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The strategy of managing the pricing processes, in particular managing the dynamics of the price of the underlying asset and its volatility, the prices of indices, shares, options, the magnitude of financial flows, in the method of calculating the company’s rating based on the available quotations of securities, is developed. The article deals with the study of pricing and calculating the volatility of European options with general local-stochastic volatility, applying Taylor series methods for degenerate diffusion processes. The application of this idea requires new approaches caused by degradation difficulties. Price approximation is obtained by solving the Cauchy problem of partial differential equations diffusion with inertia, and the volatility approximation is completely explicit, that is, it does not require special functions. If the payoff of options is a function of only x, then the Taylor series expansion does not depend on t and an analytical expression of the fundamental solution is considerably simplified. Applied an approach to the pricing of derivative securities on the basis of classical Taylor series expansion, when the stochastic process is described by the diffusion equation with inertia (degenerate parabolic equation). Thus, the approximate value of options can be calculated as effectively as the Black-Scholes pricing of derivative securities. On the basis of the solved problem, it is possible to calculate their turns step-by-step. This enables to predict the dynamics of the pricing of derivatives and to create a strategy of behavior at options according to the passage of the process. For each approximation, price volatility is calculated , which makes it possible to take into account all changes in the market and to calculate possible situations. The step-by-step finding of the change in yield and volatility in the relevant analysis enables us to make informed strategic decisions by traders in the financial markets. Ivan Burtnyak (Ukraine), Anna Malytska (Ukraine) BUSINESS PERSPECTIVES LLC “СPС “Business Perspectives” Hryhorii Skovoroda lane, 10, Sumy, 40022, Ukraine www.businessperspectives.org Taylor expansion for derivative securities pricing as a precondition for strategic market decisions Received on: 7th of December, 2017 Accepted on: 17th of January, 2018 INTRODUCTION Modeling and forecasting of pricing dynamics by using financial instrument are an important element of investment activity, as they have a high level of financial leverage and fulfillment of obligations is realized in the future. Derivative prices tend to change and predicting their behavior is becoming more and more complicated. Stochastic processes that are described by the equation of diffusion with inertia are widely used in the theory of mass service, in particular in the theory of queues. Often, such processes occur in financial markets with the pricing of European and Asian options. The theory of pricing derivatives and the study of the behavior of volatility for the analysis of profitability are necessary for flexibility in the adoption of managerial strategic decisions by managers. The validity of strategic decisions allows managers to make step-by-step additional investments in order to maintain strategic positions of the company in the stock market. Typically, a high level of volatility gives managers more opportunities to change their decisions in the future. Volatility is important for trad© Ivan Burtnyak, Anna Malytska, 2018 Ivan Burtnyak, Ph.D. (Economics), Associate Professor, Department of Economic Cybernetics, Vasyl Stefanyk Precarpathian National University, Ukraine. Anna Malytska, Ph.D. (Physics and Mathematics), Associate Professor, Department of Mathematical and Functional Analysis, Vasyl Stefanyk Precarpathian National University, Ukraine. pricing strategy, price dynamics management, European options, profitability, volatility