Χρήση μορφοκλασματικών μεθόδων για τη μελέτη χρονολογικών δεδομένων στα χρηματοοικονομικά
Use of fractal methods for the study of time series financial data
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Abstract
This thesis examines the concept of fractal methods and their use in the analysis of time series data. The term fractal describes a certain category of geometrical objects, which are characterized by some rather interesting attributes such as self-similarity and the existence of one or more defining parameters, which are called dimensions. Those dimensions are called fractal dimensions and can take values that are not necessarily integer. A mono-parametric fractal is characterized by its fractal dimension, which is also called Hausdorff dimension. Other fractal dimension estimation methods also exist, and the most important of them is the box counting dimension. Another important concept inspired by the fractal theory is the Hurst exponent, which is a method for estimating the degree of correlation between the observations of a time series. For the cases where more than one fractal dimension exists other methods have been defined that allow their study. The most important of them are the local Holder exponent and the multifractal spectrum. Multifractal and a generalization of Brownian motion called fractional Brownian motion are the tools that form the basis for a very important financial mode called the Multifractal Model of Asset Returns. All those concepts are examined using empirical data from the Athens Stock Market Exchange.