Burtniak I. Spectral methods of studying the behavior of stock market volatility.

The dissertation is devoted to the establishment and substantiation of theoretical and methodological provisions for the development and implementation of a complex of economic and mathematical models for analyzing and forecasting volatility of stock market instruments aimed at achieving macroeconomic stabilization and dynamic stock market development.A model for finding the price of options created by diffusion is developed, which increases powerfully to find the size of a market stock portfolio and to determine the value of internal volatility in the market at any given time, as well as to investigate the dynamics of the stock market and to monitor financial flows through the expansion of the system of Bessel functions first kind. This greatly facilitates the statistical estimation of their parameters in the process of monitoring the pricing of derivatives and the study of the behavior of volatility for the analysis of profitability and the adoption of strategic management decisions for the implementation of transactions in the stock market.The methods of calculating the approximate price of options with the help of instruments of spectral analysis, singular and regular wave theory in the case of the influence of rapidly and slowly operating factors are developed. By combining methods from the spectral theory of singular and regular perturbations, one can approximate the price of derivative financial instruments, as a schedule by its own functions.A model for finding stock indexes corresponding to the dynamics of the stock market and the size of financial flows, which are described by Kolmogorov's processes, is developed. Such a model allows finding the prices of derivatives and their volatility, as well as minimizing speculative changes in pricing, analyzing the processes in the stock market and taking concrete steps to improve the situation with regard to optimizing financial strategies, increasing the accuracy of the forecast and adopting sound management strategic decisions by the participants on stock market.