The evaluation of derivatives of double barrier options of the Bessel processes by methods of spectral analysis

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 main aim of the paper is to measure hedging efficiency using the Short Put Ladder strategy formed by barrier options in the equity market. The researchers hedge full protection against price’s drop, combining the European down and knock-in put options with the lowest exercise price and vanilla or barrier put options with the higher exercise prices. The authors chose the analyzed alternatives according to the requirement of the zero-cost strategy. The aim of the investigated hedging variants is to secure the minimum constant selling price for the underlying asset’s price drop. Theoretical results of this approach were applied in the equity market, i.e., SPDR S&P 500 ETF. The authors analyzed and compared all hedging variants to each other, however, only the selected techniques were presented in the paper. The findings reveal that the barrier options used for managing the equity risk produce significant reductions of that risk. The right combination of options with the strike prices and the barrier levels wisely selected plays a significant role in risk elimination. Finally, according to the findings, the recommendations for potential investors are introduced.

The stable development of Ukraine as the agricultural state actualizes a complex of economic, organizational and legal issues, which are concerned with an implementation of the land relationship. The maximum usage of rental tools is the most effective among the existing budget filling mechanisms. The aim of the article is to conduct a research of land relationship by mechanisms of improving the agricultural lands rent management. The object of a study is the interaction of state institutions at different levels of land lease management. The basis of the study is a cognitive method in the patterns of development of the land relationship. Therefore, in the article, the alternative version of the organizational and economic mechanism for the implementation of land relations was proposed with the aim to improve the existing practice that will facilitate the additional financing of local self-government authorities. At the state regulation level, it is proposed to create an informational electronic database, which should display cadastral numbers and location of land plots, as well as information about land plot owners. Measures of control should be fulfilled by such state authorities as State Geo Cadastre and Ministry of Justice of Ukraine. In order to follow a principle of openness, it has been proved that this database should be public. As a result, methodological and organizational tools are based on the algorithm of lease relationship management as the main source of budget filling for local self-government authorities and main tasks for implementation of administrative-territorial reform that were declared by the government.