Micro-Foundation Using Percolation Theory of the Finite-Time Singular Behavior of the Crash Hazard Rate in a Class of Rational Expectation Bubbles

Abstract

We present a plausible micro-founded model for the previously postulated power law finite time singular form of the crash hazard rate in the Johansen-Ledoit-Sornette model of rational expectation bubbles. The model is based on a percolation picture of the network of traders and the concept that clusters of connected traders share the same opinion. The key ingredient is the notion that a shift of position from buyer to seller of a sufficiently large group of traders can trigger a crash. This provides a formula to estimate the crash hazard rate by summation over percolation clusters above a minimum size of a power $s^a$ (with $a>1$) of the cluster sizes $s$, similarly to a generalized percolation susceptibility. The power $s^a$ of cluster sizes emerges from the super-linear dependence of group activity as a function of group size, previously documented in the literature. The crash hazard rate exhibits explosive finite-time singular behaviors when the control parameter (fraction of occupied sites, or density of traders in the network) approaches the percolation threshold $p_c$. Realistic dynamics are generated by modelling the density of traders on the percolation network by an Ornstein-Uhlenbeck process, whose memory controls the spontaneous excursion of the control parameter close to the critical region of bubble formation. Our numerical simulations recover the main stylized properties of the JLS model with intermittent explosive super-exponential bubbles interrupted by crashes.