Applying Fractal Theory: Solving the Geometric Challenge of Price Change and Scaling in Economics

Abstract

<i>Background</i>: Price changes in economics present significant geometric challenges due to sharp discontinuities, which cannot be efficiently described by continuous processes like Brownian motion. Traditional models often rely on linear assumptions, yet financial data frequently exhibit irregular, complex patterns. Fractal theory, a mathematical framework, offers a more accurate way to describe these fluctuations by revealing the underlying self-similar structures in price changes and scaling phenomena. This study explores the use of fractal geometry to gain deeper insights into market behavior. <i>Objective</i>: The objective is to demonstrate that an alternative model, constructed based on geometric scaling assumptions, offers a more accurate description of price changes in competitive markets. <i>Method</i>: The study combined the scaling principle from fractal geometry with a stable Levy model to formulate an integrated model. The logarithmic transformation of the model was applied over successive price changes to observe the behavior of market prices. <i>Result</i>: The scaling principle asserts that no specific time interval (such as a day or a week) holds inherent significance in competitive markets. Instead, these time features are compensated or arbitrated away, supporting the idea that market behavior is self-similar across different time scales. <i>Conclusion</i>: The scaling principle provides a more reliable framework for modeling price changes and is recommended for consideration in economic analyses.