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

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

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, but there is not always an analytical formula for the image of the solution, so we assume that the processes are Markov processes.
Bessel processes are most often used in the theory of mass service, in particular in the theory of queues. Stochastic Bessel processes are of different type with respect to the problems on their own values and their own responsibilities when spectrum of Bessel is continuous or has a finite number of values and continuous part is presenting density through the functions of Whittaker's and Laguerre polynomials considered in many papers and most detailed by Linetsky (2004), where the boundary value problem on the interval 0, x with boundary conditions imposed on the first derivative of the density function for x is considered.
Bessel processes play an important role in financial mathematics, because by nature closely related to the geometric Brownian motion model and processes CIR (Cox-Ingersoll-Ross), they allow a clear representation of the density transition including bond prices and options and facilitate statistical evaluation of process parameters (Cox et al., 1985). Under certain characteristics of the diffusion process, Bessel operator never becomes zero. Several works of Yor (1984) were devoted to these cases. In this paper we consider the cases where derivatives of Bessel processes flow may become zero.
Under these conditions is explained, the excess rate of growth of stock portfolio is determined and the excess rate of growth of the market portfolio provide which a measurement of the internal market volatility at any given time (Linetsky, 2007).
The paper by Coffman et al. (1998) describes the application of Bessel processes in the financial markets and the relation with the integral of the geometric Brownian motion model. Going-Jaeschke and Yor (2003), review was conducted on the pricing of Asian options using Bessel processes.
The diffusion with the Bessel operator was investigated by Davydov and Linetsky (2003), but under other boundary conditions, and in orthogonal systems of functions.

1.
In this paper, we consider one-dimensional diffusion with Bessel process with a drift, which is zero (there is a number of processes of this type where the drift is not equal to zero, but the research can be reduced to processes with zero drift). Such processes are used in solving economic problems in finding short-term interest rates, credit spreads and stochastic volatility of derivatives (Going-Jaeschke et al., 2003).
The problem of this type is considered for the first time. For such problems, there is an unsolved issue when Bessel diffusion has nonlocal volatility dependent on various factors, but such an issue is only partially solved for ordinary diffusion process-es generated by the Brownian motion. Therefore, new approaches and analysis are already required for solving problems of this type.
The approach developed by us can be applied to the research for the pricing of barrier options generated by Bessel processes. To do this, we must consider the financial flows generated by Bessel diffusion processes by expanding the distribution density (the Green's function of the corresponding problem) into the Fourier-Bessel series. Such a development gives ample opportunities for the application of this theory, because the Bessel functions and their derivatives are tabulated and well-studied. We can calculate the size of a market share portfolio and determine the level of internal volatility in the market at any given time, and examine the dynamics of the stock market.
The spectral method is applied to derivative financial instruments, pricing through the presentation of the price of the derivative asset , u t x neutral to the risk of waiting for some function from the future value of the main process X . Process X may represent many economic processes. For example, the size of stocks, the price of the index, a reliable short percentage, etc.
Let us consider the process for which the operator L has the form where p is constant, which is called index, 0. To study L on eigenvalues and eigenfunctions, under certain boundary conditions, we consider the Bessel equation.
The solution of equation (2), except for the partial values of p , is not expressed through elementary functions (in finite form), these non-elementary functions are called Bessel functions. They are widely used in economics, technics and physics. Since the Euler-Bessel equation is linear, its general solution can be written as follows: where 1 2 , v v are two arbitrary linearly independent partial solutions of Euler-Bessel equation, and  (2), is as follows: for integral , p we find another partial solution cos , sin which is the Bessel function of the second kind, which is indefinite for 0 x using the L'Hopital rule, we find the boundary for 0 x and by this number we define the function at zero  Let v x be a solution (2), then the function v x will also be a solution of the equation of this sort Equation (13) is the Bessel equation with the parameter .
Any solution of equation (2), which is the Bessel function, has an infinite set of positive roots, which are close to the roots of the function sin , x the form , n k n , const n -integer (similarly for negative roots, as they are symmetrically relative to the origin of coordinates), if 0, n k then they are simple roots and form a denumerable set.
Bessel functions are alternating sign-rows, so the evaluation can be made using the Leibniz lemma, by which we can determine the accuracy of the approximation.
To find eigenfunctions and eigenvalues let us, consider such a boundary value problem So, we consider the Sturm-Liouville problem. This problem has a single solution. We impose problem situation (6)

Since (2) has its integral
then on the basis of properties of the Bessel function, problem (8) has the solution Keeping in mind the boundary conditions, we have .
Since the Bessel function of the second kind 0 Y x is infinite at zero, then 2 0, C so 1 0 , v x C J x meeting the boundary conditions, we observe 0 0 0. J x We receive the roots of this equation x Let us consider the Bessel process described and the boundary condition where K is strike, the process is homogeneous, From the Sturm-Liouville theory it follows that 1 .
x K e The financial flows are as follows: We calculated the decomposition of the financial flow in the system of Bessel functions p J of the first kind, but the distribution of flows is set by Green's function of the corresponding problem. Therefore, it is convenient for calculating to expand the Green's function in the Bessel system. The process we consider corresponds with heterogeneous boundary value problem

2.
Since the problem of evaluation and research of two-dimensional barrier options is reduced to the study and solution of the boundary value problem, (Burtnyak et al., 2013, 2016).