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Drawdowns preceding rallies in the Brownian motion model

Author

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  • Olympia Hadjiliadis
  • Jan Vecer

Abstract

We study drawdowns and rallies of Brownian motion. A rally is defined as the difference of the present value of the Brownian motion and its historical minimum, while the drawdown is defined as the difference of the historical maximum and its present value. This paper determines the probability that a drawdown of a units precedes a rally of b units. We apply this result to examine stock market crashes and rallies in the geometric Brownian motion model.

Suggested Citation

  • Olympia Hadjiliadis & Jan Vecer, 2006. "Drawdowns preceding rallies in the Brownian motion model," Quantitative Finance, Taylor & Francis Journals, vol. 6(5), pages 403-409.
  • Handle: RePEc:taf:quantf:v:6:y:2006:i:5:p:403-409
    DOI: 10.1080/14697680600764227
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    Citations

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    Cited by:

    1. Ola Mahmoud, 2015. "The Temporal Dimension of Risk," Papers 1501.01573, arXiv.org, revised Jun 2016.
    2. Aleksandar Mijatovic & Martijn R. Pistorius, 2011. "On the drawdown of completely asymmetric Levy processes," Papers 1103.1460, arXiv.org, revised Sep 2012.
    3. Cui, Zhenyu & Nguyen, Duy, 2016. "Omega diffusion risk model with surplus-dependent tax and capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 150-161.
    4. Zhenyu Cui, 2014. "Omega risk model with tax," Papers 1403.7680, arXiv.org.
    5. Mijatović, Aleksandar & Pistorius, Martijn R., 2012. "On the drawdown of completely asymmetric Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3812-3836.
    6. Zhenyu Cui, 2013. "Stochastic areas of diffusions and applications in risk theory," Papers 1312.0283, arXiv.org.
    7. Rafa{l} {L}ochowski, 2009. "Truncated Variation, Upward Truncated Variation and Downward Truncated Variation of Brownian Motion with Drift - their Characteristics and Applications," Papers 0912.4533, arXiv.org, revised Dec 2011.
    8. David Landriault & Bin Li & Hongzhong Zhang, 2014. "On the Frequency of Drawdowns for Brownian Motion Processes," Papers 1403.1183, arXiv.org.
    9. 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.

    More about this item

    Keywords

    Drawdowns; Brownian motion model; Rallies;

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