scholarly journals A Finite Difference Scheme for Pricing American Put Options under Kou's Jump-Diffusion Model

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jian Huang ◽  
Zhongdi Cen ◽  
Anbo Le
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Diffusion Model ◽  
Finite Difference Scheme ◽  
Jump Diffusion ◽  
Uniform Mesh ◽  
Put Options ◽  
American Put Options ◽  
Jump Diffusion Model ◽  
American Put

We present a stable finite difference scheme on a piecewise uniform mesh along with a penalty method for pricing American put options under Kou's jump-diffusion model. By adding a penalty term, the partial integrodifferential complementarity problem arising from pricing American put options under Kou's jump-diffusion model is transformed into a nonlinear parabolic integro-differential equation. Then a finite difference scheme is proposed to solve the penalized integrodifferential equation, which combines a central difference scheme on a piecewise uniform mesh with respect to the spatial variable with an implicit-explicit time stepping technique. This leads to the solution of problems with a tridiagonal M-matrix. It is proved that the difference scheme satisfies the early exercise constraint. Furthermore, it is proved that the scheme is oscillation-free and is second-order convergent with respect to the spatial variable. The numerical results support the theoretical results.

A parameter-uniform finite difference scheme for singularly perturbed parabolic problem with two small parameters

2021 ◽  
Author(s):  
Tesfaye Aga Bullo ◽  
Guy Aymard Degla ◽  
Gemechis File Duressa
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Variable Parameter ◽  
Accurate Solution ◽  
Singularly Perturbed ◽  
Finite Difference Approximation ◽  
Parabolic Problems ◽  
Difference Approximation ◽  
Uniform Mesh

A parameter-uniform finite difference scheme is constructed and analyzed for solving singularly perturbed parabolic problems with two parameters. The solution involves boundary layers at both the left and right ends of the solution domain. A numerical algorithm is formulated based on uniform mesh finite difference approximation for time variable and appropriate piecewise uniform mesh for the spatial variable. Parameter-uniform error bounds are established for both theoretical and experimental results and observed that the scheme is second-order convergent. Furthermore, the present method produces a more accurate solution than some methods existing in the literature.   

scholarly journals A Robust Spline Collocation Method for Pricing American Put Options

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Zhongdi Cen ◽  
Anbo Le ◽  
Aimin Xu
Keyword(s):  
Maximum Principle ◽  
Collocation Method ◽  
Discrete Maximum Principle ◽  
Original Form ◽  
Uniform Mesh ◽  
Spline Collocation ◽  
Put Options ◽  
Spline Collocation Method ◽  
American Put Options ◽  
American Put

In this paper a robust numerical method is proposed for pricing American put options. The Black-Scholes differential operator in the original form is discretized by using a quadratic spline collocation method on a piecewise uniform mesh for the spatial discretization and the implicit Euler scheme for the time discretization. The position of collocation points is chosen so that the spline difference operator satisfies the discrete maximum principle, which guarantees that the scheme is maximum-norm stable. The error estimation is derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets. It is proved that the scheme is second-order convergent with respect to the spatial variable and first-order convergent with respect to the time variable. Numerical results demonstrate that the scheme is stable and accurate.

scholarly journals SCHEMES CONVERGENT Ε-UNIFORMLY FOR PARABOLIC SINGULARLY PERTURBED PROBLEMS WITH A DEGENERATING CONVECTIVE TERM AND A DISCONTINUOUS SOURCE

2015 ◽  
Vol 20 (5) ◽  
pp. 641-657 ◽  
Author(s):  
Carmelo Clavero ◽  
Jose Luis Gracia ◽  
Grigorii I. Shishkin ◽  
Lidia P. Shishkina
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Euler Method ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Interior Layer ◽  
Convective Term ◽  
Singularly Perturbed Problems ◽  
First Order

We consider the numerical approximation of a 1D singularly perturbed convection-diffusion problem with a multiply degenerating convective term, for which the order of degeneracy is 2p + 1, p is an integer with p ≥ 1, and such that the convective flux is directed into the domain. The solution exhibits an interior layer at the degeneration point if the source term is also a discontinuous function at this point. We give appropriate bounds for the derivatives of the exact solution of the continuous problem, showing its asymptotic behavior with respect to the perturbation parameter ε, which is the diffusion coefficient. We construct a monotone finite difference scheme combining the implicit Euler method, on a uniform mesh, to discretize in time, and the upwind finite difference scheme, constructed on a piecewise uniform Shishkin mesh condensing in a neighborhood of the interior layer region, to discretize in space. We prove that the method is convergent uniformly with respect to the parameter ε, i.e., ε-uniformly convergent, having first order convergence in time and almost first order in space. Some numerical results corroborating the theoretical results are showed.

MHD flows on irregular boundary over a diverging channel with viscous dissipation effect

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Iyyappan G. ◽  
Abhishek Kumar Singh
Keyword(s):  
Heat Transfer ◽  
Finite Difference ◽  
Difference Scheme ◽  
Skin Friction ◽  
Viscous Dissipation ◽  
Finite Difference Scheme ◽  
Uniform Mesh ◽  
Irregular Boundary ◽  
Content Type ◽  
Diverging Channel

Purpose The purpose of this paper is to analyse the force convection laminar boundary layer flow on irregular boundary in diverging channel with the presence of magnetic field effects. Effects of various fluid parameters such as suction/injection, viscous dissipation, magnetic parameter and heat source/sink on velocity and temperature profiles are numerically analyzed. Moreover, numerically investigated on skin-friction and heat transfer coefficients when suction/injection occur. Design/methodology/approach The governing coupled partial differential equations are transformed to dimensionless form using non-similarity transformations. The non-dimensional partial differential equations are linearized by quasi-linearization technique and solved by varga's algorithm with numerical finite difference scheme on a non-uniform mesh. Findings The computation results are presented in terms of temperature, heat transfer and skin friction coefficients; these are useful for determining surface heat requirements. It was found that, in finite difference scheme for non-uniform mesh with quasi-linearization technique method gives smoothness of solution compared to finite difference scheme for uniform mesh, and this evidence is graphically represented in Figure 2. Originality/value The impacts of viscous dissipation (Ec) and magnetic parameter (Ha) on temperature profiles, skin friction and heat transfer are analyzed, which determine the heat generation/absorption to ensure the MHD flow of the laminar boundary layer on irregular boundary over a diverging channel.

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