On the drawdown of completely asymmetric Levy processes
File(s)
Accepted version
Author(s)
Mijatovic, A
Pistorius, MR
Type
Journal Article
Abstract
The drawdown process $Y=\bar{X} - X$ of a completely asymmetric L\'{e}vy
process $X$ is given by $X$ reflected at its running supremum $\bar{X}$.In this
paper we explicitly express the law of the sextuple
$(\tau_a,\bar{G}_{\tau_a},\underline{X}_{\tau_a},\bar{X}_{\tau_a},Y_{\tau_a-},Y_{\tau_a}-a)$
in terms of the scale function and the L\'evy measure of $X$, where $\tau_a$
denotes the first-passage time of $Y$ over the level $a>0$, $\bar{G}_{\tau_a}$
is the time of the last supremum of $X$ prior to $\tau_a$ and $\underline{X}$
is the running infimum of $X$. We also explicitly identify the distribution of
the drawup $\hat{Y}_{\tau_a}$ at the moment $\tau_a$, where $\hat{Y} =
X-\underline{X}$, and derive the probability of a large drawdown preceding a
small rally. These results are applied to the Carr & Wu \cite{CarrWu} model for
S&P 500.
process $X$ is given by $X$ reflected at its running supremum $\bar{X}$.In this
paper we explicitly express the law of the sextuple
$(\tau_a,\bar{G}_{\tau_a},\underline{X}_{\tau_a},\bar{X}_{\tau_a},Y_{\tau_a-},Y_{\tau_a}-a)$
in terms of the scale function and the L\'evy measure of $X$, where $\tau_a$
denotes the first-passage time of $Y$ over the level $a>0$, $\bar{G}_{\tau_a}$
is the time of the last supremum of $X$ prior to $\tau_a$ and $\underline{X}$
is the running infimum of $X$. We also explicitly identify the distribution of
the drawup $\hat{Y}_{\tau_a}$ at the moment $\tau_a$, where $\hat{Y} =
X-\underline{X}$, and derive the probability of a large drawdown preceding a
small rally. These results are applied to the Carr & Wu \cite{CarrWu} model for
S&P 500.
Date Issued
2012-11
Citation
Stochastic Processes and Their Applications, 2012, 22 (11), pp.3812-3836
ISSN
0304-4149
Publisher
ELSEVIER SCIENCE BV
Start Page
3812
End Page
3836
Journal / Book Title
Stochastic Processes and Their Applications
Volume
22
Issue
11
Copyright Statement
Copyright © 2012 Elsevier Ltd. All rights reserved. NOTICE: this is the author’s version of a work that was accepted for publication in Stochastic Processes and Their Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Stochastic Processes and Their Applications, 122(11), 2012. DOI:10.1016/j.spa.2012.06.012
Identifier
http://arxiv.org/abs/1103.1460
Publication Status
Published