Critical Transitions in financial models: Bifurcation- and noise-induced phenomena

Abstract

A so-called Critical Transition occurs when a small change in the input of a system leads to a large and rapid response. One class of Critical Transitions can be related to the phenomenon known in the theory of dynamical systems as a bifurcation, where a small parameter perturbation leads to a change in the set of attractors of the system. Another class of Critical Transitions are those induced by noisy increments, where the system switches randomly between coexisting attractors. In this thesis we study bifurcation- and noise-induced Critical Transitions applied to a variety of models in finance and economy. Firstly, we focus on a simple model for the bubbles and crashes observed in stock prices. The bubbles appear for certain values of the sensitivity of the price based on past prices, however, not always as a Critical Transition. Incorporating noise to the system gives rise to additional log-periodic structures which precede a crash. Based on the centre manifold theory we introduce a method for predicting when a bubble in this system can collapse. The second part of this thesis discusses traders' opinion dynamics captured by a recent model which is designed as an extension of a mean-field Ising model. It turns out that for a particular strength of contrarian attitudes, the traders behave chaotically. We present several scenarios of transitions through bifurcation curves giving the scenarios a market interpretation. Lastly, we propose a dynamical model where noise-induced transitions in a double-well potential stand for a company shifting from a healthy state to a defaulted state. The model aims to simulate a simple economy with multiple interconnected companies. We introduce several ways to model the coupling between agents and compare one of the introduced models with an already existing doubly-stochastic model. The main objective is to capture joint defaults of companies in a continuous-time dynamical system and to build a framework for further studies on systemic and individual risk.