important for the analysis of random dynamical systems [@arnold; @GAKN], one of the main objectives in mathematical finance is the pricing of (path-dependent) European-type derivatives, which means to compute real numbers $ {\mathbf{E}}\Phi(X)$ where $\Phi : C([0,T]; {\mathbb{R}}^d) \rightarrow {\mathbb{R}}$ is the discounted payoff of the derivative. Since the integrability of the random constants in the error bounds is an open problem, the above pathwise convergence rates do not imply weak or strong convergence rates. Nevertheless, if $\Phi$ is bounded and continuous and if ${ \overline{X}}^{\gamma}=({ \overline{X}}^{\gamma}_t)_{ t \in [0,T]}$ is the piecewise linear interpolation of the $\gamma$-Itô-Taylor scheme (standard, modified or reflected) then $$\mathbf{E} \Phi({ \overline{X}}^{\gamma}) \longrightarrow  \mathbf{E} \Phi(X)$$ for $n \rightarrow \infty$, so for bounded and continuous pay-offs (e.g. put options) one obtains at least the convergence of the corresponding standard Monte Carlo estimators for the option price. The same is true for barrier options with payoff of the form $$\Phi(X)= \phi(X_T) \mathbf{1}_{ \{  K_1 \leq |X_t| \leq K_2, \,\,  t \in [0,T] \} }$$ with $0 \leq K_1 \leq K_2 < \infty$, if $\phi$ is bounded and continuous and the law of $\sup_{t \in [0,T]} |X_t|$ and $\inf_{t \in [0,T]} |X_t|$ has a density with respect to 