Monotone Finite-Difference Schemes With Second Order Approximation Based on Regularization Approach for the Dirichlet Boundary Problem of the Gamma Equation

Le Minh Hieu,Dang Ngoc Hoang Thanh,Truong Thi Hieu Hanh

doi:10.1109/access.2020.2978594

Abstract

We investigate the initial boundary value problem for the Gamma equation transformed from the nonlinear Black-Scholes equation for pricing option to a quasilinear parabolic equation of second derivative. Furthermore, two-side estimates for the exact solution are also provided. By using regularization principle, the unconditionally monotone second order approximation finite-difference scheme on uniform and nonuniform grids is generalized, in that the maximum principle is satisfied without depending on relations of the coefficients and grid parameters. By using the difference maximum principle, we acquired two-side estimates for difference solution for the arbitrary non-sign-constant input data. Finally, we also provide a proof for a priori estimate. It can be confirmed that the two-side estimates for difference solution are completely consistent with the differential problem. Otherwise, the maximal and minimal values of the difference solution is independent from the diffusion and convection coefficients.

Highlights

Over the last decades, financial engineers and mathematicians have paid special attention to the valuation of derivative financial instruments
Since being introduced by Fischer Black and Myron Scholes in 1973, the Black-Scholes model based on partial differential equation has been widely employed in modern mathematical finance and become a common-sense approach for pricing options as well as many other financial securities [25]
Two-side estimates provide a manner to prove the nonnegativity of the exact solution, but it is helpful to find out sufficient conditions based on the input data if the nonlinear problem is parabolic

Summary

INTRODUCTION

Financial engineers and mathematicians have paid special attention to the valuation of derivative financial instruments. In the case of nonlinearity, the estimates provide an approach to confirm the nonnegativity of the exact solution This feature is very important in physical problems, as well as to find conditions of the input data to let the problem being parabolic or elliptic, for example, investigating the Gamma equation in financial mathematics to model pricing of options. The acquired results are generalized to the construction of monotone finite-difference schemes of second-order of local approximation on uniform and non-uniform grids for a given equation. The construction of such schemes is based on the appropriate choice of the perturbed coefficient, to [39]. A priori estimate of the approximate solution in the grid norm C depending on the initial and boundary conditions only is proved

MAXIMUM PRINCIPLE FOR DIFFERENCE SCHEMES WITH VARIABLE SIGN INPUT DATA

DIFFERENCE SCHEME

APPROXIMATION ERROR

NUMERICAL TEST

TRANSFORMATION OF THE SCHRODINGER EQUATION TO THE BLACK-SCHOLES EQUATION

CONCLUSION