Journal of Applied Probability

On the frequency of drawdowns for Brownian motion processes

David Landriault, Bin Li, and Hongzhong Zhang

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 52, Number 1 (2015), 191-208.

Dates
First available in Project Euclid: 17 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1429282615

Digital Object Identifier
doi:10.1239/jap/1429282615

Mathematical Reviews number (MathSciNet)
MR3336855

Zentralblatt MATH identifier
1317.60106

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60J65: Brownian motion [See also 58J65] 91B24: Price theory and market structure

Keywords
Drawdown frequency Brownian motion

Citation

Landriault, David; Li, Bin; Zhang, Hongzhong. On the frequency of drawdowns for Brownian motion processes. J. Appl. Probab. 52 (2015), no. 1, 191--208. doi:10.1239/jap/1429282615. https://projecteuclid.org/euclid.jap/1429282615


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