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

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