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

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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

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

Acta Mathematicae Applicatae Sinica , Volume 37 (3) – Aug 5, 2021

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Publisher
Springer Journals
Copyright
Copyright © The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2021
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-021-1017-9
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Aug 5, 2021

Keywords: discrete hedging; transaction costs; hedging error; diverging speed; 91G10

References