Economics and Finance: q-Statistical Stylized Features Galore

Abstract

The Boltzmann–Gibbs (BG) entropy and its associated statistical mechanics were generalized, three decades ago, on the basis of the nonadditive entropy $S_q$ ($q\in \mathcal{R}$), which recovers the BG entropy in the $q\to 1$ limit. The optimization of $S_q$ under appropriate simple constraints straightforwardly yields the so-called q-exponential and q-Gaussian distributions, respectively generalizing the exponential and Gaussian ones, recovered for $q=1$. These generalized functions ubiquitously emerge in complex systems, especially as economic and financial stylized features. These include price returns and volumes distributions, inter-occurrence times, characterization of wealth distributions and associated inequalities, among others. Here, we briefly review the basic concepts of this q-statistical generalization and focus on its rapidly growing applications in economics and finance.

1. Introduction

Exponential and Gaussian functions ubiquitously emerge within linear theories in mathematics, physics, economics and elsewhere. To illustrate in what sense they are linear, let us focus on three typical mathematical situations, namely an ordinary differential equation, a partial derivative equation and an entropic optimization.

Consider the following ordinary differential equation:

$$\frac{dy}{dx}=ay[y\left(0\right)=1].$$

(1)

The solution is the well-known exponential function:

$$y=e^{ax}.$$

(2)

Consider now the following partial derivative equation:

$$\frac{\partial p(x,t)}{\partial t}=D\frac{{\partial }^{2}p(x,t)}{\partial x^2}[D>0;t\ge 0;p(x,0)=\delta \left(x\right)],$$

(3)

where $\delta \left(x\right)$ is the Dirac delta function. The solution is the well-known Gaussian distribution:

$$p(x,t)=\frac{1}{\sqrt{2\pi Dt}}{e}^{-x^2/2Dt}.$$

(4)

Let us finally consider the following entropic functional:

$$S_{BG}=-k\int dxp\left(x\right)lnp\left(x\right)(k>0)$$

(5)

with the constraint:

$$\int dxp(x)=1,$$

(6)

where BG stands for Boltzmann–Gibbs; k is a conventional positive constant (usually $k=k_B$ in physics, and $k=1$ elsewhere). If we optimize the functional (5) with the constraint (6) and:

$$\langle ϵ(x)\rangle \equiv \int dxp(x)ϵ(x)=u(u\in \mathcal{R})$$

(7)

$ϵ\left(x\right)$ being bounded below, we obtain:

$$p\left(x\right)=\frac{e^{-\beta ϵ\left(x\right)}}{\int dx{e}^{-\beta ϵ\left(x\right)}},$$

(8)

where $\beta \equiv 1/kT$ is the Lagrange parameter associated with Constraint (7); T is the absolute temperature within BG statistical mechanics (necessarily $T>0$ if $ϵ\left(x\right)$ is unbounded from above; but both $T>0$ and $T<0$ possibilities exist if $ϵ\left(x\right)$ is bounded also from above). The probability distribution $p\left(x\right)$ corresponds to the celebrated BG weight, where $Z\equiv \int dx{e}^{-\beta ϵ\left(x\right)}$ is usually referred to as the partition function. Two particular cases emerge frequently. The first of them is $ϵ\left(x\right)=x(x\ge 0)$ with $u\equiv \langle x\rangle$, hence $p\left(x\right)=\frac{e^{-x/\langle x\rangle }}{\langle x\rangle }$, thus recovering solution (2). The second one is $ϵ\left(x\right)=x^2$ with $u\equiv \langle x^2\rangle$, hence $p\left(x\right)=\frac{e^{-x^2/2\langle x^2\rangle }}{\sqrt{2\pi \langle x^2\rangle }}$, thus recovering solution (4). Therefore, basic cases connect $S_{BG}$ with the solutions of the linear Equations (1) and (3). In addition to that, let us make explicit in what sense $S_{BG}$ is itself linear. We consider a system $(A+B)$ constituted by two probabilistically independent subsystems A and B. In other words, we consider the case where the joint probability of $(A+B)$ factorizes, i.e., $p^{(A+B)}(x,y)=p^{\left(A\right)}\left(x\right)p^{\left(B\right)}\left(y\right)[\forall (x,y)]$. We straightforwardly verify that the functional $S_{BG}$ is additive in the sense of Penrose [1], namely that:

$$S_{BG}(A+B)=S_{BG}\left(A\right)+S_{BG}\left(B\right).$$

(9)

In the present brief review, we shall address a special class of nonlinearities, namely those emerging within nonextensive statistical mechanics, q-statistics for short [2,3,4,5,6].

Equation (1) is now generalized into the following nonlinear one:

$$\frac{dy}{dx}=a{y}^q[y\left(0\right)=1;q\in \mathcal{R}].$$

(10)

Its solution is:

$$y=e_q^{ax},$$

(11)

where the q-exponential function is defined as $e_q^z\equiv {[1+(1-q)z]}_{+}^{1/(1-q)}(e_1^z=e^z)$, with ${[1+(1-q)z]}_{+}=1+(1-q)z$ if $z>0$ and zero otherwise. Its inverse function is the q-logarithm, defined as ${ln}_{q}z\equiv \frac{z^{1-q}-1}{1-q}({ln}_{1}z=lnz)$. To avoid any confusion, let us mention that many other q-deformations of the exponential and logarithmic functions have been introduced in the literature for a variety of purposes; among them, we have for instance Ramanujan’s q-exponential function, unrelated to the present one.

Equation (3) is now generalized into the following nonlinear one (referred to in the literature as the porous medium equation [7,8,9]):

$$\frac{\partial p(x,t)}{\partial t}=D_q\frac{{\partial }^2{\left[p(x,t)\right]}^{2-q}}{\partial x^2}[D_q(2-q)>0;D_1\equiv D;q<3;t\ge 0;p(x,0)=\delta \left(x\right)].$$

(12)

Its solution generalizes Equation (4) and is given by:

$$p(x,t)=\frac{1}{\sqrt{\pi A_q}}{e}_q^{-x^2/\{A_q{\left[\left(D_q\right)t\right]}^{\frac{2}{3-q}}\}}$$

(13)

with:

$$A_q=\left\{\begin{array}{cc}\frac{\sqrt{q-1}\Gamma \left(\frac{1}{q-1}\right)}{\Gamma \left(\frac{3-q}{2(q-1)}\right)}\hfill & \mathrm{if}1<q<3,\hfill \\ 2\hfill & \mathrm{if}q=1,\hfill \\ \frac{\sqrt{1-q}\Gamma \left(\frac{5-3q}{2(1-q)}\right)}{\Gamma \left(\frac{2-q}{1-q}\right)}\hfill & \mathrm{if}q<1.\hfill \end{array}\right.$$

(14)

Before going on, let us mention that solution (13) implies that $x^2$ scales like $t^{\frac{2}{3-q}}$, hence normal diffusion for $q=1$, anomalous sub-diffusion for $q<1$ and super-diffusion for $1<q<3$, which has recently been impressively validated (within a $2\%$ experimental error) in a granular medium [10]. The important connection between the power-law nonlinear diffusion (12) and the entropy $S_q$ described here below was first established by Plastino and Plastino in [11], where they considered a more general evolution equation that reduces to (12) in the particular case of vanishing drift (i.e., $F\left(x\right)=0,\forall x$). The Plastino–Plastino Equation [11] $\frac{\partial p(x,t)}{\partial t}=-\frac{\partial }{\partial x}\left[F\left(x\right)p(x,t)\right]+D_q\frac{{\partial }^2{\left[p(x,t)\right]}^{2-q}}{\partial x^2}$ with $F\left(x\right)=-dV\left(x\right)/dx$ generalizes the porous medium equation in the same sense that the linear Fokker–Planck equation generalizes the classical heat equation. The above nonlinear Equations (10) and (12) have been addressed here in order to provide some basic mathematical structure to approaches of various economic- and financial-specific features presented later on.

Let us now focus on the entropic functional $S_q$ upon which nonextensive statistical mechanics is based. It is defined as follows:

$$S_q\equiv k\frac{1-\int dx{\left[p\left(x\right)\right]}^q}{q-1}=k\int dxp\left(x\right){ln}_q\frac{1}{p\left(x\right)}=-k\int dx{\left[p\left(x\right)\right]}^q{ln}_{q}p\left(x\right)=-k\int dxp\left(x\right){ln}_{2-q}p\left(x\right)$$

(15)

with $S_1=S_{BG}$. If we optimize this functional with the constraints (6) and:

$${\langle ϵ\left(x\right)\rangle }_q\equiv \frac{\int dx{\left[p\left(x\right)\right]}^qϵ\left(x\right)}{\int dx{\left[p\left(x\right)\right]}^q}=u_q(u_q\in \mathcal{R};u_1=u)$$

(16)

we obtain [4]:

$$p\left(x\right)=\frac{e_q^{-{\beta }_qϵ\left(x\right)}}{\int dx{e}_q^{-{\beta }_qϵ\left(x\right)}}=\frac{e_q^{-{\beta }_q^{\prime }[ϵ\left(x\right)-u_q]}}{\int dx{e}_q^{-{\beta }_q^{\prime }[ϵ\left(x\right)-u_q]}}({\beta }_1={\beta }_1^{\prime }=\beta ).$$

(17)

As before, two particular cases emerge frequently. The first of them is $ϵ\left(x\right)=x(x\ge 0)$ with $u_q\equiv {\langle x\rangle }_q$; hence, $p\left(x\right)$ recovers the form of (11). The second one is $ϵ\left(x\right)=x^2$ with $u_q\equiv {\langle x^2\rangle }_q$; hence, $p\left(x\right)$ recovers the form of solution (13). Finally, if we consider $S_q$ itself for two independent subsystems A and B, we straightforwardly verify the following nonlinear composition law:

$$\frac{S_q(A+B)}{k}=\frac{S_q\left(A\right)}{k}+\frac{S_q\left(B\right)}{k}+(1-q)\frac{S_q\left(A\right)}{k}\frac{S_q\left(B\right)}{k},$$

(18)

hence

$$S_q(A+B)=S_q\left(A\right)+S_q\left(B\right)+\frac{1-q}{k}{S}_q\left(A\right)S_q\left(B\right).$$

(19)

We then say that $S_q$ is nonadditive for $q\ne 1$. Entropic additivity is recovered if $(1-q)/k\to 0$, which can occur in two different circumstances: $q\to 1$ for fixed k or $k\to \infty$ for fixed q. Since k always appears in physics in the form $kT$, the limit $k\to \infty$ is equivalent to $T\to \infty$. This is, by the way, the basic reason for which, in the limit of high temperatures or low energies, Maxwell–Boltzmann statistics, Fermi–Dirac, Bose–Einstein and q-statistics asymptotically coincide.

The above q-generalized thermostatistical theory has been useful in the study of a considerable number of natural, artificial and social systems (see [12]). Theoretical and experimental illustrations in natural systems include long-range-interacting many-body classical Hamiltonian systems [13,14,15,16,17,18,19,20] (see also [21,22]; the study of the long-range version of [23] would surely be interesting), dissipative many-body systems [24], low-dimensional dissipative and conservative nonlinear dynamical systems [25,26,27,28,29,30,31], cold atoms [32,33,34], plasmas [35,36], trapped atoms [37], spin-glasses [38], power-law anomalous diffusion [39,40], granular matter [10], high-energy particle collisions [41,42,43,44,45,46], black holes and cosmology [47,48], chemistry [49], earthquakes [50], biology [51,52], solar wind [53,54], anomalous diffusion in relation to central limit theorems and overdamped systems [55,56,57,58,59,60,61,62,63,64], quantum entangled systems [65,66], quantum chaos [67], astronomical systems [68,69], thermal conductance [70], mathematical structures [71,72,73,74,75,76] and nonlinear quantum mechanics [77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96], among others. Illustrations in artificial systems include signal and image processing [97,98] and (asymptotically) scale-free networks [99,100,101]. In the realm of social systems, from now on, we focus on economics and financial theory [102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118].

2. Applications in Economics and Finance

2.1. Prices and Volumes

Time series of prices $p_t$ (say of stocks, commodities, etc.), where t runs along chosen units (say seconds or minutes, or days, or years) are conveniently replaced by their corresponding returns (or logarithmic returns), defined as follows:

$$r_t=ln\frac{p_{t+1}}{p_t}\simeq \frac{p_{t+1}-p_t}{p_t}(t=0,1,2,\dots ).$$

(20)

Returns do not depend on the specific currency of the prices and fluctuate around zero; in addition to that, their definition cancels systematic inflation. The distribution of returns usefully characterizes the price fluctuations. See an illustration in Figure 1, from [103] (see also [104,118]). The amounts of the corresponding transactions are currently referred to as volumes: see, for example, Figure 2.

2.2. Volatilities

The volatility characterizes the size (standard deviation) of the fluctuations of returns. The volatility smile characterizes the correction of empirical volatilities with regard to a Gaussian-based expectation: see an illustration in Figure 3 (from [103]). To be more explicit, let us assume that we are handling the following Gaussian distribution $\propto e^{-\mathcal{B}{r}^2}$, where $\mathcal{B}$ characterizes univocally the volatility. To discuss the probability distribution of quantities such as $\mathcal{B}$, Queiros introduced [108] the q-log normal probability function:

$$p_q\left(x\right)=\frac{1}{Z_q{x}^q}{e}^{-\frac{{[{ln}_{q}x-\mu ]}^2}{2{\sigma }^2}}(x>0),$$

(21)

where $Z_q$ is a normalizing factor and $(q,\mu ,\sigma )$ are parameters. The $q=1$ particular case corresponds to the standard log-normal function. See Figure 4 for illustrative examples of this function. See also Figure 5 for a real financial application.

2.3. Inter-Occurrence Times

We can see in Figure 6 two typical time series of price returns, together with a chosen threshold $Q=-0.037$, which corresponds to an average inter-occurrence time $R_Q=70$ [106,107]. The quantity $R_Q$ monotonically increases with $|-Q|$ in each one of the examples shown in Figure 7 and Figure 8. For a fixed value of $R_Q$, we verify that $p_Q\left(r\right)\propto e_{q_{threshold}}^{-{\beta }_{threshold}r}$, with:

$$q_{threshold}=1+q_{0}ln(R_Q/2)(q_0\simeq 0.168).$$

(22)

See the illustrations in Figure 9 and Figure 10. The fact that we have analytically $p_Q\left(r\right)$ enables us to straightforwardly obtain an explicit expression for the risk function $W_Q(t;\Delta t)$, which is defined as the probability of having once again a fluctuation larger than $|-Q|$ within an interval $\Delta t$ at time t after the last large fluctuation. It can be shown [106,109] that:

$$W_Q(t;\Delta t)\equiv \frac{{\int }_t^{t+\Delta t}{p}_Q\left(r\right)dr}{{\int }_t^{\infty }{p}_Q\left(r\right)dr}=1-{\left[1+\frac{{\beta }_{threshold}(q_{threshold}-1)\Delta t}{1+{\beta }_{threshold}(q_{threshold}-1)t}\right]}^{\frac{q_{threshold}-2}{q_{threshold}-1}}=1-\frac{e_{\tilde{q}}^{-({\beta }_{threshold}/\tilde{q})(t+\Delta t)}}{e_{\tilde{q}}^{-({\beta }_{threshold}/\tilde{q})t}}$$

(23)

with $\tilde{q}\equiv 1/(2-q_{threshold})$. See Figure 11.

2.4. Wealth

Wealth inequality within a given country is a classical and most important matter, which can be characterized within q-statistics as shown in [105]: see Figure 12 and Figure 13. The larger the index $q_{inequality}$ is, the larger the inequality. As we verify, the U.K. and Germany are more egalitarian countries than the U.S. and Brazil. In addition to that, inequality appears to increase in the U.S. and Brazil, at least during the years indicated in the plots.

Another index that characterizes the wealth of a country and its inequalities is associated with the prices of the land. See in Figure 14 (from [105]) an illustration for Japan, where $q_{landprice}=2.136$.

3. Conclusions and Perspectives

We have described a variety of financial and economic properties with a plethora of q-indices, such as $q_{return},q_{volume},q_{volatility},q_{inter},q_{threshold},q_0,\tilde{q},q_{inequality},q_{landprice}$. For a given system, how many independent indices should we expect? The full answer to this question remains up to now elusive. It seems however that only a few of them are essentially independent, all of the others being (possibly simple) functions of those few. Such an algebraic structure was first advanced and described in [119] and has been successfully verified in the solar wind [53] (see also [6] and the references therein) and elsewhere; it has recently been generalized [120,121] and related to the Moebius group. The central elements of these algebraic structures appear to constitute what is currently referred to in the literature as q-triplets [122]. The clarification and possible verification of such structures constitutes nowadays an important open question, whose further study would surely be most useful.

Another crucial question concerns the analytic calculation from first principles of some or all of the above q-indices. This is in principle possible (as illustrated in [63,64,65,66]), but it demands the complete knowledge of the microscopic model of the specific class of the complex system. For the full set of the q-indices shown in the present overview, such models are not available, even if they would be very welcome.

Let us finally emphasize that many other statistical approaches exist for the quantities focused on in the present overview. However, as announced in the title of this paper, this is out of the present scope. The present paper is one among various others belonging to the same Special Issue of the journal Entropy. The entire set of articles is expected to enable comparisons between these many approaches.