Excess Demand Financial Market Model.

Recently we reported on an application of the Tsallis non-extensive statistics to the S&P500 stock index @mike1. There we argued that the statistics are applicable to a broad range of markets and exchanges where anamolous (super) diffusion and 'heavy' tails of the distribution are present, as they are in the S&P500. We have characterized the statistics of the underlying security as non-extensive, and now we seek to generalize to the non-extensive statistics the excess demand models of investors that drive the price formation in a market.

In the past decade, there have been many models (@cont1 [@chowdhury1]) proposed that attempt to capture the dynamics and statistics of market participants that drive the price changes in a market. These range from minority game models, multi-agent models, and spin models of the bias of investors. They all have in common the fact that they are models for the excess demand. The instantaneous excess demand $\phi (t)$ can be defined as the mismatch in the number of buyers $N_{+}(t)$ and sellers $N_{-}(t)$ for a number of shares at time $t$. The hallmark for the success of an interacting investors market model has been the ability of a given model in reproducing the stylized facts of real markets. These are the heavy tails (power-law) of the distributions, anomalous (super) diffusion, and therefore statistical dependence (long-range correlations) of subsequent price changes (@econophysics).

Recently the Tsallis non-extensive statistics were applied to analyzing the price changes and intra-day statistical dynamics of the S&P500 stock index (@mike1). This study characterized the statistics of the price changes as being well-modeled by the non-extensive statistics. It was also argued that the statistics are applicable to a broad range of markets and exchanges where anamolous (super) diffusion and power-law tails of the distribution are present, as found in the S&P500 (@econophysics).  

In this paper we examine the demand-side in light of these recent findings and outline a method by which one can obtain many-investor models within the context of the Tsallis non-extensive statistics using the maximum entropy approach. We review the maximum entropy approach to be utilized. The non-extensive, least-biased probability density (PDF) $P(z,t)$ of an underlying (say, continuous) observable $z(t)$ is obtained by maximizing an incomplete information-theoretic measure equivalent to the Tsallis entropy $S_{q}$ ( @tsallis1 [@wang1]) and subject to the known (or assumed) observables of the system as constraints $$\begin{aligned}
\left\langle S\right\rangle _{q} &=&S_{q}=-{\frac{1}{1-q}}\left( 1-\int
P(z,t)^{q}\,dx\right) ,  \nonumber \\
\left\langle z\right\rangle _{q} &=&\int z\text{ }P(z,t)^{q}\,dx.\end{aligned}$$ The maximization of the entropy with $\alpha $ a Lagrange multiplier is $$\delta \left\langle S_{q}\right\rangle -\delta \lbrack \alpha \left\langle
z\right\rangle _{q}]\equiv 0,$$ and the maximization yields the least biased probability distribution given the constraints. In the nonextensive statistics this is a $q$-parametrized power-law Tsallis distribution (@tsallis1) $$P(z,t)=\frac{\{1+\alpha (q-1)\text{ }z\}^{\frac{-1}{q-1}}}{Z_{q}},$$ and $q$ is the degree of non-extensivity or equivalently the incompleteness of the information measure. In the limit of $q\rightarrow 1$ we recover the Gibbs-Boltzmann extensive statistics and the distribution becomes an exponential. The normalization is $Z_{q}(t)$, the partition function, and $%
\alpha (t)$ is a possibly time-dependent Lagrange multiplier associated with the constraints. Again, the constraints will be the known (or assumed) observables of interest, and are presumed to capture the statistical behavior of our many-investor system.

We initially wish to model the fluctuating variables of the number of investors $N_{\pm }(t)$ buying ($+$) and selling ($-$) in the market. We write the excess demand as $\phi (t)=N_{+}(t)-N_{-}(t)$ and note that it is proportional to the instantaneous price $x(t).$ That is, we can write approximately $$x(t)=\frac{\phi (t)}{\lambda }=\frac{N_{+}(t)-N_{-}(t)}{\lambda },\text{ }
\label{supply1}$$ where $\lambda $ is the market depth, or the excess demand needed to move the price by one dollar. The market depth is assumed to evolve slowly during the time scales considered, and will be taken to be a constant. The observables of interest are the means and variances of the fluctuating variables. We can write the entropy and observables as statistical averages about the means

$$\begin{aligned}
\left\langle S\right\rangle _{q} &=&S_{q}=-{\frac{1}{1-q}}\left(
1-\sum\limits_{N_{+},N_{-}=0}^{N}P(N_{+},N_{-},t)^{q}\,\right) ,  \nonumber
\\
\left\langle (N_{+}-\overline{N}_{+})\right\rangle _{q}
&=&0=\sum\limits_{N_{+},N_{-}=0}^{N}(N_{+}-\overline{N}_{+})\text{ }%
P(N_{+},N_{-},t)^{q}\,,  \nonumber \\
\left\langle (N_{+}-\overline{N}_{+})^{2}\right\rangle _{q}
&=&\sum\limits_{N_{+},N_{-}=0}^{N}(N_{+}-\overline{N}_{+})^{2}\text{ }%
P(N_{+},N_{-},t)^{q}\,=\eta _{+}^{2}(t)_{q},  \nonumber \\
\left\langle (N_{-}-\overline{N}_{-})\right\rangle _{q}
&=&0=\sum\limits_{N_{+},N_{-}=0}^{N}(N_{-}-\overline{N}_{-})\text{ }%
P(N_{+},N_{-},t)^{q}\,,  \nonumber \\
\left\langle (N_{-}-\overline{N}_{-})^{2}\right\rangle _{q}
&=&\sum\limits_{N_{+},N_{-}=0}^{N}(N_{-}-\overline{N}_{-})^{2}\text{ }%
P(N_{+},N_{-},t)^{q}\,=\eta _{-}^{2}(t)_{q},\end{aligned}$$

subject to the additional constraint of normalization $1=\sum%
\limits_{N_{+},N_{-}=0}^{N}P(N_{+},N_{-},t)\,$. We then seek the least biased probability distribution given these observables. We maximize the entropy given the constraints $$\delta \left\langle S\right\rangle _{q}-\delta \left\{ \beta
(t)[\left\langle (N_{+}-\overline{N}_{+})^{2}\right\rangle _{q}+\left\langle
(N_{-}-\overline{N}_{-})^{2}\right\rangle _{q}]\right\} \equiv 0,$$ and here $\beta (t)$ is a time dependent Lagrangian multiplier and can be shown to be proportional to the inverse of the variance @mike1. The maximization yields the power-law least biased probability distribution $$P(N_{+},N_{-},t)=\frac{\left( 1+(q-1)\left[ \beta (t)[(N_{+}-\overline{N}%
_{+})^{2}+(N_{-}-\overline{N}_{-})^{2}\right] \right) ^{\frac{-1}{q-1}}}{%
Z_{q}},$$ where $Z_{q}=\sum\limits_{N_{+},N_{-}=0}^{N}P(N_{+},N_{-},t)\,$ is the partition function and is related to the normalization. We can therefore obtain all the observables of interest by performing statistical averages with respect to the distribution. The evaluation of the Lagrange multiplier(s) proceeds as in the extensive statistics case ( @curado1) $$\begin{aligned}
-\frac{\partial }{\partial \beta }\frac{Z_{q}^{1-q}-1}{1-q} &=&\left\langle
(N_{+}-\overline{N}_{+})^{2}\right\rangle _{q}+\left\langle (N_{-}-\overline{%
N}_{-})^{2}\right\rangle _{q}  \nonumber \\
&=&\eta _{+}^{2}(t)_{q}+\eta _{-}^{2}(t)_{q},  \label{observe1}\end{aligned}$$ and in the limit as $q\rightarrow 1$ we recover the standard extensive statistics expression $-\frac{\partial lnZ_{1}}{\partial \beta }%
=\left\langle \vartheta \right\rangle _{1}$.

The demand-side model then provides us with a statistical description of the fluctuating variables ($N_{+},N_{-}$), the difference of which can be related to the price by $x(t)=\frac{N_{+}(t)-N_{-}(t)}{\lambda }$. This model has the power-law behavior of real markets as a consequence of the pseudo-additive nature of the entropy of the subsystems. That is to say, the number of investors buying and selling at time $t$ in the 'subsystems' of buyers and sellers are not statistically independent, and the incremental changes in time of the number of buyers and sellers will be correlated in time. It is also known that the price changes $dx(t)$ exhibit anomalous diffusion and correlations in time and therefore by $dx(t)=\frac{d\phi (t)}{%
\lambda }$ so will the change in the excess demand $d\phi (t)$. The composition of the entropy of the two subsystems must be written from the joint probability decomposition (suppressing the time parameter) $$P(N_{+},N_{-})=P(N_{+}\mid N_{-})P(N_{-})$$ which gives the pseudo-additive entropy $$\begin{aligned}
S_{q}(N_{+},N_{-}) &=&S_{q}(N_{-})+S_{q}(N_{+}\mid
N_{-})+(1-q)S_{q}(N_{-})S_{q}(N_{+}\mid N_{-}),  \nonumber \\
S_{q} &=&-ln_{q}P=-\frac{P^{1-q}-1}{1-q}.\end{aligned}$$ It can be shown (@rajagopal1) that the Tsallis entropy satisfies this condition, and the resulting probability will be of the power-law form derived above.

In order to simplify the solution, let us symmetrize the range of the distribution. We define $n_{\pm }=N_{\pm }-\frac{N}{2}$ such that the variable $n_{\pm }$ now has the symmetrical range $-\frac{N}{2}\leq n_{\pm
}\leq \frac{N}{2}$ and the excess demand becomes $\phi =n_{+}-n_{-}$. We next pass to the continuum limit of the number of investors. Alternatively, we could have started our derivation from this approximation. Also, due to the observation that most trades involve a small fraction of the overall number of investors ( $N_{\pm }-\overline{N}_{\pm }\ll N$) we relax the range to $N\rightarrow \infty $. The distribution $P(n_{+},n_{-},t)$ is now taken to be symmetric and continuous, and is given by a similar power-law form as before $$P(n_{+},n_{-},t)=\frac{\left( 1+(q-1)\left[ \beta (t)[(n_{+}-\overline{n}%
_{+})^{2}+(n_{-}-\overline{n}_{-})^{2}\right] \right) ^{\frac{-1}{q-1}}}{%
Z_{q}}.$$ We integrate to obtain the partition function $$\begin{aligned}
Z_{q}(t) &=&\int\limits_{-\infty }^{\infty }\int\limits_{-\infty }^{\infty
}P(n_{+},n_{-},t)dn_{+}dn_{-} \\
&=&\frac{\pi \text{ }}{\beta (t)(2-q)},\end{aligned}$$ where the range of $q$ must now be restricted to $1<q<2$ to insure normalization. We note that $\beta (t)Z_{q}(t)=c_{q\text{ }}$, a time independent constant of the process dependent on the non-extensivity parameter $q$. This is useful in that we can relate the Lagrange multiplier parameter $\beta (t)$ (and therefore the inverse variance) to the partition function for all times considered. The range of $q$ is chosen as $1\leq
q\leq q_{max}$ consistent with the requirements that the distribution is normalizable for all times and that the regular variance remain finite.

The regular variances $\eta _{\pm }^{2}(t)$ can be computed readily from the distribution $$\begin{aligned}
\left\langle (n_{\pm }-\overline{n}_{\pm })^{2}\right\rangle
&=&\int\limits_{-\infty }^{\infty }\int\limits_{-\infty }^{\infty }(n_{\pm }-%
\overline{n}_{\pm })^{2}P(n_{+},n_{-},t)dn_{+}dn_{-} \\
&=&\frac{1}{2\text{ }\beta (t)\text{ }(3-2q)},  \nonumber\end{aligned}$$ and the range of $q$ must be further restricted to $1<q<1.5$ to assure convergence of the integral and therefore a finite variance.

The power-law distribution $P(n_{+},n_{-},t)$ has a time evolution that satisfies a two dimensional Fokker-Planck partial differential equation ( @zanette1 [@tsallis1]), and we assume a linear drift term $\alpha (n_{\pm
})=a_{\pm }$ $n_{\pm }$ consistent with the linear drift of the price distribution (@mike1) $$\begin{aligned}
\frac{\partial }{\partial t}P(n_{+},n_{-},t) &=&-\frac{\partial }{\partial
n_{+}}[\alpha (n_{+})\text{ }P(n_{+},n_{-},t)]-\frac{\partial }{\partial
n_{-}}[\alpha (n_{-})P(n_{+},n_{-},t)]  \nonumber \\
&&+\frac{D}{2}\left( \frac{\partial }{\partial n_{+}^{2}}+\frac{\partial }{%
\partial n_{-}^{2}}\right) [P^{2-q}(n_{+},n_{-},t)].\end{aligned}$$ This subsequently implies the underlying stochastic differential equations ( @gardiner1) for the number of buyers and sellers

$$\begin{aligned}
dn_{+} &=&\alpha (n_{+})dt+\sqrt{DP^{1-q}(n_{+},n_{-},t)}\text{ }dW_{+}(t), 
\nonumber \\
dn_{-} &=&\alpha (n_{-})dt+\sqrt{DP^{1-q}(n_{+},n_{-},t)}\text{ }dW_{-}(t).
\label{sde1}\end{aligned}$$

Here $dW_{\pm }(t)=\xi _{\pm }(t)dt$ , where $W_{\pm }(t)$ is the Wiener process and $\xi _{\pm }(t)$ is a delta correlated noise such that $%
\left\langle \xi _{\sigma }(t)\xi _{\sigma }(t^{\prime })\right\rangle
=\delta _{\sigma \sigma ^{\prime }}\delta (t-t^{\prime })$. If we examine the change in the excess demand we obtain the relationship to the change of price $$\begin{aligned}
d\phi  &=&dn_{+}-dn_{-} \\
&=&\lambda \text{ }dx(t)\end{aligned}$$ The solutions of these SDE's can be obtained by using Ito's change of variables and performing the integration over time $$n_{\pm }(t)=n_{\pm }(t_{o})e^{a_{\pm
}(t-t_{o})}+\int\limits_{t_{o}}^{t}e^{a_{\pm }(t-t^{\prime })}\sqrt{%
DP^{1-q}(n_{+},n_{-},t)}dt^{\prime },$$ and we thus have the desired result for the formation of the instantaneous excess demand $\phi (t)$, and therefore the instantaneous price $x(t)$$$\begin{aligned}
\phi (t) &=&n_{+}(t)-n_{-}(t)  \nonumber \\
&=&\lambda \text{ }x(t).\end{aligned}$$ We note that the solutions of these stochastic equations depends on the knowledge of the time evolution of the probability $P(n_{+},n_{-},t)$. One must then solve the Fokker-Planck equation simultaneously with the SDE's. Equivalently one can obtain the time evolution of $\beta (t)$ analytically and therefore the evolution of $P(n_{+},n_{-},t)$ as outlined in (@mike1).

The stochastic differential equations (SDEs) in Eq.(\[sde1\]) can be seen to be a non-extensive SDE generalizations of the Cont and Bouchaud model ( @cont1), in the continuum approximation. Moreover, they are of the statistical feedback form (@mike1 [@borland1]), in that the microscopic stochastic dynamics are coupled to the macroscopic probability distribution $%
P(n_{+},n_{-},t)$. The microscopic stochastic equations and the macroscopic Fokker-Planck equation are known to be equivalent descriptions of a given random process. In this case of anomalous diffusion of the excess demand, the statistical dependence of subsequent demand changes are explicitly represented in the SDE. Hence this statistical model of the excess demand reproduces the stylized facts of real markets, namely power-law distributions and anomalous diffusion as a consequence of the pseudo-additivity of the system entropy.

We have in this article generalized a model of excess demand and price formation to the case of the non-extensive statistics of C. Tsallis. We state the need to go beyond the extensive Gibbs-Boltzmann statistics in modeling real markets from the point of view of the pseudo-additivity of the entropy of statistically dependent subsystems, and the decomposition of the non-factorizable joint probability. This approach to statistically dependent subsystems and statistically correlated variables has been applied directly to the price statistics of real markets (@mike1 [@mike2; @borland1; @borland2; @ramos1]) and shows promise in modeling the stylized facts of these markets, such as the power-law behavior and anomalous diffusion of price changes. We are working to examine this model given data sets of investor demand and market depths from a real market (such as the S&P500) that is known to exhibit the anomalous diffusion and power-law behavior of price changes. We are also recasting this model of investor demand in terms of spins (bias), and moving beyond this number statistics model to an interacting many-investor spin model of markets.

M.D. Johnson and F. Michael acknowledge support from the NSF through grant number DMR99-72683.

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{'source': 'arXiv_out/cond-mat0207376'}