Is the Approximation Rate for European Pay-offs in the Black–Scholes Model Always 1/ $\sqrt{n}$

Abstract

For a Borel-function $f:\mathbb{R} \to \mathbb{R}$, we consider the approximation of a random variable f(W ₁) with $\mathbb{E}f^{2}(W_{1})<\infty$ by stochastic integrals with respect to the Brownian motion $W = (W_{t})_{t \in[0, 1]}$ and the geometric Brownian motion, where the integrands are piecewise constant within certain deterministic time intervals. In earlier papers it has been shown that under certain regularity conditions the optimal approximation rate is 1/ $\sqrt{n}$, if one optimizes over deterministic time-nets of cardinality n. We will show the existence of random variables f(W ₁) such that the approximation error tends as slowly to zero as one wishes.