Mathematical Analysis of Financial Bubbles through Fractal Dimensionality

Purpose. Financial bubbles and crises are complex phenomena related to the deviation of asset prices from their fundamental value, which requires the development of new approaches to their analysis. The aim of the paper is to investigate financial bubbles through fractal analysis techniques, including fractal dimension calculation, to identify anomalies and predict market crashes. Methods. The methods used include RS analysis (Hurst method) to assess the persistence of time series, Box-count method to determine the fractal dimension and multifractal analysis to study the local features of market dynamics. Novelty. The novelty of the study lies in the comprehensive application of fractal analysis for early detection of financial bubbles, as well as in comparing the effectiveness of different methods. Results. The results show that the increase in fractal dimension correlates with the periods of bubble formation, and its sharp changes can serve as indicators of crises, which is confirmed by the examples of the dot-com crisis (2000) and the financial crisis (2008). Practical relevance. The significance of the work lies in the possibility of using the proposed approach to create systems of early warning of crises, algorithmic trading and risk management.