Qualitatively Stable Nonstandard Finite Difference Scheme for Numerical Solution of the Nonlinear Black–Scholes Equation

Mohammad Mehdizadeh Khalsaraei,Hamid Mohammad Sedighi,Ali Shokri,Zahra Mohammadnia,Efthymios G Tsionas

doi:10.1155/2021/6679484

Mohammad Mehdizadeh Khalsaraei, Hamid Mohammad Sedighi + Show 3 more

Open Access

https://doi.org/10.1155/2021/6679484

Copy DOI

Abstract

In this paper, we use a numerical method for solving the nonlinear Black–Scholes partial differential equation of the European option under transaction costs, which is based on the nonstandard discretization of the spatial derivatives. The proposed scheme, in addition to the unconditional positivity, is stable, consistent, and monotone. In order to illustrate the efficiency of the new method, numerical results have been performed by four models.

Highlights

Financial mathematics is a branch of applied mathematics that deals with financial markets
An option on an underlying asset traded on financial markets is an agreement that gives its holder the option to buy or sell the asset mentioned in the contract on a specified date at a certain and predetermined price. e date specified in the contract calls to the maturity date or expiration date, and the price specified in the contract is called the strike price or exercise price. e biggest advantage of an option contract is limited loss and unlimited profits
Numerical methods based on the standard finite difference approach are consistent with the original differential equation and guarantee convergence of the discrete solution to the exact one, but they impose severe restrictions on the time step, and essential qualitative properties of the solution are not transferred to the numerical solution

Summary

Introduction

Financial mathematics is a branch of applied mathematics that deals with financial markets. Scholes and developed by Robert Merton [1] It is represented by a partial differential equation in the following form: Ut. where U U(S, t) is the value of the option, S is the current underlying asset price, t t t is the time, T > 0 is the expiration date of the option, r ≥ 0 is the risk-free interest rate, and σ0 is the volatility ( called implicit volatility). In [22], combination of a sixthorder finite difference scheme in space and a third-order strong stability preserving Runge–Kutta in time has been used to obtain numerical solutions of the linear and nonlinear European put option models. Ey proposed a numerical method that involves transforming the free boundary problem for a nonlinear Black–Scholes equation into the so-called Gamma variational inequality with a new variable depending on the Gamma of the option.

Nonstandard Finite Difference Scheme

Construction of the New NSFD

Analysis of the New Method

Numerical Results

Discussion and Conclusion