How Fast Does It Diverge? Discrete Hedging Error with Transaction Costs

In the present paper, we focus on the diverging behavior of discrecte hedging error with transaction costs. We added the hedging cost to the error directly. The main idea is to divide the hedging error into two parts: the pure hedging error and transaction cost of rebalance. The later part will be diverged when hedging number n goes to infinity. Firstly we show an upper bound of diverging part, which is O(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\sqrt{n})$$\end{document} of rebalancing number n, then we prove both the upper bound and the lower bound of discrete hedging error with transaction costs are of n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{n}$$\end{document} order, finally we give an approximation of hedging error to determine the coefficient in front of n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{n}$$\end{document}. The main technique in the proof is Itô’s formula, L’Hopital’s rule and three important lemmas in [Yuri, Kabanov, Mher, Safarian. Markets with Transaction Costs. Springer-Verlag, Berlin, Heidelberg, 2009]. The numerical result support our theoretical conclusion.