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Physica A 588 (2022) 126527
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Contents lists available at ScienceDirect
Physica A
journal homepage: www.elsevier.com/locate/physa
A three-state opinion formationmodel for financial markets
Bernardo J. Zubillaga a, André L.M. Vilela a,b, Chao Wang a,c,∗, Kenric P. Nelson a,d,
H. Eugene Stanley a
a Center for Polymer Studies and Department of Physics, Boston University, Boston, MA, 02115, USA
b Física de Materiais, Universidade de Pernambuco, Recife, PE, 50100-010, Brazil
c School of Economics and Management, Beijing University of Technology, Beijing, 100124, China
d Electrical and Computer Engineering, Boston University, Boston, MA, 02215, USA
a r t i c l e i n f o
Article history:
Received 29 May 2019
Received in revised form 8 August 2021
Available online 20 October 2021
Keywords:
Econophysics
Sociophysics
Phase transition
Critical phenomena
Monte Carlo simulation
a b s t r a c t
We propose a three-state microscopic opinion formation model for the purpose of
simulating the dynamics of financial markets. In order to mimic the heterogeneous
composition of the mass of investors in a market, the agent-based model considers two
different types of traders: noise traders and noise contrarians. Agents are represented
as nodes in a network of interactions and they can assume any of three distinct
possible states. The time evolution of the state of an agent is dictated by probabilistic
dynamics that include both local and global influences. A noise trader is subject to local
interactions, tending to assume the majority state of its nearest neighbors, whilst a noise
contrarian is subject to a global interaction with the behavior of the market as a whole,
tending to assume the state of the global minority of the market. The model exhibits
the typical qualitative and quantitative features of real financial time series, including
distributions of returns with heavy tails, volatility clustering and long-time memory for
the absolute values of the returns. The distributions of returns are fitted by means of
coupled Gaussian distributions, quantitatively revealing transitions between leptokurtic,
mesokurtic and platykurtic regimes in terms of a non-linear statistical coupling which
describes the complexity of the system.
© 2021 Elsevier B.V. All rights reserved.
1. Introduction
Over the course of the past few decades, there has been widespread interest in the implementation of tools and
ethods of statistical mechanics for the purpose of investigating the behavior and dynamics of social and economic
ystems [1–23]. Statistical physics studies the complex behavior of macroscopic physical systems in terms of the basic
nteractions between their numerous fundamental components. A hallmark of complexity is the emergence of collective
r cooperative effects which are not reducible to the behavior of any individual component. Financial markets are complex
ystems composed of millions of investors worldwide, partaking of commercial activities, interacting amongst themselves
or the purposes of buying and selling financial assets such as stock, bonds, options and commodities. The convergence
f individual decisions between different financial agents has repercussions on the behavior of macroscopic observables
uch as the time series of prices of said assets.
Therefore, one of the fundamental questions on financial systems is the formulation of simple models of microscopic
ynamics capable of reproducing the essential quantitative and qualitative statistical features of real financial time series.
∗ Corresponding author at: School of Economics and Management, Beijing University of Technology, Beijing, 100124, China.
E-mail address: chaowanghn@vip.163.com (C. Wang).
ttps://doi.org/10.1016/j.physa.2021.126527
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https://doi.org/10.1016/j.physa.2021.126527
B.J. Zubillaga, A.L.M. Vilela, C. Wang et al. Physica A 588 (2022) 126527
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his task suggests an examination of the possible underlying processes and dynamics of opinion formation in a market,
hich ultimately drive the financial decisions of purchase or sale of assets. This process involves not merely financial
onsiderations, but also psychological, emotional and social factors.
The study of microscopic models of opinion formation, aided by computer simulations, may shed light on the essential
nteraction laws between agents in a financial market, which can give rise to the salient behavior of macroscopic financial
bservables as a consequence of those microscopic interactions, such as heavy-tailed return distributions and volatility
lustering [3–10]. Microscopic models of opinion formation do not pretend to mimic human opinion formation dynamics
n an absolutely rigorous fashion. Rather, they attempt to reduce such processes to very basic interaction laws with very
ew parameters in an attempt to understand possible candidates for the fundamental mechanisms at play that give rise
o such macroscopic complexity.
Ising-like systems have been proposed in the past as frameworks for agent-based models for financial systems: One
uch model, proposed by Bornholdt, considers the Ising model with global interactions that couple the agents/spins to
he global magnetization of the system as well as local interactions between each agent and its nearest neighbors [4].
etastable states ensue in this model below its critical temperature as a consequence of the competition between
ocal and global interactions. Sznajd-Weron and Weron proposed a microscopic model of price formation, in which a
ne dimensional chain of spins is endowed with dynamical rules for outward flow of information and a heterogeneous
omposition of agents [7]. Other examples include the models by Krawiecki, Holyst and Helbing, whose Ising-like mean
ield model implements heat bath dynamics in such a way that the interactions between the agents change randomly
ver time [8]. A similar model implementing heat bath dynamics with stochastic and time-varying interactions considers
nly nearest neighbor interactions on non-trivial topologies of Barabási–Albert networks [9]. These simple models are
apable of reproducing principal features of real financial time series such as long-term correlations and scaling.
The majority-vote model is another such widely studied agent-based model in statistical mechanics [11–16]. It was
ormulated in order to study the dynamics of opinion formation in a society. The three-state version of the model assumes
hat individuals are nodes placed in a social network in any one of three possible states/opinions, interacting with their
earest neighbors, exerting influence on them and being influenced by them in return. In this model, with probability
− q, the agent will agree with the majority state of its neighbors or dissent from it with probability q, also known as
he noise parameter. The three-state majority-vote model (MVM3) exhibits a second-order phase transition in a square
attice network at the critical noise qc ≈ 0.118. [12–14]
The dynamics of financial markets ensues not only from rational action on the part of investors but also from emotional
ehaviors, which are a reflection of the rich social psychology of the influences and interactions between agents in a
etwork of financial investors. An example of such a phenomenon is herding, whereby individuals tend to follow the
pinion or behavior of their neighbors [24,25]. Herding behavior has been suggested to play an essential role in human
ehavior and in the animal kingdom, in situations that range from collective behaviors of flocks of birds and schools
f fish to riots, strikes, sporting events and opinion formation (where often coherencein social imitation manifests as
nformational cascades [26]). In the context of financial markets, the field of behavioral finance has identified herding
s a key to the understanding of the collective irrationality of investors [27]. The market herd behavior is carried out
y so-called noise traders. They typically follow the trends of their neighbors and have a propensity to overreact to the
rrival of news when buying or selling.
Other agents in a market seem to follow the trends of the global minority as an investment strategy. Thus they tend to
uy when noise traders drive the prices down and they tend to sell when noise traders drive the prices up. We shall refer
o these agents as noise contrarians [3–6,28–31]. Rational decision-making on their part tends to drive prices towards
he values suggested by the analysis of the fundamentals of an asset, such as expected profits of the company, expected
nterests rates, dividends, future plans of the company, etc.
Based on the MVM3, we propose a generalization of the opinion formation model for financial markets introduced
y Vilela et al. into a three-state model, including a third opinion state [10]. In this work, we shall classify investors in
market as either noise traders or noise contrarians, a simple scheme that models a heterogeneous composition of the
ass of investors. Noise traders follow the local minority via herding behavior, while noise contrarians follow the global
inority, consistent with fundamental analysis. Agents are represented as nodes in a square lattice and they take any one
f three possible states, with their opinion being influenced by the states of their neighbors or the predominant opinion
f the system. In this context, the noise parameter q represents market anxiety, and it induces agents to not follow their
tandard financial strategy. In this way, the social psychology of opinion formation dynamics serves as a basis for our
icroscopic market model.
This paper is organized as follows. In Section 2, we describe the model and dynamics of the agents. In Section 3, we
resent the results of the simulations and numerical calculations with their corresponding discussions, followed by our
oncluding remarks in Section 4.
. The model
Financial agents are represented by nodes in a regular 2D lattice network of size N = L × L with periodic boundary
onditions. Each agent assumes any one of three available states s ∈ {1, 2, 3} at any instant of time. In a financial context,
such states could represent, for example, a desire to purchase a unit of an asset, to sell a unit of an asset or to remain
neutral/inactive.
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B.J. Zubillaga, A.L.M. Vilela, C. Wang et al. Physica A 588 (2022) 126527
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λ
Fig. 1. Illustration of the local influence of four neighboring individuals λ, θ , γ , and ω on the central individual i for a square lattice network of
inancial interactions.
Table 1
Scheme for a subset of possible relative majority and minority states in the neighborhood
of an individual i.
(λ, θ, γ , ω) Majority state(s) Minority state(s)
(1, 1, 1, 1) 1 2 and 3
(1, 1, 1, 2) 1 2 and 3
(1, 1, 1, 3) 1 2 and 3
(1, 1, 2, 3) 1 2 and 3
(1, 1, 2, 2) 1 and 2 3
(1, 1, 3, 3) 1 and 3 2
In order to mimic the heterogeneous composition of agents in a real-world financial market, we consider two types
of agents in this model: noise traders and noise contrarians. Let f represent the fraction of noise contrarian agents in the
model and 1−f is the fraction of noise traders. In this context, we initially discuss the definitions of majority and minority
states in our model and the intrinsic dynamics of each agent and its economic strategy in the following subsections.
2.1. Influence, majority and minority states
Prior to stating the dynamical probabilistic rules on which our three-state microscopic market model is grounded, it is
necessary, for the sake of clarity, to remove a potential source of ambiguity in the language typically used in the literature
concerning MVM3. Indeed, the formal name three-state majority-vote model involves the word majority, where perhaps
the word plurality would be more appropriate and illuminating.
It is widely accepted, in the context of the political science of electoral systems, that a plurality vote, also termed relative
majority, represents a situation where an opinion or proposition enrolls more votes than any other option, but does not
necessarily hold more than half of all votes numbered [32]. In the context of the three-state majority-vote model, the
term majority (minority) is really meant to be understood as relative majority (minority) (instead of the standard simple
majority (minority). Henceforth, we shall use the standard convention adopted by the existing literature on the MVM3,
whereby the term majority replaces the term plurality for the influence and voting scheme.
In the MVM3, an individual manifests one of three different choices s ∈ {1, 2, 3}, following the dominant opinion of
its neighboring individuals with probability (1 − q), and choosing any different opinion with a probability proportional
to q. Defining the dominant opinion of the immediate neighborhood of an individual is essential for the statement of the
dynamics of the three-state majority-vote model. Thus, we define the local influence on an individual i as said agent’s
interaction with lattice sites directly connected to him, quantified as ki. On the other hand, the global influence on agent
i stands for the interaction of agent i with the totality of the individuals in the socio-financial network.
In Fig. 1, we illustrate the local influence over an individual i given by the opinions of the four neighboring individuals
, θ , γ , and ω. Consider the case where {sλ, sθ , sγ , sω} = {1, 1, 2, 3}. In this case, the majority state is the opinion s = 1,
whereas two minority states influence the agent i: one individual in the state s = 2, and one with s = 3. In this situation,
the individual i will tend to follow the majority state 1 with probability (1 − q), and he will adopt any one of the two
minority states available, 2 or 3, with probability q/2. We illustrate a subset of different scenarios of this local influence
in Table 1.
2.2. Noise trader agents
The state of a noise trader is updated according to the following probabilistic prescription, in accordance with the three-
state majority-vote model [12–14]. The behavior of a noise trader is given by its tendency to follow the local majority,
i.e., by its tendency to agree with the state of the majority of its nearest neighbors with probability 1 − q, or to dissent
from it with probability q. Let i represent a noise trader agent. In the event of a single local majority state, agent i will
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dopt it with probability 1− q and either of the two local minority states may be adopted by i with probability q/2 each.
n the event of a tie between two local majority states, agent i may assume any of those two states with probability
1 − q)/2 each and the state of the local minority with probability q.
Let ki,s represent the number of nearest neighbors of noise trader i that find themselves in state s ∈ {1, 2, 3}, with
i,1 + ki,2 + ki,3 = 4 for square lattice networks, as illustrated in Fig. 1. The aforementioned rules for the update of the
tate of noise trader i can be summarized thus:
P(1|ki,1 > ki,2; ki,3) = 1 − q,
P(1|ki,1 = ki,2 > ki,3) = (1 − q)/2,
P(1|ki,1 < ki,2 = ki,3) = q,
P(1|ki,1; ki,2 < ki,3) = q/2.
(1)
he probabilities for the remaining two states (2 and 3) follow easily from the symmetry operations of the C3ν group. It
s worth noticing that the condition P(1|{ki}) + P(2|{ki}) + P(3|{ki}) = 1 holds for any configuration {ki} ≡ {ki,1; ki,2; ki,3}
f the local neighbors, as it should for the update probabilities to be normalized.
For the model to make sense, one must have 1− q > q/2, i.e., the probability for a noise trader to align with the local
ajority hasto be greater than the probability of aligning with any of the two local minorities. This condition implies
hat the model is defined for values of the noise parameter in the range 0 ≤ q ≤ 2/3.
.3. Noise contrarian agents
The behavior of the noise contrarian agents, also known as fundamentalist agents, is defined by their tendency to follow
he global minority, i.e., by their tendency to agree with the state of the minority out of all N traders in the system with
robability 1 − q, or to dissent from it with probability q. This strategy is a non-local interaction between the agent and
he market as a whole. The probabilistic rules of update for the state s of noise contrarian agent are described as follows.
et j be a noise contrarian agent. In the event of a single global minority state, agent j will adopt it with probability 1− q
nd each of the two global majority states will be adopted by j with probability q/2. In the event of a tie between two
lobal minority states, agent j shall assume any of those two states with probability (1 − q)/2 each and the state of the
lobal majority with probability q. Finally, in the event of a three-way tie, agent j will assume any state with probability
/3.
Let Ns represent the total number of traders in the market that find themselves in state s ∈ {1, 2, 3}, where
N1 + N2 + N3 = N . In this case, the rules for the update of the state of noise contrarian agent j can be summarized
as follows:
P(1|N1 < N2;N3) = 1 − q,
P(1|N1 = N2 < N3) = (1 − q)/2,
P(1|N1 > N2 = N3) = q,
P(1|N1;N2 > N3) = q/2,
P(1|N1 = N2 = N3) = 1/3.
(2)
The probabilities for the remaining two states (2 and 3) follow easily from the symmetry operations of the C3ν group.
It is worth noticing that the condition P(1|{N}) + P(2|{N}) + P(3|{N}) = 1 also holds for any global state configuration
{N} ≡ {N1,N2,N3}.
2.4. The order parameter
For the purposes of this model, we shall adopt the magnitude of the magnetization, defined in analogy to the three-state
Potts model, as an order parameter for this system. The magnetization is to be taken as a vector with components: Ms for
s ∈ {1, 2, 3}. Its magnitude M is thus given by M =
√
M21 + M
2
2 + M
2
3 and its components are calculated according to:
Ms =
√
3
2
[
Ns
N
−
1
3
]
. (3)
With this definition, it can be shown that the components of the magnetization are not independent of one another. In
fact, it holds that M1 + M2 + M3 = 0.
3. Numerical results and discussion
We now present the numerical results of Monte Carlo simulations of the stochastic dynamics introduced in the previous
section on a square lattice of size N = 100 × 100 with periodic boundary conditions. In each simulation, each node is
randomly assigned to be a noise contrarian or a noise trader with probability f or 1 − f , respectively. Time is measured
in units of Monte Carlo Steps (MCS). In one MCS, a total of N attempts are made to change the states of the agents. In
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Fig. 2. Snapshots of Monte Carlo simulations of the model on a square network of agents with periodic boundary conditions. Each dot represents an
agent and the three available states are depicted in black, red and yellow. The noise parameter for these simulations is q = 0.12 and the fractions of
oise contrarians are equal to f = 0.0 for (a) and f = 0.5 for (b), (c) and (d) where we have black, red and yellow as the majority state, respectively.
Fig. 3. Time evolution of the fraction of agents in each of the three states ns with s = 1 (black), 2 (red) and 3 (yellow) for q = 0.12. The magnitude
f the order parameter M of the system is shown in blue. The fraction of noise contrarians are: f = 0.00 and f = 0.50 for (a) and (b), respectively.
ach attempt, a node is randomly selected in the network and its state is updated according to Eqs. (1), if the node is a
oise trader, or Eqs. (2), if the selected node is a noise contrarian.
The initial state of the system is random, assigning to each agent any one of the three available states with equal
robability. In each simulation, a total of 20000 MCS are performed, of which the first 10000 MCS are discarded as
hermalization time of the system. A total of 100 Monte Carlo simulations were performed for every set of parameters
q, f ) considered. Therefore, effectively, a total of 106 MCS were recorded for each pair of parameters (q, f ) from all runs. In
he absence of noise contrarians (f = 0), it is clear that the dynamics reduces to the MVM3, which, on a 2D square lattice,
xhibits a critical point at qc ≈ 0.118 in the thermodynamic limit (N → ∞). For q < qc , the system exhibits an ordered
hase with the presence of large clusters of agents that share the same state, which results in spontaneous magnetization.
or q > qc , thermal fluctuations fully destroy the order and the order parameter, M , vanishes in the thermodynamic limit.
In Fig. 2 we present a visual representation of individual snapshots of the states of the system for two concentration
f noise contrarians f = 0.0 and 0.5 to get a sense of the microscopic behavior of the system. In the absence of noise
ontrarians [Fig. 2(a)], the system clearly exhibits clusters of ordered opinions near criticality [q ∼ qc(f = 0) ∼ 0.118].
he introduction of noise contrarians to the system induces the destruction of clusters of local order [Fig. 2(b), 2(c) and
(d)]. For high values of f the global interactions of noise contrarians may induce an order of their own as they collectively
eek the state of the global minority en masse. This enables oscillatory behaviors in the populations of the states of the
ystem, with cyclical regimes driven by a tendency of noise contrarians to flee from a global majority state into a global
inority state. In Fig. 2(b) we see a metastable state where the state s = 1 (black) is the predominant state of the system.
y running it further we check that the majority state can also oscillate into majority state s = 2 (red) and s = 3 (yellow).
By calculating the fraction of agents in each state ns(t) = Ns(t)/N and the magnitude of the order parameter M(t) as a
unction of time, we find the typical time evolution of the relative populations shown in Fig. 3 for q = 0.12. In Fig. 3(a),
e show the typical critical behavior of the order parameter and its components. Given the long-range correlations
stablished in the system in the vicinity of the critical point, large fluctuations of the order parameter and its respective
omponents are observed. The oscillatory pattern of the majority can be verified in the behavior of ns(t) shown in Fig. 3(b)
or f = 0.50. Recalling that the fluctuations of the order parameter diverge at the critical point in the thermodynamic limit,
he destruction of local clusters of ordered opinion by increasing the number of noise contrarians drives the system away
rom criticality, thereby reducing the magnitude of the relative time fluctuations of the magnetization, which, in the
ontext of a financial interpretation of this system, may suggest an increase of market stability.
At an instant of time t , we will relate the time variations of the magnitude of the magnetization M(t) of this model
o the logarithmic returns r(t) of a financial asset (such as a stock or an index fund, for example), understanding each of
the three possible opinions or states s ∈ {1, 2, 3} of the agents as intentions to buy a unit of that asset, to sell a unit of
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Fig. 4. Time series of logarithmic returns for a noise parameter q = 0.16 and concentrations of noise contrarians given by f = 0.00 and f = 0.50.
In the absence of noise contrarians, the time series clearly exhibits volatility clustering.
that asset or to do neither or remain inactive, for example. We define the logarithmic return at time t in terms of the
magnetization
r(t) ≡ ln[M(t)] − ln[M(t − 1)], (4)
wherebyM(t), being a positive-definite quantity, can be interpreted as a measure of the price of the asset. We shall also
define the volatility of the asset as a measure of the dispersion of the relative variations of the prices locally in time:
v(t) ≡ |r(t)|, (5)
i.e., we define the volatility to be the absolute value of the logarithmic return. We now study the statistical properties of
the time series of returns generated by the model.
In Fig. 4, we present an example of time series of returns produced by this model for q = 0.16 and for two different
oncentrations of noise contrarians: f = 0.00 and f = 0.50. As shown in the figure, the presence of noise contrarians
serves to noticeably decrease the size of the fluctuations in the returns.
Qualitatively, it is also visually clear that the time series in the absence of noise contrarians exhibits volatility clustering,
well known feature of real time series of returns first identified by Mandelbrot, whereby large (small) fluctuations tend
o follow by large (small) fluctuations. This suggests that a measure of nonlinear correlations, i.e., correlations of volatility,
hould exhibit this effect of memory and structure of the time series [1,3,33]. Volatility clusters are an essential feature
f real financial time series, particularly prominent during times of bubbles and financial crashes, often a reflection of the
ollective mechanism of herding behavior driving the market, which has been shown to contradict the basic assumptions
f perfectly rational behavior of investors in financial systems [27].
In order to quantify the effect of volatility clustering on the long-term memory of the volatility, we define its
utocorrelation function thus:
ρ(τ ) ≡
∑T−τ
t=1 [|r(t)| − ⟨|r|⟩][|r(t + τ )| − ⟨|r|⟩]∑T
t=1[|r(t)| − ⟨|r|⟩]2
. (6)
Fig. 5 exhibits the behavior of the autocorrelation function of the absolute values of the log-returns for different
densities of noise contrarians, a value of q = 0.16, a simulation of time T = 15000 MCS. Also, for comparison purposes,
we show the autocorrelation function of daily log-returns of the closing values of the S&P500 from to Dec 12, 1958 to
June 25, 2018, for a total of 15000 days. As is visually clear from the plots, there is qualitative agreement between real
data and the simulations, the curve of the S&P500 lying somewhere between that of f = 0.00 and f = 0.10.
As in the two-state model proposed by Vilela et al. [10] and the Bornholdt model [4,6], we observe that the time series
of the volatility exhibit long-range memory, in agreement with the main features of non-stationary, real time series. In
effect, as has been observed empirically in real financial data since Mandelbrot’s initial observations, the autocorrelation
of the volatility is known to decrease typically as a power law, suggesting a lack of a characteristic time scale for the time
series of absolute values of returns.
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Fig. 5. Autocorrelation function of the volatility for a noise parameter q = 0.16 and various concentrations of noise contrarians: f = 0.00, 0.10,
.20, 0.30, 0.40, 0.50. Also shown is the autocorrelation function of the daily volatility of the closing values of the S&P500 from to Dec 12, 1958 to
une 25, 2018, for a total of 15 000 days. The dashed lines correspond to fits to power laws.
Fig. 6. Autocorrelation function of the logarithmic returns for a noise parameter q = 0.16 and two concentrations of noise contrarians: f = 0.00
and 0.50. Also shown is the autocorrelation function of the daily returns of the closing values of the S&P500 from to Dec 12, 1958 to March 22,
1959 for a total of 100 days.
It is obvious from Fig. 5 that a deviation from this scale invariant regime eventually sets in after a time lag of about
τ ≈ 103, as evidenced by the stretched exponential decay that follows. This decay is a reflection of the finite sizes of the
data samples presented for the time series of both the simulations and the S&P500.
As a manner of comparison, we present the fits (in dashed lines) of the autocorrelation function for f = 0.00 and the
empirical data of the S&P500 to power laws, i.e., ρ(τ ) ∼ τ−η , over the same time periods. The exponents of the fits are
η = −0.0273±0.0005 for the simulation with f = 0.00 and η = −0.0664±0.0005 for the daily volatility of the S&P500,
which shows agreement in the order of the rate of decay of correlations between simulations and real data.
A calculation of the autocorrelation function for the time series of returns produced by the model suggests that the
returns are essentially uncorrelated, a feature that is consistent with the efficient market hypothesis [3]. This is shown in
Fig. 6, where the autocorrelation function of the logarithmic returns are plotted for simulations of the dynamics with two
different concentrations of noise contrarians, f = 0.00 and f = 0.50 and the autocorrelation function of the daily returns
of the S&P500 is also shown for comparison. The agreement between real financial data and the simulation in the absence
of noise contrarians is very clear. In the case of f = 0.50, it is clear that, although uncorrelated in the long run, in the short
term there is a decaying antipersistent oscillation, the origin of which can be clearly tracked to the cyclical behavior of
the system caused by a large fraction of noise contrarians constantly seeking to occupy the instantaneous global minority
state.
We now proceed to study the distribution of returns of the time series generated by our model. As noted before, Fig. 4
clearly illustrates the decrease of the magnitude of the fluctuations of the returns as the number of noise contrarians is
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Fig. 7. Distributions of logarithmic returns for a noise parameter q = 0.16 and various concentrations of noise contrarians: f = 0.00, 0.10,
.20, 0.30, 0.40, 0.50.
Fig. 8. Distributions of volatility for a noise parameter q = 0.16 and various concentrations of noise contrarians: f = 0.00, 0.10, 0.20, 0.30, 0.40, 0.50.
its of the curves to symmetric coupled exponential distributions defined in Eq. (7) are presented in dashed lines.
ncreased. This effect is captured by the tails of the return distributions, as shown in Fig. 7. It is immediately apparent
hat an increase in the fraction of noise contrarians produces a reduction in the heaviness of the tails, greater fluctuations
n the returns becoming less likely. In effect, for high values of f , the distribution loses its fat tails and becomes Gaussian,
n accordance with the results of the overly simplistic and widespread model of financial markets given by Geometric
rownian Motion, used typically in Black–Scholes calculations of derivative pricing [3].
In order to quantify the transition of the return distributions from a heavy-tailed leptokurtic regime into a Gaussian
esokurtic regime and, possibly, into a compact-support platykurtic regime, we consider distributions from the symmetric
oupled exponential family [34,35]. Such family of distributions is defined by:
Pσ ,κ,α(r) ≡
[
Z(σ , κ, α)
(
1 + κ
⏐⏐⏐ r
σ
⏐⏐⏐α) 1+κακ
+
]−1
, (7)
where σ , κ and α are the parameters of the function and (a)+ ≡ max(0, a). We shall refer to the shape parameter κ as the
nonlinear statistical coupling and to σ as the scale parameter in an interpretation of nonextensive statistical mechanics [36]
which has been used as a model of financial markets [37,38] and other complex systems.
This family of distributions is characterized by the fact that it is capable of capturing the aforementioned transition
in a very natural way. Indeed, when κ > 0, the function exhibits a heavy-tail decay. When κ = 0, the function is a
generalized Gaussian distribution. When −1 < κ < 0, the function is a distribution with compact support. Moreover,
if α = 2, the function is a coupled Gaussian distribution: with κ = 0, the Gaussian distribution; and withκ > 0, the
8
B.J. Zubillaga, A.L.M. Vilela, C. Wang et al. Physica A 588 (2022) 126527
f
G
S
c
r
a
b
f
n
e
f
a
i
o
o
e
t
s
t
e
i
i
d
e
o
c
r
A
s
n
m
t
a
Fig. 9. Coupling and scale parameters of the return distributions for a noise parameter q = 0.16 and various concentrations of noise contrarians:
= 0.00, 0.02, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50. A dashed horizontal line marks the value of the nonlinear coupling of a
aussian distribution κ = 0.
tudent’s t distribution with the degree of freedom being the reciprocal of κ , ν = 1/κ . Therefore, the nonlinear statistical
oupling is capable of providing a numerical measure of the transition from a leptokurtic to a platykurtic state.
We fix α = 2 and use the coupled Gaussian distribution to fit the histograms of the absolute values of the logarithmic
eturns, which measure the volatility of the time series, and focus on the behavior of the nonlinear coupling parameter as
function of the number of noise contrarians. With this prescription, the heavy tails of the coupled Gaussian distribution
ehave as Pσ ,κ,2(r) ∼ |r|−(1+κ)/κ ∼ |r|−(1+ν). Fig. 8 shows the distribution of the absolute values of the logarithmic returns
or different concentrations of noise contrarians and its fit to the function defined by Eq. (7).
In Fig. 9, we present the values of the nonlinear coupling parameter κ and the scale σ of these fits. Notice that the
onlinear coupling parameter κ of the coupled Gaussian distribution monotonically decreases with f , providing numerical
vidence for the progressive loss of the heavy tails as the distribution approaches a Gaussian for high values of f . In
act, for values of f ≥ 0.4, κ ≈ 0, indicating that the distribution is approximately Gaussian. Further, κ , although
pproximately zero, attains negative values for a higher fraction of noise contrarians, suggesting that the dynamics may
ndeed produce distributions of compact support in the antipersistent oscillatory regime present with very high fractions
f noise contrarians. Therefore, the nonlinear statistical coupling is capable of capturing the complexity of the behavior
f the system, numerically measuring the change from a heavy tailed regime to a non-heavy tailed regime.
The scale parameter σ , which is a generalization of the standard deviation within the context of the coupled
xponential family, decreases for values of f > 0.05. This is consistent with the increase in the fraction of noise contrarian
raders reducing the variation in the log-returns. So the noise contrarian traders decrease variation by reducing both the
cale and the shape or coupling of the log-return distribution. However, it is clear that, for f < 0.05, the broadness of
he distribution has a remarkably different behavior. This can be qualitatively appreciated in Figs. 7 and 8, where a visual
xamination of the f = 0.00 curve relative to the rest indeed suggests a heavier tail exponent but a smaller scale, which
s approximately the knee of the log–log plot of the distributions. This suggests that near f = 0.05, where σ peaks, there
s a change in the behavior of the scale of the distribution, even though the statistical coupling is strictly monotonically
ecreasing with f .
Further confirmation of the transition from a leptokurtic to a mesokurtic regime is provided by the behavior of the
xcess kurtosis, γ2 ≡ ⟨r4⟩/⟨r2⟩2 − 3, of the return distribution itself, as shown in Fig. 10, as a measure of the heaviness
f its tail. Recall that the excess kurtosis of a Gaussian distribution is γ2 = 0. It is clear from the figure that an increasing
oncentration of noise contrarians tends to remove the heavy tails of the distribution, driving the transition into a Gaussian
egime for high values of f .
Fig. 11 shows the distribution of logarithmic returns for different values of q in the absence of noise contrarians.
n interesting feature of the MVM3 dynamics is that, even in the absence of noise contrarians (f = 0.00), the relative
izes in the fluctuations of the order parameter as representations of the returns of an asset can be controlled by the
oise parameter q alone. As is to be expected, in an ordered regime with q ≪ qc , the fluctuations in the spontaneous
agnetization of the system are very small, therefore leading to a state of very small returns and volatility. Proximity to
he critical point, where fluctuations in the magnetization diverge in the thermodynamic limit, suggests a transition from
low variance regime to a more turbulent state with large fluctuations, as can be seen in the figure.
9
B.J. Zubillaga, A.L.M. Vilela, C. Wang et al. Physica A 588 (2022) 126527
e
h
t
t
a
d
w
t
a
c
t
c
t
t
p
e
f
q
i
Fig. 10. Excess kurtosis of the return distributions for a noise parameter q = 0.16 and various concentrations of noise contrarians. The value of the
xcess kurtosis of a Gaussian distribution is γ2 = 0.
Fig. 11. Distributions of logarithmic returns in the absence of noise contrarians f = 0.00 for q = 0.10, 0.12, 0.14, 0.16, 0.18, 0.20.
A more comprehensive picture of the behavior of the model emerges when looking at Fig. 12, in which we present the
eat maps of the fitting parameters κ and σ for different pairs of values (q, f ) of the model’s parameters. A first feature
hat one can extract from the heat map of the nonlinear statistical coupling is the monotonic decrease of κ , and therefore
he transition from a leptokurtic regime to a platykurtic regime, as one increases the fraction of noise contrarians for
ny fixed value of the noise parameter, q. Indeed, κ becomes negative for high enough values of f , suggesting that the
istributions not only lose their fat tails as they approach a Gaussian regime, they also eventually transition into a state
here they exhibit compact supports (κ < 0), as discussed previously for Fig. 9. It is clear from the heat map that the
ransition, for fixed q, from heavy tails to a complete loss of tails with increasing fraction of noise contrarians occurs faster
long the f axis for smaller values of q. These observations suggest that the nonlinear statistical coupling is capable of
apturing the degree of complexity of the system, as reflected by behavior of the tails of the distributions.
The heat map of the scale parameter is also consistent with our discussion of Fig. 9, showing, for fixed values of q,
hat σ increases rapidly for small values of f , reaches a maximum, and consequently decreases for large fractions of noise
ontrarians. So even though for small values of f the scale parameter (as a measure of the distribution’s width) is small,
he distributions still exhibit heavy tails, as evidenced by the nonlinear coupling κ . Notice that the scale parameters seem
o consistently peak around the same values of f for fixed values of q. However, the peak of this transition in the scale
arameter is wider for small values of q than it is for high values of q.
In the absence of noise contrarians f = 0.00, the scale parameter σ increases monotonically with q. This result is
vident in Fig. 11, where around criticality, q ≈ 0.12, the spread of the distribution grows noticeably with increasing q. For
= 0.01, this situation is slightly reversed. However, as can be seen from the heat map of Fig. 12, for values of f ≥ 0.02 and
≥ 0.02, the scale parameter resumes its monotonic increase with q for fixed values of f , suggesting a change of regime
n the behavior of the scale parameter around f ≈ 0.02. Indeed, for any fixed macroscopically relevant concentration of
10
B.J. Zubillaga, A.L.M. Vilela, C. Wang et al. Physica A 588 (2022) 126527
o
n
o
a
4
a
i
s
r
o
d
g
n
Fig. 12. Heat map of (a) the nonlinear statistical coupling κ and (b) the scale parameter σ as a function of the noise parameter q and the percentage
f noise contrarians f .
oise contrarians (f ≳ 0.02), the scale parameter increases with q. Overall, the heat map provides evidence for the role
f the noise parameter as a measure of social temperature capable of turning the dynamics of thesystem more turbulent
s it increases.
. Conclusion
In this work, we propose a three-state, agent-based, microscopic market model with stochastic dynamics. It features
heterogeneous population of traders comprised of two categories: noise traders and noise contrarians. Noise traders
nteract locally with their nearest neighbors, tending to agree with the state of the local majority. Noise contrarians are
ubject to global interactions with the market as a whole and they tend to follow the state of the global minority. By
elating changes in the order parameter of this system to price fluctuations in the market, the simulation of the dynamics
f the model is capable of reproducing the main qualitative and quantitative features of real financial time series, such
istributions of returns with long tails, volatility with long-term memory and volatility clustering.
The logarithmic returns of the simulation fit a coupled Gaussian distribution, which is parametrized by the scale or
eneralized standard deviation and the shape or nonlinear statistical coupling. For macroscopically relevant fractions of
oise contrarians (f > 0.02), an increase in the noise contrarians decreases both the scale and the shape of the distribution.
11
B.J. Zubillaga, A.L.M. Vilela, C. Wang et al. Physica A 588 (2022) 126527
I
s
t
c
n this same region, increasing the probability of dissent q also increases the scale and shape of the returns. For a very
mall fraction of noise contrarians (f < 0.02), the nonlinear statistical coupling κ attains a maximum in the vicinity of
he critical noise parameter (q ≈ 0.12), whilst the behavior of the scale parameter becomes more complex, evidencing a
hange of regime of the log-return distributions around f ≈ 0.02.
The model’s simplicity is capable of shedding light into the potential mechanisms at play behind the social psychology
of decision making and opinion formation in a financial market.
CRediT authorship contribution statement
Bernardo J. Zubillaga: Conceptualization, Methodology, Software, Validation, Investigation, Formal analysis, Writing.
André L.M. Vilela: Conceptualization, Methodology, Software, Validation, Investigation, Formal analysis, Writing. Chao
Wang: Conceptualization, Methodology, Software, Validation, Investigation, Formal analysis, Writing. Kenric P. Nelson:
Investigation, Formal analysis, Writing. H. Eugene Stanley: Investigation, Formal analysis, Writing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgments
The authors acknowledge financial support from Brazilian and Chinese institutions and funding agents UPE, Brazil
(PFA2018, PIAEXT2018) FACEPE, Brazil (APQ-0565-1.05/14, APQ-0707-1.05/14), CAPES, Brazil, CNPq, Brazil, National
Natural Science Foundation of China (61603011, 72071006), the International Postdoctoral Exchange Fellowship Program,
China (20170016) and Beijing Social Science Foundation of China (16JDGLC005). The Boston University Center for
Polymer Studies is supported by NSF, USA Grants PHY-1505000, CMMI-1125290, and CHE-1213217, by DTRA, USA Grant
HDTRA1-14-1-0017, and by DOE, USA Contract DE-AC07-05Id14517.
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	A three-state opinion formation model for financial markets
	Introduction
	The model
	 Influence, majority and minority states
	Noise trader agents
	Noise contrarian agents
	The order parameter
	Numerical results and discussion
	Conclusion
	CRediT authorship contribution statement
	Declaration of competing interest
	Acknowledgments
	References