Alpha-stable paradigm in financial markets

Statistical models of financial data series and algorithms of forecasting and investment are the topic of this research. The objects of research are the historical data of financial securities, statistical models of stock returns, parameter estimation methods, effects of self-similarity and multifractality, and algorithms of financial portfolio selection. The numerical methods (MLE and robust) for parameter estimation of stable models have been created and their efficiency were compared. Complex analysis methods of testing stability hypotheses have been created and special software was developed (nonparametric distribution fitting tests were performed and homogeneity of aggregated and original series was tested; theoretical and practical analysis of self-similarity and multifractality was made). The passivity problem in emerging markets is solved by introducing the mixed-stable model. This model generalizes the stable financial series modeling. 99% of the Baltic States series satisfy the mixed stable model proposed. Analysis of stagnation periods in data series was made. It has been shown that lengths of stagnation periods may be modeled by the Hurwitz zeta law (insteed of geometrical). Since series of the lengths of each run are not geometrically distributed, the state series must have some internal dependence (Wald-Wolfowitz runs test corroborates this as-sumption). The inner series dependence was tested by the Hoel criterion on the order of the Markov chain. It has been concluded that there are no zero order Markov chain series or Bernoulli scheme series. A new mixed-stable model with dependent states has been proposed and the formulas for probabilities of calculating states (zeros and units) have been obtained. Methods of statistical relationship measures (covariation and codifference) between shares returns were studied and algorithms of significance were introduced.