The Order Barrier for the  \(L^1\) -approximation of the Log-Heston SDE at a Single Point

We study the \(L^1\) -approximation of the log-Heston SDE at the terminal time point by arbitrary methods that use an equidistant discretization of the driving Brownian motion. We show that such methods can achieve at most order \( \min \{ \nu , \tfrac{1}{2} \}\) , where \(\nu \) is the Feller index of the underlying CIR process. As a consequence Euler-type schemes are optimal for \(\nu \ge 1\) , since they have convergence order \(\tfrac{1}{2}-\epsilon \) for \(\epsilon >0\) arbitrarily small in this regime.