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Working Paper Series
_______________________________________________________________________________________________________________________
National Centre of Competence in Research Financial Valuation and Risk Management
Working Paper No. 716
Information Percolation in Centralized Markets
Daniel Andrei Julien Cujean
First version: January 2011 Current version: July 2011
This research has been carried out within the NCCR FINRISK project on
“Dynamic Asset Pricing”
___________________________________________________________________________________________________________
Information Percolation in Centralized Markets∗
Daniel Andrei† Julien Cujean‡
July 14, 2011
PRELIMINARY AND INCOMPLETE. COMMENTS WELCOME
Abstract
We explore the effects of information propagation in a centralized finan-cial market. Specifically, we embed search frictions within the Grossman andStiglitz (1980) framework, relying on information percolation as modeled inDuffie, Malamud, and Manso (2009). First, we show that information percola-tion produces a positive autocorrelation in stock returns. Second, we introducea rumor among the population of investors which we let to propagate duringtwo periods. As trading approaches its final round and that the fundamentalvalue is close to be revealed, we let the rumor die out and show that it producesa strong price reversal toward the fundamental value. Importantly, informa-tion percolation introduces heterogeneity in individual precision among agents.This leads to drastically different investment strategies: agents who have beenmore efficient at gathering signals will tend to act as market makers. Whereas,agents who collected a lesser amount of signals will tend to be trend followers.
∗The authors are grateful for comments received from Rui Albuquerque, Philippe Bacchetta,Snehal Banerjee, Nittai Bergman, Hui Chen, Darrell Duffie, Bernard Dumas, Andrei Jirnyi, Harri-son Hong, Ron Kaniel, Leonid Kogan, Arvind Krishnamurthy, Semyon Malamud, Gustavo Manso,Jianjun Miao, Dimitris Papanikolaou, Rémy Praz, Sergio Rebelo, Costis Skiadas, Jules van Bins-bergen, and seminar participants at Boston University, Kellogg School of Management and MITSloan. Financial support from the Swiss Finance Institute and from NCCR FINRISK of the SwissNational Science Foundation is gratefully acknowledged.†Swiss Finance Institute, University of Lausanne, Ecole des Hautes Etudes Commerciales. E-
mail: [email protected], www.danielandrei.net‡Swiss Finance Institute, Ecole Polytechnique Fédérale de Lausanne. E-mail:
[email protected], www.juliencujean.com
1
1 Introduction
The concept of rational-expectations equilibrium has been proposed to illustrate howmarkets aggregate information that is initially dispersed across individual investors1.While centralized markets contribute to centralize information that is dispersedlyheld by market participants, they only do imperfectly so. In particular, the pres-ence of noise trading and idiosyncratic shocks slows down information aggregationthrough prices. As noted by Grossman and Stiglitz (1980), “this is perhaps lucky,for were it (the price) to do it perfectly, an equilibrium would not exist”. As a result,the Efficient-Market Hypothesis fails and prices slowly drift toward the fundamentalvalue. This fact has alternatively been viewed as an illustration of stock marketsbeing beauty contests. Pursuing Keynes’ idea, Allen, Morris, and Shin (2006) arguethat investors try to forecast the forecasts of others instead of assets’ fundamentalvalue. Doing so, investors end up chasing the crowd. More generally, depending onthe amount of noise in their private signals, investors tend to either follow the trendor to speculate. Whether or not this phenomenon is susceptible to produce pricedrift has been carefully analyzed in Cespa and Vives (2009) and Banerjee, Kaniel,and Kremer (2009).
In the present paper, we attempt to introduce social interactions within theGrossman and Stiglitz (1980) framework. In particular, we assume that marketsare centralized, but that information is not. Accordingly, agents have to search foreach other’s private information. Search dynamics are modeled based on the recenttheoretical developments in the Information Percolation literature2. We show that,although markets are centralized, social interactions still have a strong effect onstock prices. On the one hand, social networks are prone to generate momentumin stock returns. On the other hand, they may help spreading a rumor throughfinancial markets, producing significant reversal once the uncertainty pertaining tothe rumor is resolved.
The rational-expectations literature focuses on the market as a whole and vastlyignores private interactions among investors. This is unfortunate as prices beingimperfect aggregators, social networks have an important role to play, in particularas an alternative channel conveying information through private discussions. Forinstance, Shiller (2000) writes: “Word-of-mouth transmission of ideas appears to bean important contributor to day-to-day or hour-to-hour stock market fluctuations...”.In the context of decentralized markets, Duffie, Malamud, and Manso (2009) stud-
1See Hayek (1945), Arrow (1974) or Grossman (1981).2See Duffie and Manso (2007), Duffie, Malamud, and Manso (2009), Duffie, Malamud, and
Manso (2010), Duffie, Giroux, and Manso (2010).
2
ies the percolation of information through a population of investors. Beyond theparticular scope of financial markets, examples of the relevance of social networks inEconomics abound: Acemoglu, Bimpikis, and Ozdaglar (2010) investigate the effec-tiveness of dynamics of communication as an aggregator of dispersed information.Burnside, Eichenbaum, and Rebelo (2010) show how social dynamics may explainthe moves on the housing market. Stein (2008) describes conversations as being acentral part of economic life. Shiller and Pound (1989) provide evidence that directinterpersonal communications are an important determinant of investors decisions3.
Empirically, few works (to the best of our knowledge) have documented therelevance of word-of-mouth communications for stock return autocorrelation. Theempirical investigation suffers from an identification problem: Due to the so-calledManski critique, a given strategy could be implemented by two individuals becausethey either met and shared their private information or because they observed thesame public news. Two important contributions are still to be mentionned: Hong,Lim, and Stein (2000) show that stocks with greater analyst coverage exhibit moremomentum. More to the point, Hong, Hong, and Ungureanu (2010) investigate,both theoretically and empirically, the relation between the serial correlation ofreturns and information diffusion based on word-of-mouth communication.
We show that, depending on the intensity at which agents meet with each other,information percolation produces momentum in stock returns. More precisely, whenthe frequency of meetings is low, noise trading overwhelms the impact of the diffusionof private information and agents’ portfolio positions tend to be mostly driven bynoise trading. As a consequence, stock returns exhibit reversal. As informationpercolation intensifies, the average private information precision goes up and stockreturns exhibit momentum. Drawing on this insight, we introduce a rumor which welet to propagate among the population of investors for several periods. As tradingapproaches its final round and that the fundamental value is close to be revealed,we let the rumor die out and show that it produces a strong price reversal towardthe fundamental value.
In another context, Hong and Stein (1999) also use gradual diffusion of informa-tion in order to explain momentum and reversal4. However, importantly, unlike thelatter contribution, we do not need to postulate different types of agents in orderto produce this effect. Neither do we prevent investors to extract information from
3For other contributions showing evidence that word-of-mouth communications play a strongrole in financial markets, see Hong, Kubik, and Stein (2004), Feng and Seasholes (2004), Brown,Ivkovic, Smith, and Weisbenner (2008), Grinblatt and Keloharju (2001), Ivkovic and Weisbenner(2005) or Cohen, Frazzini, and Malloy (2008).
4Other famous behavioral contributions include Barberis, Shleifer, and Vishny (1998) andDaniel, Hirshleifer, and Subrahmanyam (1998).
3
prices. On the contrary, we assume that agents start off being identical individ-uals, and behave exactly as in the Grossman and Stiglitz (1980) setup5. As theystart meeting with each other, they progressively become informationally heteroge-nous. This ultimately leads to drastically different investment strategies: Agentswho have been more efficient at gathering signals tend to act as market makers.Whereas, agents who collected a lesser amount of signals tend to be trend followers.
The paper is structured as follows: In Section 2, we derive a noisy rationalequilibrium in which agents meet and share their private information. In Section3, we introduce a rumor. Section 4 concludes. For convenience, we report allcomputations in the Appendix.
2 The Benchmark Model
2.1 Setup
There is a risky asset with payoff realized at date 1. The payoff is U , a randomnormal variable with mean 0 and precision H. There is a continuum of investorsi ∈ [0, 1]. Each investor i is endowed at time 0 with quantities of the risky as-sets represented by X i. Investors have exponential utility function with commoncoefficient of absolute risk aversion 1/γ. The aggregate per capita supply of therisky asset, X0 =
∫ 1
0X idi, is normally and independently distributed with mean 0
and precision Φ. Trading takes place in 4 trading sessions which are held at timesτt = t/4, t = 0, ..., 3. The asset payoff is realized and consumption takes place attime 1 after the last trading session.
Immediately prior to each trading session, each investor i obtains a private signalabout the asset payoff, Zi
t , with
Zit = U + εit, (1)
where εit is distributed normally and independently of U , has mean zero, precisionS, and is independent of εkt , if k 6= i of j 6= t.
New liquidity traders are assumed to enter the market in each trading session.The incremental net supply of these traders6 is Xt, which have zero mean and preci-
5For other explanations of momentum and reversal with rational agents, see Albuquerque andMiao (2010), Holden and Subrahmanyam (2002), Berk, Green, and Naik (1999), Johnson (2002),Vayanos and Woolley (2010), Makarov and Rytchkov (2009), Biais, Bossaerts, and Spatt (2010) orWang (1993).
6We could assume, as in He and Wang (1995) for instance, an AR(1) noise trading process. Still,we prefer to assume that noise trading follows a random walk in order to isolate the momentumeffect. Introducing an AR(1) process for noise trading would only reinforce our momentum results.
4
sion Φ. All the information concerning the present setup is grouped for conveniencein Table 1.
Variable Mean Precision CommentsFundamental U 0 H
Per capita supply X0 =∫ 1
0 Xidi 0 Φ at time t = 0
Liquidity traders Xt 0 Φ at times t = 1, 2, 3
Private signals Zit = U + εit (for εit) 0 S at times t = 0, 1, 2
Table 1: Random variables in the benchmark model
The precision S, is assumed to be the same for all investors and uniformlybounded. It follows that the population average of the precision of private signalsis equal to S.
Let Pt denote the equilibrium risky asset price and Dit the risky asset demand
for investor i. We denote by Ft the common information available to all investorsat date t, and by F i
t the total information available to investor i at date t. Thefollowing theorem describes the risky asset prices and investor asset demands ateach date in a noisy rational expectations equilibrium.
Theorem 1. There exists a partially revealing rational expectations equilibrium in the 4trading session economy in which the risky asset price, Pt, and individual asset demands,Dit, for t = 0, .., 3, are given by:
Pt =1
Kt
(Kt − S (t+ 1)) µt + SU (t+ 1)− 1
γ
t∑j=0
Xj
, (2)
Dit = γ
t∑j=0
[SZij +
Xj
γ
]− SU (t+ 1)
, (3)
where
µit ≡ E[U | F it
]=
1
KtS
t∑j=0
[Zij + γ2SΦQj
]
µt ≡ E[U | Ft
]=
1
Kt − S (t+ 1)γ2S2Φ
t∑j=0
Qj
Qj ≡ U −1
γSXj
Kt ≡ Var−1[U | F it
]= H + S
(1 + γ2SΦ
)(t+ 1)
(4)
5
The optimal trading strategy of the individual investor i at time t = 1, 2, 3 is given by
∆Dit ≡ Di
t − Dit−1 = γ
(SZit − SU +
Xt
γ
)= γS
(Zit − Qt
) (5)
Proof. See Appendix A. Q.E.D.
The trading strategy of investor i in periods t = 1, 2, 3 depends on the differencebetween his private signal in period 1 and the normalized price signal, weighted byhis private signal precision and his risk aversion. If the precision is high, he will takea larger position. If the risk aversion is high (γ is low), he will take a smaller po-sition. The sign of each investor’s position depends on how his private informationcompares with public information. In such a setting, stock returns exhibit positiveautocorrelation provided that the quality of information shows sufficient improve-ment across periods. Hence, in order to obtain a significant amount of momentum,one needs to assume exogenously large improvement in precision over time. Tosee this, let us consider the covariance between the first and second period pricedifferences, cov
(P1 − P0, P2 − P1
):
− HS (γ2SΦ + 1)2
γ2Φ (H + γ2S2Φ + S) (H + 2S (γ2SΦ + 1))2 (H + 3S (γ2SΦ + 1))
This covariance is always negative, irrelevant of the calibration. As a con-sequence, in a standard rational-expectations setting, price returns tend to gen-erally exhibit reversal. The same argument applies for the next period covari-ance, cov
(P2 − P1, P3 − P2
). We borrow the intuition from Makarov and Rytchkov
(2009): In a standard full information setting, investors being risk averse requirea higher compensation in the form of a risk premium for holding a higher share ofthe risky asset. This results into a positive relation between expected returns andnoisy supply and a negative relation between prices and noisy supply. If the noisysupply is assumed to be i.i.d., as in the present model, realized returns are negativelyautocorrelated. In a partially revealing equilibrium, the same mechanism is at play,although slightly obscured by private information7.
If we consider different precisions at each date S for the private signals, it thenturns out that a very simple condition needs to be satisfied in order for stock returnsto exhibit momentum:
(S1 − S0)(1 + γ2S0Φ
)> H
7See Makarov and Rytchkov (2009) for a detailed derivation of the conditions for negative serialcorrelation to obtain.
6
Accordingly, in order to obtain momentum in a standard rational-expectationsframework, we need to increase the average precision of private information, at theexpense of losing in realism. This fact has been carefully analyzed in Cespa andVives (2009).
While significant surge in precision over time appears difficult to sustain exoge-nously, this arises as a natural phenomenon if we let investors chat with each other.In particular, information percolation delivers a convenient and intuitive mechanismto produce that effect: As agents meet with each other and gather others’ signals,they become more and more precise as time goes by.
Grossman (1981) proposed the concept of rational expectations equilibrium tocapture the idea that stock prices aggregate information that is initially dispersedacross investors. However, because of noise trading and idiosyncratic shocks, it onlydoes imperfectly so. Hence, even though centralized markets contribute to centralizeinformation, social networks thus still have an important role to play, in particularas an alternative channel conveying information through private discussions. For in-stance, Stein (2008) or Acemoglu, Bimpikis, and Ozdaglar (2010) have emphasizedthe importance of communication in social networks. In the next subsections, weintroduce social dynamics among investors and show how it allows to naturally pro-duce an improvement in investors’ precision over time, thus generating momentum.
2.2 Introducing Information Percolation
For the remainder of this section, we shall assume that each investor i only receives asingle private signal Zi
0 prior to the first trading session. Still, we let investors receiveadditional signals at t = 1, 2, 3, yet in a very specific way that we shall explain in thissubsection. From date t = 0 onward, we let agents meet with each other and sharethe private signal Zi
0 that they received at date t = 0. It is important to emphasizethat the signals they give to each other are not exactly the signals they were initiallyendowed with at date t = 0, for these signals have already been accounted for in theprice at date t = 0. Rather, initial signals should be seen as being reshuffled amongthe population of agents8.
Although trading is centralized, we assume that information is not. Agents haveto search for each other’s information. There are at least four ways to motivate
8One might be concerned that, doing so, we are introducing some persuasion bias in the termsof Demarzo, Vayanos, and Zwiebel (2003). This is, however, not true: First, at each point intime, agents share their initial signal and the signals that they have gathered by means of word-of-mouth communications. Since we rule out overlapping meetings between agents, the signalsgathered from a meeting are always different. Second, at each trading session, private signals arewiped out through aggregation. As a consequence, agents consider signals accruing from furthermeetings as being genuinely new. Put differently, there is no double accounting of signals.
7
this assumption: As suggested by Hong, Kubik, and Stein (2004), two possibilitiesinclude word-of-mouth learning and enjoyment from talking with friends about themarket9. Although we pursue a rational explanation, two more behavioral motiva-tions involve limited attention and news being only understandable when explainedby friends, as suggested by Hong, Hong, and Ungureanu (2010). Related to thelimited attention story, The Economist notes10: “In the face of this torrent of in-formation, it is perhaps unsurprising that investors rely on whom they know, ratherthan what they know.” Given a friend network or degree of sociability, some agentsmay be more successful in matching other agents and may, thus, end up gatheringmore signals than others. This produces a natural source of heterogeneity amongagents: In our model, while they all start off being identical agents, they becomeinformationally heterogeneous as soon as they start meeting with each other.
We now formalize this idea using the setup described in Duffie, Malamud, andManso (2009) and the references therein. We fix a probability space (Ω,F , P ) anda nonatomic space (A,A, α) of agents which represents the continuum describedabove. As in the previous subsection, all agents benefit from information about therandom variable U . Each agent initially gets a signal. For almost every pair (i, j)
of agents, their signal sets are disjoint.Any particular agent is matched to other agents at each of a sequence of Pois-
son arrival times with a mean arrival rate λ, which is common across agents. Ateach meeting time, the matched agent is randomly selected from the population ofagents. We assume that, for almost every pair of agents, this matching procedure isindependent. There is a zero probability that the set of agents that i has met beforetime t overlaps with the set of agents that j has met before time t.
By the joint Gaussian assumption and by induction in the number of prior meet-ings of each of a pair of currently matched agents, it is enough for the purpose ofupdating the agents’ conditional expectation of U that agent i tells his counterpartyat any meeting at time t his or her current conditional mean µit of U and the totalnumber Nit of signals that played a role in calculating the agent’s current conditionaldistribution of U . This number of signals is initially 1, and is then incremented ateach meeting by the number Njt of signals that similarly influenced the informationabout U that had been gathered by his counterparty j by time t.
The cross-sectional distribution µt of information precision (i.e. N) at time t isdefined, at any set B of positive integers, as the fraction µt (B) = α (i : Nit ∈ B) /qof agents whose precisions are currently in the set B and where q is the total number
9The authors specify that there might be great distinctions to be made between these two socialinteraction mechanisms.
10We refer to Too much information - Buttonwood, The Economist, 14 July 2007.
8
of agents (which is fixed because there are no entries nor exits). The cross-sectionalprecision distribution satisfies the following Boltzmann equation
ddtµt = λµt ∗ µt − λµt
= λn−1∑m=1
µt (n−m)µt (m)− λµt (6)
where ∗ denotes the discrete convolution product.Since agents are assumed to be initially endowed with a single signal, the initial
distribution of signals is a Dirac mass at 1, i.e. µ0 = δ1. This feature has the majoradvantage of leading to a closed-form solution for the cross-sectional distribution oftypes. The latter is simply given by
µt (n) = e−nλt(eλt − 1
)n−1. (7)
The social dynamics in (7) admit a strikingly simple solution. The simplicity of(7) is particularly compelling when considering the rich heterogeneity among agentsthat it is prone to generate. Producing information heterogeneity with tractablesocial dynamics is a delicate task. For instance, Hong and Stein (1999) build asimple social network which does not produce any heterogeneity in the amount ofinformation across agents. Similarly, Hong, Hong, and Ungureanu (2010) make useof a social network borrowed from the Epidemics literature and which brings abouttwo classes of agents while staying admittedly difficult to manipulate11. Rather, theinformation percolation mechanism delivers both tractable cross-sectional dynamicsand heterogeneity among agents. Moreover, while we only restrict our attention toa particular type of initial distributions (a Dirac mass), this mechanism is is flexibleand remains tractable for a vast class of initial distributions. Figure 1 illustrates theevolution of the cross-sectional distribution µt (n) of signals through time.
In order to solve for the equilibrium, we shall work with the incremental signalsy that have accrued between time t and time T rather than the total amount ofsignals n at a given date t. Notice that both may be equivalently used; we chooseto use distributions of increments because of its convenience. Hence, it will proveuseful to determine the cross-sectional distribution πt (y) of the incremental signalsy that kicked in between t and T . The following purpose is implemented in theproposition below.
11The authors have to restrict their analysis to a network comprised of a restricted number ofagents.
9
1
2
3
4...0.05
0.10
0.15
0.20
0.25
0.30
0.35
12
34...0.05
0.10
0.15
0.20
0.25
0.30
0.35
Figure 1: Evolution of the probability density function of the number ofsignals through time when agents are initially endowed with one signal.Each graph depicts µ at time t = 1 and t = 2 , respectively, with λ = 1.
Proposition 2. The cross-sectional distribution π of incremental signals that accruedduring time t and T satisfies
πT = eλ(∫ Tt µsds−(T−t))
or equivalently solves the differential equation
dπt = −λπt + λπt ∗ µt
with initial condition given by the Dirac measure π0 = δ0 at zero.
Proof. See Appendix B. Q.E.D.
This distribution satisfies the same dynamics as the probability density of theposterior type at time t of a given agent’s type who was randomly drawn withdensity π (·, 0) at time zero in Duffie and Manso (2007) except that the distributionalways starts at a Dirac mass δ0 in order to reflect that the counter has been resetto zero at the beginning of the relevant period. Therefore, initially, at time t = 0, πand µ are the same distributions expect that π has a Dirac mass at 0 while µ has aDirac mass at one. The distribution πt also admits a closed-form solution:
πt (y) = e−µt(y)(eλt−1)−µT (y)(eλT−1)
y−λ(T−t). (8)
We now have all the necessary elements at hand to compute the equilibriumwhen the social dynamics described above are involved. Some additional notationis required. An important statistic for the present economy is the cross-sectionalaverage of additional signals at time t, Ωt =
∑n∈N πt (n)n. Furthermore, let kt =
kjtj=0 denote an investor type, i.e. the whole history of signals he has gathered up
10
to and including t. By construction, we have k0 = k3 = 1. At times t = 2 and t = 3,it follows from Gaussian theory that the average of the kt private signals receivedby agent i, denoted hereafter Zi
t,kt, is sufficient in the estimation of U . The demand
at time t of an individual investor of type kt is denoted by Dit,k. Proceeding with
similar notations, let us denote by F it,k the information that agent i of type kt has
at his disposal at date t. The following theorem is the analogue of Theorem 1 whenagents meet and share their private information.
Theorem 3. There exists a partially revealing rational expectations equilibrium in the 4trading session economy in which the risky asset price, Pt, and individual asset demands,Dit,k, for t = 0, .., 3, are given by:
Pt =Kt −HKt
U −t∑
j=0
1 + γ2SΦΩj
γKtXj ,
Dit,k = γ
t∑j=0
(kjSZ
ij,kj− SΩjU +
Xj
γ
)−∆Ki
t,kPt
(9)
where
µit,k ≡ E[U | F it,k
]=
1
Kit,k
t∑j=0
[kjSZ
ij,kj
+ γ2S2Ω2jΦQj
]Qt ≡ U −
1
γSΩtXt
(10)
and
Kit,k ≡ Var−1
[U | F it,k
]= H +
t∑j=0
[kjS + γ2S2ΦΩ2
j
]Kt ≡
∑k∈N
πt (k)Kit,k = H +
t∑j=0
[ΩjS + γ2S2ΦΩ2
j
] (11)
The difference ∆Kit,k ≡ Ki
t,k−Kj represents the precision (dis)advantage of agent i with re-spect to the average precision of all investors. The optimal trading strategy of the individualinvestor i at time t = 1, 2, 3 is given by
∆Dit,k ≡ Di
t,k − Dit−1,k = γ
[ktS
(Zit,kt − Pt
)− SΩt
(U − Pt
)+Xt
γ−∆Ki
t−1,k∆Pt
](12)
Proof. See Appendix C. Q.E.D.
Theorem 3 contains two important results, one of which is summarized in thefollowing lemma.
Lemma 4. Investors are myopic with respect to their future amount of signals.
11
Intuitively, investors’ demand in (12) does not depend on the future number ofsignals. That is, investors simply do not take into account the uncertainty withrespect to the signals they will gather, or equivalently the number of other investorsthey will meet in the future.
Second, the last term of the optimal strategy as per equation (12) of Theorem 3carries a signification that is central to the implication of noisy rational expectationsequilibria with information percolation. In particular, introducing information per-colation produces heterogeneous precisions among agents through the distribution πof signals amount across the population of agents. Analogous to Brennan and Cao(1996) in the general case of heterogeneous precisions, optimal demands contain thefollowing term
−∆Kit−1,k
(Pt − Pt−1
).
In the case of the latter reference, this simply means that agents who are more precisethan the average agent tend to be contrarians while agents who are less precise tendto be trend followers. When information percolation prevails, it means that investorswho have been more efficient at gathering signals, liquidate their positions. On theother hand, investors who have gathered a lesser amount of signals tend to buy, thusbeing trend followers.
This result bear some similarity with the setup of Hong and Stein (1999). Bet-ter informed investors act as the “newswatchers” from their model, while poorlyinformed investors act as the “momentum traders”. This is because poorly informedinvestors adjust more heavily on the public price signal, and thus they become trend-chasers. Note however that, unlike Hong and Stein (1999), we do not artificiallycreate these two different investor types. It arises naturally from the informationpercolation mechanism. Neither do we restrict “newswatchers” to learn from prices,or “momentum traders” to have access to private information.
Importantly, this reveals a crucial implication of information percolation: Start-ing within an homogenous agents setting, information percolation first producesinformationally heterogeneous agents which ultimately leads to drastically differentinvestment strategies. In the present case, because ∆Ki
0,k = 0, investors only tradeon price variations from period t = 1 and thereafter. Our model may alternativelybe viewed as one with a time-varying representative agent12.
Two papers verify empirically this result. First, in Brennan and Cao (1996),similar trading behavior on price variations comes from an accumulated information(dis)advantage. The main difference is that, in our case, the accumulation comesfrom the quantity of signals and not from their relative precision. Brennan and Cao
12We thank Sergio Rebelo for pointing out this interpretation.
12
(1996) assume initial heterogeneity in the information precision, whereas investorsin the present model start off being all identical agents and become heterogeneousas they search for each other. This has a strong economic appeal: For instance,if the present setup were to be applied to the home bias puzzle with which thelatter reference is concerned, we would not need to assume that home investors haveinitial higher precision. In contrast to Brennan and Cao (1996), we could simplyassume that agents are initially identical but, due to home networks being moredense, agents have more chances to meet home investors than foreign investors, thusfocusing their precision on local concerns.
Second, Feng and Seasholes (2004) obtain the same implication by using a modelsimilar to Brennan and Cao (1996), on a sample of individual brokerage accountsfrom the Chinese stock market. One of their results is that investors who live near theheadqurters of a firm, and thus are able to obtain better private information, havenet trades that are negatively correlated with stock returns. While “far” investorshave net trades that are positively correlated with stock returs.
2.3 Price Drift
The fact that information percolation increases precision over time produces a sig-nificant amount of price drift, an implication to which we now turn. As in Cespa andVives (2009), we concentrate on the covariance between the first and second periodreturns and the second and third period returns. This allows to capture the signof the serial comovements in returns. We then normalize these covariances in orderto obtain autocorrelations. The latter covariances are exposed in the propositionbelow.
Proposition 5. The covariances between the first and second period returns and the secondand third period returns are respectively given by
cov(P1 − P0, P2 − P1
)=
1
K1
1∑j=0
Ωj
(γ2SΦΩj + 1
)− 1
K0
(γ2SΦ + 1
)×
1
K2
2∑j=0
Ωj
(γ2SΦΩj + 1
)− 1
K1
1∑j=0
Ωj
(γ2SΦΩj + 1
) S2
H
+
(1
K2− 1
K1
) (γSΦ + 1
γ
)2 (1K1− 1
K0
)+ 1K1
(γSΦΩ1 + 1
γ
)2
1
Φ.
(13)
13
and
cov(P3 − P2, P2 − P1
)=
1
K2
2∑j=0
Ωj
(γ2SΦΩj + 1
)− 1
K1
1∑j=0
Ωj
(γ2SΦΩj + 1
)×
( (1K3− 1
K2
)∑2j=0 Ωj
(γ2SΦΩj + 1
)+ 1K3
Ω3
(γ2SΦΩ3 + 1
) )S2
H
+
(1
K3− 1
K2
) (1K2− 1
K1
)∑1j=0
(γSΦΩj + 1
γ
)2
+ 1K2
(γSΦΩ2 + 1
γ
)2
1
Φ.
(14)
Proof. See Appendix D. Q.E.D.
By substituting the average precisions from (11) into Proposition 5, we obtain asimple condition for momentum to arise in the first period:
(Ω1 − 1)(1 + γ2SΦ
)> H
Since the cross-sectional average of signals at time 1, Ω1 is strictly increasing inthe meeting intensity λ, there must be a treshold λ? such that for λ > λ? weobtain momentum, and for λ < λ? we obtain reversal. Figure 2 shows that, for ourcalibration, this threshold is approximately λ? = 1.
Consistent with the impact of information percolation exposed in Subsection 2.2,gathering a large amount of signals significantly improves the precision over time.Since this phenomenon is best captured by the intensity λ at which agents meetwith each other, we plot serial autocorrelations in return between period t = 0, 1,t = 1, 2 and t = 2, 3 against the average meeting in Figure 2 below.
0.5 1.0 1.5 2.0 2.5 3.0
Λ
-0.3
-0.2
-0.1
0.1
0.2
0.3
Ρ
Figure 2: The solid thick line represents the first period autocorrelationwhile the dashed line represents the second period autocorrelation. Thecalibration is H = S = Φ = 1 and γ = 1
3 .
14
As is apparent from Figure 2, we observe that for a low meeting intensity, returnsexhibit reversal. This case has been referred to the Keynesian equilibrium in Cespaand Vives (2009). This happens when the improvement in precision across time istoo low. Specifically, when agents do not meet at all, prices are martingales andexhibit no drift. This result is reminiscent of Kyle (1985) and extensions thereof,where informed traders only receive a private signal at date t = 0. It is howevernot strictly comparable to the usual Kyle (1985) setup because, in such setting, themarket maker, whose role is explicitely determined, is widely assumed to be riskneutral.
For low meeting frequencies, the situation is akin to the setup of Theorem 1.Because this setup compares to standard rational-expectations equilibria, followingthe discussion of Subsection 2.1, returns exhibit reversal. As information percolationintensifies, meetings accelerate and the average precision goes up. This eventuallyoffsets and dominates the former effect, thus producing momentum. That is, in-formation percolation eventually produces momentum, a phenomenon that wouldhardly arise within a classical rational-expectations equilibrium except maybe forsome far-fetched calibration.
Crucially, notice that this figure is strikingly in line with empirical evidences,particularly so with Figure 1 in Hong, Lim, and Stein (2000). The latter referencedocuments a pronounced nonlinear and non-monotonic relation between firm size13
and momentum. The study tend to cluster the analysis on the part of the samplefor which the relation between momentum and size is decreasing. In Figure 2, thiscorresponds to λ > 2. Importantly, unlike Hong and Stein (1999) and because weallow for supply shocks, our model is able to reproduce this pattern over their wholesample range. As a result, consistent with Hong, Lim, and Stein (2000), we find thatfirms for which information slowly diffuses exhibit reversal. Finally, not only are weable to reproduce the whole empirical pattern, but we do this within a rationalexpectations framework and with a single parameter λ.
2.4 Volume and Price Volatility
Information percolation carries other implications for asset pricing, particularly re-garding trading volume and price volatility. The information flow generated byword-of-mouth communications strongly impacts the way individuals build theirportfolios: Information percolation increases the speed at which individuals tilt theirportfolios toward speculation or market making. Thus volume is closely related to
13The authors use a measure of analyst coverage corrected for firm size in order to proxy for theextent of information diffusion.
15
the information flow. That information percolation accelerates portfolio adjustmentsshould be particularly compelling when looking at the trading volume. In order tosee how the diffusion of information affects trading volume, we first consider thesetting of Theorem 1. In this setting, agents rebalance their portfolios according to(5). In particular, it is apparent that ∆Di
t is driven by the extent of the noise εit inagent i’s private signal and noise trading Xt. It reflects both the speculative andthe market-making part of an investor’s portfolio. Following He and Wang (1995),we define the volume in the benchmark setting as V B
t ≡∫ 1
0| ∆Di
t |di. The expectedvolume V B
t in the Benchmark model is thus given by
V Bt = E
[∫ 1
0
| ∆Dit |di
]=
√2
π
(γ2S +
1
Φ
). (15)
The expected trading volume in the Benchmark setup is decreasing in the riskaversion and the precision of noise trading and increasing the average precision ofprivate information. Importanly, as in He and Wang (1995), although (15) containsthe integral over the continuum of agents, it is actually never used in the computa-tions. That is, aggregation does not affect the determinants of trading volume. Thisis due to information precision S being identical across individuals. Since He andWang (1995) also assume, as in the Benchmark setup, that the precision in privateinformation is homogenous across the population of agents, aggregation does notbite either when computing expected trading volume. However, when informationpercolation prevails, this turns out to be no longer true. To see this, first notice thatthe expression (12) for ∆Di
t,k in Theorem 3 can be rewritten as
∆Dit,k =
Kit,k
Kt
Xt + γSktεit,k + γ2SΦΩt
(Kit,k
Kt
− 1
)Xt (16)
+ γ
(Kit,k
Kt
−Kit−1,k
Kt−1
)HU +
(Kit,k
Kt
−Kit−1,k
Kt−1
)t−1∑j=0
(1 + γ2SΦΩj
)Xj
where εit,k is the average noise in investor i’s private signal. From (16), it is apparent
that the ratioKit,k
Ktbetween investor i’s individual precision and the precision of the
average agent plays an important role. In particular, if we shut down the informationpercolation channel and let each individual receive a single signal, then kt = Ωt =Kit,n+k
Kt=
Kit−1,k
Kt−1= 1 and εit,k = εit. As a consequence, only the two first terms in (16)
remain and we recover the asset demand that prevails in the Benchmark setup.Second, it will prove convenient for aggregation purposes to notice that agent i’s
16
individual precision Kit,k may be written as
Kit,n+k = Ki
t−1,n + kS + γ2S2ΦΩ2t
where k ∼ πt is the increment of new signals that accrued between time t − 1
and time t while n ∼ µt−1 is the total number of signals gathered up to time t− 1.Disentangling an agent’s total current amount n+k of signals in that manner allowsto separate the aggregation of signals that have been gathered up to time t−1 fromthe aggregation of those which just accrued at date t.
Now, we can use the fact that the random variables U , Xj appearing in (16) are
independent and that for X ∼ N (0, σ2), E [|X|] =√
2πσ, and directly obtain the
expression for the expected trading volume when information percolation is involved
Vt =
√2
π×∑n∈N
∑k∈N
πt (k)µt−1 (n) (17)
×
√√√√√√ γ2k2S +(Kit,n+k
Kt(1 + γ2SΦΩt)− γ2SΦΩt
)21Φ
+(Kit,n+k
Kt− Ki
t−1,n
Kt−1
)2 (γ2H + 1
Φ
∑t−1j=0 (1 + γ2SΦΩj)
2) .
As mentioned above, the striking feature of the expected trading volume in (17) isthat aggregation matters. This yields a novel implication regarding the behavior ofthe trading volume in a rational expectations equilibrium with respect to the work ofHe and Wang (1995). Because the aggregation of agents is driven by the diffusion ofinformation which is itself parametrized by λ, we plot the expected trading volumeas a function of λ for times t = 1, 2 in the figure below. More precisely, we plotthe Benchmark-adjusted expected trading volume Vt − V B
t . In Figure 3, the blueline represents the time t = 1 expected volume which is monotically increasing in λ.This confirms that the diffusion of information contributes to significantly increasethe volume due to the acceleration in trading strategies. In particular, from (12)one observes that price variations do not yet play a role in agents’ portfolio rebal-ancing during the first period. The reason is that they only initially receive a singlesignal and thus ∆Ki
0,k = 0. The next period, as information spreads out throughword-of-mouth communications, agents start to trade on price variations and thevolume exhibits a non-monotonic pattern as seen from the red line. Specifically,for sufficiently low or large information diffusion intensity (say λ < 1 and λ > 4),the time t = 1 and time t = 2 volumes are very close. This is intuitive: For lowsearch intensities, information percolation takes more time to take off and, thus, theamounts of signal gathered respectively at time t = 1 and t = 2 are not tremen-
17
1 2 3 4Λ
5
10
15
20
25
30
35
V - V B
Figure 3: The solid thick line represents the first period expected trad-ing volume while the dashed line represents the second period expectedtrading volume. The calibration is H = S = Φ = 1 and γ = 1
3 .
dously different. For high search intensities, a large fraction of the population hasalready gathered a significant amount of signals at time t = 1 and the incrementalchange in portfolio position is accordingly of small magnitude. The most interest-ing pattern in the trading volume appears for intermediate information diffusionintensities. In the latter case, the information is sufficiently spread over the wholepopulation to allow individuals who are better informed to take advantage of theirsuperior knowledge in order to speculate on price variations. This result is in linewith findings of Holden and Subrahmanyam (2002).
Turning to price volatility, in order to keep the same structure, we consider the
volatility σt ≡√var
(∆Pt
)of price differences. The latter is easily shown to be
σt =
(1 + γ2SΦΩt
γKt
)21
Φ+
(1
Kt
− 1
Kt−1
)2(H +
1
γ2Φ
t−1∑j=0
(1 + γ2SΦΩj
)2
)(18)
We plot the volatility of the price differences for time t = 1, 2 in the Figure 4.In the first period, the volatility of the price difference is increasing in the infor-
mation diffusion intensity. The reason is that the price at time t = 0 reflects thetrading of homogenous investors who only receive a single signal. The price at timet = 1 reflects a correction induced by information percolation. As the intensity ofcommunications among investors increases, a larger fraction of investors gathered asizable amount of signals and, as a result, the price correction is stronger. Thus,the volatility of the price difference increases. During the second period, volatilityexhibits the opposite pattern. As information further diffuses between time t = 2
18
1 2 3 4 5 6Λ
0.5
1.0
1.5
Σ
Figure 4: The solid thick line represents the first period price differ-ence volatility while the dashed line represents the second period pricedifference volatility. The calibration is H = S = Φ = 1 and γ = 1
3 .
and t = 3, the price further corrects. Still, if the intensity was already high at timet = 2, the price adjustment at time t = 3 is of small magnitude. This temporal pat-tern in price difference volatility also matches Holden and Subrahmanyam (2002)’sresults.
3 Residual Uncertainty, or Rumors
We now proceed to introduce a rumor14 about the final payoff U . The way wemodel the rumor strongly relates to the residual uncertainty component in He andWang (1995) and Cespa and Vives (2009). In particular, we introduce additionaluncertainty in the private signals that agents are initially endowed with and thatthey communicate to each other. That is, their signals are not centered on thefinal payoff U but on some random variable U + V where V denotes the rumor.This error is common across agents and, although agents are aware of it, they donot learn about it, at least for some time. The rumor, thus, brings some marketopaqueness into the benchmark setting of Section 2. However, unlike He and Wang(1995) and Cespa and Vives (2009), the rumor is not part of the final payoff, but onlypertains to the private signals agents gather. It can be clearly interpreted as somecommon noise in the private signals of investors. Then, by means of the informationpercolation mechanism, the rumor propagates and circulates through the markets,up to some point when it vanishes.
14As defined by Peterson and Gist (1951), a rumor is viewed as “an unverified account or expla-nation of events circulating from person to person and pertaining to an object, event, or issue inpublic concern.”
19
This timing in the resolution of uncertainty makes another important differencebetween our setting and the latter references: In the present model, we let the rumordie out prior to the terminal date. As a result, the additional uncertainty broughtabout by the rumor is partially revealed before U is resolved. More precisely, theuncertainty resolution is modeled as follows: we let the initial signals Zi
0 to bereshuffled and shared across the population of agents through information percola-tion until time t = 2. At time t = 3, we let agents receive a private signal Zi
3 = U+εi3
centered on the fundamental value U . As a result, investors realize on average thatthe information they have been sharing so far was biased and, thus, correct theirbeliefs. On the one hand, this mechnanism shares some common features with theinformational cascades literature initiated by Bikhchandani, Hirshleifer, and Welch(1998), namely i) erroneous mass behavior and ii) fragility. Unlike the informationalcascade mechanism, the first aspect is not strategic and is only produced by the factthat signals contain an error. The second aspect stems from the occurrence of anunusual signal which, in the present case, overturns the effect of information per-colation. It is worth mentioning that this effect could also be caused by a publicsignal. Hence, we could alternatively use a public signal Y3 in order to reverse i).On the other hand, the timing of the rumor and the way it produces reversal bothrelate to the advance information mechanism in Albuquerque and Miao (2010). Inthe latter contribution, advance information produces reversal when it materializesor, equivalently, when it becomes stale information. That is, reversal occurs in thesetting of Albuquerque and Miao (2010) when advance information loses of its value(because it becomes public). Yet, while advance information carries useful informa-tion about future fundamental values, our notion of rumor eventually turns out tobe unrelated to the value of the fundamental.
He and Wang (1995) have shown that this type of uncertainty offers a natu-ral mean to enhance trading volume, yet under certain conditions. If the residualuncertainty is too large and, as a result, markets too opaque, V may overwhelmpotential gains from speculation. We now want to investigate how additional uncer-tainty in the form of the rumor described feedbacks with information percolation.In particular, we want to gauge whether both rumor and information propagationare amenable to a phase of momentum followed by a second phase of reversal.
3.1 Equilibrium Computation
We now consider the Benchmark setup and proceed to introduce a rumor. As in thebenchmark model, there is a risky asset with payoff realized at date 1. There are 4trading sessions which are held at times τt = t/4, t = 0, ..., 3. Imediately prior to
20
trading sessions 0, 1, and 2, each investor i obtains a private signal about the assetpayoff, Zi
t , withZit = U + V + εit, t = 0, 1, 2 (19)
The extra random variable V has zero mean, precision ν, and is independent ofall other variables from the model. This random variable makes the total amountof private information sufficient to infer U + V , but not the true asset payoff U .Therefore, there exists an unobservable component of the private information thatis never revealed and makes the setup similar to the general case exposed in He andWang (1995), and Cespa and Vives (2009). Hence, two things are worth mentioningabout this economy. First, even if there is an infinite number of private signals attimes 0, 1 and 2, the uncertainty about the value of the stock remains until the endof the economy. Second, given that investors are aware of all of them having thiscomponent in their private signals, V can be interpreted as a rumor pertaining tothe final payoff.
Still, we shall let this rumor exist only at times 0, 1 and 2. That is, at time 3,investors receive non-biased private signals about the asset payoff:
Zi3 = U + εi3 (20)
As a consequence, at time 3 the rumor is partially resolved, by making the privatesignals centered on the true fundamental. By applying the Law of Large Numbers,at time 3 agents are right on average about the final payoff of the asset. Notice thatthis does not imply that some agents know more than others, but simply that agentsknow that at time 3 the rumor will die out. Intuitively, as an asset approaches somepayoff date, some rumors pertaining to its fundamental value simply disappear.
All the other elements of the model are as in the benchmark case. Finding aclosed form solution, although very complicated, is possible for 2 or 3 periods, butthe problem becomes intractable for more than 3 periods. We resort therefore tonumerical methods. However, even if the model is solved numerically, we prefer tokeep some analytical structure for the coefficients. This will prove to be beneficialwhen interpreting the equilibrium solution.
We first begin by assuming as usual that prices are linear functions of the ran-dom variables. Then, we normalize the price signals to find the Qts, which areinformationnally equivalent to the prices. We assume the following structure for thenormalized price signals:
Qt = U + ιtV −1
γSΩt
(21)
where ιt = 1 for t = 0, 1, 2 and ι3 = 0. Since there is no rumor at time 3, we fix
21
Ω3 = 1. The other 3 coefficients, Ω0,1,2, are to be found numerically. In fact, it turnsout that it is sufficient to solve for them in order to obtain a complete solution tothe equilibrium. They solve 3 non-linear equations that are easily found. Denotingby θt the coefficient of the private signal Zi
t in the optimal demand of investor i, Dit,
for t = 0, 1, 2, it can be proven that the equations to be solved by the Ωts are
θ2 =γS (Ω0 + Ω1 + Ω2)
3
θ1 =γS (Ω0 + Ω1)
2
θ0 = γSΩ0
(22)
If ν →∞, e.g., if the rumor has infinite precision and is therefore equal to zero,all the Ωts are equal to one. Otherwise, they lie between 0 and 1. For most ofthe calibrations that we use, Ωt is decreasing in time, except at time 0, when it isjumping to 1.
The individual conditional precision at time t is the same for all the investorsand is equal to
Kt = H + Λt
((t+ 1)S + γ2S2Φ
t∑j=0
Ω2j
)(23)
withΛt =
ν
ν + (t+ 1)S + γ2S2Φ∑t
j=0 Ω2j
(24)
One can easily check that, when ν →∞, Λt is equal to 1, and that Λt is decreasingas t is increasing, except at time 4, when it is jumping to 1. Thus, Λt can beinterpreted as a dampening factor which limits the increase in conditional precisionas long as there is a rumor in the private signals. Once the rumor disappears, thereis a massive increase in the conditional precision, driven by the sudden shift in Λt.This sudden increase occurs because investors’ private information is now centeredon the true fundamental.
It can be verified that, for t = 0, 1, 2, Kt = H + ν (1− Λt). Thus, during therumor periods, the conditional precision of the investors is still increasing, but onlyat a slower rate. Every new period of trading is beneficial for learning, but at alower marginal rate than what obtains in the case of no rumor.
The conditional expectation at each period is equal to
µit =Λt
Kt
t∑j=0
(SZi
j + γ2S2ΦΩ2jQj
)(25)
22
There are two differences as compared to the standard case. First, investors useless of the public information available at time t (the history of prices, reflectedin the Qjs). The effect comes from the presence of Ωj in the second term of thesum. Moreover, since Ωj is decreasing in time during the rumor periods, the effectis stronger as the number of trading sessions increases. Observe that, at time t = 3,Ω3 jumps to 1, producing a substantial shift in expectations. Second, the term Λt,being smaller than one and decreasing during the rumor periods, is reinforcing theeffect. It is again a dampening factor, making investors use less of all the availableinformation that they have, when they form their conditional expectation.
During the rumor periods, investors know that on average there exists an errorin their learning. As a result, they optimally modify their learning behavior to takethis into account. Intuitively, they become more prudent regarding the availableinformation at each period. Once the rumor is revealed on average (the privatesignals are centered on the true fundamental), the learning mechanism suddenlyshifts to what is observed in a standard rational expectations equilibrium with noresidual uncertainty.
In separate calculations, we added a periodic public signal in the investors infor-mation set. We supposed that the public signal is centered on the true fundamental,U . It turns out that the coefficient of this public signal has exactly the same formas that in the standard setup with no rumor15. Since Kt is smaller due to the Ωsand the Λs, the investors give a much higher weight to it as it is the case in a stan-dard setup. This result might offer an alternative explanation for why investors mayover-react to information such as public announcement, news, etc16.
Let us analyze now the optimal demand of the investors. It is given by
Dit =
γ∑t
j=0
(S
∑tj=0 Ωj
t+1Zij − SΩjQj
)t = 0, 1, 2
Di2 + SZi
3 − SQ3 t = 3(26)
According to this optimal demand, the optimal trading strategy of the individualinvestor at time t = 1, 2, 3, Di
t − Dit−1, is given by
15More precisely, it is equal to N/Kt, where N is the precision of the public signal.16Allen, Morris and Shin (2006) stated that higher order beliefs might push agents to give a
larger weight to public information. Even if the higher order belief mechanism is present in ourpaper as well, we have a somewhat different view, by arguing that it depends on which publicinformation. If investors are aware of some rumor in their private information, they might rely lesson prices as public signals and more on financial news, public announcements, etc.
23
∆Dit =
γ
[∑t−1j=0 S
(∑tj=0 Ωjt+1 −
∑jk=0 Ωkj+1
)Zij + S
∑tj=0 Ωjt+1 Zit − SΩtQt
]t = 1, 2
SZi3 − SQ3 t = 3
(27)
The first term in the square brackets would be equal to zero in a standardsetup. However, here it supports that investors trade at each trading session ontheir previous private information, and in a meaningful way. During the rumorperiods, the coefficients of Zi
j is negative (because Ωj is decreasing in time, itsaverage is also decreasing in time). That is, investors unwind to some extent thepositions that they build following the past private information. With the arrivalof the new information, investors have a better knowledge of what the rumor mightbe, and thus re-adjust their portfolios. It is somewhat like they were overshootingin the past and now they are adjusting for that. This feature is well documented inHe and Wang (1995), where they show that the presence of the residual uncertaintyincreases the trading volume.
In period 3, once the rumor is revealed on average, investors switch back to astandard trading strategy. That is, they stop trading on previous signals.
Finally, the equilibrium price of the asset satisfies
Pt =SΩt + γ2S2ΦΩ2
tΛtΓtKt
(U + V
)+
t−1∑j=0
(SΩj + γ2S2ΦΩ2
jΛjΓt
Kt
Qj
)− 1 + γ2SΦΩtΛtΓt
γKt
Xt
(28)
with Γt a coefficient that is found numerically once the Ωjs are known. This coeffi-cients takes the value 1 as ν →∞, and is again decreasing in time during the rumorperiods.
3.2 Information Percolation, Momentum and Reversal
The agents’ optimal investment strategy described in the previous section was point-ing to a feature that will become obvious now. Namely, we saw that the presenceof a rumor pushes investors to re-adjust their past trades. Hence, since we are in ageneral equilibrium setting, these successive portfolio adjustments generate reversalin asset returns. It appears that this is truly the case: asset returns in this settingare strongly negatively autocorrelated. Moreover, at date 3, when aggregate privatesignals reveal the true value of the fundamental, the reversal is even stronger.
At this point, we emphasize once again that neither the rumor nor the funda-
24
mental are completely revealed at time 3. What happens is that investors’ privatesignals do not contain the common noise that we refer to as the rumor, and in-vestors are aware of that fact. As a consequence, a significant adjustment towardthe fundamental value of the asset price occurs at time 3.
The question that we address now is whether the information percolation is stillable to produce significant momentum, despite the fact that, in the present setup,returns are strongly negatively autocorrelated. In order to do so, we embed oncemore the search dynamics exposed in Subsection 2.2 in the present model. However,given the presence of the rumor, computations are more tedious. We prefer to leavetechnical details in Appendix E, and show here some very preliminary results.
In Figure 5, we build a graph similar to Figure 2 in the benchmark case. Theonly additional parameter with respect to the benchmark case is the precision of therumor ν. We find it convenient for the exposition to build 3 graphs, for 3 differentvalues of ν. If ν is large, that is, if the rumor is negligible, we approach the bench-mark case. This can be seen in the left panel. When ν is getting smaller, and thusthe rumor getting “large”, stock returns tend to be more negatively serially corre-lated. However, once the information percolation mechanism is involved, the firstperiod autocorrelation tends to be positive, while the second period autocorrelationremains negative. That is, stock returns clearly exhibit momentum and reversal:the momentum is generated by the diffusion of the information, while the reversal isgenerated by the exctinction of the rumor. This can be seen in the middle panel. Ifthe rumor gets too large, then the negative autocorrelation gets stronger and offsetsthe information diffusion effect. As a result, momentum almost disappears, as seenin the right panel.
We build impulse response functions to better investigate the effect of informa-tion percolation. A similar methodology has been used in Hong and Stein (1999) andDaniel, Hirshleifer, and Subrahmanyam (1998). At each trading session, we com-pute the average price, conditional on the fundamental values U and V .We consideran initial date “-1” at which everybody is symmetrically informed only with priorsregarding U , V . Since all agents are identical at this date and have a prior of zerofor the liquidation value of the risky asset, P−1 = 0. The asset is liquidated at date4, thus P4 = U . Obviously, the parameter λ will have an impact on the averagepath of the prices from P−1 to P4.
As can be seen from Figure 6, in the absence of information percolation, the priceconverges slowly toward the fundamental. In this numerical exercise we fixed therumor at V = 0.5. Without information percolation the price does not overshoot thefundamental value. Once the information percolation intensifies, a hump-shapped
25
0.5
1.0
1.5
2.0
2.5
3.0
Λ
-0.3
-0.2
-0.1
0.0
0.1
0.2
Ρ
Ν=
10
0.5
1.0
1.5
2.0
2.5
3.0
Λ
-0
.3
-0
.2
-0
.1
0.0
0.1
0.2
Ρ
Ν=
5
0.5
1.0
1.5
2.0
2.5
3.0
Λ
-0
.3
-0
.2
-0
.1
0.0
0.1
0.2
Ρ
Ν=
3
Fig
ure
5:The
solid
thicklin
erepresents
thefirst
period
autocorrelationwhile
theda
shed
linerepresents
thesecond
period
autocorrelation.
The
calib
ration
isH
=S
=Φ
=1an
dγ
=1 3.The
rearethreepa
nels,e
achwithadiffe
rent
valueofν.
26
æ
æ æ æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
ææ
æ
æ
-1 1 2 3 4Time
0.2
0.4
0.6
0.8
1.0
1.2
Average Price
Figure 6: Average cumulative impulse response as a function of themeeting intensity λ. The dashed line is for the case λ = 0, the dottedline for λ = 1 and the solid line for λ = 2.5. In this graph, U = 1 andV = 0.5.
pattern appears in the average price path. The rumor propagates into the marketand pushes prices to overshoot the fundamental value. A reversal takes place atdate 3, when the rumor is on average revealed.
4 Conclusion
This paper attempts to show that the diffusion of information through financialmarkets has a significant impact on the dynamics of asset prices, even in a central-ized market setting. Since, in such a setting, prices do not completely reveal thefundamental, there is certainly a benefit for investors to meet and communicate withtheir peers.
As a first step, we show that the percolation of information produces price drift.The mechanism is simple and intuitive: Investors’ precision in their estimate ofthe fundamental increases gradually through time, by gathering informations fromtheir peers. This increase in the conditional precision tends to accelerate the priceconvergence to its path toward the fundamental, thus producing momentum in as-set returns. Additionnally, information percolation appears to be a natural wayof generating heterogeneity in investments strategies. Each investor starts with oneprivate signal and randomly meets their peers. As time shifts to the the next tradinground, the diffusion of information produces heterogeneity in the number of signalsthat each agent gathers. Some of them become better informed, and act as con-trarians, whereas others, who collected less information, implement trend-followingportfolio strategies.
27
Since the information percolation mechanism appears to be a natural way fornews to propagate through financial markets, we address a second question. Whatif the informations that spread through the market are not completely accurate inthe aggregate, e.g., what if they contain a rumor? We add such a feature to thebenchmark model, and we show that returns actually continue to exhibit momentum,yet to a lesser extent. Crucially, the momentum phase is followed by a significantreversal, in line with the common under- and over-reaction patterns in stock returnswidely documented.
At this point, many fascinating extensions are pending, some of which are workin progress.
28
A Proof of Theorem 1We provide the proof for a two trading session economy, that is, we eliminate for simplicity the datest = 2 and t = 3. Once the equilibrium quantities are written in a recursive form, as in Brennan andCao (1996), or in He and Wang (1995), it is straightforward to derive the full recursive equilibriumsolution. The model is solved backwards, starting from date 1 and then going back to date 0.First, conjecture that prices in period 0 and period 1 are
P0 = β0U − γ0X0 (29)P1 = ϕ1U − γ01X0 − γ1X1. (30)
Consider the normalized price signal in period zero
Q0 =1
β0P0 = U − γ0
β0X0. (31)
Q0 is informationally equivalent to P0. Replace X0 from (31) into (30) to obtain
P1 = β1U + ξ01Q0 − γ1X1. (32)
Notice that the coefficients of U and V have changed. We can now normalize the price signal inperiod one
Q1 =1
β1
(P1 − ξ01Q0
)= U − γ1
β1X1 (33)
We shall make the following assumptions (see Admati (1985)):
γ0
β0=
1
γS0,γ1
β1=
1
γS1. (34)
These assumptions will be verified once the solution is obtained.
A.1 Period 1The total wealth of the investor i at date t = 1 is
W i1 = XiP0 + Di
0
(P1 − P0
)+ Di
1
(U − P1
)(35)
At date t = 1, the information of investor i is Zi1, Zi0, Q1 and Q0. With this information athand, investor i will try to forecast U . The state variables corresponding to investor i are therefore
Precision → 1H
1S
1S
1Φ
1Φ
Variable ↓ U εi1 εi0 X1 X0 E [·]U 1 0 0 0 0 0
Zi1 1 1 0 0 0 0
Zi0 1 0 1 0 0 0
Q1 1 0 0 - 1γS 0 0
Q0 1 0 0 0 - 1γS 0
With this information at hand, it is straightforward to calculate
K1 = V ar−1[U | Zi1, Zi0, Q1, Q0
](36)
µi1 = E[U | Zi1, Zi0, Q1, Q0
](37)
by using the projection theorem:
29
Theorem 6. Consider a n-dimensional normal random variable
(θ, s) ∼ N([
µθµs
],
[Σθ,θ Σθ,sΣs,θ Σs,s
]).
The conditional density of θ given s is normal with conditional mean
µθ + Σθ,sΣ−1s,s (s− µs)
and variance-covariance matrixΣθ,θ − Σθ,sΣ
−1s,sΣs,θ
provided Σs,s is non-singular.
The optimal demand of trader i in period 1 has a simple expression (from the normality ofdistribution assumption in conjunction with the exponential utility function):
Di1 = γK1
(µi1 − P1
). (38)
Integrate the optimal demands to get the total demand. Follow the convention used by Admati(1985) that implies
∫ 1
0Zij = U , a.s., and replace U − 1
γS1X1 to obtain
D1 = γ(S(Zi0 + Zi1
)+ γ2S2Φ
(Q0 + Q1
)−K1P1
). (39)
Integrate the optimal demands to get the total demand. Follow the convention used by Admati(1985) that implies
∫ 1
0Zij = U a.s., and replace U − 1
γS X1 to obtain
D1 = γ(S(2 + γ2SΦ
)U + γ2S2ΦQ0 − γ2S2ΦX1 −K1P1
). (40)
In order to clear the market, we need to have D1 = X0 + X1. Once we impose market clearing,we can use the conjectured equation (32) to get the undetermined coefficients. The solution isshown below.
β1 =S(1 + γ2SΦ
)K1
,
ξ01 =S(1 + γ2SΦ
)K1
γ1 =1 + γ2SΦ
γK1
(41)
From these solutions, we can verify one of the two equalities assumed in (34), namely γ1β1
= 1γS .
A.2 Period 0The problem of the investor is now
maxDi0E[−e−
1γ W
i | Zi0, Q0
](42)
where, as stated before, the final wealth is given by
W i = XiP0 + Di0
(P1 − P0
)+ Di
1
(U − P1
). (43)
Replace Di1 from (38) into this equation and observe that the investor needs to estimate U ,
P1 and µi1, after observing Zi0 and Q0. This is a quadratic form of a multivariate normal variable.Accordingly, we use the following convenient theorem:
30
Theorem 7. Consider a random vector z ∼ N (0,Σ). Then,
E[ez′Fz+G′z+H
]= |I − 2ΣF |−
12 e
12G′(I−2ΣF )−1ΣG+H
The matrices F , G and H have complicated forms and are not exposed here for simplicity.They are replaced in the objective function, which is then differentiated with respect to Di
0 andequalized to zero in order to solve for the optimal demand. Once the optimal demand is found, itcan be integrated to solve for P0. This allows to finally find the undetermined coefficients of P0:
β0 =S(1 + γ2SΦ
)K0
,
γ0 = γSΦ +1
γ.
(44)
At this point, the remaining equality conjectured in (34), namely γ0β0
= 1γS is verified as well.
Once we know the coefficients, we can write the solution in a recursive form as done in Theorem1.
B Proof of Proposition 2In order to prove Proposition 2, we compute the distribution of new signals that have been gatheredbetween time 0 and time T , i.e. during the time period T . Denote by Xti the number of newsignals accruing if a meeting occurs at time ti and observe that it is distributed as
Xti ∼ µ (ti, ·)
where the distribution µ (t, x) satisfies the differential equation in (6). It is reminded that, in thelatter differential equation, λ represents the intensity of the Poisson meeting process and ∗ theconvolution product. That is, the number N (T ) of meetings that have taken place between time0 and T is a Poisson counter with intensity λ. The total number YT of new signals that have beengathered between time 0 and T is accordingly given by
∑N(T )i=1 Xti . We now proceed to characterize
its distribution. Notice that the Xti ’s are independent but not identically distributed, which makesthis task somewhat more difficult. Yet, this problem is very similar to the accumulated loss claimproblem with time-dependent claim amount in the actuarial sciences field (e.g. see Buehlmann(2005)). Our problem also somewhat compares to Duffie, Malamud, and Manso (2010).
A first step is to observe that YT , conditional on the set of times 0 ≤ t1 ≤ t2 ≤ ... ≤ tN(T ) ≤ Tat which a meeting has occurred during T and the total number of meetings N (T ) (that is,conditioning on the whole trajectory ANT of the Poisson process), is distributed as
YT |ANT ∼∫RN−1
µ(YtN − YtN−1
, tN)dµ(YtN−1
− YtN−2, tN−1
)...dµ (Yt1 − 0, t1)
≡∫RN−1
µ (XtN , tN ) dµ(XtN−1
, tN−1
)...dµ (Xt1 , t1) .
In order to bridge this with Duffie, Malamud, and Manso (2010), we observe that the distributionµ (Xti , ti) of the increment can be expressed as a translation T of the type measure µ and theincrement x. Hence, as in the latter reference, the distribution above may be written as
YT |ANT ∼ ΓN(T )i=1 µti
where, for any probability measures α1, ..., αk, we write Γki=1 = α1 ∗ α2 ∗ ... ∗ αk.Now, as a second step, observe that each ti in the sequence of meetings 0 ≤ t1 ≤ t2 ≤ ... ≤
31
tN(τ) ≤ T conditional on N (T ) is uniformly distributed over T . Accordingly, we have that
YT |N (T ) ∼ ΓN(T )i=1
1
T
∫ T
0
µtidti =
1
TN(T )
(∫ T
0
µsds
)∗N(T )
where ∗n denotes the n−fold convolution.A third and final step is to notice that N (T ) is a Poisson(λ) counter and thus
YT ∼∞∑k=0
e−λT(λT )
k
k!
1
T k
(∫ T
0
µsds
)∗k=
∞∑k=0
e−λTλk
k!
(∫ T
0
µsds
)∗k.
Using the fact that, by Taylor expansion, ex is equivalently written as ex =∑∞k=0
xk
k! , we thus canwrite
πT = eλ(∫ T0µsds−T).
Equivalently, πt satisfies the differential equation
dπt = −λπt + λπt ∗ µt
with initial condition given by the Dirac measure δ0 at zero.
C Proof of Theorem 3As for Theorem 1, we proceed to prove Theorem 3 by means of a simpler example. The proofthen follows by induction. We consider two dates t = 0, 1 and two types of agents i = 1, 2. Agentsof type 1 receive a private signal at time t = 0 while agents of type 2 receive two private signalsat t = 0. There is a proportion λ of agents of type 2. Both types receive a single signal at datet = 1. The remaining elements of the economy are identical to the Benchmark economy of Section2. Because we use the same theorems and the same steps as in Appendix A, we do not repeatthem. Rather, we directly expose the key elements of the solutions of the particular case treatedhere.
The conditional precisions for each agent type are
K0,1 = H + S + γ2S2ΦΩ20
K1,1 = H + 2S + γ2S2ΦΩ20 + γ2S2Φ
and
K0,2 = H + 2S + γ2S2ΦΩ20
K1,2 = H + 3S + γ2S2ΦΩ20 + γ2S2Φ
with Ω0 = (1− λ)+2λ, that is the cross-sectional average 1+λ. Given these two investor categories,there exists an average K:
K0 = H + SΩ0 + γ2S2Ω20
andK1 = H + S (1 + Ω0) + γ2S2Ω2
0 + γ2S2Φ.
The conditional expectation for each agent type are
µi0,1 =1
K0,1
(SZi0,1 + γ2S2ΦΩ2
0Q0
)µi1,1 =
1
K1,1
(SZi0,1 + SZi1 + γ2S2ΦΩ2
0Q0 + γ2S2ΦQ1
)
32
and
µi0,2 =1
K0,2
(2SZi0,1 + Zi0,2
2+ γ2S2Ω2
0Q0
)
µi1,2 =1
K1,2
(2SZi0,1 + Zi0,2
2+ SZi1 + γ2S2ΦΩ2
0Q0 + γ2S2ΦQ1
)
with
Q0 = U − 1
γSΩ0X0
Q1 = U − 1
γSX1.
There also exists an average conditional average given by
µ0 =1
K0 − SΩ0
(γ2S2ΦΩ2
0Q0
)µ1 =
1
K1 − S (1 + Ω1)
(γ2S2ΦΩ2
0Q0 + γ2S2ΦQ1
).
The optimal demands are given by
Di0,1 = γ
(SZi0,1 − SΩ0U +
1
γX0
)−∆K0,1P0
Di1,1 = γ
(SZi0,1 + SZi1 − SΩ0U − SU +
1
γ
(X0 + X1
))−∆K1,1P1
and
Di0,2 = γ
(2SZi0,1 + Zi0,2
2− SΩ0U +
1
γX0
)−∆K0,2P0
Di1,2 = γ
(2SZi0,1 + Zi0,2
2+ SZi1 − SΩ0U − SU +
1
γ
(X0 + X1
))−∆K1,2P1
where
∆K0,1 ≡ K0,1 −K0 = S (1− Ω0)
∆K1,1 ≡ K1,1 −K1 = ∆K0,1
∆K0,2 ≡ K0,2 −K0 = S (2− Ω0)
∆K1,2 ≡ K1,2 −K1 = ∆K0,2.
The equilibrium prices are given by
P0 =1
K0
[(K0 − SΩ0) µ0 + SΩ0U −
1
γX0
]P1 =
1
K1
[(K1 − S (1 + Ω0)) µ1 + S (1 + Ω0) U − 1
γ
(X0 + X1
)].
The optimal strategy for investors of type 1 satisfies
∆Di1,1 ≡ Di
1,1 −Di0,1 = γ
(S(Zi1 − P1
)− S
(U − P1
)+X1
γ−∆K0,1∆P1
)
33
while the optimal strategy for investors of type 2 satisfies
∆Di1,2 ≡ Di
1,2 −Di0,2 = γ
(S(Zi1 − P1
)− S
(U − P1
)+X1
γ−∆K0,2∆P1
).
Now, proceeding by induction and denoting by k the investor type, i.e. the number of privatesignals that the investor has, we can write
Kit,k ≡ Var−1
[U | F it,k
]= H +
t∑j=0
[kS + γ2S2ΦΩ2
j
].
with Ω0 =∑k∈N π0 (k) k and Ω1 = 1 and where π is defined in Proposition 2. The average precision
satisfies
Kit ≡ Var−1
[U | Ft
]= H +
t∑j=0
[S + γ2S2ΦΩ2
j
].
The individual conditional expectation of U is
µit,k ≡ E[U | F it,k
]=
1
Kt,k
t∑j=0
(kSZ
i
j,k + γ2S2ΦΩ2jQj
)
where Zi
j,k denotes the average of the private signals received by agents of type k and
Qj ≡ U −1
γSΩjXj
and
µt ≡1
Kj − SΩj
t∑j=0
γ2S2ΦΩ2jQj .
The individual optimal demands are of the form
Dit,k = γ
t∑j=0
[kSZ
i
j,k − SΩjU +1
γXj
]−∆Kj,kPj
.
The price obeys
Pt =1
Kt
(Kt − SΩt) µt +
t∑j=0
[SΩjU −
1
γXj
]and the optimal strategy for investors of type k is
∆Dit,k ≡ Di
1,k − Di0,k = γ
(S(Zi1 − P − 1
)− S
(U − P − 1
)+
1
γX1 −∆K0,k∆P1
).
By induction on the number of trading sessions, we obtain Theorem 3.
34
D Proof of Proposition 5Using the recursive form of the price in Theorem 3, we can write the prices difference betweent = 0 and t = 1 as
P1 − P0 =
(...) + 1K1
(r2S2ΦΩ2
1
(U − 1
γSΩ1X1
))+(SΩ1U − 1
γ X1
) +(
1K1− 1
K0
)(...) +
(γ2S2ΦΩ2
0
(U − 1
γSΩ0X0
))+(SΩ0U − 1
γ X0
) and the prices difference between t = 1 and t = 2 as
P2 − P1 =
(...) + 1K2
(γ2S2ΦΩ2
2
(U − 1
γSΩ2X2
))+(S2Ω2U − 1
γ X2
) +(
1K2− 1
K1
)(...) +
(∑1j=0 γ
2S2ΦΩ2j
(U − 1
γSΩjXj
))+(∑1
j=0 SΩjU −∑1j=0
1γ Xj
) This yields the following covariance
cov(P1 − P0, P2 − P1
)=
1
K1
1∑j=0
Ωj(γ2SΦΩj + 1
)− 1
K0Ω0
(γ2SΦΩ0 + 1
) S2
H
×
1
K2
2∑j=0
Ωj(γ2SΦΩj + 1
)− 1
K1
1∑j=0
Ωj(γ2SΦΩj + 1
)+
(1
K2− 1
K1
)1
Φ
(γSΦΩ0 + 1
γ
)2 (1K1− 1
K0
)+ 1K1
(γSΦΩ1 + 1
γ
)2
.
Now, considering the price difference between t = 2 and t = 3, we have that
P3 − P2 = (...) +1
K3
(γ2S2
3Φ3Ω23
(U − 1
γS3Ω3X3
)+S3Ω3U − 1
γ X3
)
+
(1
K3− 1
K2
) (...) +∑2j=0
(γ2S2ΦΩ2
j
(U − 1
γSjΩjXj
))+∑2j=0
(SΩjU − 1
γ Xj
) Accordingly, we obtain the following covariance
cov(P3 − P2, P2 − P1
)=
1
K2
2∑j=0
Ωj(γ2SΦΩj + 1
)− 1
K1
1∑j=0
Ωj(γ2SΦΩj + 1
)×
( (1K3− 1
K2
)∑2j=0 Ωj
(γ2SΦΩj + 1
)1H
+ 1K3
Ω3
(γ2SΦΩ3 + 1
)1H
)S2
+
(1
K3− 1
K2
)(
1K2− 1
K1
)∑1j=0
(γSΦΩj + 1
γ
)2
+ 1K2
(γSΦΩ2 + 1
γ
)2
1
Φ.
35
E Information Percolation in the Rumor Case[TO BE COMPLETED]
36
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