On discrete time hedging in d-dimensional option pricing models

TL;DR

This paper analyzes the error in approximating continuous-time stochastic integrals in d-dimensional diffusion models using discrete adjustments, showing an $n^{-1/2}$ convergence rate under certain conditions.

Contribution

It provides a theoretical analysis of the approximation error for discrete-time hedging in multi-dimensional models, optimizing over non-uniform time nets.

Findings

Approximation error converges at rate $n^{-1/2}$.

Optimal non-uniform time-net selection improves approximation.

Results apply to discretely adjusted portfolios in finance.

Abstract

We study the approximation of certain stochastic integrals with respect to a d-dimensional diffusion by corresponding stochastic integrals with piece-wise constant integrands. In finance this corresponds to replacing a continuously adjusted portfolio by discretely adjusted one. The approximation error is measured with respect to $L^2$ and it is shown that under certain assumptions the approximation rate is $n^{-1/2}$ when one optimizes over deterministic but not necessarily equidistant time-nets.