Log-Periodic Power Law Autonomous Stock Market Model

Abstract

For the empirically well explored LPPLS phenomenon (log-periodic power law with singularity) is known and used as a market crash precursor, we propose an autonomous dynamical theoretical model. First, we explain a non-oscillative trend with singularity. For that, we consider a long-term excessive investment demand we’ve derived from constant leverage investor model. Due to the trend deviation suppression (and due to the noise agents activity that balance trend followers buy or sell actions), we formally write supply side of the market balance equation that in our view is proportional to the trend deviation calculated as a second-order tailor term based on the non-oscillative return rate derivative normalized on gauss volatility deviation. We derive from the parameters of the best constant leverage strategy this conglomerate looks like a logarithmic derivative of the smoothed return rate. Due to the supply-demand balance we come to a sequence of two-four differential equations, first leads to hyperbolic time dependence for the return rate, the next leads to a reciprocal power-law for the price. Then we come to a log-periodic oscillations in a second-order equation.