Physica A 391 (2012) 5532–5545

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Physica A

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Formation of share market prices under heterogeneous beliefs andcommon knowledgeYuri Biondi a, Pierpaolo Giannoccolo b, Serge Galam c,∗

a Pôle de Recherches en Économie et Gestion (PREG), École Polytechnique and CNRS, Boulevard Victor, 32, F-75015 Paris, Franceb Department of Economics, University of Bologna, 40100 Bologna, Italyc Centre de Recherche en Épistémologie Appliquée (CREA), École Polytechnique and CNRS, Boulevard Victor, 32, F-75015 Paris, France

a r t i c l e i n f o

Article history:Received 28 October 2011Received in revised form 28 May 2012Available online 21 June 2012

Keywords:Opinion dynamicsSociophysicsEconophysicsReaction–diffusionBubble formationMarket dynamicsMarket price formation

a b s t r a c t

Financial economic models often assume that investors know (or agree on) thefundamental value of the shares of the firm, easing the passage from the individual tothe collective dimension of the financial system generated by the Share Exchange overtime. Our model relaxes that heroic assumption of one unique ‘‘true value’’ and deals withthe formation of share market prices through the dynamic formation of individual andsocial opinions (or beliefs) based upon a fundamental signal of economic performance andposition of the firm, the forecast revision by heterogeneous individual investors, and theirsocial mood or sentiment about the ongoing state of the market pricing process. Marketclearing price formation is then featured by individual and group dynamics that make itscollective dimension irreducible to its individual level. This dynamic holistic approach canbe applied to better understand the market exuberance generated by the Share Exchangeover time.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Advances in heterogeneous agents modeling from economics [1,2] and complex systems dynamics in sociophysics [3,4]call for an understanding of theworking of the financialmarket based upon the collective and dynamic properties of systemsfeatured by interacting parts and structures. These elements can be atoms or macromolecules in a physical context, as wellas investors, firms or regulated Exchanges in a socio-economic context. These approaches aim then to analyze the propertiesof socio-economic systems over time by focusing on interactions, relationships and the overall architecture of those systems[5,6].

Drawing upon these advances, this paper integrates the phenomenon of opinion dynamics studied by sociophysics [7,8]to an economic dynamicmodel ofmarket price formation over time through hazard, ignorance and interaction [9]. The studyof opinion dynamics has been a long and intensive subject of research among physicists working in sociophysics [10–16]:we apply here the Galam sequential probabilistic majority model of opinion dynamics [16–18].

During the last decades, financial market analysis has assisted in the proliferation of financial economic models thatrelax received assumptions of full knowledge, individual rationality and market efficiency. However, many models remainsomewhat tied to an equilibrium approach to the formation of share market prices over time. This approach entails apricing rule that satisfies all the market orders simultaneously passed by all investors in the purpose of maximizing theirexpected utilities. This approach actually implies a peculiar understanding of the market coordination between individual

∗ Corresponding author.E-mail addresses: yuri.biondi@polytechnique.edu (Y. Biondi), pierpaolo.giannoccolo@unibo.it (P. Giannoccolo), serge.galam@polytechnique.edu

(S. Galam).

0378-4371/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2012.06.015

Y. Biondi et al. / Physica A 391 (2012) 5532–5545 5533

investors. This coordination is supposed to be achieved in a solitarymoment beyond time and context [19]when all investorscontemplate the past, present and future of the business firm and univocally agree on its fundamental value of reference.

Once this unanimous consensus achieved, they performmarket transactions at that price, which does not change unlessthe fundamental value of the firm does change [20]. Therefore, the share market price is supposed to incorporate (all theavailable information on) the fundamental value of the firm at every instant [21,22]. The share market price becomes asufficient statistics of the fundamental value of the firm [23], and investors are then supposed to know (or agree with) thefundamental value of its shares, even though the current market price may diverge from this ‘‘true value’’ in some waysover time. The understanding and the modeling of market pricing, and the dynamics of individual and collective opinions,are then driven by this assumption of uniqueness of the value of the firm.

Our model relaxes this heroic assumption of the market price as the best evidence of the ‘‘true value’’, and deals withthe formation of share market price of one firm through the dynamic formation of individual and social opinions (or beliefs)based upon a fundamental signal Ft on the economic performance and position of the firm, the market clearing price ofeach share pt , and a social mood (or sentiment) mt on the ongoing state of market pricing process. Accordingly, individualinvestors are assumed to form their personal opinions—which orient their financial decisions to sell or hold, and buy orwait— in a fundamentally interactive context [24]. At every instant k, each investor i does form its opinions respectively on theevolution of corporate fundamentals and the market clearing price that is continuously changed by achieved transactionsthrough the Share Exchange.

Nothing can assure one investor about the permanent alignment between his opinion on the evolving fundamentals,its opinion on the current market price, and the market price itself [25]; nor can it be sure that the market order — whichit passes through the Share Exchange according to those opinions — may be eventually satisfied. In this dynamic setting,the formation of share prices critically depends on both the interactive formation of social opinions among investors, andtheir common knowledge of corporate fundamentals over time. Every investor strives then to revise its price expectationsEt(pt+1)|i according to the dynamics of the fundamental signal Ft and the social market sentiment mt . In this way, wedevelop a theoretical model of financial price formation over socio-economic space and time based upon the combination ofindividual, group, and collective levels of analysis. Both individual idiosyncratic investment strategies, the formation of socialopinions through investor group interaction, and collective coordination through the Share Exchange and the provisionof fundamental signals, play distinctive roles in our framework of analysis. Indeed we are in line with recent theoreticaladvances aiming to formalize the co-evolution of market prices and corporate fundamentals through socio-economicinteraction in a consistent and parsimonious way, while capturing some stylized facts from empirical evidence [26–34].

2. Definition of variables and timing

According to our model, the formation of share market price over time depends on the dynamic formation of individualand social opinions (or beliefs) based upon a fundamental signal Ft on the economic performance and position of the firm, themarket clearing price of each share pt , and a socialmood (or sentiment)mj

t about themarket pricing. These three dimensions(or layers, or orders) correspond to three different rhythms of change, that is, three different timings:• Ft,h, the fundamental signal, has the slowest rhythm or the largest lag (duration). This means that Ft can be constant for

t + h periods; it lasts for h periods;• pt , the market clearing price, when it exists, changes at each period t;• mj

t,k, the social mood, has the quickest rhythm or the shortest lag. At each period t , its value is the final result of kinteractions; each mood lasts indeed for 1

k periods.

Two distinctive forces drive the market clearing price formation through time. From one side, ongoing market pricingis submitted to individual guesses and intentions, hopes and fears, subsumed by the social mood mj

t,k and its quickestinteractions; from another side, it is concernedwith the slowest history of reporting and disclosure that, in principle, may bepartly public, consistent, and conventionally agreed. This general system (which is no longer an equilibrium)1 consists in anddepends upon the coherence and universal diffusion of relevant and reliable knowledge through a price system (providingmarket information) and an accounting system (disclosing firm-specific, fundamental information) publicly determined andannounced.

In particular, the fundamental signal is assumed to be common knowledge among all investors:• Ft is the fundamental signal about the economic performance and position generated by the business firm over time; it

is fundamentally related to the firm’s share price, but agents do not know (or agree on) the working of this relationship;• Ft can be positive or negative and is exogenous to the model;• By assumption

Ft ≥ 0;

• Each agent applies an individual weight ϕi ∈ [0; 1] to this signal, related to its personal confidence degree on it, fromϕi = 0 (no confidence at all) to ϕi = 1 (full confidence); this implies that all agents agree on the direction (sign) of thefundamental signal, but disagree on its material impact on the share price;

• In some specifications of the model, Ft may influence the social moodmjt,k;

1 Our analysis distinguishes system and equilibrium as distinctive concepts.

5534 Y. Biondi et al. / Physica A 391 (2012) 5532–5545

The social mood (ormarket sentiment) captures the group interaction that generates the collective opinion on the currentstate of market pricing:

• mjt,k ∈ [0; 1] is the mood of group j at time t , resulting from k group interactions (steps) starting frommj

t,k=0;• At each time t,mj

t,k=0 ∈ [0; 1] exists and is exogenous to the model; in fact,mjt,k=0 may be endogenous to the model; in

particular, it may depend on F .

Themarket clearing price (when it exists) is generated by thematching of aggregate supply and demand, which are basedupon heterogeneous price expectations by individual agents:• pt is the market clearing price at time t;• By assumption, pt ≥ 0;• Et(pt+1)|

ji is the price expectation at time t by agent i belonging to the group j on themarket clearing price at period t+1.

Individual investors have both group and individual heterogeneities regarding the formation of their expectations, whichare then based upon individual and social opinions (or beliefs). In particular:• Investors are distinguished between actual and potential shareholders. Analytically, they belong then to two groups

j = S,D, where S denotes supply by potential sellers (actual shareholders), while D denotes demand by potential buyers(potential shareholders);

• In each group j, the number of agents is normalized to one, with i ∈ [0; 1];• In each group j, every agent i is characterized by an individual weight ϕi ∈ [0; 1] that is applied to the fundamental

signal Ft ;• In each group j, agents are further characterized by the social moodmj

t,k that constitutes themarket sentiment expressedby group j at time t; its weight results from k inter-individual interactions between t − 1 and t .

3. The formation of individual expectations

It has been advocated that the two broad categories of chartism and fundamentalism account for most of possibleinvestment strategies [35]. On this basis, every agent forms its price expectation according to the following genericfunction [36,37]:

Et(pt+1)|ji = pt + mj

t,k (pt − pt−1) − βji

Et−1(pt)|

ji − pt

+ γ jϕiFt (1)

with j = S (Supply), D (Demand); i, ϕi ∈ [0, 1] ;mjt,k ∈ [0, 1] ; β

ji ∈ [0; 1] ; γ j > 0, and

εji,t ≡

Et−1(pt)|

ji − pt

. (2)

This equation comprises four elements. The first element is the past clearing price pt . The second element is the marketsignal (or price trend) that is weighted by the social opinion mj

t,k of the group j at time t , expressing the group’s ongoingmarket confidence. The third element is the individual forecast revision that consists of the difference between the past priceexpectation and the current realized price. This revision is weighted by β

ji which may include both group and individual

heterogeneities. The forth element denotes the formation of an individual opinion by investor i (belonging to the groupj) based upon available fundamental information Ft , which is common knowledge for both groups and all the individualinvestors, and is weighted then by the individual parameter ϕi.

This structure of individual expectations follows the dual structurewhich the sharemarket process is embedded in: Fromthe cognitive viewpoint, investors are confrontedwith fundamental information from the business firm (they invest in) fromone side, and the market pricing from another side. From the financial viewpoint, they are confronted with dividends andnet earnings generated by the business firm, and the capital gains and losses involved in the market trading (see Ref. [38]for further details). The firm’s side is subsumed here by the factor F , while the market’s side is captured by the price trendt−1∆t (p).

Following Galam’s specification of the formation of social opinions [17,18], for each side of the Share Exchange (potentialbuyers who currently do not hold shares, and potential sellers who hold them) we can define a generic function of theirsocial mood mj

t,k as follows:

mjt,k = f

mj

t,k=0, kjt , Ft,h

, (3)

where the fundamental signal F can influence kjt . For each group j = S,D,mjt,k defines then the density at time t of individual

investors who are confident in the market signal or trend (m → 1), while 1−mjt,k defines the density at time t of investors

who distrust that market signal (m → 0). The initial valuemjt,k=0 can be exogenous or endogenous to the model setting. In

particular, it can depend on F and its history.On this basis, at each time t , we assume that individual investors interact within each group j by subgroups of various

given sizes for a series of successive k sub-periods, in order to generate the group opinion for that time t . This subgroup

Y. Biondi et al. / Physica A 391 (2012) 5532–5545 5535

interaction is an analytical tool that enables ourmodel to capture the density distribution of market beliefs and its evolutionover time. The subgroups can then be understood as special fields that frame and shape individual opinions. These fieldsmayrelate to tools, methodologies and emerging mindsets and worldviews shared by some investors, as well as to the financialintermediation structure comprising asset management funds, consultants and brokers with various sizes and influenceabilities Let us take the example of groups of size 3 and 4. In particular, for groups of size 3, the density after k successiveupdates is

mjt,k = (mj

t,k−1)3+ 3(mj

t,k−1)2(1 − mj

t,k−1), (4)

where mjt,k−1 is the proportion of agents who, at time t , are confident in the market signal at a distance of (k − 1) updates

from the initial proportionmjt,0 at the same time t . For groups of size 4, the same density writes,

mjt,k = (mj

t,k−1)4+ 4(mj

t,k−1)3{1 − mj

t,k−1} + 6(qj)(mjt,k−1)

2{1 − mj

t,k−1)2, (5)

where the last term includes the tie case contribution (with two ‘‘believers’’ confronted with two ‘‘distrusters’’) weightedwith the probability qj that defines the common degree of confidence in themarket trend. Accordingly, in case the group hasdoubts, the local four agents become either distrusters with probability (1− qj), establishing their commonmood (density)down to 0 or believers with probability qj pushing their common mood up to 1.

For a mixture of group sizes n with the probability distribution an under the constraintL

n=1 an = 1 — where L is thelargest group size and n refers to the group size, the above equations become

mjt,k =

Ln=1

an

nj=N[ n

2 +1]Cnj (m

jt,k−1)

j(1 − mjt,k−1)

(n−j)+ (qj)V (n)Cn

n2(mj

t,k−1)n2 (1 − mj

t,k−1)n2

, (6)

where Cnj ≡

n!(n−j)!j! ,N

n2 + 1

≡ Integer Part of

n2 + 1

, and V (n) ≡ N

n2

− N

n−12

. This implies V (n) = 1 for n even

and V (n) = 0 for n odd. The proportion of distrusters is then 1 − mjt,k.

It is worth emphasizing that the Galam model of opinion dynamics tangles up three main mechanisms to producea threshold opinion dynamics among two competing choices within an ensemble of investors. The first mechanism isexogenous and combines all effects which act directly and individually on the investor to influence its own personal choice,here to trust or distrust the current market price. This mechanism determines the initial share mj

t,k=0 of investors whoare respectively confident in, or distrusting of that trend. The two other mechanisms are endogenous to the ensemble ofinteracting investors.

The second mechanism embeds a social mimetic effect using a local majority rule: investors confront their actual choicewith the ones of a small group of other investors and update their personal choices following the choice which was locallymajoritywithin that group. At the collective global level, for groups of odd sizes, this interactive process produces a thresholddynamics for which the tipping point is located at precisely fifty percent: the choice which starts with an initial support ofmore than fifty percent of investors will drive the market along its direction.

The thirdmechanism ismore subtle and depends on the occurrence of a local doubtwithin an even size group of investorswhich have to settle their group opinion. If such a doubt occurs, all the involved investors adopt just themoodwhich followsthe prevailing common belief about the market price trend. Accordingly, within an ensemble of investors, withmj

t,k percentof them trusting themarket trend, a local doubting group of even sizemay decide to either trust that trendwith a probabilityof (qj) or distrust it with a probability of (1 − qj).

The breaking contribution of this leading common belief is to unbalance drastically the threshold dynamics by placingthe tipping point (TP) at a value which can be as low as 15% for the choice which goes along the common belief qj, and ashigh as 85% for the choice which contradicts that common belief [17,18]. This tipping point TP is a function of the degree ofcommon belief qj and the group size n.

For the case of a group of size 4 used in this work, we have 23% and 77% for the tipping points related to respectivelyqj = 0 (complete distrust in the market trend) and qj = 1 (complete trust). This implies that, at every step k, the groupmood always goes either towards 0 for any mj

t,k=0 < 0.23 (mjt,k ⇒ 0 with increasing k), or towards 1 when mj

t,k=0 > 0.77(mj

t,k ⇒ 1 with increasing k).However, for tipping points between these extreme values, the model determines whether the common belief under

doubt qj dominates the group moodmt,k as follows,

TP(qj) =−1 + 6qj −

13 − 36qj(1 − qj)

6(−1 + 2qj)(7)

with mjt,k ⇒ 0 when mj

t,k=0 < TP(qj) and mjt,k ⇒ 1 when mj

t,k=0 > TP(qj). For the special value mjt,k=0 = TP(qj) we have

mjt,k = TP(qj) for any k. The case qj = 1/2— implying a common belief perfectly balanced between trust and distrust - yields

5536 Y. Biondi et al. / Physica A 391 (2012) 5532–5545

TP(1/2) = 1/2, in between the two extreme cases TP(1) =5−

√13

6 ≈ 0.23 and TP(0) =1+

√13

6 ≈ 0.77. For a combinationof sizes, the TP can still be calculated using Eq. (7) through numerical computation with 0 ≤ TP(q) ≤ 1 depending on theirvarious proportions.

This third mechanism especially illustrates how a common belief shared by some groups of investors can shapesubstantially the working of the market pricing over time by influencing the ongoing formation of the collective moodm [9]. This mechanism also sheds light on the framing and shaping role plaid by aggregating operators (so-called marketmakers) such as asset management funds, consultants and brokers with various sizes and influence abilities.

4. The formation of the market clearing price

The formation of the market clearing price p∗

t+1 over time depends on the aggregation of individual bids of demand andsupply at each period t . In particular, every shareholder (j = S) i wishes to sell if p∗

t+1 ≥ Et(pt+1)|Si , while every potential

buyer (j = D) i wishes to buy if p∗

t+1 ≤ Et(pt+1)|Di . By assuming uniform distribution of individual investors within each

group j = S,D, the individual price expectation Et(pt+1)|ji of investor i belonging to group j can be rewritten as a function

of expectations expressed by investors i = 0 and i = 1 defined as follows:

εt |j0 ≡

Et−1(pt)|

j0 − pt

εt |

j1 ≡

Et−1(pt)|

j1 − pt

.

Individual price expectation by investor i may then be described as follows:

Et(pt+1)|ji = pt + mj

t,k (pt − pt−1) −

β

j0 (1 − ϕi) ε

j0,t + β

j1ϕiε

j1,t

+ ϕiγ

jFt . (8)

Aggregated demand and supply are now defined by the focal prices of four representative agents with i = 0 andi = 1 ∀j = S,D. By defining:

P jt ≡ max arg

Et(pt+1)|

ji=0 ; Et(pt+1)|

ji=1

P jt ≡ min arg

Et(pt+1)|

ji=0 ; Et(pt+1)|

ji=1

,

the aggregate functions of supply xSt+1 and demand xDt+1 integrate individual bids as follows:xSt+1 =

p∗t+1

PSt

1

PSt − PS

t

dx

xDt+1 =

PDt

p∗t+1

1

PDt − PD

t

dx

(9)

or, equivalently:

xSt+1 =

0 if p∗

t+1 ≤ PSt

p∗

t+1 − PSt

PSt − PS

t

if PSt < p∗

t+1 < PSt

1 if p∗

t+1 ≥ PSt

(10)

xDt+1 =

1 if p∗

t+1 ≤ PDt

PDt − p∗

t+1

PDt − PD

t

if PDt < p∗

t+1 < PDt

0 if p∗

t+1 ≥ PDt .

(11)

The necessary condition for the existence of a market clearing price p∗

t+1 (implying that both demand and supply aredifferent from zero) is

PSt ≤ p∗

t+1 ≤ PDt . (12)

This condition implies two different scenarios:

Y. Biondi et al. / Physica A 391 (2012) 5532–5545 5537

1. if PDt ≤ PS

t , there is no matching between demand and supply; therefore, no exchange transactions occur, and the ShareExchange does not fix any updated clearing price at period t; at the next period t+1, investors will then observe a specialno-clearing price pNCgenerated by the market-making process according to some external rule or device;

2. if PDt > PS

t , there ismatching, and themarket clearing price pC is defined as the price thatmakes demand equal to supply.2

On this basis, the market clearing price at period t is

p∗

t+1 =

pNC if PD

t ≤ PSt

pC if PDt > PS

t .(13)

Let assume that the no-clearing price pNC is fixed according to the following rule:

pNC = pt + ϵ, (14)

where ϵ is the smallest tick value available on the Share Exchange. Furthermore, concerning the clearing price pC , demandis equal to supply if

pC − PSt

PSt − PS

t

=PDt − pC

PDt − PD

t

, implying that (15)

pC =

PDt

PSt − PS

t

+ PS

t

PDt − PD

t

PDt − PD

t

+

PSt − PS

t

. (16)

Therefore, the market clearing price p∗

t+1 at time t is:

p∗

t+1 =

pNC = pt + ϵ if PD

t ≤ PSt

pC =

PDt

PSt − PS

t

+ PS

t

PDt − PD

t

PDt − PD

t

+

PSt − PS

t

if PDt > PS

t .(17)

5. The dynamics of the market clearing price

In order to analyze the dynamics of the market clearing price (when it exists, i.e., p∗

t+1 = pC ) over time, let us define∀j = S,D:

Pj (n) ≡

tn=1

−β

j0

n pt−n + mj

t−n (pt−n − pt−n−1) − pt−n+1

Fj (n) ≡

tn=0

−β

j0

n γ jFt−n

Lj (P (n) , F (n)) ≡

β

j1 − β

j0

· Pj (n) + Fj (n)

j=S,D

βj1 − β

j0

· Pj (n) + Fj (n)

−1

Mj(P (n) , F (n)) ≡

β

j1 − β

j0

· Pj (n) + Fj (n) .

Accordingly,

Et(pt+1)|ji = pt + mj

t,k (pt − pt−1) + βj0 · Pj (n) + ϕi · Mj(·). (18)

The four representative agents are then described as follows:

∀j = S,D with ϕi = 0 : Et(pt+1)|ji=0 = pt + mj

t,k (pt − pt−1) + βj0 · Pj (n)

∀j = S,D with ϕi = 1 : Et(pt+1)|ji=0 = pt + mj

t,k (pt − pt−1) + βj1 · Pj (n) + Fj (n) .

2 The Walrasian auction is included by this scenario when the whole share offer is satisfied.

5538 Y. Biondi et al. / Physica A 391 (2012) 5532–5545

By computation, the market clearing price function can be rewritten as follows:

p∗

t+1 = pt +

j=S,D

mj

t (pt − pt−1) + Pj (·)

L¬j (·)

+

MD(·)

LS (·)

ifMj(·) > 0 ∀j

MS(·)

LD (·)

ifMj(·) < 0 ∀j

j=S,D

Mj(·)

L¬j (·)

ifMD(·) > 0and MS(·) < 0

0 ifMD(·) < 0and MS(·) > 0.

(19)

Accordingly, the pattern of market clearing price p∗

t+1 is based on the historical price pt by adding two further elements.The first element comprises (for both j = S and j = D) two sub-elements:

• the numerator,mjt,k (pt − pt−1)+Pj (n), is independent fromsignal Ft anddependent on the price trend∆t (p∗)weighted3

by the current group moodmjt and its weighted past series Pj (n).

• the denominator, Lj (·), depends on both Fj (n)which represents theweighted fundamental signal trend series, and Pj (n)which represents the weighted market price trend series; for each group j, this sub-element weights the contribution ofthe price trend series to the formation of the market clearing price at time t .

The second element depends on both weighted past series Fj (·) and Pj (·). In particular, ifMj(·) is positive (negative) forboth groups, then this element increases (decreases) themarket clearing price.Moreover, ifMj(·) is negative for shareholders(j = S) while it is positive for potential investors (j = D), then the divergence between groups is mutually balanced on themarketplace. On the contrary, ifMj(·) is positive for shareholders while negative for potential investors, then the divergencemakes the whole element equal to zero.

In sum, the formation of share prices over time depends respectively on the dynamics of the fundamental signal F fromone side, and the dynamics of the clearing market price p from another side. Both dynamics are shaped by the ongoingevolution of individual and group opinions (and related bids) captured by the structure of the model.

6. The results of the model

Previous literature has identified stylized facts of market price series dynamics such as fat-tailed distribution of absolutereturns that experience long-memory and persistence, and the distribution of their autocorrelation function that decaysrapidly to zero, both distributions being then described by power laws. These facts are interesting and relate to somestatistical measurements of properties of the price series alone. They refer indeed to a technical notion of financial marketefficiency. Even though this paper does not purport to address these facts specifically, our model is consistent with them.Under appropriate calibration, it can reproduce power laws of absolute returns and their autocorrelation function, bothshowing exponents of the same magnitude as empirical series of share market prices [39–42].

Nevertheless, our model further raises the different question of whether the formation of the market prices over timeevolves in linewith overarching fundamentals. This question refers to a substantial notion of financial market efficiency thatis less investigated by current streams of research. The distinction between technical and substantial notions of financialmarket efficiency have been investigated extensively [43–45].

Descriptive analyses may be empirically satisfying, and may even forecast future states of the financial system well, butthey do not provide any theoretical explanation of why market price fluctuations arise. However, our model links suchfluctuations to shifting socio-economic conditions of an underlying ensemble of investors and structures. This theoreticalapproachmay constitute the beginning of a usefulmodel to better understandmarket pricemovements over socio-economictime and space. In particular, our model quantizes the possible strategies of individual investors into two polar behaviorsshaped by two distinctive signals: the market trend, and the fundamental price of reference. Both signals belong to theinformation set and modify the payoffs available to every investor. While the fundamental price is exogenous to the model,the other conditions are endogenously generated through the dynamics of interacting agents that influence each other.On this basis, the model can help to explain whether and whenever the market price movements are in line with thefundamental price dynamics.

Our model cannot be solved analytically and thus requires some numerical treatment to extract its main results. Thisgoal is implemented here by using a series of characterizing cases which exhibit the major results of the model. For thispurpose, let us assume that β

ji = β and γ j

= γ ∀j = S,D and ∀i. This specification implies that group heterogeneity iscaptured by the group mood mj

t,k and leads to the following statement: If βji = β and γ j

= γ ∀j = S,D and ∀i, the group

3 Remember thatmjt,k → 1 implies full weight to this information in order to build individual price expectations. The moodm can be influenced by the

fundamental signal F , and is the final result of the dynamic interaction within the group j for k steps occurring between t − 1 and t .

Y. Biondi et al. / Physica A 391 (2012) 5532–5545 5539

mood mjt,k subsumes all the group heterogeneities between demand and supply; then, there exists only one market clearing price

case instead of the four cases defined above.In particular, the individual price expectation function becomes:

Et(pt+1)|ji = pt + mj

t,k (pt − pt−1) +

tn=1

(−β)n

pt−n − mj

t−n (pt−n − pt−n−1) − pt−n+1

+ ϕi

tn=0

(−β)n (γ Ft−n)

or Et(pt+1)|

ji = pt + mj

t,k (pt − pt−1) + Pj (n) + ϕiγjFj (n) .

Concerning the formation of the market clearing price, for βji = β ∀i, j, Lj (·) = 2. Therefore, closer are β

ji ∀i, j, closer

is Lj (·) to 2, implying that the whole first element of Eq. (19) tends to become independent from the fundamental signalseries Fj (n). Furthermore, when β

j1, β

j0 = β j,Mj(·) = Fj (n) ∀j: Closer are β

j0 and β

j1 ∀j, closer is Mj(·) to Fj (n) that is

independent from the market price trend series Pj (n). Therefore, this specification clearly distinguishes the dual structureof the market clearing price dynamics which is driven by two distinct factors: the market signal or trend ∆t (p∗) weightedby the evolution of groups’ market sentiments, and the fundamental signal F . The market clearing price becomes:

p∗

t+1 = pt +12

j=S,D

mj

t,k (pt − pt−1) + Pj (n)

+

F (n)2

(20)

where

Pj (n) ≡

tn=1

(−β)npt−n + mj

t−n (pt−n − pt−n−1) − pt−n+1

F (n) ≡

tn=0

(−β)n (γ Ft−n)

.

Accordingly, the dynamics of the market clearing price (when it exists) is denoted as follows:

∆t+1p∗

≡ pt+1 − pt = f

∆t

p∗

,mj

t,k

+ g (F (n)) . (21)

This price pattern comprises two different elements. The first element, f∆t (p∗) ,mj

t,k

, is a group factor that depends

on the market signal ∆t (p∗) weighted by the group mood mjt,k that is collectively assigned to the market price trend by

group j at time t . The second element, g (F (n)) =F(n)2 , depends on the weighted trend of the fundamental signal Ft ,

with Fj (n) = FD (n) = FS (n) in this particular specification. Consequently, if F (n) is positive (negative), then g (F (n))proportionally increases (decreases) the market clearing price at time t .

This specification of the model polarizes the formation of market prices over time between market sentiment andfundamental signal, which interact over time in a non trivial way. Market sentiment may then reduce, amplify or contradictthe impact of the ongoing fundamental signal on market pricing according to occurred configurations generated by theirinteracting dynamics. This theoretical result describes the endogenous formation of market prices, and improves onapproaches that assume their exogenous existence in order to model market exuberance.

The rest of the section presents various possible scenarios exhibited by this reduced form of our model. The evolution ofthe fundamental signal is exogenous to the model and is simulated as an external parameter, according to some stochasticdynamics. Such simulations are illustrations of the model and should not be confused with a Monte Carlo simulation of thesystem. Here only one fundamental signal series is simulated, while all other mechanisms obey their respective equationsaccording to the chosen set of scenario parameters.

More precisely, let define the fundamental price series based upon the accumulation of fundamental signals as follows:

pFt = pFt−1 + Ft−1 = p0 +

t−1n=0

Fn, (22)

where, by assumption, the initial share price p0 = pt=0 and the initial fundamental price pFt=0 are equal, and

Ft ≥ 0(implying that pFt ≥ 0). By construction, the reduced form of our model aligns fundamental price and share market pricedynamics when all the parameters are symmetric between demand and offer and group interaction is absent (k = 0). Thiscase illustrates a perfectly balanced financial systemwheremS

t,k=0 = mDt,k=0 = 1/2 and, for groups of size 4, qS = qD = 1/2.

The tipping point TP is then equal to 1/2 independently of the group size. This set of heroic assumptions implies astraightforward formation of individual and group opinions perfectly balanced between chartism and fundamentalism,demand and supply, actual and potential shareholders at every points of time {k, t, h}. This also implies that no group

5540 Y. Biondi et al. / Physica A 391 (2012) 5532–5545

Fig. 1. Series of fundamental signal pFt and market clearing price pt,k as a function of time with 0 ≤ t ≤ 500 without group interaction (k = 0).Ft = U(0; 1) − U(0; 1) follows a stochastic pattern ∀t ≥ 1. The parameters are set to values β

ji = β = 0.5, γ j

= γ = 1, ϵ = 0.01, p0 = 10,mSt,k=0 =

0.4,mDt,k=0 = 0.4, εt=0|

ji = 0.1 · (U(0; 1) − U(0; 1)), ∀j = S,D and ∀i.

dynamics is involved in market pricing over time. In this context, our model shows a market price formation patternthat remains ever and ever in line with the evolution of fundamentals generated by the underlying business firm, thosefundamentals being common knowledge between all investors.

In the following, we shall illustrate several scenarios at variance with this case. Let assume that the fundamental signalexperiences a random pattern: Ft = U(0; 1) − U(0; 1) ∀t ≥ 1. For sake of comparability, all scenarios are simulated underthe same stochastic fundamental signal pattern, and assume: β

ji = β = 0.5; γ j

= γ = 1; ϵ = 0.01; p0 = 10; F0 =

0;mSt,k=0 = 0.4;mD

t,k=0 = 0.4; εt=0|ji = 0.1 · (U(0; 1) − U(0; 1)), ∀j = S,D and ∀i.

6.1. Illustrative scenarios

On the basis of the baseline set of parameters, absent group interaction (k = 0), the reduced form of ourmodel generatesa market price dynamics related to the exogenously selected stochastic pattern of fundamental signals (Fig. 1).

The overall co-evolution of both series shows not only temporal sequences (series of periods t) when market pricesconnects with fundamental prices, but also sequences when the formers spontaneously moves away from the latter,endogenously generating market exuberance and market disconnection. This latter phenomenon is the most insightful,since it shows as the market price dynamics may persistently remain apart from the fundamentals dynamics of reference(from the statistical viewpoint, the respective averages are then significantly different for a long while).

Following existing empirical literature [39], we define returns on the market price series as the relative price change atvarious time scales, and absolute returns as follows:

|r∆t (t)| =|pt+∆t − pt |

ptwith ∆t = 1. (23)

Accordingly, we analyze the complementary cumulative distribution (density function) of absolute returns:

P(|r∆t (t)| > y) ∼ ay−x with ∆t = 1. (24)

Available empirical evidence shows that, at different time scales, this distribution follows a power law with exponent2 < x ≤ 4 that is outside the stable Levy regime requiring x < 2. Following [39], we verify that market price series ofbaseline scenario as Fig. 1 fits a power law with exponent x1 = 1.10561 for lower orders, and x2 = 3.30276 for higherorders of absolute returns, in line with empirical evidence (Fig. 2).

We further analyze the autocorrelation function Ci(t) of those returns (sampled at ∆t = 1), defined as follows:

Ci(t) =

⟨rt rt+i⟩ − ⟨pt⟩2

⟨p2t ⟩ − ⟨pt⟩2

for 1 < i < 1000. (25)

Accordingly, we analyze the complementary cumulative distribution (density function) of the above autocorrelationfunction:

P(Ci(t) > y) ∼ ay−x. (26)

Available empirical evidence shows that, at different time scales, this distribution follows a power law with exponent0.1 < x < 0.3. Following [39], we verify the autocorrelation function distribution of the market price series of baselinescenario as Fig. 1 fits a power law with exponent x3 = 0.177136, in line with empirical evidence (Fig. 3).

Y. Biondi et al. / Physica A 391 (2012) 5532–5545 5541

Fig. 2. Log–Log plot of the complementary cumulative distribution of absolute returns |r∆t (t)| =|pt+∆t −pt |

ptwith∆t = 1 for the baseline scenario as Fig. 1.

Power-law regression fit gives an exponent of x1 = 1.10561 in the lower order region, and x2 = 3.30276 in the higher order region.

Fig. 3. Log–Log plot of the autocorrelation function sampled at ∆ = 1 for returns Ci(t) =

(⟨rt rt+i⟩−⟨pt ⟩2)

⟨p2t ⟩−⟨pt ⟩2

for 1 < i < 1000; power-law regression fit gives

an exponent of x3 = 0.177136, in line with empirical evidence. Note that, after some hundred periods, the correlations reach the noise level.

Furthermore,we analyze the impact of group interaction (market sentiment dynamics) based upon this baseline scenario.In particular, the group interaction concerns groups of size 3 in Fig. 4, and various characteristic configurations for groupsof size 4 with qD = qS = 0.3 in Fig. 5, qD = qS = 0.7 in Fig. 6, qD = 0.7, qS = 0.3 in Fig. 7. By symmetry the same Fig. 7holds for qD = 0.3, qS = 0.7. In case of group indeterminacy, the probability qj decides whether the group belief tends totrust the market or not. In particular, since 0.23 < mt,k=0 < 0.77, the group moodmt,k tends to 1 if TP(qj) > mt,k=0 (mooddominance), while it tends to 0 if TP(qj) < mt,k=0 (common belief dominance). Under our assumptions, TP(qj = 0.3) = 0.36and TP(qj = 0.70) = 0.64, while mt,k=0 = 0.4 for both populations of shareholders (j = S) and investors (j = D) in allperiods t .

We perform simulations for periods from t = 1 to t = 500, with various k fixed respectively at 0, 3 and 6. In allsimulations, the market clearing price and the fundamental signal do change at the same rhythm t , while the market

5542 Y. Biondi et al. / Physica A 391 (2012) 5532–5545

Fig. 4. Fundamental signal pFt and market clearing price pt,k for respectively k = (0, 3, 6) as a function of time with 0 ≤ t ≤ 500 using groups of size 3. Ftfollows a stochastic trend (same as the one in Figs. 1 and 5–7). The parameters are set to values β

ji = β = 0.5, γ j

= γ = 1, ϵ = 0.01, p0 = 10,mSt,k=0 =

0.4,mDt,k=0 = 0.4, εt=0|

ji = 0.1 · (U(0; 1) − U(0; 1)), ∀j = S,D and ∀i.

Fig. 5. Fundamental signal pFt and market clearing price pt,k for respectively k = (0, 3, 6) as a function of time with 0 ≤ t ≤ 500 using groups of size 4with qD = qS = 0.3. Ft follows a stochastic trend (same as the one in Figs. 1, 4, 6 and 7). The parameters are set to values β

ji = β = 0.5, γ j

= γ = 1, ϵ =

0.01, p0 = 10,mSt,k=0 = 0.4,mD

t,k=0 = 0.4, εt=0|ji = 0.1 · (U(0; 1) − U(0; 1)), ∀j = S,D and ∀i.

sentiment changes at its rhythm k from 0 (no steps, implying no change from the initial value) to 6 (six steps betweent − 1 and t). It is worth emphasizing that the whole population for both shareholders (j = S) and investors (j = D) beingnormalized to one, the number and size of groups are treated as proportions and probabilities. Therefore the actual total sizeof the population is irrelevant providing it is large enough to allow the use of probabilities. Figs. 4–7 illustrate the resultsusing those probabilities.

The simulations show that changes in the market sentiment (captured by the interactive steps k) exacerbate the marketexuberance around the path provided by the fundamental price pFt over time. The evolvingmarket sentiment either amplifiesor reduces the relative change driven by the fundamental signal. In some cases or sequences, market sentiment may evengenerate a peculiar market price dynamics that is somewhat disconnected from fundamentals in a non-trivial way. Thislatter result is counterintuitive, implying that the formation of a market clearing price over time is not sufficient to assurethat market pricing is aligned on the fundamental price that arises from fundamentals that are common knowledge amongheterogeneous market participants. It may be interesting to further analyze in which conditions such a case emerges. Insum, the resulting market price does under- or over-valuate the shares relative to their fundamental price. This result is inline with theoretical and empirical analyses of market exuberance discussed by Refs. [46–48] among others.

7. Conclusive remarks

Our model considers the imbrications of three different interlocked levels which are all essential to the dynamics of theprice formation. These levels change at different rhythms. From one side, the fundamental signal has the slowest rhythm orthe largest duration h, while, from another side, the social mood is driven by the quickest rhythm or the shortest durationk. In-between stands the market clearing price that changes at each period of the basic time unit t .

This hierarchically ordered frame sheds a novel light on the complexity of the interplay between ‘‘objective’’ andsubjective driverswhich intervene in security price formation. On one hand, themodel reproduces some expected behaviors,

Y. Biondi et al. / Physica A 391 (2012) 5532–5545 5543

Fig. 6. Fundamental signal pFt and market clearing price pt,k for respectively k = (0, 3, 6) as a function of time with 0 ≤ t ≤ 500 using groups of size 4with qD = qS = 0.7. Ft follows a stochastic trend (same as the one in Figs. 1, 4, 5 and 7). The parameters are set to values β

ji = β = 0.5, γ j

= γ = 1, ϵ =

0.01, p0 = 10,mSt,k=0 = 0.4,mD

t,k=0 = 0.4, εt=0|ji = 0.1 · (U(0; 1) − U(0; 1)), ∀j = S,D and ∀i.

Fig. 7. Fundamental signal pFt and market clearing price pt,k for respectively k = (0, 3, 6) as a function of time with 0 ≤ t ≤ 500 using groups of size 4with qD = 0.7, qS = 0.3. Ft follows a stochastic trend (same as the one in Figs. 1 and 4–6). The parameters are set to values β

ji = β = 0.5, γ j

= γ =

1, ϵ = 0.01, p0 = 10,mSt,k=0 = 0.4,mD

t,k=0 = 0.4, εt=0|ji = 0.1 · (U(0; 1) − U(0; 1)), ∀j = S,D and ∀i. The same Figure holds for qD = 0.3, qS = 0.7.

like the long-term connection between market pricing and overarching fundamentals, but, on another hand, it is furthercapable to single out cases where totally unexpected behavior is observed when the market price disconnects fromfundamental signal over time. Both results are obtained endogenously, while the same mechanisms are at work in thoseopposite situations. In this way, our theoretical model points out some underlying conditions that can help to explainwhether and whenever the market price movements evolve in tune with fundamental price dynamics.

Financial economicmodels often assume that investors know (or agree on) the fundamental value of the firm’s securitiesthat are traded, easing the passage from the individual to the collective dimension of the financial system generated bythe Share Exchange over time. Our model relaxes that heroic assumption of one unique ‘‘true value’’ and deals with theformation of share market prices through the dynamic formation of individual and social opinions (or beliefs) based upona fundamental signal of economic performance and position of the firm, the forecast revision by heterogeneous individualinvestors, and their social mood or sentiment about the ongoing state of the market pricing process. Market clearing priceformation is then featured by individual and group dynamics that make its collective dimension irreducible to its individuallevel.

This dynamic holistic approach provides a better understanding of the market exuberance generated by the ShareExchange over time. This exuberance depends not only on individual biases or mistakes, but also on dynamic and collectivedimensions that arise from the interaction of individuals among them and with evolving collective structures over time.Our model captures this collective dimension through the evolution of available common knowledge on the economicperformance and position of the firm (fundamental or firm-specific information), as well as through the evolving socialmood or sentiment on the current state of the market (or the industry, or the whole economy). While the former can berelated to information release by accounting for reporting and disclosure, the latter can be related to investors’ confidenceand financial analysts’ consensus, and their respective evolution over socio-economic time and space where the financialmarket is embedded.

5544 Y. Biondi et al. / Physica A 391 (2012) 5532–5545

Appendix

For sake of completeness, we provide here various other versions of the market price equation. Starting from Eq. (19), letus define:

λt ≡

MSMS

+MD

= LS (P (n) , F (n))−1

(1 − λt) ≡

MDMS

+MD

= LD (P (n) , F (n))−1

with Mj(·) =

β

j1 − β

j0

· Pj (n) + Fj (n) = −

β

j1ε

j1,t − β

j0ε

j0,t

+ γ jFt .

Or, equivalently:

λt =

βS1 − βS

0

tn=1

−βS

0

n pt−n + mS

t−n (pt−n − pt−n−1) − pt−n+1

+

tn=0

−βD

0

n γ DFt−n

j=S,D

βj1 − β

j0

tn=1

−β

j0

n pt−n + mj

t−n (pt−n − pt−n−1) − pt−n+1

+

tn=0

−β

j0

n γ jFt−n

and

(1 − λt) =

βD1 − βD

0

tn=1

−βD

0

n pt−n + mD

t−n (pt−n − pt−n−1) − pt−n+1

+

tn=0

−βD

0

n γ DFt−n

j=S,D

βj1 − β

j0

tn=1

−β

j0

n pt−n + mj

t−n (pt−n − pt−n−1) − pt−n+1

+

tn=0

−β

j0

n γ jFt−n

.

Therefore, the market clearing equation can be rewritten as follows:

p∗

t+1 = pt + λtmDt (pt − pt−1) + (1 − λt)mS

t (pt − pt−1) + λtβD0 εD

0,t

+ (1 − λt)

βS0ε

S0,t

+

λt

βD1 εD

1,t − βD0 εD

0,t

+ γ D

t Ft

ifMj > 0

(1 − λt)

βS1ε

S1,t − βS

0εS1,t

+ γ S

t Ft

ifMj < 0

λt

βD1 εD

1,t − βD0 εD

0,t

+ γ D

t Ft+

(1 − λt)

βS1ε

S1,t − βS

0εS1,t

+ γ S

t Ft ifMD > 0

and MS < 0

0 ifMD < 0and MS > 0.

Or, equivalently:

p∗

t+1 = pt + λtmDt (pt − pt−1) + (1 − λt)mS

t (pt − pt−1)

+ λt

tn=1

−βD

0

n pt−n + mD

t−n (pt−n − pt−n−1) − pt−n+1

+ (1 − λt)

tn=1

−βS

0

n pt−n + mS

t−n (pt−n − pt−n−1) − pt−n+1

+

λtMD(·)

ifMj > 0 ∀j

(1 − λt)MS(·)

ifMj < 0 ∀j

λtMD(·)

+ (1 − λt)

MS(·)

ifMD > 0and MS < 0

0 ifMD < 0and MS > 0.

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