Convergence rate for the hedging error of a path-dependent example

Abstract

We consider a Brownian functional $F=g({\int }_0^T\eta (s)d{W}_s)$ with $g\in L_2(\gamma )$ and a singular deterministic $\eta$. We deduce the $L_2$-convergence rate for the approximation $F^{(n)}=EF+{\int }_0^T{\varphi }^{(n)}(s)d{W}_s$ for a class of piecewise constant predictable integrands ${\varphi }^{(n)}$ from the fractional smoothness of $g$ quantified by Besov spaces and the rate of singularity of $\eta$.