Modeling and forecasting of financial instruments dynamics using econometrics models and fractal analysis

The task of forecasting the dynamics of changes in the rates of financial instruments is relevant, since its solution would reduce risks and increase the profitability of operations in financial markets. According to the classical concept of the nature of markets, their pricing processes are stochastic and cannot be predicted. In recent years the general trend in scientific research is econo-physics - the science in the convergence of economics and physics that uses the approaches typical for physical phenomena investigation to analyze economic systems. One of these approaches is fractal analysis. It is based on the fractal market hypothesis, which states that the dynamics of changes in prices of financial instruments is subject to power laws and their prediction is possible. The fractal of financial series is expressed in their property to maintain the trend of changes (so called long memory) for a long time. There are modifications of econometric models that take into account the fractal properties of time series. Their use in domestic markets has not been studied enough: there are few examples of successful predictions obtained using such models; there are no works that focus on the comparison of fractal and non-fractal models on sufficiently large arrays of price data, and it does not allow us to say with certainty about the superiority of models that take into account the fractal properties of series. The main idea of the study is to compare the prediction accuracy of econometric models and their fractal extensions using the same data. The purpose of the research is to verify the hypothesis that taking into account the fractal of financial series, while all other things are equal, allows us to obtain better and more accurate forecasts of financial instruments. The following approaches and methods have been used: fractal analysis and econometric models (ARIMA, GARCH), as well as their modifications, taking into account the property of long memory. For the first time, the accuracy of a large number of similar fractal and non-fractal models has been compared with differences in the nuances of fractal parameter estimation on the same data, the problems of determining the boundaries of financial series sections with different fractal properties and selecting a series (initial or transformed) for evaluating the operator of fractional differentiation of the ARFIMA model have been formulated. We have obtained the following results. The hypothesis of the best predictive ability of fractal models is not rejected, since in most cases the forecast accuracy of ARFIMA and ARFIMA-GARCH was higher than the forecast accuracy of ARIMA and GARCH. The hypothesis of a variable fractal structure of financial series has been confirmed, as evidenced by the variable graphs of local fractal dimensions. The hypothesis of the advantage of models with long memory over models with short memory has not been rejected, since the ARIMA model has demonstrated the least accuracy of forecasts. The hypothesis of the alteration of the fractal properties of time series in the transformation has been confirmed, that has been proven with different values of the Hurst exponent and different graphics of local dimensions. We did not manage to confirm the following hypotheses: the use of local fractal indicators instead of global ones to calculate the fractional differentiation parameter increases the accuracy of forecasts; with the growth of the forecast length, its accuracy decreases; sections of series with long memory are better modeled and forecasted. Possible directions for further research have been formulated: to develop a method for determining the boundaries of local areas of financial series with different fractal properties; to formulate recommendations on which series (initial or transformed) should be evaluated by the operator of fractional differentiation of the ARFIMA model; to verify unconfirmed hypotheses with a change in the research methodology; to study various fractal modifications of GARCH models and identification of conditions for their applicability.