https://doi.org/10.1007/PL00011108
Power law distributions and dynamic behaviour of stock markets
Department of Physics, Trinity College, Dublin 2, Ireland
Corresponding author: a richmond@tcd.ie
Received:
25
July
2000
Published online: 15 April 2001
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
where 
is the usual Gaussian white
noise, i.e.: 
and 
. 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
where 
and Z is a normalization factor. This is explored for the simple
case where 
and fluctuations characterised by the amplitude 
when it readily yields for 
,
a distribution function with power law tails, viz:
.
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
.
PACS: 05.10.Gg – Stochastic analysis methods (Fokker-Planck, Langevin, etc.) / 89.65.Gh – Economics, business, and financial markets
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2001
