In this paper we will present an effective numerical method for the Black‐Scholes equation with t... more In this paper we will present an effective numerical method for the Black‐Scholes equation with transaction costs for the limiting price u(s, t;a). The technique combines the Rothe method with a two‐grid (coarse‐fine) algorithm for computation of numerical solutions to initial boundary‐values problems to this equation. Numerical experiments for comparison the accuracy ant the computational cost of the method with other known numerical schemes are discussed.
This work deals with numerical solution of option pricing Heston model. We propose a new transfor... more This work deals with numerical solution of option pricing Heston model. We propose a new transformation of the independent variables to remove the mixed derivative. As a result, the rectangular domain, on which the model is defined, alters to right-angle trapezoid. Then, we construct positivity preserving finite difference schemes on seven point stencil. Results from numerical simulations are presented.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Journal of Computational and Applied Mathematics, Feb 1, 2015
ABSTRACT We construct a first order in time and second order in space, positivity preserving nume... more ABSTRACT We construct a first order in time and second order in space, positivity preserving numerical method for a generalized Hoggard–Whalley–Wilmott, Leland’s model. We develop the hyperbolic–parabolic operator splitting method, using a kernel based algorithm for the parabolic part and van Leer flux limiter approach for the hyperbolic sub-problem. Properties of the proposed algorithms are discussed. Various numerical examples confirm the efficiency of the proposed method and verify the theoretical statements
This paper provides a numerical investigation for European options under parabolic-ordinary syste... more This paper provides a numerical investigation for European options under parabolic-ordinary system modeling markets to liquidity shocks. Our main results concern construction and analysis of fourth order in space compact finite difference schemes (CFDS). Numerical experiments using Richardson extrapolation in time are discussed.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
In this manuscript we present two-grid algorithms for the American option pricing problem with a ... more In this manuscript we present two-grid algorithms for the American option pricing problem with a smooth penalty method where the variational inequality, associated with the optimal stopping time problem, is approximated with a nonlinear Black-Scholes equation. In order to compute the numerical solution of the latter unconstrained problem we must solve a system of nonlinear algebraic equations resulting from the discretization by e.g. the finite difference or the finite element method. We propose two-grid algorithms as we first solve the nonlinear system on a coarse grid with mesh size h c and further a linearized system on a fine grid with mesh size h f , satisfying h f = O((h c) 2 k), k = 1, 2,. . ., where k is the number of Newton iterations. Numerical experiments illustrate the computational efficiency of the algorithms.
ABSTRACT In this paper we consider a pricing model that predicts increased implied volatility wit... more ABSTRACT In this paper we consider a pricing model that predicts increased implied volatility with minimal assumptions beyond those of the Black-Scholes theory. It is described by systems of nonlinear Black-Scholes equations. We propose a two-grid algorithm that consists of two steps: solving the coupled Partial Differential Equations (PDEs) problem on a coarse grid and then solving a number of decoupled sub-problems on a fine mesh by using the coarse grid solution to linearise each PDE of the system. Numerical experiments illustrate the efficiency of the method and validate the related theoretical analysis. Loading... Numerical Methods and ApplicationsNumerical Methods and Applications Look Inside Other actions Reprints and Permissions Export citation About this Book Add to Papers In this paper we consider a pricing model that predicts increased implied volatility with minimal assumptions beyond those of the Black-Scholes theory. It is described by systems of nonlinear Black-Scholes equations. We propose a two-grid algorithm that consists of two steps: solving the coupled Partial Differential Equations (PDEs) problem on a coarse grid and then solving a number of decoupled sub-problems on a fine mesh by using the coarse grid solution to linearise each PDE of the system. Numerical experiments illustrate the efficiency of the method and validate the related theoretical analysis. Loading... Numerical Methods and ApplicationsNumerical Methods and Applications Look Inside Other actions Reprints and Permissions Export citation About this Book Add to Papers In this paper we consider a pricing model that predicts increased implied volatility with minimal assumptions beyond those of the Black-Scholes theory. It is described by systems of nonlinear Black-Scholes equations. We propose a two-grid algorithm that consists of two steps: solving the coupled Partial Differential Equations (PDEs) problem on a coarse grid and then solving a number of decoupled sub-problems on a fine mesh by using the coarse grid solution to linearise each PDE of the system. Numerical experiments illustrate the efficiency of the method and validate the related theoretical analysis. Loading... Numerical Methods and ApplicationsNumerical Methods and Applications Look Inside Other actions Reprints and Permissions Export citation About this Book Add to Papers In this paper we consider a pricing model that predicts increased implied volatility with minimal assumptions beyond those of the Black-Scholes theory. It is described by systems of nonlinear Black-Scholes equations. We propose a two-grid algorithm that consists of two steps: solving the coupled Partial Differential Equations (PDEs) problem on a coarse grid and then solving a number of decoupled sub-problems on a fine mesh by using the coarse grid solution to linearise each PDE of the system. Numerical experiments illustrate the efficiency of the method and validate the related theoretical analysis Share
We consider the numerical valuation of European options in a market subject to liquidity shocks. ... more We consider the numerical valuation of European options in a market subject to liquidity shocks. Natural boundary conditions are derived on the truncated boundary. We study the fully implicit scheme for this market model, by use of different algorithms, based on the Newton and the Picard iterations at each time step. To validate the efficiency of the time-stepping and the theoretical results, various appropriate numerical experiments are performed.
ABSTRACT We present an algorithm for approximate solutions to certain nonlinear model equations f... more ABSTRACT We present an algorithm for approximate solutions to certain nonlinear model equations from financial mathematics, using kernels techniques (fundamental solution, Green’s function) for the linear Black-Scholes operator as a basis of the computation. Numerical experiments for comparison the accuracy of the algorithms with other known numerical schemes are discussed. Finally, observations are given.
Studies in computational intelligence, Aug 8, 2020
In this work we present a numerical study of a non-linear space-fractional parabolic system, desc... more In this work we present a numerical study of a non-linear space-fractional parabolic system, describing thermoelastic waves propagation in fluid saturated porous media with non-local Darcy law. We use implicit cell-centered finite difference method for numerical discretizations, combined with \(L1-2\) formula for the fractional derivative approximation. We apply Newton’s method to compute the finite difference solution. We solve a linear system of algebraic equation in block-matrix form on each iteration, using Schur complement. Numerical experiments attest good attributes of the proposed numerical method.
In this paper, we consider a class of non-linear models in mathematical finance, where the volati... more In this paper, we consider a class of non-linear models in mathematical finance, where the volatility depends on the second spatial derivative of the option value. We study the convergence and realization of the constructed, on a fitted non-uniform meshes, implicit difference schemes. We implement various Picard and Newton iterative processes. Numerical experiments are discussed.
This chapter concerns the numerical pricing of European options for markets with liquidity shocks... more This chapter concerns the numerical pricing of European options for markets with liquidity shocks. We derive and analyze high-order weighted compact finite difference schemes (WCFDS). Numerical simulations for the price and Greeks, using WCFDS combined with Richardson extrapolation in time are presented.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
This paper concerns efficient \(\sigma \) - weighted (\(0<\sigma <1\)) time-semidiscretizat... more This paper concerns efficient \(\sigma \) - weighted (\(0<\sigma <1\)) time-semidiscretization quasilinearization technique for numerical solution of Richards’ equation. We solve the classical and a new \(\alpha \) - time-fractional (\(0<\alpha <1\)) equation, that models anomalous diffusion in porous media. High-order approximation of the \(\alpha =2(1-\sigma )\) fractional derivative is applied. Numerical comparison results are discussed.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
In this paper we will present an effective numerical method for the Black‐Scholes equation with t... more In this paper we will present an effective numerical method for the Black‐Scholes equation with transaction costs for the limiting price u(s, t;a). The technique combines the Rothe method with a two‐grid (coarse‐fine) algorithm for computation of numerical solutions to initial boundary‐values problems to this equation. Numerical experiments for comparison the accuracy ant the computational cost of the method with other known numerical schemes are discussed.
This work deals with numerical solution of option pricing Heston model. We propose a new transfor... more This work deals with numerical solution of option pricing Heston model. We propose a new transformation of the independent variables to remove the mixed derivative. As a result, the rectangular domain, on which the model is defined, alters to right-angle trapezoid. Then, we construct positivity preserving finite difference schemes on seven point stencil. Results from numerical simulations are presented.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Journal of Computational and Applied Mathematics, Feb 1, 2015
ABSTRACT We construct a first order in time and second order in space, positivity preserving nume... more ABSTRACT We construct a first order in time and second order in space, positivity preserving numerical method for a generalized Hoggard–Whalley–Wilmott, Leland’s model. We develop the hyperbolic–parabolic operator splitting method, using a kernel based algorithm for the parabolic part and van Leer flux limiter approach for the hyperbolic sub-problem. Properties of the proposed algorithms are discussed. Various numerical examples confirm the efficiency of the proposed method and verify the theoretical statements
This paper provides a numerical investigation for European options under parabolic-ordinary syste... more This paper provides a numerical investigation for European options under parabolic-ordinary system modeling markets to liquidity shocks. Our main results concern construction and analysis of fourth order in space compact finite difference schemes (CFDS). Numerical experiments using Richardson extrapolation in time are discussed.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
In this manuscript we present two-grid algorithms for the American option pricing problem with a ... more In this manuscript we present two-grid algorithms for the American option pricing problem with a smooth penalty method where the variational inequality, associated with the optimal stopping time problem, is approximated with a nonlinear Black-Scholes equation. In order to compute the numerical solution of the latter unconstrained problem we must solve a system of nonlinear algebraic equations resulting from the discretization by e.g. the finite difference or the finite element method. We propose two-grid algorithms as we first solve the nonlinear system on a coarse grid with mesh size h c and further a linearized system on a fine grid with mesh size h f , satisfying h f = O((h c) 2 k), k = 1, 2,. . ., where k is the number of Newton iterations. Numerical experiments illustrate the computational efficiency of the algorithms.
ABSTRACT In this paper we consider a pricing model that predicts increased implied volatility wit... more ABSTRACT In this paper we consider a pricing model that predicts increased implied volatility with minimal assumptions beyond those of the Black-Scholes theory. It is described by systems of nonlinear Black-Scholes equations. We propose a two-grid algorithm that consists of two steps: solving the coupled Partial Differential Equations (PDEs) problem on a coarse grid and then solving a number of decoupled sub-problems on a fine mesh by using the coarse grid solution to linearise each PDE of the system. Numerical experiments illustrate the efficiency of the method and validate the related theoretical analysis. Loading... Numerical Methods and ApplicationsNumerical Methods and Applications Look Inside Other actions Reprints and Permissions Export citation About this Book Add to Papers In this paper we consider a pricing model that predicts increased implied volatility with minimal assumptions beyond those of the Black-Scholes theory. It is described by systems of nonlinear Black-Scholes equations. We propose a two-grid algorithm that consists of two steps: solving the coupled Partial Differential Equations (PDEs) problem on a coarse grid and then solving a number of decoupled sub-problems on a fine mesh by using the coarse grid solution to linearise each PDE of the system. Numerical experiments illustrate the efficiency of the method and validate the related theoretical analysis. Loading... Numerical Methods and ApplicationsNumerical Methods and Applications Look Inside Other actions Reprints and Permissions Export citation About this Book Add to Papers In this paper we consider a pricing model that predicts increased implied volatility with minimal assumptions beyond those of the Black-Scholes theory. It is described by systems of nonlinear Black-Scholes equations. We propose a two-grid algorithm that consists of two steps: solving the coupled Partial Differential Equations (PDEs) problem on a coarse grid and then solving a number of decoupled sub-problems on a fine mesh by using the coarse grid solution to linearise each PDE of the system. Numerical experiments illustrate the efficiency of the method and validate the related theoretical analysis. Loading... Numerical Methods and ApplicationsNumerical Methods and Applications Look Inside Other actions Reprints and Permissions Export citation About this Book Add to Papers In this paper we consider a pricing model that predicts increased implied volatility with minimal assumptions beyond those of the Black-Scholes theory. It is described by systems of nonlinear Black-Scholes equations. We propose a two-grid algorithm that consists of two steps: solving the coupled Partial Differential Equations (PDEs) problem on a coarse grid and then solving a number of decoupled sub-problems on a fine mesh by using the coarse grid solution to linearise each PDE of the system. Numerical experiments illustrate the efficiency of the method and validate the related theoretical analysis Share
We consider the numerical valuation of European options in a market subject to liquidity shocks. ... more We consider the numerical valuation of European options in a market subject to liquidity shocks. Natural boundary conditions are derived on the truncated boundary. We study the fully implicit scheme for this market model, by use of different algorithms, based on the Newton and the Picard iterations at each time step. To validate the efficiency of the time-stepping and the theoretical results, various appropriate numerical experiments are performed.
ABSTRACT We present an algorithm for approximate solutions to certain nonlinear model equations f... more ABSTRACT We present an algorithm for approximate solutions to certain nonlinear model equations from financial mathematics, using kernels techniques (fundamental solution, Green’s function) for the linear Black-Scholes operator as a basis of the computation. Numerical experiments for comparison the accuracy of the algorithms with other known numerical schemes are discussed. Finally, observations are given.
Studies in computational intelligence, Aug 8, 2020
In this work we present a numerical study of a non-linear space-fractional parabolic system, desc... more In this work we present a numerical study of a non-linear space-fractional parabolic system, describing thermoelastic waves propagation in fluid saturated porous media with non-local Darcy law. We use implicit cell-centered finite difference method for numerical discretizations, combined with \(L1-2\) formula for the fractional derivative approximation. We apply Newton’s method to compute the finite difference solution. We solve a linear system of algebraic equation in block-matrix form on each iteration, using Schur complement. Numerical experiments attest good attributes of the proposed numerical method.
In this paper, we consider a class of non-linear models in mathematical finance, where the volati... more In this paper, we consider a class of non-linear models in mathematical finance, where the volatility depends on the second spatial derivative of the option value. We study the convergence and realization of the constructed, on a fitted non-uniform meshes, implicit difference schemes. We implement various Picard and Newton iterative processes. Numerical experiments are discussed.
This chapter concerns the numerical pricing of European options for markets with liquidity shocks... more This chapter concerns the numerical pricing of European options for markets with liquidity shocks. We derive and analyze high-order weighted compact finite difference schemes (WCFDS). Numerical simulations for the price and Greeks, using WCFDS combined with Richardson extrapolation in time are presented.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
This paper concerns efficient \(\sigma \) - weighted (\(0<\sigma <1\)) time-semidiscretizat... more This paper concerns efficient \(\sigma \) - weighted (\(0<\sigma <1\)) time-semidiscretization quasilinearization technique for numerical solution of Richards’ equation. We solve the classical and a new \(\alpha \) - time-fractional (\(0<\alpha <1\)) equation, that models anomalous diffusion in porous media. High-order approximation of the \(\alpha =2(1-\sigma )\) fractional derivative is applied. Numerical comparison results are discussed.
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
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