Spectral study of options based on CEV model with multidimensional volatility
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Author(s)Ivan BurtnyakLink to ORCID Index: https://orcid.org/0000-0002-9440-1467
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Anna MalytskaLink to ORCID Index: https://orcid.org/0000-0002-5811-9288
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DOIhttp://dx.doi.org/10.21511/imfi.15(1).2018.03
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Article InfoVolume 15 2018, Issue #1, pp. 18-25
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208 Downloads
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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.
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JEL Classification (Paper profile tab)G11, G13, G32
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References19
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Tables0
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Figures0
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The evaluation of derivatives of double barrier options of the Bessel processes by methods of spectral analysis
Ivan Burtnyak ,
Anna Malytska
doi: http://dx.doi.org/10.21511/imfi.14(3).2017.12
Investment Management and Financial Innovations Volume 14, 2017 Issue #3 pp. 126-134 Views: 912 Downloads: 253 TO CITE АНОТАЦІЯThe 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. -
Finding the derivative price using the Vasicek model with multidimensional stochastic volatility
Ivan Burtnyak ,
Anna Malytska
doi: http://dx.doi.org/10.21511/dm.17(4).2019.02
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.
