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The paper deals with the spectral methods to calculate the value of the double barrier option generated by the Bessel diffusion process. This technique enables us to calculate the option price in the form of a Fourier-Bessel series with the corresponding ratio. The autors propose a simple method to estimate options using the Green’s expansion function for boundary value problem for a singular parabolic equation. Thus, the accuracy of the estimation coincides with the accuracy of the convergence of the FourierBessel series. In this paper, the authors use the spectral theory to calculate the price of derivatives of financial assets considering that the processes described by Markov and can be considered in Hilbert spaces. In this work, the authors use the diffusion process to find derivatives prices by introducing them through the Bessel functions of first kind. T ey also examine the Sturm-Liouville problem where the boundary conditions utilize the Bessel functions and their derivatives. All assumptions lead to analytical formulae that are consistent with the empirical evidence and, when implemented in practice, reflect adequately the passage of processes on stock markets. The authors also focus on the financial flows generated by Bessel diffusion processes which are presented in the system of Bessel functions of the first order under the condition that the linear combination of the flow and its spatial derivative are taken into account. Such a presentation enables us to calculate the market value of a share portfolio, provides the measurement of internal volatility in the market at any given time, and allows us to investigate the dynamics of the stock market. The splitting of Green’s function in the system of Bessel functions is presented by an analytical formula which is convenient for calculating the price level of options. Ivan Burtnyak (Ukraine), Anna Malytska (Ukraine) BUSINESS PERSPECTIVES LLC “СPС “Business Perspectives” Hryhorii Skovoroda lane, 10, Sumy, 40022, Ukraine www.businessperspectives.org The evaluation of derivatives of double barrier options of the Bessel processes by methods of spectral analysis Received on: 10th of July, 2017 Accepted on: 21st of August, 2017 The study of the nature and functioning of the fundamentals of options was performed by many scientists, among which should be noted: Merton (1973), Cox et al. (1976), Lorig (2014). General theory developed different approaches to the evaluation of derivatives and one double barrier option depending on assumption stochastic processes they describe, the greatest theoretical achievement in the field of derivatives was to create an option pricing model, known worldwide as a Black-Scholes model (Black & Scholes, 1973). To estimate the value of options, a binomial model was created, which was constructed by Sharpe (1985), which was later generalized by Cox et al. (1985). Recently, a number of new models of pricing options have appeared, © Ivan Burtnyak, Anna Malytska, 2017 Ivan Burtnyak, Ph.D. in Economics, Associate Professor, Department of Economic Cybernetics, Vasyl Stefanyk Precarpathian National University, Ukraine. Anna Malytska, Ph.D. in Physics and Mathematics, Associate Professor, Department of Mathematical and Functional Analysis, Vasyl Stefanyk Precarpathian National University, Ukraine. This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International license, which permits re-use, distribution, and reproduction, provided the materials aren’t used for commercial purposes and the original work is properly cited. spectral theory, barrier option, financial flows, Bessel diffusion process, Bessel functions, Green’s function, singular parabolic operator, infinitesimal operator Keywords