An elementary model of price dynamics in a financial market: Distribution, Multiscaling & Entropy

Stylized facts of empirical assets log-returns $Z$ include the existence of (semi) heavy tailed distributions $f_Z(z)$ and a non-linear spectrum of Hurst exponents $\tau(\beta)$. Empirical data considered are daily prices of 10 large indices from 01/01/1990 to 12/31/2004. We propose a stylized model of price dynamics which is driven by expectations. The model is a multiplicative random process with a stochastic, state-dependent growth rate which establishes a negative feedback component in the price dynamics. This 0-order model implies that the distribution of log-returns is Laplacian $f_Z(z) \sim \exp(-\frac{|z|}{\alpha})$, whose single parameter $\alpha$ can be regarded as a measure for the long-time averaged liquidity in the respective market. A comparison with the (more general) Weibull distribution shows that empirical daily log returns are close to being Laplacian distributed. The spectra of Hurst exponents of both, empirical data $\tau_{emp}$ and simulated data due to our model $\tau_{theor}$, are compared. Due to the finding of non-linear Hurst spectra, the Renyi entropy (RE) $R_\beta(f_Z)$is considered. An explicit functional form of the RE for an exponential distribution is derived. Theoretical REs of simulated asset return trails are in good agreement with the RE estimated from empirical returns.