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On the Frequency of Drawdowns for Brownian Motion Processes

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  • David Landriault
  • Bin Li
  • Hongzhong Zhang

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 focus on the side of severity by studying the first drawdown over certain pre-specified 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 $n$-th drawdown time, the distribution of the running maximum and the value process at the $n$-th 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.

Suggested Citation

  • David Landriault & Bin Li & Hongzhong Zhang, 2014. "On the Frequency of Drawdowns for Brownian Motion Processes," Papers 1403.1183, arXiv.org.
    • Handle: RePEc:arx:papers:1403.1183
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    References listed on IDEAS

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    1. Alexei Chekhlov & Stanislav Uryasev & Michael Zabarankin, 2005. "Drawdown Measure In Portfolio Optimization," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(01), pages 13-58.
    2. Raphaël Douady & A.N. Shiryaev & Marc Yor, 2000. "On Probability Characteristics of "Downfalls" in a Standard Brownian Motion," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01477104, HAL.
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    6. Hongzhong Zhang & Olympia Hadjiliadis, 2010. "Drawdowns and Rallies in a Finite Time-horizon," Methodology and Computing in Applied Probability, Springer, vol. 12(2), pages 293-308, June.
    7. Pospisil, Libor & Vecer, Jan & Hadjiliadis, Olympia, 2009. "Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2563-2578, August.
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    9. Peter Carr & Hongzhong Zhang & Olympia Hadjiliadis, 2011. "Maximum Drawdown Insurance," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(08), pages 1195-1230.
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    12. Zhang, Hongzhong & Leung, Tim & Hadjiliadis, Olympia, 2013. "Stochastic modeling and fair valuation of drawdown insurance," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 840-850.
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