BOLTZMANN-TYPE MODELS FOR PRICE FORMATION IN THE PRESENCE OF BEHAVIORAL ASPECTS

. We introduce and discuss a new kinetic system for a ﬁnancial market composed by agents that may belong to two diﬀerent trader populations, whose behavior determines the price dynamic of a certain stock. Our meso-scopic description is based on the microscopic Lux–Marchesi model [16, 17], and share analogies with the recent kinetic model by Maldarella and Pareschi [18], from which it diﬀers in various points. In particular, it takes into account price acceleration, as well as a microscopic binary interaction for the exchange between the two populations of agents. Various numerical simulations show that the model can describe realistic situations, like regimes of boom and crashes, as well as the invariance of the large-time behavior with respect to the number of agents of the market.


1.
Introduction. Agent-based models represent a broad class of mathematical models which have been recently considered to describe various phenomena of economic dynamics. It is nowadays clear that this description can reproduce various features of financial markets, like volatility clustering [14,15,16,17,18,19,26] and fat tails of returns [2,3,4,5,6,7,9]. This relatively new research field borrows several methods and tools from classical statistical mechanics, where the emerging complex behavior arises from relatively simple rules as a consequence of binary interactions among a large number of agents [21,22].
Starting from the microscopic dynamics, kinetic models for trading can be derived with the tools of classical kinetic theory of fluids [7,10,11,20,24], where kinetic econophysics has been treated in the framework of Boltzmann-like equation for Maxwell-type molecules. In contrast with microscopic dynamics, where the large-time behavior of the system can be often studied only empirically through computer simulations, in many cases kinetic models based on integro-differential and/or partial differential equations allow to find analytically general information on the model and its asymptotic behavior.
In most of these market models, the population of agents shares the same rules with respect to trading. Only very recently, at the cost of increased complexity of classical kinetic theory [22], the whole evolution process is here based on microscopic binary interactions between agents. For this reason, the model will result as a useful prototype of collisional models in which there is more than one class of agents involved.
The choice of a microscopic transition law allows to understand (at a microscopic level) the importance of the opinion variable in the whole process of the price formation. Indeed, it is shown that a weak role of the opinion variable in the exchange between classes, drives the price towards upper and lower bounds, which can be explicitly given in terms of the initial state of the system. In other words, a weak role of the opinion variable is such that both boom and crashes in the market are not possible.
The paper is organized as follows. The model will be introduced in Section 2. Here, the evolution equation for the price of the stock, as well as the evolution equations for the number densities of the two populations of agents are written in the form of a kinetic system of correlated equations. In particular, we will outline the main differences among the present model and the previous ones. In Section 3 we will discuss the basic role of the opinion variable, and its consequences on fluctuations of the price. Finally, Section 4 collects various numerical simulations, which describe how the choice of the relevant parameters of the system can lead to boom, crashes and fluctuations of the price.
2. The model. The aim of the pioneering Lux-Marchesi model [16,17] was to show that the scaling laws that are observed in financial markets can arise from mutual interaction between agents. In particular, Lux and Marchesi stressed the clear difference between the statistical properties of the model input, i.e. the normal noise that makes the stock fundamental price evolve in time, and the output, that is the price dynamics produced by the operations between agents.
The agents are divided in two categories: the fundamentalists, who believe in the existence of a reference price P F for the traded stocks (and so, they sell stocks if the price P (t) > P F and they buy if P (t) < P F ), and the chartists, that are noise traders whose behavior is dictated by herding and historical prices. While the total number of agents N remains fixed in time, the number of fundamentalists and chartists, n F and n C , are allowed to vary. This is done by assuming that at each time step each agent can change its own category with a given transition probability. In this way, an internal dynamics within the two classes of agents is established.
The effect of the two classes of agents on the price is very different. While fundamentalists have a stabilizing effect on the market, as their operations drive the price towards the reference one, chartists have a destabilizing effect and can create bubbles and crashes. Furthermore, chartists agents are divided in two subcategories, optimists, who believe that the price will rise and hence always buy stocks, and pessimists, who believe that the price will decrease and so, on the contrary, always sell stocks.
The main features of Lux-Marchesi model have been merged in [18] into the framework of kinetic theory, with the goal to introduce a kinetic description both for the behavior of the microscopic agents and for the price, and then to exploit the tools furnished by kinetic theory to get more insight about the way the microscopic dynamic of each trading agent can influence the evolution of the price. The model introduced in [18] is still able to describe several phenomena like the presence of booms, crashes, and cyclic oscillations of the market price. The equilibrium behavior has been studied in a suitable asymptotic regime which originates from the Boltzmann equations a system of Fokker--Planck equations for the chartist's opinion dynamics and the price formation. However, in reason of the intrinsic complexity of the interactions, various interesting aspects of the kinetic system remain not fully studied and understood.
In what follows, still resorting to the classical methods of kinetic theory, we introduce a model which is reminiscent of both Lux-Marchesi [16,17] and Maldarella-Pareschi [18] ideas. At difference with the model introduced in [18], the resulting exchange mechanism between classes is here introduced at a microscopic level, by means of a binary interaction between agents. Moreover, we will resort to the analysis in [1,8] to model the behavior of the chartists.
Assuming continuous parameters for the number of fundamentalists and chartists, let f (w, t) denote the probability to have a number w ∈ IR + of fundamentalist agents at time t > 0, and let c(v, x, t), with v ∈ IR + , denote the distribution function of the number of chartist agents. Note that, while the density of fundamentalists depends only on the time, the density of chartists is also depending on the opinion variable x ∈ [−1, 1] [25]. This opinion variable describes the individual propensity of agents to sell or buy. Conventionally, we will assume that the x > 0 corresponds to the buyer-state, and x < 0 to the seller-state. When x = 0 an agent is considered in a neutral-state.
According to the classical kinetic theory, the knowledge of the density functions f (w, t) and c(x, v, t) allows to compute all averaged quantities of interest. Among them, the mass densities of fundamentalists and chartists The mean number of fundamentalists agents (respectively chartists) is obtained by taking the mean with respect to the number variables and By taking the mean with respect to the opinion variable, we obtain the average investment propensity X(t) of chartists, defined by Other mean quantities will result of interest in what follows. By taking the means with respect to the opinion variable (respectively the number variable) only, we obtain the marginal distribution densities of chartists and Likewise, by taking the mean with respect to the number variable only, we obtain the local mean number of chartists The total mean number n(t) of agents at time t ≥ 0 results by summing up n C (t) + n F (t). Then ρ F (t) = n F (t)/n(t) and ρ C (t) = n C (t)/n(t) gives the percentages of chartists and fundamentalists participating to the market. We will speak of conservative in the mean population a market in which n(t) = n 0 is constant. Together with the densities of agents, and given a certain stock, the principal object of investigation is the evolution of its price. To this aim, let us introduce the distribution function h(p, t) of prices of this stock, which is such that h(p, t)dp denotes the probability that the price belongs to the interval (p, p + dp). Starting from h(p, t) one can then define the effective market price P (t). The effective market price is defined as the mean price of the stock In order to write the kinetic system of equations for the distribution functions f (w, t), c(x, w, t) and h(p, t) we have to introduce the rules of variations of the microscopic variables x, v, w, p. Following [17], we will assume that the microscopic variation of the price p is related both to the percentages of fundamentalists and chartists and to the mean investment propensity (2.5) by the formula In (2.10) the parameter β characterizes the price speed evaluation, while γ denotes the reaction strength of fundamentalists to deviations from the fundamental value P F of the price. Moreover η is a random variable with zero mean and variance σ 2 . Thus, the term ηp(t) represents a risky component, which induces random deviations of the price. In (2.10) it is supposed that chartist agents buy or sell the same number of units, expressed by the parameter µ C . According to the kinetic theory of gases [22], the evolution in time of the density price, where the variation of p is given by (2.10), obeys to the linear in p kinetic equation (in weak form) In (2.11) ϕ(p) is a smooth function of p, and · denotes mathematical expectation.
Choosing ϕ(p) = p one obtains that the expected value for the price of the stock satisfies the differential equation [17] (2.12) The time variation of the price is linked to the percentages n C (t) and n F (t) of chartists and fundamentalists, as well as to the average investment propensity X(t).
Thus, the kinetic equation (2.11) needs to be coupled with the evolution equations of f (w, t) and c(x, v, t). As before, these evolution equations are constructed from the microscopic dynamics of the number variables v and w, and of the opinion variable x.
In what follows we assume that these microscopic dynamics could depend on the expected value of the price P (t) towards its first and second derivatives. Concerning the dynamics of the opinion variable of chartists, following [18,25] we will assume the following law of variation (2.13) In (2.13), the post-interaction opinion x * is modified essentially by three different contributions. The first one is related to the distance of the individual opinion from the average investment propensity X(t). Its intensity is measured by a parameter α 1 ∈ [0, 1], and by the function H(x), which characterizes the herding behavior. The simplest choice is to fix H(x) = 1, which means that all chartists are influenced in the same way by the average investment propensity. A more realistic choice [18] is given by (2.14) Note that, according to [25], in this case extremal opinions for which |x| = ±1 are not influenced by the average investment propensity. The second contribution measures the variation of the individual opinion consequent to the market dynamical state. The size of this variation is expressed by the second parameter α 2 ∈ [0, 1]. We will here assume that the function Φ(Ṗ ,P ) is defined by Within this simple assumption, chartists agents are pushed to buy in presence of a local growing convex price P (t), or to sale in presence of a local decreasing concave price. In the remaining cases, there is no variation of the opinion variable consequent to the variation of the price. Last, the third contribution is related to a possible change of opinion due to external effects. This is represented by the random variable θ, assumed with zero mean and variance τ 2 . The function D(x), defined by describes the diffusive behavior [25]. We remark that, as in the case of the herding behavior, the choice (2.16) is such that the opinions close to be extremal are less influenced by external events than the neutral ones.
Other choices of functions are of course possible. Note however that in order to preserve the bounds for x it is essential that D(x) vanishes in x = ±1.
The chartist's interaction (2.13) differs in a substantial way from the interaction in [18], where the mechanism of opinion formation for chartists was given as a function of the derivative of the logarithm of the price, and it is consistent with the remark of [1,8], in which even chartists agents use their knowledge of a sufficiently long time history of the price to decide what to do. Clearly, our choice in (2.15) can be modified in many ways, including the possibility to generalize the function considered in [18] to include the acceleration of the logarithm of the price.
We are now in a position to define the microscopic law of variation of the number of chartists and fundamentalists. For any given pair (v, w) of numbers of the two classes of agents, we assume that the new pair (v * , w * ) will depend both on the transition probabilities p CF and p F C to change class (from chartists to fundamentalists and viceversa), and on the local opinion of the chartists agents. The starting point of our rule of transition is based on the assumption that a chartist is close to become a fundamentalist when he has not a better strategy than a fundamentalist, and this eventually happens when his opinion is approximately neutral. Thus, changes from chartists to fundamentalists will be allowed only when their opinion is not too far from the neutral one. This neutrality will be expressed by a positive parameter λ ≤ 1. Likewise, we will assume that the fundamentalists that change class will enter the class of chartists with opinion close to the neutral one. We will denote the measure of this interval by the parameter δ. Consequently, the general microscopic law of variation of the pair (v, w) reads namely the conservation (in the mean) of the total number of agents. Among the possible choices of the transition rates between the classes of agents, we will assume in the following that these rates could depend of the acceleration of the price. A possible law for this is the following where 0 < χ ≤ 1 is a statistical weight. Within this assumption, to mimic the resistance to move towards a risk strategy in a turbulent market, the probability of changing from fundamentalist to chartist decreases when the price acceleration increases.
Finally the time evolution of the distributions c(x, v, t) of chartist agents and f (w, t) of fundamentalists is given by the kinetic system (in weak form) We remark that, choosing the test functions ϕ = 1 in (2.20) and (2.21) shows that the densities m C (t) and m F (t) of chartists and fundamentalists remain probability densities at any subsequent time. Analogously, choosing ϕ(x, v) = v in (2.20) and ϕ(w) = w in (2.21) we obtain Hence, thanks to (2.18), we conclude that the total mean number of agents is conserved in time, so that n C (t) + n F (t) = n 0 . Furthermore, let us set ϕ = x in equation (2.20). This allows to compute the evolution in time of the average investment propensity X(t). In the case in which in (2.13) the herding function is assumed constant, H = 1, the evolution equation for X(t) simplifies and takes the form 3. The importance of the opinion of chartists. The kinetic system described in Section 2 aims to describe the time-evolution of the price of a stock in a system of agents which play according to different rules. In particular, the chartists strategy is deeply linked to the formation of a personal opinion which would help to decide whether or not it is opportune to buy the stock itself. The basic hypothesis behind the construction of our model, with respect to the previous ones existing in the literature [16,18] is the combined rules of the opinion formation of chartists, and the new idea of using acceleration of the price to measure both the change of opinion and the change of class (from chartists to fundamentalists and viceversa).
Despite our simple choices of interactions, which are described by (2.13) and (2.17), to obtain an analytic precise description of the evolution of the densities h(p, t), f (w, t) and c(x, v, t) seems to be prohibitive.
Nevertheless, some insight into the time behavior of the price induced by the kinetic system can be done. The main result here will be to clarify the importance of the opinion of chartists in order to drive the system towards extremal situations, which are typically known as booms (the mean price increases unlimited) and crashes (the mean price collapses towards zero value).
To start with, let us assume that the constants λ and δ which are responsible of the passage from chartists to fundamentalists and vice-versa are taken equal to unity. In this case, independently of their personal opinion, a percentage of chartists is moved to the other class. In this case, equations (2.22) simplify and give Note that, by definition (2.19), p CF = χ − p F C , so that equation (3.1) can be rewritten as Therefore, when the right-hand side of equation (3.2) is non-negative, the mean density n F (t) starts to increase. On the other hand Owing to the conservation law n F (t) + n C (t) = n 0 gives the lower bound or, what is the same Therefore, in case λ = δ = 1, the percentage of fundamentalists is always greater than a fixed positive valueρ. Using this information on equation (2.12), and considering that the mean opinion X(t) ≥ −1, we obtain that at any time t > 0, P (t) can not be smaller than Consequently, the crash of the price is prevented. Likewise, if the initial percentage of fundamentalists is bigger or equal than 2/5, so that by the previous argument it remains bounded below by 2/5, and γ > 3t C /2, so that Consequently, using that X(t) ≤ 1 in (2.12) one obtainṡ where the coefficient of P (t), by (3.4) is negative. Therefore P (t) remains always below the upper value and there is no possibility of an indefinite growth (boom-state) of the price. This simple analysis enlightens the importance of the parameters λ and δ in the large-time behavior of the model, and at the same time the rule of the presence of fundamentalists, which guarantee that the price evolution is contained in a bounded interval, whit the lower value bounded away from zero.

Numerical experiments.
To study the evolution of price, we have performed a series of kinetic Monte Carlo simulations for our Boltzmann-type system. In this rather basic simulations, known as direct simulation Monte Carlo (DSMC) method or Bird's scheme, agents are randomly and non-exclusively selected for interactions, and modify opinion according to the respective trade rules. One time step corresponds to N such interactions, with N denoting the number of agents. In all our experiments the group is composed by N = 10.000 agents with an initial equal percentage of chartists and fundamentalists (i.e. ρ C = ρ F = 0.5). All agents possess initially a unit wealth, and all chartist agents possess initially a neutral opinion (x = 0).
The simulation steps are as follows. One agent is randomly selected. If the selected agent is a chartist, it changes the investment propensity according to the change of opinion rules (2.13). In the case in which the selected agent is a fundamentalist it buys (x = 1) when the average price increases (Ṗ > 0) or sells (x = −1) when the price decreases (Ṗ < 0) with respect to the fundamental price P F .
To compute a stock approximation of the long-time behavior we noticed it is appropriate to carry out the simulation for about 10 3 time steps.
Owing to structure of the kinetics equations, and to the exact results obtained in Section 3, we tested the eventual formation of the boom and crash states, by acting on the exchange rules between chartists and fundamentalists, in such a way the number of density of chartists tends to dominate the number density of fundamentalists. A boom effect is then obtained by fixing the parameters β = 4, t C = 0.01, α 1 = 0.1, α 2 = 0.5, γ = 0.1, λ = 0.1, δ = 0.01 and µ = 0.5. The boom price behavior is displayed in Fig.(4.1).
The last experiment refers to the case of predominance of the fundamentalist agents. After the discussion of Section 3 it is known that this situation would prevent the formation of critical situations. Indeed, the time evolution of the average price is shown to produce damped oscillations which converge towards the fundamental price value P F . In Fig.(4.3) the average price and the density of the chartist agents are reported. In this case the choice of the parameters is β = 0.1, t C = 1, α 1 = 0.1,

5.
Conclusions. In this paper we analyzed a new kinetic version of the multiagent system introduced by Maldarella and Pareschi [18] in agreement with the model introduced by Lux and Marchesi [15,16,17] to study the evolution of price in presence of two different human behaviors, represented by fundamentalists and chartists reactions to the market evolution. At difference with the kinetic description of the opinion formation considered by Maldarella and Pareschi [18], based on price velocity, we analyzed the effects of price acceleration. Also, a microscopic description of the migration from one population to the other has been introduced. It is show both theoretically and numerically that the formation of critical situations is linked to the opinion variable which characterizes the behavior of chartists agents, and to the parameters of migration, which depends on price acceleration. In the case in which migration towards chartists agent is predominant, one shows that the extremal phenomena of boom and crash can be observed for suitable values of the parameters characterizing the model itself. On the contrary, when migration towards chartists is not predominant, so that the percentage of fundamentalists          remains bounded away from zero, these extremal phenomena are prevented. Numerical simulations enlighten the various behaviors of the price observed in the previous models, together with their stability with respect to the mean number of agents present in the market.