Mesoscopic modelling of financial markets
Abstract
We derive a mesoscopic description of the behavior of a simple financial market where the agents can create their own portfolio between two investment alternatives: a stock and a bond. The model is derived starting from the LevyLevySolomon microscopic model [14, 15] using the methods of kinetic theory and consists of a linear Boltzmann equation for the wealth distribution of the agents coupled with an equation for the price of the stock. From this model, under a suitable scaling, we derive a FokkerPlanck equation and show that the equation admits a selfsimilar lognormal behavior. Several numerical examples are also reported to validate our analysis.
Keywords: wealth distribution, powerlaw tails, stock market, selfsimilarity, kinetic equations.
1 Introduction
In recent years, physicists have been growing more and more interested in new interdisciplinary areas such as sociology and economics, originating what is today named socioeconomical physics [1, 3, 4, 6, 9, 13, 14, 20, 24, 29]. This new area in physics borrows several methods and tools from classical statistical mechanics, where complex behavior arises from relatively simple rules due to the interaction of a large number of components. The motivation behind this is the attempt to identify and characterize universal and nonuniversal features in economical data in general.
A large part of the research in this area is concerned with powerlaw tails with universal exponents, as was predicted more than one century ago by Pareto [4, 16, 23]. In particular, by identifying the wealth in an economic system with the energy of a physical system, the application of statistical physics makes it possible to understand better the development of tails in wealth distributions. Starting from the microscopic dynamics, mesoscopic models can be derived with the tools of classical kinetic theory of fluids [1, 5, 6, 7, 13, 21, 22, 25].
In contrast with microscopic dynamics, where behavior often can be studied only empirically through computer simulations, kinetic models based on PDEs allow us to derive analytically general information on the model and its asymptotic behavior. For example, the knowledge of the largewealth behavior is of primary importance, since it determines a posteriori whether the model can fit data of real economies.
In some recent papers, the explicit emergence of power laws in the wealth distribution, with Pareto index strictly larger than one, has been proved for open market economies where agents can interact through binary exchanges together with a simple source of speculative trading [1, 5, 22, 25].
The present work is motivated by the necessity to have a more realistic description of the speculative dynamics in the above models. To this end, we derive a mesoscopic description of the behavior of a simple financial market where a population of homogeneous agents can create their own portfolio between two investment alternatives: a stock and a bond. The model is closely related to the LevyLevySolomon (LLS) microscopic model in finance [14, 15]. This model attempted to construct from simple rules complex behavior that could then mimic the market and explain the price formation mechanism. As a first step towards a more realistic description, we derive and analyze the model in the case of a single stock and under the assumption that the optimal proportion of investments is a function of the price only. In principle, several generalizations are possible (different stocks, heterogeneous agents, a timedependent optimal proportion of investments, …), and we leave them for future investigations.
In our nonstationary financial market model, the average wealth is not conserved and this produces price variations. Let us point out that, even if the model is linear since no binary interaction dynamic between agents is present, the study of the large time behavior is not immediate. In fact, despite conservation of the total number of agents, we don’t have any other additional conservation equation or entropy dissipation. Although we prove that the moments do not grow more than exponentially, the determination of an explicit form of the asymptotic wealth distribution of the kinetic equation remains difficult and requires the use of suitable numerical methods.
A complementary method to extract information on the tails is linked to the possibility to obtain particular asymptotics which maintain the characteristics of the solution to the original problem for large times. Following the analysis in [5], we shall prove that the Boltzmann model converges in a suitable asymptotic limit towards a convectiondiffusion equation of FokkerPlanck type for the distribution of wealth among individuals. Other FokkerPlanck equations were obtained using different approaches in [1, 27, 18].
In this case, however, we can show that the FokkerPlanck equation admits selfsimilar solutions that can be computed explicitly and which are lognormal distributions. One is then led to the conclusion that the formation of Pareto tails in the wealth distribution observed in [1, 5] is a consequence of the interplay between the conservative binary exchanges having the effect of redistributing wealth among agents and the speculative trading causing the growth of mean wealth and social inequalities.
The rest of the paper is organized as follows. In the next section, we introduce briefly the microscopic dynamic of the LLS model. The mesoscopic model is then derived in Section 3 and its properties discussed in Section 4. These properties justify the asymptotic procedures performed in section 5. The model behavior together with its asymptotic limit is illustrated by several numerical results in section 6. Some conclusions and remarks on future developments are then made in the last section.
2 The microscopic dynamic
Let us consider a set of financial agents who can create their own portfolio between two alternative investments: a stock and a bond. We denote by the wealth of agent and by the number of stocks of the agent. Additionally we use the notations for the price of the stock and for the total number of stocks.
The essence of the dynamic is the choice of the agent’s portfolio. More precisely, at each time step each agent selects which fraction of wealth to invest in bonds and which fraction in stocks. We indicate with the (constant) interest rate of bonds. The bond is assumed to be a riskless asset yielding a return at the end of each time period. The stock is a risky asset with overall returns rate composed of two elements: a capital gain or loss and the distribution of dividends.
To simplify the notation, let us neglect for the moment the effects due to the stochastic nature of the process, the presence of dividends, and so on. Thus, if an agent has invested of its wealth in stocks and of its wealth in bonds, at the next time step in the dynamic he will achieve the new wealth value
(1) 
where the rate of return of the stock is given by
(2) 
and is the new price of the stock.
Since we have the identity
(3) 
we can also write
(4)  
(5) 
Note that, independently of the number of stocks of the agent at the next time level, it is only the price variation of the stock (which is unknown) that characterizes the gain or loss of the agent on the stock market at this stage.
The dynamic now is based on the agent choice of the new fraction of wealth he wants to invest in stocks at the next stage. Each investor is confronted with a decision where the outcome is uncertain: which is the new optimal fraction of wealth to invest in stock? According to the standard theory of investment each investor is characterized by a utility function (of its wealth) that reflects the personal risk taking preference [12]. The optimal is the one that maximizes the expected value of .
Different models can be used for this (see [15, 29]), for example, maximizing a von NeumannMorgenstern utility function with a constant risk aversion of the type
(6) 
where is the risk aversion parameter, or a logarithmic utility function
(7) 
As they don’t know the future stock price , the investors estimate the stock’s next period return distribution and find an optimal mix of the stock and the bond that maximizes their expected utility . In practice, for any hypothetical price , each investor finds the hypothetical optimal proportion which maximizes his/her expected utility evaluated at
(8) 
where and is estimated in some way. For example in [15] the investors expectations for are based on extrapolating the past values.
Note that, if we assume that all investors have the same risk aversion in (6), then they will have the same proportion of investment in stocks regardless of their wealth, thus .
Once each investor decides on the hypothetical optimal proportion of wealth that he/she wishes to invest in stocks, one can derive the number of stocks he/she wishes to hold corresponding to each hypothetical stock price . Since the total number of shares in the market , is fixed there is a particular value of the price for which the sum of the equals . This value is the new market equilibrium price and the optimal proportion of wealth is .
More precisely, following [15], each agent formulates a demand curve
characterizing the desired number of stocks as a function of the hypothetical stock price . This number of share demands is a monotonically decreasing function of the hypothetical price . As the total number of stocks
(9) 
is preserved, the new price of the stock at the next time level is given by the socalled market clearance condition. Thus the new stock price is the unique price at which the total demand equals the supply
(10) 
This will fix the value in (1) and the model can be advanced to the next time level. To make the model more realistic, typically a source of stochastic noise, which characterizes all factors causing the investor to deviate from his/her optimal portfolio, is introduced in the proportion of investments and in the rate of return of the stock .
3 Kinetic modelling
We define , , the distribution of wealth , which represents the probability for an agent to have a wealth . We assume that at time the percentage of wealth invested is of the form , where is a random variable in , and is distributed according to some probability density with zero mean and variance . This probability density characterizes the individual strategy of an agent around the optimal choice . We assume to be independent of the wealth of the agent. Here, the optimal demand curve is assumed to be a given monotonically nonincreasing function of the price such that .
Note that given the actual stock price satisfies the demandsupply relation
(11) 
where denotes the mathematical expectation of the random variable and has been normalized
More precisely, since and are independent, at each time , the price satisfies (see Figure 1)
(12) 
with
(13) 
being the mean wealth and by construction,
At the next round in the market, the new wealth of the investor will depend on the future price and the percentage of wealth invested according to
(14) 
where the expected rate of return of stocks is given by
(15) 
In the above relation, represents a constant dividend paid by the company and is a random variable distributed according to with zero mean and variance , which takes into account fluctuations due to price uncertainty and dividends [15, 11]. We assume to take values in with so that and thus negative wealths are not allowed in the model. Note that equation (15) requires estimation of the future price , which is unknown.
The dynamic is then determined by the agent’s new fraction of wealth invested in stocks, , where is a random variable in and is distributed according to . We have the demandsupply relation
(16) 
which permits us to write the following equation for the future price:
(17) 
Now
(18) 
thus
(19)  
(20) 
This gives the identity
(21) 
Using equation (12) we can eliminate the dependence on the mean wealth and write
(22)  
Remark 3.1
The equation for the future price deserves some remarks.

Equation (22) determines implicitly the future value of the stock price. Let us set
Then the future price is given by the equation
for a given . Note that
so the function is strictly increasing with respect to . This guarantees the existence of a unique solution
(23) Moreover, if and , the unique solution is and the price remains unchanged in time.
For the average stock return, we have
(24) where
(25) Now the right hand side of (24) has nonconstant sign since . In particular, the average stock return is above the bonds rate only if the (negative) rate of variation of the investments is above a certain threshold

In the constant investment case , with constant, then we have and
which corresponds to a dynamic of growth of the prices at rate . As a consequence, the average stock return is always larger then the constant return of bonds:
By standard methods of kinetic theory [2], the microscopic dynamics of agents originate the following linear kinetic equation for the evolution of the wealth distribution
(26) 
The above equation takes into account all possible variations that can occur to the distribution of a given wealth . The first part of the integral on the right hand side takes into account all possible gains of the test wealth coming from a pretrading wealth . The function gives the probability per unit time of this process.
Thus is obtained simply by inverting the dynamics to get
(27) 
where the value is given as the unique fixed point of (17).
The presence of the term in the integral is needed in order to preserve the total number of agents
The second part of the integral on the right hand side of (26) is a negative term that takes into account all possible losses of wealth as a consequence of the direct dynamic (14), the rate of this process now being . In our case, the kernel takes the form
(28) 
The distribution function , together with the function , characterizes the behavior of the agents on the market (more precisely, they characterize the way the agents invest their wealth as a function of the actual price of the stock).
Remark 3.2
In the derivation of the kinetic equation, we assumed for simplicity that the actual demand curve which gives the optimal proportion of investments is a function of the price only. In reality, the demand curve should change at each market iteration and should thus depend also on time. In the general case where each agent has a wealthdependent individual strategy, one should consider the distribution of agents having a fraction of their wealth invested in stocks.
4 Properties of the kinetic equation
We will start our analysis by introducing some notations. Let be the space of all probability measures on and by
(29) 
we mean the space of all Borel probability measures of finite momentum of order , equipped with the topology of the weak convergence of the measures.
Let , be the class of all real functions on such that and is Hölder continuous of order ,
(30) 
the integer and the number be such that , and denote the th derivative of .
Clearly the symmetric probability density which characterizes the stock returns belongs to for all since
Moreover, to simplify computations, we assume that this density is obtained from a given random variable with zero mean and unit variance. Thus of variance is the density of . By this assumption, we can easily obtain the dependence on of the moments of . In fact, for any ,
Note that equation (26) in weak form takes the simpler form
(31) 
By a weak solution of the initial value problem for equation (26) corresponding to the initial probability density , , we shall mean any probability density satisfying the weak form (31) for and all , and such that for all ,
(32) 
The form (31) is easier to handle, and it is the starting point to study the evolution of macroscopic quantities (moments). The existence of a weak solution to equation (26) can be seen easily using the same methods available for the linear Boltzmann equation (see [28] and the references therein for example).
From (31) follows the conservation of the total number of investors if . The choice is of particular interest since it gives the time evolution of the average wealth which characterizes the price behavior. In fact, the mean wealth is not conserved in the model since we have
(33) 
Note that since the sign of the right hand side is nonnegative, the mean wealth is nondecreasing in time. In particular, we can rewrite the equation as
(34) 
From this we get the equation for the price
(35) 
where is given by (22) and
Now since from (24) it follows by the monotonicity of that
using (34) we have the bound
(36) 
From (12) we obtain immediately
which gives
(37) 
Remark 4.1
For a constant , we have the explicit expression for the growth of the wealth (and consequently of the price)
(38) 
Analogous bounds to (36) for moments of higher order can be obtained in a similar way. Let us consider the case of moments of order , which we will need in the sequel. Taking , we get
(39) 
Moreover, we can write
where, for some ,
Hence,
From
and
we have
Since
(40) 
we finally obtain the bound
(41) 
where
and , , and are suitable constants.
We can summarize our results in the following
Theorem 4.1
Let the probability density , where for some . Then the average wealth is increasing exponentially with time following (36). As a consequence, if is a nonincreasing function of , the price does not grow more than exponentially as in (37). Similarly, higher order moments do not increase more than exponentially, and we have the bound (41).
5 FokkerPlanck asymptotics and selfsimilar solution
The previous analysis shows that in general it is difficult to study in detail the asymptotic behavior of the system. In addition, we must take into account the exponential growth of the average wealth. In this case, one way to get information on the properties of the solution for large time relies on a suitable scaling of the solution. As is usual in kinetic theory, however, particular asymptotics of the equation result in simplified models (generally of FokkerPlanck type) whose behavior is easier to analyze. Here, following the analysis in [5, 22] and inspired by similar asymptotic limits for inelastic gases [8, 25], we consider the limit of large times in which the market originates a very small exchange of wealth (small rates of return and ).
In order to study the asymptotic behavior of the distribution function , we start from the weak form of the kinetic equation
(42) 
and consider a secondorder Taylor expansion of around ,
where, for some ,
Inserting this expansion into the collision operator, we get
where
Recalling that , , and , we can simplify the above expression to obtain
Now we set
which implies that satisfies the equation
Now we consider the limit of very small values of the constant rate . In order for such a limit to make sense and preserve the characteristics of the model, we must assume that
(44) 
First let us note that the above limits in (22) imply immediately that
(45) 
We begin by showing that the remainder is small for small values of .
Since and ,
(46) 
Hence
By the inequality
and (40), we get
As a consequence of (44)–(45), from this inequality it follows that converges to zero as if
remains bounded at
any fixed time , provided the same bound holds at time
. This is guaranteed by inequality (41)
since in the asymptotic limit defined by
(44).
Next we write
where
Then, using the above expansion from (44) in (22), we obtain
(47) 
with
(48) 
Now, sending under the same assumptions, we obtain the weak form