Approximate solution of nonlinear Black–Scholes equation via a fully discretized fourth-order method

Azadeh Ghanadian; Taher Lotfi; Azadeh Ghanadian; Taher Lotfi

doi:10.3934/math.2020060

AIMS Mathematics

2020, Volume 5, Issue 2: 879-893. doi: 10.3934/math.2020060

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Research article

Approximate solution of nonlinear Black–Scholes equation via a fully discretized fourth-order method

Azadeh Ghanadian ,
Taher Lotfi ^,

Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran

Received: 06 October 2019 Accepted: 11 December 2019 Published: 06 January 2020
MSC : 41A15, 65M20

In this work, a new fourth-order finite difference (FD) approximation (for both structured and unstructured grid of nodes) is contributed and equipped with the fourth-order Runge–Kutta scheme to tackle the financial nonlinear partial differential equation (PDE) of Black–Scholes. This timedependent PDE problem is converted to a set of ordinary differential equations (ODEs). It is proved that under several criteria the procedure is time stable. Computational illustrations presented here, show that our approach is fast and accurate. The proposed technique reduces the computational cost, when more accurate results are requested.
- fourth-order,
- Black-Scholes equation,
- transaction costs,
- time-dependent,
- FD weights
Citation: Azadeh Ghanadian, Taher Lotfi. Approximate solution of nonlinear Black–Scholes equation via a fully discretized fourth-order method[J]. AIMS Mathematics, 2020, 5(2): 879-893. doi: 10.3934/math.2020060

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Abstract

In this work, a new fourth-order finite difference (FD) approximation (for both structured and unstructured grid of nodes) is contributed and equipped with the fourth-order Runge–Kutta scheme to tackle the financial nonlinear partial differential equation (PDE) of Black–Scholes. This timedependent PDE problem is converted to a set of ordinary differential equations (ODEs). It is proved that under several criteria the procedure is time stable. Computational illustrations presented here, show that our approach is fast and accurate. The proposed technique reduces the computational cost, when more accurate results are requested.

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