Power law distributions and dynamic behaviour of stock markets

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 < 0 and D≈ | J|.