Power law distributions and dynamic behaviour of stock markets

A simple agent model is introduced by analogy with the mean field approach to the Ising model for a magnetic system. Our model is characterised by a generalised Langevin equation $\dot{\varphi} = F\left( {\varphi} \right) + G\left( {\varphi} \right) \hat{\eta} \left( {t} \right)$ where $\hat{\eta}\left( {t}\right) $ is the usual Gaussian white noise, i.e.: $\left\langle {\hat{\eta}\left( {t}\right) \hat{\eta}\left( {{t} ^{\prime }}\right) }\right\rangle =2D\delta \left( {t-{t}^{\prime }}\right) $ and $\left\langle {\hat{\eta}\left( {t}\right) }\right\rangle =0$. Both the associated Fokker Planck equation and the long time probability distribution function can be obtained analytically. A steady state solution may be expressed as $P\left( {\varphi }\right) =\frac{{1}}{Z}{\rm exp}\{-\Psi \left( {\varphi }\right) -\ln G(\varphi )\}$ where $\Psi \left( {\varphi }\right) =-\frac{{1}}{{D}}\int\limits^{\varphi } {F/\left( {G}\right) ^{2}{\rm d}\varphi }$ and Z is a normalization factor. This is explored for the simple case where $F\left( {\varphi }\right) =J\varphi +b\varphi ^{2}-c\varphi ^{3}$ and fluctuations characterised by the amplitude $G\left( {\varphi }\right) =\varphi +\varepsilon $ when it readily yields for $\varphi \gg \varepsilon $, a distribution function with power law tails, viz: $P\left( {\varphi }\right) =\frac{{1}}{Z{|\varphi |^{1-\frac{{J}}{{D}}}}} {\rm exp}\{\left( {2b\varphi -c\varphi ^{2}}\right) /D\}$. The parameter c ensures convergence of the distribution function for large values of φ. It might be loosely associated with the activity of so-called value traders. The parameter J may be associated with the activity of noise traders. Output for the associated time series show all the characteristics of familiar financial time series providing J< 0 and $D\approx |J|$