We consider a Brownian functional $F=g\bigl(\int_0^T η(s) dW_s\bigr)$ with $g \in L_2(γ)$ and a singular deterministic $η$. We deduce the $L_2$-convergence rate for the approximation $F^{(n)} = E F + \int_0^T φ^{(n)}(s) dW_s$ for a class of piecewise constant predictable integrands $φ^{(n)}$ from the fractional smoothness of $g$ quantified by Besov spaces and the rate of singularity of $η$.
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