Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations

Darae Jeong,Dongsun Lee,Hyeongseok Hwang,Seungsuk Seo,Yongho Choi,Junseok Kim

doi:10.1155/2015/359028

Darae Jeong, Dongsun Lee + Show 4 more

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https://doi.org/10.1155/2015/359028

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Journal: Discrete Dynamics in Nature and Society	Publication Date: Jan 1, 2015
Citations: 25	License type: CC BY 3.0

Affiliation: Korea University

Abstract

We briefly review and investigate the performance of various boundary conditions such as Dirichlet, Neumann, linear, and partial differential equation boundary conditions for the numerical solutions of the Black-Scholes partial differential equation. We use a finite difference method to numerically solve the equation. To show the efficiency of the given boundary condition, several numerical examples are presented. In numerical test, we investigate the effect of the domain sizes and compare the effect of various boundary conditions with pointwise error and root mean square error. Numerical results show that linear boundary condition is accurate and efficient among the other boundary conditions.

Highlights

We investigate the effect of the domain sizes and compare the effect of various boundary conditions with pointwise error and root mean square error
We briefly review and perform a comparison study for Dirichlet, Neumann, linear, and partial differential equation (PDE) boundary conditions (BCs) for the Black-Scholes (BS) partial differential equations
To obtain an approximation of the option value, one can compute a solution of the BS equations (1) and (2) using a finite difference method (FDM) [2,3,4,5,6,7,8], finite element method [9,10,11], finite volume method [12,13,14], a fast Fourier transform [15,16,17], and their optimal BC [18]

Summary

Introduction

We briefly review and perform a comparison study for Dirichlet, Neumann, linear, and partial differential equation (PDE) boundary conditions (BCs) for the Black-Scholes (BS) partial differential equations. To obtain an approximation of the option value, one can compute a solution of the BS equations (1) and (2) using a finite difference method (FDM) [2,3,4,5,6,7,8], finite element method [9,10,11], finite volume method [12,13,14], a fast Fourier transform [15,16,17], and their optimal BC [18]. The purpose of this paper is to investigate the effects of several BCs on the numerical solutions for the BS PDE (1) and (2).

Numerical Solution

Numerical Experiments

Conclusion