Fokker–Planck equation of distributions of financial returns and power laws

Abstract

Our purpose is to relate the Fokker-Planck formalism proposed by [Friedrich et al., Phys. Rev. Lett. 84, 5224 (2000)] for the distribution of stock market returns to the empirically well-established power law distribution with an exponent in the range 3 − 5.We show how to use Friedrich et al.'s formalism to predict that the distribution of returns is indeed asymptotically a power law with an exponent µ that can be determined from the Kramers-Moyal coefficients determined by Friedrich et al. However, with their values determined for the U.S. dollar-German mark exchange rates, the exponent µ predicted from their theory is found around 12, in disagreement with the often-quoted value between 3 and 5. This could be explained by the fact that the large asymptotic value of 12 does not apply to real data that lie still far from the stationary state of the Fokker-Planck description.Another possibility is that power laws are inadequate. The mechanism for the power law is based on the presence of multiplicative noise across time-scales, which is different from the multiplicative noise at fixed time-scales implicit in the ARCH models developed in the Finance literature.

Patterns in high-frequency FX data: discovery of 12 empirical scaling laws

“…In finance, there is one scaling law that has been widely reported (Müller et al [1990], Mantegna and Stanley [1995], Galluccio et al [1997], Guillaume et al [1997], Ballocchi et al [1999], , Corsi et al [2001], Di Matteo et al [2005]): the size of the average absolute price change (return) is scale-invariant to the time interval of its occurrence. This scaling law has been applied to risk management and volatility modelling (see Ghashghaie et al [1996], Gabaix et al [2003], Sornette [2000], Di Matteo [2007]) even though there has been no consensus amongst researchers for why the scaling law exists (e.g., Bouchaud [2001], Barndorff-Nielsen and Prause [2001], Farmer and Lillo [2004], Lux [2006], Joulin et al [2008]). …”

Nonlinear Fokker-Planck Equation in the Model of Asset Returns

“…experimental data. A more detailed analysis of the power laws u(∆ξ) ∼ (∆ξ) −(1+µ) made in [12] shows that the exponents µ found in the framework of the FPE formalism are not in good agreement with experimentally obtained values.…”

“…Following [1,12,13], we use quadratic approximation for the diffusion coefficient (7) and linear drift (8). Our formalism can be also applied to the NFPE with more general dependence of coefficients on ∆ξ.…”

“…We consider the feedback caused by reciprocal effects of assets at the financial market. In accordance with [8] and by analogy between the exchange rates and the hydrodynamic turbulence [1,12] let us modify the FPE (1) assuming that its coefficients D (1) and D (2) depend on the order parameter described by the first moment…”

“…In accordance with [1,12], where linear form of the drift V ∆ξ (∆ξ, τ ) =α∆ξ gives close approximations of empirical data, we take modified drift coefficient (10) as −α∆ξ +κ ∞ −∞ (∆ξ − y)u(y, τ )dy that results in −α∆ξ − κX u (τ ), α =α − κ, that, in turn, gives (8).…”