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

TL;DR

This paper analyzes the convergence rate of hedging errors for a class of path-dependent Brownian functionals, linking the rate to the smoothness of the payoff function and the singularity of the integrand.

Contribution

It derives the $L_2$-convergence rate for approximations of Brownian functionals with singular integrands, based on fractional smoothness and Besov space analysis.

Findings

Convergence rate depends on the fractional smoothness of the payoff function.

Singularity of the integrand affects the approximation accuracy.

Provides explicit bounds for the hedging error rate.

Abstract

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