

The imitation mechanism we develop here is based onthe concept of social learning: our agents base themselveson the experience of traders they link with and, particu-larly, on the expectation of traders they link with.

In the economic literature the concept of expectationhas a central position. Expectations drive individual be-haviours and individual behaviours determine the eco-nomic outcome, i.e., prices and trading. “Therefore, a mar-ket, like other social environments, may be viewed as anexpectations feedback system”[23].

In an order driven market, it is not necessarily the casethat this kind of imitation leads to larger coordination andin turn to larger price fluctuations. In fact, even if theguru expects a price increase/decrease, he himself and/orthe agents that follow his advice may submit limit ordersto buy/sell instead of market orders, and the immediate

3 Our market is populated by agents with naive tradingstrategies, called noise traders. In this way we are close to thetradition of Zero-Intelligence (ZI) traders as in ([17–19])

impact of their trades may be negligible or too small to af-fect prices. As it is well documented ([24,25]) the impactof limit orders, specially in case of synchronization andimitation, should reduce price fluctuations and not am-plify them. In fact the mechanism describing the impactof limit orders on prices would run as follows. A marketorder on one side of the book attract compensating limitorders, so when there are more market orders in one sideof the book, one expects more limit orders and cancella-tions on the same side as well. Intuitively, if the guru splitshis order and buys or sells using market orders for a longperiod of time, this will attract, thank to the imitatione!ect, compensating limit orders on the same side of thebook and their e!ect should re-equilibrate prices and notto generate large price fluctuations. So the impact of a buytrade following other buy traders should be smaller thanthe impact of a sell trade following buy traders. [25] showthat events happening on the same side of the book arelong-range correlated, but the signed correlation function(that assigns an opposite sign to limit and market orderson the same side of the book) is short ranged, showingthat the compensating e!ect alluded to above is indeede!ective. The long term impact of these imitation leadtrades is also not clear a priori. For example, if imitationresults in an accumulation of limit orders near the bid/askthey may dampen future price fluctuations instead of am-plifying them.

Since limit orders tend to re-equilibrate price fluctu-ations with an impact which is not often clear a priori,to favour market order submission of gurus and of morelinked traders we assume that our investors are able topredict the impact of imitation on stock prices when theyform their expectation. In this way we assume that themore imitated agents, recognizing their ability to influenceprices, have an higher possibility to submit market or-ders. From an economical point of view these gurus couldbe considered as ‘more informed’ agents who place mar-ket orders for immediate execution. However this simplemechanism does not underestimate limit orders. In fact inthe model there are ‘less informed’ traders with few or nofollowers who place limit orders on both sides of the orderbook. This simple rule is able to better incorporate theimportance of herding into our model and its influence ongenerating trading synchronization and large price fluctu-ations.

In the last part of our study we investigate the correla-tion between herding, volatility, bid-ask spread and orderflows. While the availability of order book data has madepossible a number of empirical studies on limit order flows,only a few papers [5] have analyzed these features is sim-ulation studies. Empirical analysis has showed fat tails inthe distribution of limit order arrivals ([26] and [27,28])and fat tail decay of order distribution in the limit orderbook [29] has also shown that fat tails in the distributionof returns emerge when large gaps are present between thebest price and the price at the next best quote (and arenot linked, as suggested by [30], to large market orders ar-rival). We show that our model is capable of reproducingthese stylized facts.

G. Tedeschi et al.: The role of communication and imitation in limit order markets 491

The rest of the paper is organized as follows. In Sec-tion 2 we describe the model; in Section 3 we present theresults of the simulations. Section 4 concludes.

2 The mathematical structure of the model

2.1 The market

The market in our model is order-driven. A populationof N traders can either place market orders, which areimmediately executed at the current best listed price, orthey can place limit orders. Limit orders are stored in theexchange’s book and executed using time priority at agiven price and price priority across prices. A transactionoccurs when a market order hits a quote on the oppositeside of the market.

Trading happens over a number of periods tk, withk = 1, . . . T . At the beginning of each period, traders makeexpectations about the price at the end of a given timehorizon ! (that we take to be the same for all traders).The future price4 expected at time tk + ! by agent i isgiven by

pitk,tk+! = ptkeri

tk,tk+!

!! (1)

where ritk,tk+! is the agent’s expectation on the spot re-

turn which, as we will see later, may be a!ected by theexpectation of other agents and ptk is the reference priceobserved by all agents at the beginning of each period.

After expectations are made, Nt trades are submittedeach period. Agents enter the market, sequentially andin a random order, determine a limit price and an ordersize, and submit their orders. If an agent expects a fu-ture price increase (decrease) he decides to buy (sell) at aprice bi

t (ait) lower (higher) than his expected future price

pitk,tk+! . Bids and asks are uniformly distributed in an

interval (pmin, pmax) around the current price, calculatedaccording to the following rules

bit ! U(pb

min, pbmax), p

bmin = pt(1 " "1

t ), pbmax = pi

tk,tk+!

(2)

ait ! U(pa

min, pamax), p

amin = pi

tk,tk+! , pamax = pt(1 + "2

t )(3)

where "1,2t are random variables uniformly distributed in

the interval (0, 1) and pt is the price at the time the orderis submitted. The price is recalculated as trading goes onas follows: pt is given by the price at which a transactionoccurs, if any; if no new transaction occurs, a proxy for pt

is given by the average of the quoted ask aqt (the lowest

ask listed in the book) and the quoted bid bqt (the highest

bid listed in the book), pt = (aqt + bq

t )/2; if no bids orasks are listed in the book a proxy for pt is given by theprevious traded or quoted price. Bids, asks and prices are

4 Equation (1) wants to reproduce a geometric Brownian mo-tion simulation. This process can use (or not) a square rootwithout considerable implications for the dynamic itself.

positive and investors can submit limit orders at any priceon a prespecified grid, defined by the tick size #5.

We assume that agents have a random demand func-tion and that the size of their order is bounded only bybudget constraints. Agents hold a finite amount of cashCi

t and stocks Sit in their portfolio. The size si of agent’s

i order is determined as follows:

– if the agent expects a price decrease he sells a randomfraction of his assets si = $tSi;

– if the agent expects a price increase he invests a ran-dom fraction of his cash in the assets equal tosi = $tCi

t/aqt if he submits a market order

si = $tCit/bi

t if he submits a limit orderwhere $t is a random variable uniformly distributed onthe interval (0, 1).

The order book is the list of all buy and sell ‘limit’ orders,with their corresponding price and volume, at a given in-stant of time. A limit order specifies the maximum (orminimum) price at which a trader is willing to buy (or sell)a certain number of shares. At a given instant of time, alllimit buy orders are below the best buy order called the‘bid price’, corresponding to pb

max (Eq. (2)), while all sellorders are above the best sell order called the ‘ask price’,corresponding to pa

min (Eq. (3)). When a new order ap-pears (say a buy order), it either adds to the book if it isbelow the ask price, or generates a trade at the ask if it isabove (or equal to) the ask price (we call all these ‘mar-ket orders’). The price dynamics is therefore the result ofthe interplay between the order book and the order flow.More in detail, if bi

t < atq or ai

t > btq the agent place a limit

order at its limit price. If bit # at

q, then the agent places amarket order to buy at the current quoted ask at

q. If thesupply available on the book at the ask price at

q is notsu"ciently large to fulfil the order, the agent buys all thequantity available at the ask and then moves on to checkthe second best ask price, iterating the process until hehas no more stocks to buy or there are no more sell ordersin the book at a price smaller than bi

t. Symmetrically, ifai

t $ btq, then the agent places a market order to sell at a

price btq. If the demand available on the book at this price

is not su"ciently large, he fills the available demand at thebid and then moves on to check the second best bid price,iterating the process until the agent has no more stock tosell or there are no more buy orders in the book at a pricegreater than ai

t. If the order can only be partially filled,the agent places its unmatched demand/supply as a limitorder. If the limit order is still unmatched at time t + ! ,it is removed from the book.

The essential details of the trading mechanism aresummarized in Table 1, showing how it depends on the bidprice bi

t, the ask price ait, the expected price level pi

tk,tk+! ,the quoted ask aq

t and the quoted bid bqt .

When agents place a market order, their cash andstocks positions are updated accordingly. When agentsplace a limit order, the cash they commit to buy and thestocks they commit to sell are also temporarily removed

5 The tick is the smallest possible change of pt.

492 The European Physical Journal B

Table 1. Summary of the trading mechanism of a typicaltrader i with bids bi

t and asks ait uniformly distributed in an

interval (pmin, pmax). The lowest ask and highest bid listed inthe book are aq

t and bqt respectively. pi

tk,tk+! is the expectedfuture price.

Position Type of order Volumebit < pi

tk,tk+! < atq BUY Limit order si = !tCi

t/bit

aqt " bi

t < pitk,tk+! BUY Market order si = !tC

it/aq

t

btq # ai

t > pitk,tk+! SELL Market order si = !tS

i

ait > pi

tk,tk+! > btq SELL Limit order si = !tS

i

from their portfolios (even if a limit order does not com-port an immediate transaction). In this way agents cannot spend money or sell stocks that have already commit-ted in the book. If the order is cancelled, the stocks andcash that were tied down in the orders are returned to thetraders who had submitted them.

2.2 The interaction network

To model how agents’ decision are influenced by their mu-tual interaction we introduce a communication structurein which nodes represents agents and the hedges are theconnective links between them. Links are directional andgo from the agent that requests advice to the agent thatprovides advice.

In general local interaction models, agent interacts di-rectly with a finite number of others in the population.The set of nodes with whom a node is linked is referredto as its neighborhoods. In our model the number of out-going links is constrained to be one. The reason beingthat in a highly connected random network synchronisa-tion could be achieved via indirect links. The e!ects ofdirect imitation are easier to be tested in a diluted net-work where indirect synchronisation is less likely to arise.

We start with the case where each agent i makes aunique link with a randomly chosen agent j. In this casethe probability that any two agents (i, j) are linked is in-dependent of both i and j. We call this model the randomattachment case. We then move to the case where eachagent i chooses to link, with a probability % (indepen-dent of i), to a fixed agent j and with probability 1 " %to a randomly chosen agent. We call this fixed agent j,who attracts the largest number of in-coming links, the“guru”. The random attachment case is recovered in thelimit % = 0. When % is equal to one all agents are linkedto the guru and the network becomes star like.

We subsequently introduce a time dependent proba-bility of attachment that evolves as

d%t

%t= &"'t. (4)

The probability %t is constrained to take values in theinterval (0, 1) and is reflected at the boundaries.

2.3 The expectation formation mechanism

At the beginning of each trading period tk, agents makeidiosyncratic expectations about the spot return, ri

tk,tk+!

in the interval (tk, tk + !). We assume that agents are notinformed and have random expectation of future returns.We also assume that agents are heterogeneous in that theyhave di!erent forecasts of the returns’ volatility, &i

t. Ex-pected returns are thus given by

ritk,tk+! = &i

tk'tk (5)

where &itk

is a positive, agent specific, constant and 't !N(0, 1) is a normal noise.

After individual expectation are generated, a consul-tation round starts during which agents sequentially, andin a random order, revise their expectation. The revisedexpected return is obtained by weighing agent i’s own ex-pectation with that of agent j to which i is linked to

ritk,tk+! = wri

tk ,tk+! + (1 " w)rjtk ,tk+! . (6)

When w is equal to zero, i trusts completely the opinionsof j, while when w is equal to one i considers exclusivelyhis own opinion and agents decisions are fully independentfrom each other. At the beginning of each period tk agentsexpectations are reset to random values. We stress thatin the model imitation is purely expectation based, andagents do not imitate the actions of others.

While agents are noise traders in our model, we assumethat they correctly anticipate the impact of herding onasset prices. In particular if an agent has several incominglinks, and w is small (in which case the agent expects tobe able to influence the decisions of others), he forecast alarger price volatility. This is incorporated in the modelby assuming that the volatility of forecasted returns isproportional to the number of incoming links and to theweights w, such that

&itk

= &i0(A + l%i,tk

(1 " w)) (7)

where l%i,tkin the percentage of existing links that point

to agent i at time tk and A is a constant parameter. Thevalues of &i

0 are chosen, with uniform probability, in theinterval (0, &0). The e!ect of this rule is to favour marketorder submission over limit order submission from agentswho have many incoming links. This happens because theinterval over which the limit price is chosen is wider when&i

tkis larger and the selected limit order are more likely

to be immediately marketable. Not only the guru is morelikely to submit a market order in this way, but also hisfollowers are, because of the expectation formation mech-anism in equation (6). A sequence of market orders in thesame direction will in turn generate large price fluctua-tions fulfilling the original expectation of the guru.

3 Simulations and results

The model is studied numerically for di!erent values ofthe parameters % and w. We focus the analysis on the sta-tistical properties of the probability distribution of stockreturns and on the auto-correlation of market volatility.

G. Tedeschi et al.: The role of communication and imitation in limit order markets 493

We also analyse some properties of the order submis-sion strategies such as the spread, the distance betweenbest price (bid/ask) and the price at the next best quoteand the trading volume.

In the simulations the number of traders is set atN = 100. Each agent is initially given a random amount ofstocks, uniformly distributed in the interval S0 % [0, 100]and an amount of cash uniformly distributed in the in-terval C0 % [0, 100]. The initial stock price is chosenat p0 = 1000. We fix ! = 200, # = 0.01, n0 = 0.01,&" = 0.01. The results reported here are the outcome ofsimulations of T = 2000 periods and Nt = 200 trades perperiods. With about 200 transactions per period, a tradingperiod would be of the order of few minutes to few hours,depending on the the liquidity of the market. Simulationsare repeated M = 100 times with a di!erent random seed.While 100 agents may appear quite small, the number ofagents in real markets, who monitor each other expecta-tions is not large. Adding a number of pure noise tradersto the system, may make the simulations more realisticbut would not change qualitatively the results, becausenoise traders transactions would, on average, o!set eachothers.

3.1 Returns and volatility

We first analyse the statistical properties of returns in anetwork with a fixed guru as we increase the (constant)probability of attachment %.

Figure 1 displays a sample path of the return timeseries for % = 0, % = 0.5 and % = 1 when w = 0.1.

Figure 2 shows the return time series when the prob-ability of attachment follows equation (4), for di!erentvalues of w.

A more precise measure of fat tails is provided by theHill exponent. In Figure 3 (upper box) we plot the hillexponent as a function of % when w = 0.5, and in theright side we plot it as a function of w for % = 0, % = 1,and % time dependent.

Empirically the tail exponent is found to take valuesbetween 2 and 4. Changing the parameters of the modelour simulations generates values for the Hill exponent inthe same range. When w = 1 or % = 0, that is in absence ofimitation, the tail exponent approaches the normal valueof 4. Nonetheless this limit value is never reached, becauseof the budged constrain.

It is clear from the figures that the introduction of afixed guru in the model leads to fat tails but is not capableof generating volatility clustering. This is due to the factthat synchronisation e!ects become more and more im-portant when % increases and this can induce large fluctu-ations at all times. Volatility clustering instead reflects thetransition from relatively quite trading period to more tur-bulent ones. To generate volatility clustering in our modelwe need to have a probability of attachment % that in-creases and decreases over time.

A simple way to check if volatility is persistent is tomeasure the autocorrelation function of absolute returnsfor di!erent time lags. Empirically it is observed that

0 500 1000 1500 2000Time

!0.7

!0.5

!0.3

!0.1

0.1

0.3

0.5

0.7

retu

rns

0 500 1000 1500 2000Time

!0.7

!0.5

!0.3

!0.1

0.1

0.3

0.5

0.7

retu

rns

0 500 1000 1500 2000Time

!0.7

!0.5

!0.3

!0.1

0.1

0.3

0.5

0.7

retu

rns

Fig. 1. Return time series for " = 0 (top) and " = .5 (center)and " = 1 (bottom) and w=0.1.

absolute returns are autocorrelated over lags of severalweeks and decay slowly to zero. In fact, several authors(see [32–37] and [7] for evidence), have shown that theautocorrelation function of absolute returns, decreases hy-perbolically with the time lag ! :

|C(!)| ! B

!#(8)

494 The European Physical Journal B

0 500 1000 1500 2000Time

!0.5

!0.3

!0.1

0.1

0.3

0.5

0.7

0.9

Ret

urns

0 500 1000 1500 2000Time

!0.5

!0.3

!0.1

0.1

0.3

0.5

0.7

0.9

Ret

urns

0 500 1000 1500 2000Time

!0.5

!0.3

!0.1

0.1

0.3

0.5

0.7

0.9

Ret

urns

Fig. 2. (Color online) Returns time series with a time depen-dent " and w=0.2 (top), w=0.5 (center) and w=0.8 (bottom).The evolution of p is described by the red line.

with ( % [0.2, 0.4] and is B a positive constant called thetail or scale parameter. In Figure 4 we plot the autocor-relation of absolute returns (top) and of return (center)for w=0.1 and % = 0 (black line), % = 1 (dotted line)and time dependent % (dashed line). We note that whilethe autocorrelation of returns is insignificant in all cases,a positive and slowly decaying autocorrelation of absolutereturns is present in the case of time dependent %. Theautocorrelation function of absolute returns is well fitted

!0.1 0.1 0.3 0.5 0.7 0.9lambda

1.5

2

2.5

3

3.5

Hill

exp

onen

t

!0.1 0.1 0.3 0.5 0.7 0.9 1.1w

1

1.5

2

2.5

3

3.5

4

Hill

exp

onen

t

Fig. 3. (Color online) Hill exponents of the returns distribu-tion as as a function of constant " for w = 0.5 (top). Hillexponents of the returns distribution as a function of w for" = 0, " = 1 and " time dependent, respectively black, dottedred and dashed green lines (bottom).

by a power law with ) = 0.32 as seen in Figure 4 (upperbox).

A more precise measure of long term memory is pro-vided by the R/S statistics. In the modified R/S anal-ysis the parameter that allows to discriminate betweenshort and long term memory is the exponent )6. If onlyshort memory is present ) converges to 1/2 while withlong memory, it converges to a value larger than 1/2. InFigure 5 we plot the modified R/S exponent ) for ab-solute returns as a function of w. We observe that, onlyfor time dependent %, ) becomes significantly larger that0.5. In this case we also observe that ) increases when wdecreases reflecting that the memory is longer when theimitation is stronger.

The attachment mechanism in equation (4) shows thatthe dynamic of creating, sustaining and destroying theguru is a plausible source of volatility clustering and longmemory. The % in our model evolves according to a re-

6 The # estimates the slope of the log of the Lo’s modifiedR/S statistic versus the log of the sample size.

G. Tedeschi et al.: The role of communication and imitation in limit order markets 495

0 10 20 30 40 50 60 70 80 90 100Lag

0

0.1

0.2

0.3

0.4

Aut

ocor

rela

tion

func

tion

of a

bsol

ute

retu

rns

0 20 40 60 80 100Lag

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

Aut

ocor

rela

tion

func

tion

of r

etur

ns

0 10 20 30 40 50 60 70 80 90 100Lag

0

0.1

0.2

0.3

0.4

Aut

ocor

rela

tion

func

tion

of a

bsol

ute

retu

rns

Fig. 4. (Color online) Autocorrelation of absolute returns(top) and of returns (center) for w=0.1 and " = 0 (black line)," = 1 (dotted red line) and time dependent " (dashed greenline). Autocorrelation of absolute returns for w=0.5 and timedependent " (black line) and the power law best fit (red line)(bottom).

flected Geometric Brownian Motion. The exact dynamicof % is not crucial but volatility clustering only arises if% does not change too quickly7. Here % is updated every

7 In the endogenous version of the model oscillations on thenumber of incoming links to the guru are generated by allow-ing agents to choose to whom to link. In fact, we implement analgorithm where agents link to the most successful agent, andnot to a fixed guru. In that contest we introduce a switching

!0.1 0.1 0.3 0.5 0.7 0.9 1.1w

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bet

a

Fig. 5. (Color online) R/S exponent for absolute returns asfunction of w for " = 0 (black line), " = 1 (dotted red line)and the time dependent case (dashed green line).

trading period, but the value of &" used for the simulationsis small (&" = 0.01). Figure 2 indeed shows that significa-tive changes in %, and the burst of volatility, occur overa few hundreds periods that correspond, approximately,to periods of months. A more precise fine tuning of theexponent ) could be achieved by optimizing the ratio be-tween the time scale at which the information networkevolves and the time scale at which the trading decision isupdated. A di!erent ratio between these two time scalescould explain the di!erent values of ) observed empiricallyin di!erent markets.

3.2 Book and order flows

In this subsection we investigate the impact of the imita-tion on the order flows when the probability of attachmentfollows equation (4) (the case that leads to volatility clus-tering).

Figure 6 (top) displays the decumulative distributionfunction (DDF) of the limit order placement distance fromthe midpoint for di!erent values of w. We observe that,increasing imitation, limit orders are placed at large dis-tance from the midpoint. Figure 6 (bottom) shows theDDF of the bid-ask spread when changing w. For smallvalues of w we observe that the DDF becomes power-law,in agreement with the empirical observation that spreadand volatility are positively correlated. These results areconsistent with the empirical analyses of [28] and [27].

Next, we investigate the relationship between fat-taildfluctuations in asset prices and trading volume and gapsizes. According to [30], power-law tails in price fluctua-tion are driven by power-low fluctuations in the volumeof transactions. The power-law decay of trading volumenonetheless is not well establishes empirically and variesfrom exchange to exchange [29] have suggested an alterna-tive explanation for the large fluctuation in returns. They

probability smaller than one, to avoid that a new guru substi-tutes himself too quickly to the previous one.

496 The European Physical Journal B

1 10 100 1000 10000distance

1

100

10000

1000000

DD

F

0 1 10 100 1000 10000spread

1

10

100

1000

10000

DD

F

Fig. 6. (Color online) DDF of limit order placement distancefrom the midpoint, respectively for w equal to 0 (black line), 0.1(red line), 0.3 (green line), 0.5 (blue line), 0.7 (yellow line), 0.9(cyan line) and 1 (magenta line) (top). DDF of bid-ask spread,respectively for w equal to 0 (black line), 0.1 (red line), 0.3(green line), 0.5 (blue line), 0.7 (yellow line), 0.9 (cyan line)and 1 (magenta line) (bottom).

show empirically that order submission typically results ina large price change, not when the order is large but whena large gap is present between the best price (ask- bid)and the price at the next best quote. Figure 7 displays theDDF of the trading volume in our model for di!erent val-ues of w. We observe that, when imitation increases, trad-ing volume also increases, but the DDF does not changeshape to become more fat tailed, thus contradicting theargument of [30]. Figure 8 shows the DDF of the firstgap on the ask side (top) and bid side (bottom) in ourmodel as a function of w. We observe that the gap dis-tribution does become power law as imitation increases.This result indicates that the formation of large gaps is theleading mechanism generating large price fluctuations inthe model. Large gaps in our model are created when im-itation is large because limit orders are placed at a widerrange of distances from the midpoint (as seen in Fig. 6).This in turn is due to the larger volatility predicted bythe Guru, via equation (7), which a!ects both the choiceof pi

tk,tk+! and of the interval (pmin, pmax).

1 10 100 1000 10000trading volume

1

10

100

1000

10000

DD

F

Fig. 7. (Color online) DDF of trading volume, respectively forw equal to 0 (black line), 0.1 (red line), 0.3 (green line), 0.5(blue line), 0.7 (yellow line), 0.9 (cyan line) and 1 (magentaline).

0 0 0 1ask gap

1

100

10000

1000000

DD

F

0 0 0 10bid gap

1

10

100

1000

10000

100000

1000000

DD

F

Fig. 8. (Color online) DDF of gap sizes for ask (top) and bid(bottom) side, respectively for w equal to 0 (black line), 0.1(red line), 0.3 (green line), 0.5 (blue line), 0.7 (yellow line), 0.9(cyan line) and 1 (magenta line).

4 Conclusion

In this paper we have investigated the impact of imitatinga fixed guru on the statistical properties of asset prices.We have shown that the model can reproduce a numberof stylised facts observed in real financial time series.

G. Tedeschi et al.: The role of communication and imitation in limit order markets 497

The model proposed in this paper is a toy model thatrelies on a number of ad-hoc exogenously imposed ruleswhose purpose is not to describe the behaviour of realagents in real markets, but to identifying the main condi-tions under which imitation leads to fat tails and volatilityclustering in a limit order market. In particular imitationof believes in itself does not generate trading synchronisa-tion and large price fluctuations unless the guru is capableof anticipating the impact of herding on the asset price.Furthermore for clustering to appear the popularity of theguru need to change, slowly, over time.

The main limitation of this study is in that the commu-nication structure is imposed exogenous on the system. Ina forthcoming paper [31] we extend the analysis allowingimitation to arise endogenoulsy via a fitness mechanismbased on agents wealth. In this more general setting wealso study under which conditions a guru may endoge-nously rise and fall over time, and how imitation a!ectsthe distribution of agents wealth. Furthermore we intro-duce a proper utility function from which to derive agent’sdemand, like the one in [5] and [38] and we perform a morecareful study of how the results depend on the system sizeand on the ratio between the time scale at which the in-formation network evolves and the time scale the tradingdecision is updated.

Gabriele Tedeschi thanks the Economics Department of CityUniversity for kind hospitality during the initial stage of thisthis project.

References

1. B. LeBaron, W.B. Arthur, R. Palmer, Journal ofEconomic Dynamics and control 23, 1487 (1999)

2. T. Lux, M. Marchesi, International Journal of Theoreticaland Applied Finance 3, 675 (2001)

3. C. Chiarella, G. Iori, Quantitative Finance 2, 346 (2002)4. M. LiCalzi, P. Pellizzari, Quantitative finance 3, 470 (2003)5. C. Chiarella, G. Iori, J. Perello, Journal of Economic

Dynamics and Control 33, 525 (2008)6. A. Kirman, G. Teyssiere, Studies in nonlinear dynamics

and econometrics 5, 281 (2002)7. Y. Liu, Journal Econometrics 99, 139 (2000)8. A.V. Bannerjee, Quarterly Journal of Economics 107, 797

(1992)9. A.V. Bannerjee, Review of Economic Studies 60, 309

(1993)10. A. Kirman, Quarterly Journal of economics 108, 137

(1993)11. A. Orlean, Journal of Economic Behavior and

Organization 28, 257 (1995)

12. R. Cont, J.P. Bouchaud, Macroeconomic Dynamics 4, 170(2000)

13. M. Raberto, S. Cincotti, S.M. Focardi, M. Marchesi,Physica A 299, 319 (2001)

14. G. Iori, Journal of Economic Behavior and Organization49, 269 (2002)

15. D. Stau!er, D. Sornette, Physica A 271, 496 (1999)16. B. LeBaron, R. Yamamoto, Physica A 383, 85 (2007)17. G.S. Becker, Journal of Political Economy 70, 1 (1962)18. D.K. Gode, S. Sunder, Journal of Political Economy 101,

119 (1993)19. D.K. Gode, S. Sunder, The Quarterly Journal of

Economics 112, 603 (1997)20. N.J. Vriend in Many-Agent Simulation and Artificial Life,

edited by E. Hillebrand, J. Stender (IOS, Amsterdam,1994), pp. 31–47

21. N.J. Vriend, Journal of Economic Behavior andOrganization 29, 263 (1996)

22. N.J. Vriend, Journal of Economic Behavior andOrganization 24, 1 (2000)

23. P. Heemeijer, C. Hommes, J. Sonnemans, J. Tuinstra,Working paper (2007)

24. F. Lillo, J.D. Farmer, Studies in Nonlinear Dynamics andEconometrics 8, 1 (2004)

25. Z. Eisler, J.-P. Bouchaud, J. Kockelkoren, Working paper(2009)

26. J.P. Bouchaud, M. Mezard, M. Potters, QuantitativeFinance 2, 251 (2002)

27. M. Potters, J.P. Bouchaud, Physica A 324, 133 (2003)28. I.I. Zovko, J.D. Farmer, Quantitative finance 2, 387 (2002)29. J.D. Farmer, L. Gillemont, F. Lillo, S. Mike, A. Sen,

Quantitative finance 4, 383 (2004)30. X. Gabaix, P. Gopikrishnan, P. Plerou, H.E. Stanley,

Nature 423, 267 (2003)31. G. Tedeschi, G. Iori, M. Gallegati, Working paper (2008)32. Z. Ding, C. Granger, R. Engel, Journal of empirical finance

1, 185 (1983)33. Z. Ding, C. Granger, Stylized facts on the temporal distri-

butional properties of daily data from speculative markets,Working Paper (1994)

34. Z. Ding, C. Granger, Journal of Econometrics 73, 185(1996)

35. M.M. Dacorogna, U.A. Muller, O.V. Pictet, C.G. De Vries,Olsen and Associates internal document UAM (1992)

36. R. Cont, M. Potter, J.P. Bouchaud, edited by Dubrulle,Graner, Sornette (Springer, 1997)

37. R. Cont, Doctoral Thesis, Universite de Paris XI, 199838. G. Bottazzi, G. Dosi, I. Rebesco, Journal of Mathematical

Economics 41, 197 (2005)39. B.M. Hill, Annals of Statistics 3, 1163 (1975)40. T. Lux, Applied Financial Economics 11, 299 (2001)41. B. Mandelbrot, Annals of Economic and Social

Mesurements 1, 259 (1972)42. A. Lo, Econometrica 59, 1279 (1991)