Complexity Research Of The Financial Time Series Based On Fractal Analysis

The financial markets are nonlinear dynamic complex systems with fractal and chaotic structures. The financial data are generally non-stationary time series exhibiting drastic fluctuation characteristics. In recent years, it has become a hot research area that applying fractal statistical methods to study the complexity and volatility in the financial markets. In this thesis, we choose the representative domestic and international financial markets as the research objects and have a comprehensive in-depth study.Taking the CS1300 Index as the research objects of stock index time series in the domestic stock markets, we first use the fractal interpolation method to analyze and predict their fluctuation law and volatility characteristics. Then we apply the fractional dimensions and the Hurst index to depict quantitatively the characteristics of the fluctuation complexity and the long-range correlations of stock index series. Finally, we make a further multifractal analysis for the stock index returns series by means of the MF-DFA and multifractal spectrum analysis methods. The results show that the stock index returns series show the long-range correlation and multifractal characteristics.In the international markets, we choose the spot prices of WTI crude oil and Brent crude oil as well as the US dollar index as the research objects to discuss the cross-correlation relationships between them. First, we use the cross-correlation statistics and the cross-correlation coefficient to explain qualitatively the cross-correlation relationships between the prices of crude oil markets and the US dollar index. Then we make a quantitative analysis by using MF-DCCA for the cross-correlations between them and confirm that the cross-correlation relationships between them have multifractal characteristics. Further, we compare their multifractal strength. At the same time, we have a multifractal analysis for the auto-correlations of each research object. Finally, we study the cause of the multifractality by the shuffled and surrogated series, and indicate that both the long-range correlations and the fat-tailed distributions play important roles in the contributions of multifractality. The dissertation is organized as follows:Chapter 1 is the introduction. The related topic background and research significance are briefly introduced. Moreover, the research status of the fractal theory and methods in the financial market analysis are described.Chapter 2 is the fractal analysis and prediction of stock index time series. Taking the CSI 300 Index as the empirical research objects of Chinese stock markets, we explore the fractal structures of Chinese stock markets and have a reasonable prediction by means of the fractal interpolation model. Then we make a further multifractal analysis for the CSI300 Index returns series by means of the MF-DFA and multifractal spectrum analysis methods. Based on the above researches, we also study the auto-correlations of the stock index returns series.Chapter 3 is devoted to the multifractal detrended cross-correlation analysis of the spot markets of the International crude oil markets and the US dollar index. We choose the Brent crude oik the WTI crude oil and the US dollar index as the research objects. First we use the descriptive statistics to have a preliminary description for their statistical characteristics. Then we apply the cross-correlation test statistics and the cross-correlation coefficients to analyze qualitatively the cross-correlations between them. Further, we use respectively the MF-DCCA and MF-DFA methods to analyze quantitatively the cross-correlations between any two objects and the auto-correlations of any single object. Finally, the causes of the multifractality between them are investigated.Chapter 4 is a summary of this thesis and a prospect for the future research.