Využití teorie fraktálů při modelování volatility

This Master thesis deals with an idea of volatility modeling. The theory of fractals and multifractality is a relatively new approach which is successfully applied in volatility simulation and prediction. The first chapter defines the basic theoretical concepts such as fractal, self-similarity, moment scaling, fractional Brownian motion and multifractality. The second chapter explains the most recognized multifractal models, which are MMAR model, MSM process and MRW model. The greatest emphasis is placed on the MSM process, since it is the most researched and the most recognized multifractal model. MSM process is very attractive for the volatility modeling, as it exhibits the main empirical properties of volatility, which are heavy tails of the probability distribution of returns, persistence and jumps in volatility, as well as multiscaling. The third part of the text deals with comparing MSM and GARCH models regarding to their simulation capabilities. Based on theoretical and empirical part of this study, we conclude that MSM process is significantly better than the GARCH model in the volatility modeling.