Ivan Burtnyak

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 Fourier-Bessel series.
In this paper, the authors use the spectral theory to calculate the price of derivatives of financial assets considering that the processes described are 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. They 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.

This article studies the derivatives pricing using a method of spectral analysis, a theory of singular and regular perturbations. Using a risk-neutral assessment, the authors obtain the Cauchy problem, which allows to calculate the approximate price of derivative assets and their volatility based on the diffusion equation with fast and slow variables of nonlocal volatility, and they obtain a model with multidimensional stochastic volatility. Applying a spectral theory of self-adjoint operators in Hilbert space and a theory of singular and regular perturbations, an analytic formula for approximate asset prices is established, which is described by the CEV model with stochastic volatility dependent on l-fast variables and r-slowly variables, l ≥ 1, r ≥ 1, l ∈ N, r ∈ N and a local variable. Applying the Sturm-Liouville theory, Fredholm’s alternatives, as well as the analysis of singular and regular perturbations at different time scales, the authors obtained explicit formulas for derivatives price approximations. To obtain explicit formulae, it is necessary to solve 2l Poisson equations.

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.

Methods of calculating the approximate price of options using instruments of spectral analysis, singular and regular wave theory in the context of influence of fast and slow acting factors are developed. By combining methods from the spectral theory of singular and regular disturbances, one can approximate the price of derivative financial instruments as a schedule of its own functions. The article uses the theory of spectral analysis and the singular and regular theory of perturbations, which are applied to the short-term interest rates described by the Vasicek model with multidimensional stochastic volatility. The approximate price of derivatives and their profitability are calculated. Applying the Sturm-Liouville theory, the Fredholm alternative, and the analysis of singular and regular disturbances in different time scales, explicit formulas were obtained for the approximation of bond prices and yields based on the development of their own functions and eigenvalues of self-adjoint operators using boundary value problems for singular and regular perturbations. The theorem for estimating the accuracy of derivatives price approximation is established. Such a technique, in contrast to existing ones, makes it possible to study the stock market dynamics and to monitor the financial flows in the market. This greatly facilitates the statistical evaluation of their parameters in the process of monitoring the derivatives pricing and the study of volatility behavior for the profitability analysis and taking strategic management decisions on the stock market transactions.

Type of the article: Research Article

Abstract
The Russian invasion of Ukraine has reshaped the functioning of local budgets, creating fiscal challenges for municipalities while simultaneously testing the resilience of the decentralization reform launched in the pre-war period. This study examines the dynamics of local public finance in Ukraine during wartime (2022–2024), focusing on revenue fluctuations, expenditure restructuring, and the emerging role of participatory budgeting as an adaptive tool for community engagement. Using official data from the Ministry of Finance, the State Statistics Service, and the Open Budget Portal, the analysis compares fiscal indicators before and during the full-scale war. Case studies of selected municipalities highlight divergent strategies between frontline and rear communities in balancing defense-related needs, social support for internally displaced persons, and development priorities. Despite reduced revenues and rising security expenditures, local budgets remained stable, largely due to the decentralization framework. In addition to domestic revenues, local budgets increasingly relied on external inflows: international grants, loans, and non-repayable donor assistance. These instruments served as a critical buffer that compensated for the wartime decline in municipal revenues and stabilized key public services. We further examined how these external resources shaped the fiscal resilience of Ukrainian municipalities under wartime conditions. Moreover, participatory budgeting, though often suspended, proved to be a potential mechanism for maintaining public trust and civic involvement under crisis conditions. The study argues that “wartime participatory budgeting” represents a novel phenomenon with implications for both crisis governance and post-war reconstruction.