  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
$\nu \geq 1$, since they have convergence order $\tfrac{1}{2}-\epsilon$ for
$\epsilon >0$ arbitrarily small in this regime.
