Critical phenomena with renormalization group analysis of a hierarchical model of financial crashes

Financial market models are able to help the investors foresee the risk of a financial market crash and reduce the probability of its occurrence. Modelling in financial markets is categorized into microscopic models and macroscopic models. The microscopic models study the mechanisms behind the market and their behaviour. These models assist in an understanding of the causes of financial market crashes. Macroscopic models find the disciplines and rules from the historical macroscopic data for the prediction of market trends, directions and crashes. They aim to provide forecasts of when the crashes should occur. Except for large fluctuations, the stock market price or index movements can be characterized by a random walk. The stock price trajectory possesses the properties of self-similarity and scale invariance, and hence mimics fractals. Furthermore, the stock price movement is a manifestation of the actions and interactions of stock traders. The hierarchical model with fractal structures, representing the interaction structure of the stock traders, is applicable to the study of the microscopic mechanisms of stock price movements. In the language of statistical mechanics, stock market crashes are viewed as critical phenomena where a crash occurs only at the critical point with a phase transition. The renormalization group is a mathematical apparatus that allows the decomposition of a macroscopic problem viewed at different scales. The renormalization group analysis of the hierarchical model finds that the time to reach the critical point of a system is a function of the interaction degree of stock traders in a power law. Moreover, the hierarchical model with the renormalization group formalism shows that the behavior of the fraction of all stock traders putting buy orders over time follows a power law coupled with log-periodic oscillations. Based on the renormalization group analysis results of the hierarchical model, a log-periodic power law model is derived by constructing an renormalization group formalism from the risk-driven model. In the log-periodic power law model, the expected time to crash is always finite. Therefore, market crashes are inevitable, but the existence of bubbles and crash risks is foreseen by the model.