Hierarchical and Ultrametric Models of Financial Crashes

Abstract

We explore the log-periodic behavior which is known to precede the critical events of some complex systems. Particularly, we consider the hierarchical model of financial crashes introduced by A. Johansen and D. Sornette, which reproduces the log-periodic power law behavior of the price before the critical point. Much attention is being paid to a problem of critical point invariance which is investigated by comparison of probability density functions of the crash times corresponding to systems with various total numbers of agents. In order to build the ultrametric modification of this model we introduce the dependence of an influence exponent on an ultrametric distance between agents. We found out that for this modification, invariance of the critical point remains true. We also introduce the new pure ultrametric model, which exhibits power law behavior modulated by decreasing-period oscillations.