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We study the $L^1$-approximation of the log-Heston SDE at equidistant timepoints by Euler-type methods. We establish the convergence order $1/2-\epsilon$ for $\epsilon >0$ arbitrarily small, if the Feller index $\nu$ ofthe underlying CIR process satisfies $\nu > 1$. Thus, we recover the standardconvergence order of the Euler scheme for SDEs with globally Lipschitzcoefficients. Moreover, we discuss the case $\nu \leq 1$ and illustrate ourfindings by several numerical examples.

Sharp $L^1$-Approximation of the log-Heston SDE by Euler-type methods