Convergence and Runtime of Crank-Nicolson Schemes for Barrier Option Pricing under Mixed Fractional Brownian Motion

Chatarina Enny Murwaningtyas (Department of Mathematics Education, Sanata Dharma University, Yogyakarta, 55281 Indonesia)

Zero : Jurnal Sains, Matematika, dan Terapan
Vol 10, No 2 (2026): Zero: Jurnal Sains Matematika dan Terapan

We examined an up-and-out barrier call under mixed fractional Brownian motion using three finite-difference time integrators: implicit Backward Euler, Crank-Nicolson (CN), and CN with a Rannacher startup. The spatial mesh was kept unchanged, while the number of time steps was increased. CN eventually produced very small price errors, but on course and moderate grids it also produced oscillations and negative values; the effect was stronger at higher volatility. Backward Euler did not show these large undershoots, although its error decreased more slowly. The Rannacher version removed the undershoots in the cases tested with little additional runtime. At N_t = 128, its relative error ranged from 8.20 x 10^-5 to 1.28 x 10^-4. For this numerical setting, the startup damping gave the most reliable moderate-grid behavior. to .