Fractal Theory And Its Applications In Financial Market Analysis

Fractals is a new subject starting from 1970s which is an active branch of nonlinear science. It focuses on many complex patterns and phenomena with certain law,which widely exist in the nature and social life. It offers us a new method to research the self-similarity objects and irregular phenomena. Some phenomena which can't be explained with Euclidean geometry could be interpreted well with fractal geometry. Fractal theory and its methodology provide people a new view and new ideas to know the world and it made our way of thinking enter into the nonlinear stage. So fractal geometry has become a powerful tool of theory for studying and depicting the complex problems in the nature,in social life and in engineering techniques.In this dissertation,Chapter one gives some preliminaries in which we explain the situation of study in our subject and simply introduce the basic concepts and theorems of the fractal geometry,such as,fractal space,iterated function systems (IFSs) and fractal interpolation functions (FIFs). In Chapter two,we research the errors of FIFs based on the changes of vertical scaling factors. The errors of the FIFs caused by the changes of vertical scaling factors are analyzed quantitatively,and the concrete error expression is presented,meanwhile the upper bound of errors is estimated. In addition,by means of the numerical experiments,the relations between the changes of vertical scaling factors and the values of FIFs are demonstrated clearly. In Chapter three,we apply fractal interpolation theory to financial market analysis,and make a fractal analysis to the series of stock prices. We construct fractal interpolation models to analyze and predict the changes of stock prices. As an example,we apply the constructed models to analyze the law of variations and forecast the trend of changes for the stock prices of Qingdao Haier,which is one of the listed companies of China. In addition,we also describe the fluctuation property and long-term correlativity by fractal dimension and Hurst index for the time series of Qingdao Haier's stock prices. Since single fractal only can describe the macroscopical situation of the changes of the series of stock prices,and it can't depict the complicated structure information of fluctuation,but multifractal spectrum can analyze the micro finance properties of stock price sequences,we use multifractal spectrum to investigate the volatility of stock index in chapter four. Taking 5min high frequency trading data of SSE (ShangHai Stock Exchange) as research objects,we discuss the volatility situations of stock index in four different cases using multifractal spectrum,and analyze the relations between parameters'changes and the fluctuations of stock index,and also forecast the trend of stock index in the short future. Chapter five,the last part of this thesis,makes a summary for this paper and prospects the future development of fractals.