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On the Frequency of Drawdowns for Brownian Motion Processes

Part of: Markov processes Stochastic processes Mathematical economics

Published online by Cambridge University Press:  30 January 2018

David Landriault ,
Bin Li  and
Hongzhong Zhang
David Landriault*
Affiliation:
University of Waterloo
Bin Li*
Affiliation:
University of Waterloo
Hongzhong Zhang*
Affiliation:
Columbia University
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada.
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada.
∗∗∗∗ Postal address: Department of Statistics, Columbia University, New York, NY 10027, USA. Email address: hzhang@stat.columbia.edu
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Abstract

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Drawdowns measuring the decline in value from the historical running maxima over a given period of time are considered as extremal events from the standpoint of risk management. To date, research on the topic has mainly focused on the side of severity by studying the first drawdown over a certain prespecified size. In this paper we extend the discussion by investigating the frequency of drawdowns and some of their inherent characteristics. We consider two types of drawdown time sequences depending on whether a historical running maximum is reset or not. For each type we study the frequency rate of drawdowns, the Laplace transform of the nth drawdown time, the distribution of the running maximum, and the value process at the nth drawdown time, as well as some other quantities of interest. Interesting relationships between these two drawdown time sequences are also established. Finally, insurance policies protecting against the risk of frequent drawdowns are also proposed and priced.

Keywords

DrawdownfrequencyBrownian motion

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J65: Brownian motion 91B24: Price theory and market structure
Type
Research Article
Information
Journal of Applied Probability , Volume 52 , Issue 1 , March 2015 , pp. 191 - 208
Copyright
© Applied Probability Trust 

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