Penentuan opsi jual Amerika pada model Heston dengan Volatilitas Stokastik dan Dividen menggunakan skema beda hingga kompak orde empat

Khoirunisa, Winda (2026) Penentuan opsi jual Amerika pada model Heston dengan Volatilitas Stokastik dan Dividen menggunakan skema beda hingga kompak orde empat. Sarjana thesis, Universitas Islam Negeri Sunan Gunung Djati Bandung.

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Abstract

This study develops a numerical method for pricing American put options under the Heston model with continuous dividends. The model is formulated as a two-dimensional parabolic PDE with a free boundary. The Front-Fixing transformation x = ln(S/sf (τ )) is applied to map the moving domain to a fixed domain, with an additional convective term α(τ ) = s′f (τ )/sf (τ ) integrated simultaneously. Spatial discretization uses a fourth-order anisotropic 9-point compact finite difference scheme (O(h4 x + h4 v)) to handle cross-diffusion due to correlation ρ and non-constant diffusion coefficients, while time integration uses RK4 (O(∆τ 4)). Theoretical analysis and empirical verification show the scheme satisfies consistency(τh = O(h4x + h4v + ∆τ 4)) and numerical stability with CFL-satisfying timesteps, despite extreme parameters (ρ = −0.9450, σ = 0.5831) yielding positive eigenvalues. Convergence is verified through grid refinement tests until error E∞ = 0.00. Simulations using parameters from SPY historical data produce stable and convergent solutions, with option values decreasing as stock price increases and increasing with higher volatility, while the optimal exercise boundary decreases smoothly toward maturity. This research provides a high-accuracy numerical framework while identifying numerical challenges for future development. Keywords: American Put Option, Heston Model, Stochastic Volatility, Free Boundary, Compact Finite Difference, RK4, Stability Analysis.

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