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

The order barrier for the $L^1$-approximation of the log-Heston SDE at a single point