Sains Malaysiana 55(4)(2026): 746-755

http://doi.org/10.17576/jsm-2026-5504-13

 

 Kaedah Pengenduran Berlebihan Berturut-Turut Merah-Hitam Bagi Penentuan Harga Opsyen Asia Aritmetik

(A Red-Black Successive Over-Relaxation Method for Arithmetic Asian Option Pricing)

 

WEI SIN KOH^1,2, SAIFUL HAFIZAH JAAMAN^1,*, ROKIAH ROZITA AHMAD¹ & JUMAT SULAIMAN³

 

¹Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

²Faculty of Business and Communications, INTI International University, 71800 Nilai, Negeri Sembilan, Malaysia

³Faculty of Science and Natural Resources, Universiti Malaysia Sabah, 88450 Kota Kinabalu, Sabah, Malaysia

 

Received: 20 March 2026/Accepted: 17 April 2026

 

Abstrak

Penentuan harga opsyen Asia aritmetik merupakan masalah yang mencabar dalam matematik kewangan kerana tiada penyelesaian bentuk tertutup wujud. Dalam kajian ini, kaedah lelaran Pengenduran Berlebihan Berturut-turut Merah–Hitam (RBSOR) dibangunkan untuk penyelesaian berangka bagi penentuan harga opsyen Asia aritmetik. Model penentuan harga yang ditadbir oleh persamaan pembezaan separa (PPS) Black–Scholes didiskretkan menggunakan Skema Beza Terhingga Crank–Nicolson yang menghasilkan suatu sistem persamaan linear. Seterusnya, kaedah lelaran RBSOR diaplikasikan untuk menyelesaikan sistem linear yang terhasil dengan cekap. Beberapa uji kaji berangka dijalankan dan keputusan yang diperoleh dibandingkan dengan kaedah lelaran Gauss–Seidel (GS), Pengenduran Berlebihan Berturut-turut (SOR) dan Gauss–Seidel Merah–Hitam (RBGS). Prestasi penghitungan dinilai berdasarkan bilangan lelaran, masa penghitungan dan ralat punca min kuasa dua (RMSE). RMSE digunakan untuk menilai kejituan setiap kaedah lelaran melalui perbandingan antara keputusan kaedah berangka dan kaedah analitik. Keputusan kajian menunjukkan bahawa kaedah RBSOR mencapai kejituan yang setanding sambil mengurangkan bilangan lelaran dan masa penghitungan dengan ketara, sekali gus membuktikan kecekapan kaedah ini dalam penentuan harga opsyen secara berangka.

Kata kunci: Kaedah SOR Merah-Hitam; kecekapan sumber; opsyen Asia aritmetik; PPS Black Scholes; skema Crank-Nicolson

 

Abstract

The valuation of arithmetic Asian options is a challenging problem in financial mathematics due to the absence of a closed-form solution. In this study, the Red–Black Successive Over-Relaxation (RBSOR) method is developed for the numerical solution of arithmetic Asian option pricing. The pricing model, governed by the Black–Scholes partial differential equation (PDE), is discretized using the Crank–Nicolson finite difference scheme, resulting in a system of linear equations. The RBSOR iterative method is then applied to efficiently solve the resulting linear system. Several numerical experiments are conducted, and the results are compared with those obtained using Gauss–Seidel (GS), Successive Over-Relaxation (SOR), and Red–Black Gauss–Seidel (RBGS) iterative methods. Computational performance is evaluated in terms of number of iterations, computational time, and root mean squared error (RMSE). RMSE is used to evaluate the accuracy of each iterative method by comparing the numerical results with the analytical solution. The findings demonstrate that the RBSOR method achieves comparable accuracy while significantly reducing iteration counts and computational time, indicating its computational efficiency for numerical option pricing.

Keywords: Arithmetic Asian option pricing; Black-Scholes PDE; Crank-Nicolson scheme; Red-Black SOR method; resource efficiency

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*Corresponding author; email: shj@ukm.edu.my