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All Journal Zero : Jurnal Sains, Matematika, dan Terapan
Artiono, Rudianto
Mathematics Department, Universitas Negeri Surabaya, Surabaya, 60231, Indonesia
Agriculture, Biological Sciences & Forestry Chemistry Mathematics Physics

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Numerical Pricing of European Options under Proportional Transaction Costs: A Semi-Discretization Approach to the Nonlinear Barles-Soner Model Noor, Dwi Maya Firanti; Artiono, Rudianto
ZERO: Jurnal Sains, Matematika dan Terapan Vol 10, No 1 (2026): Zero: Jurnal Sains Matematika dan Terapan
Publisher : UIN Sumatera Utara

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30829/zero.v10i1.28084

Abstract

The classical Black–Scholes model assumes a frictionless market, which often leads to the undervaluation of option premiums when transaction costs are present. This study prices European call options under proportional transaction costs using the nonlinear Barles–Soner framework and a semi-discretization–based numerical approach. Using historical stock data from PT XYZ (an anonymized Indonesian equity), the results show that transaction costs significantly increase effective volatility and generate systematic deviations from classical Black–Scholes prices. In particular, option premiums increase by IDR 392.33 and IDR 776.66 for transaction cost parameters of 0.015 and 0.030, respectively, compared with the frictionless benchmark. These findings confirm that ignoring transaction costs leads to substantial underpricing and that the proposed framework provides a more realistic and conservative valuation for hedging and risk management in emerging markets.
Crank-Nicolson Finite Difference Pricing of European Call Options under the Black-Scholes Model Adawiyyah, Robiyatul; Artiono, Rudianto
ZERO: Jurnal Sains, Matematika dan Terapan Vol 10, No 1 (2026): Zero: Jurnal Sains Matematika dan Terapan
Publisher : UIN Sumatera Utara

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30829/zero.v10i1.28482

Abstract

This study aims to determine the price of European call options using the Crank–Nicolson finite difference method in the Black–Scholes model with stock data from XYZ Company for the period January 2025 to December 2025. Annual volatility is calculated based on historical closing price data, while numerical option prices are obtained through the Crank–Nicolson finite difference scheme and compared it with the Black–Scholes analytical solution as a reference. The results show that the Crank–Nicolson method produces a call option price of 596.08, while the Black–Scholes analytical solution gives a value of 612.50. The relative difference between the two methods is 2.68%, which indicates a good level of accuracy for the numerical method used. These findings indicate that the Crank–Nicolson finite difference method is capable of providing a stable and accurate numerical approach to determining the price of European call options. In practical terms, the results of this study contribute to the application of numerical-based option pricing models in emerging markets, particularly in conditions of dynamic volatility, where analytical approaches may have limitations in implementation
Co-Authors Adawiyyah, Robiyatul Noor, Dwi Maya Firanti