Power law distributions and dynamic behaviour of stock markets

P. Richmond
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P. Richmond: Department of Physics, Trinity College, Dublin 2, Ireland

The European Physical Journal B: Condensed Matter and Complex Systems, 2001, vol. 20, issue 4, 523-526

Abstract: Abstract: 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 = F ϕ + G ϕ t where t is the usual Gaussian white noise, i.e.: t t ′ = 2Dδ t-t ′ and t = 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 ϕ = exp{ - Ψ ϕ - ln G(ϕ)} where Ψ ϕ = - F/ G dϕ and Z is a normalization factor. This is explored for the simple case where F ϕ = Jϕ + bϕ2 - cϕ3 and fluctuations characterised by the amplitude G ϕ = ϕ + ɛ when it readily yields for ϕ≫ɛ, a distribution function with power law tails, viz: P ϕ = exp{ 2bϕ-cϕ2 /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

Keywords: PACS. 05.10.Gg Stochastic analysis methods (Fokker-Planck; Langevin; etc.) – 89.65.Gh Economics; business; and financial markets (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1007/PL00011108

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