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

Our purpose is to relate the Fokker–Planck formalism proposed by [Friedrich et al., Phys. Rev. Lett. 84 (2000) 5224] 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 to be 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.