Implementasi Metode Multigrid dalam Menentukan Harga Opsi Beli Tipe Eropa

Anggraini, Anggi (2026) Implementasi Metode Multigrid dalam Menentukan Harga Opsi Beli Tipe Eropa. Other thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Opsi merupakan salah satu instrumen keuangan yang digunakan dalam aktivias investasi, sehingga penentuan harga opsi yang akurat menjadi hal yang penting. Model Black-Scholes merupakan salah satu model yang banyak digunakan dalam penentuan harga opsi karena memiliki solusi analitik yang jelas. Dalam penerapannya, penyelesaian model ini sering dilakukan secara numerik menggunakan metode beda hingga, yang mengubah persamaan Black-Scholes menjadi sistem persamaan linear. Namun, kondisi batas yang tidak halus, seperti pada fungsi payoff dapat menimbulkan galat numerik yang relatif besar dan memperlambat proses konvergensi solusi. Tugas Akhir ini mengimplementasikan metode multigrid untuk meningkatkan efisiensi perhitungan harga opsi beli tipe Eropa. Persamaan Black-Scholes didiskritkan menggunakan skema beda hingga implisit dan diselesaikan secara mundur waktu (backward time). Kinerja metode multigrid kemudian dibandingkan dengan metode Jacobi, Gauss Seidel, dan Successive Over-Relaxation berdasarkan kecepatan konvergensi, waktu komputasi, dan akurasi solusi terhadap solusi analitik. Hasil numerik menunjukkan bahwa seluruh metode menghasilkan tingkat akurasi yang sebanding pada toleransi konvergensi yang sama, baik pada titik tertentu maupun pada seluruh domain harga saham. Dari sisi efisiensi, metode multigrid mampu mencapai kondisi konvergen dengan jumlah iterasi dan waktu komputasi yang jauh lebih sedikit. Analisis sensitivitas menunjukkan bahwa peningkatan suku bunga bebas risiko dan volatilitas menyebabkan kenaikan harga opsi. Evaluasi menggunakan norm-2 error juga menunjukkan bahwa solusi numerik metode multigrid sangat dekat dengan solusi analitik pada seluruh domain perhitungan. Dengan demikian, metode multigrid terbukti efektif untuk mempercepat proses perhitungan tanpa mengurangi ketelitian hasil.
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Options are one of the financial instruments widely used in investment activities; therefore, accurate option pricing is of great importance. The Black–Scholes model is one of the most commonly used models for option pricing due to its well-established analytical solution. In practice, this model is often solved numerically using finite difference methods, which transform the Black–Scholes equation into a system of linear equations. Non-smooth boundary conditions, such as those arising from the payoff function, may lead to relatively large numerical errors and slow down the convergence process. This undergraduate thesis implements the multigrid method to improve the computational efficiency of pricing European call options. The Black–Scholes equation is discretized using an implicit finite difference scheme and solved using a backward time-stepping approach. The performance of the multigrid method is then compared with the Jacobi, Gauss–Seidel, and Successive Over-Relaxation (SOR) methods in terms of convergence speed, computational time, and solution accuracy relative to the analytical solution. The numerical results show that all methods produce comparable levels of accuracy when the same convergence tolerance is applied, both at specific points and across the entire stock price domain. In terms of efficiency, the multigrid method achieves convergence with significantly fewer iterations and much shorter computational time. Sensitivity analysis indicates that increases in the risk-free interest rate and volatility lead to higher option prices. Furthermore, evaluation using the norm-2 error demonstrates that the multigrid numerical solution is very close to the analytical solution over the entire computational domain. Therefore, the multigrid method is proven to be effective in accelerating the computation process without sacrificing solution accuracy.

Item Type:	Thesis (Other)
Uncontrolled Keywords:	Opsi, model Black-Scholes, metode beda hingga, multigrid, metode iteratif. Option, Black-Scholes model, finite difference method, multigrid, iteratif method.
Subjects:	Q Science > QA Mathematics > QA371 Differential equations--Numerical solutions Q Science > QA Mathematics > QA401 Mathematical models. Q Science > QA Mathematics > QA431 Finite differences.
Divisions:	Faculty of Science and Data Analytics (SCIENTICS) > Mathematics > 44201-(S1) Undergraduate Thesis
Depositing User:	Anggi Anggraini
Date Deposited:	27 Jan 2026 07:43
Last Modified:	27 Jan 2026 07:43
URI:	http://repository.its.ac.id/id/eprint/130509

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