IDEAS home Printed from https://ideas.repec.org/p/arx/papers/0912.4533.html
   My bibliography  Save this paper

Truncated Variation, Upward Truncated Variation and Downward Truncated Variation of Brownian Motion with Drift - their Characteristics and Applications

Author

Listed:
  • Rafa{l} {L}ochowski

Abstract

In the paper "On Truncated Variation of Brownian Motion with Drift" (Bull. Pol. Acad. Sci. Math. 56 (2008), no.4, 267 - 281) we defined truncated variation of Brownian motion with drift, $W_t = B_t + \mu t, t\geq 0,$ where $(B_t)$ is a standard Brownian motion. Truncated variation differs from regular variation by neglecting jumps smaller than some fixed $c > 0$. We prove that truncated variation is a random variable with finite moment-generating function for any complex argument. We also define two closely related quantities - upward truncated variation and downward truncated variation. The defined quantities may have some interpretation in financial mathematics. Exponential moment of upward truncated variation may be interpreted as the maximal possible return from trading a financial asset in the presence of flat commission when the dynamics of the prices of the asset follows a geometric Brownian motion process. We calculate the Laplace transform with respect to time parameter of the moment-generating functions of the upward and downward truncated variations. As an application of the obtained formula we give an exact formula for expected value of upward and downward truncated variations. We give also exact (up to universal constants) estimates of the expected values of the mentioned quantities.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:0912.4533
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/0912.4533
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Olympia Hadjiliadis & Jan Vecer, 2006. "Drawdowns preceding rallies in the Brownian motion model," Quantitative Finance, Taylor & Francis Journals, vol. 6(5), pages 403-409.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, August.
    2. David Landriault & Bin Li & Hongzhong Zhang, 2014. "On the Frequency of Drawdowns for Brownian Motion Processes," Papers 1403.1183, arXiv.org.
    3. Tommaso Proietti, 2024. "Ups and (Draw)Downs," CEIS Research Paper 576, Tor Vergata University, CEIS, revised 03 May 2024.
    4. Hongzhong Zhang & Olympia Hadjiliadis, 2012. "Drawdowns and the Speed of Market Crash," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 739-752, September.
    5. 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.
    6. 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.
    7. Zhenyu Cui, 2014. "Omega risk model with tax," Papers 1403.7680, arXiv.org.
    8. Ola Mahmoud, 2015. "The Temporal Dimension of Risk," Papers 1501.01573, arXiv.org, revised Jun 2016.
    9. Zhang, Gongqiu & Li, Lingfei, 2023. "A general method for analysis and valuation of drawdown risk," Journal of Economic Dynamics and Control, Elsevier, vol. 152(C).
    10. Zhang, Xiang & Li, Lingfei & Zhang, Gongqiu, 2021. "Pricing American drawdown options under Markov models," European Journal of Operational Research, Elsevier, vol. 293(3), pages 1188-1205.
    11. Zhenyu Cui, 2013. "Stochastic areas of diffusions and applications in risk theory," Papers 1312.0283, arXiv.org.
    12. Vladimir Petrov & Anton Golub & Richard Olsen, 2019. "Instantaneous Volatility Seasonality of High-Frequency Markets in Directional-Change Intrinsic Time," JRFM, MDPI, vol. 12(2), pages 1-31, April.
    13. 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.
    14. Aleksandar Mijatovic & Martijn R. Pistorius, 2011. "On the drawdown of completely asymmetric Levy processes," Papers 1103.1460, arXiv.org, revised Sep 2012.
    15. Long Bai & Peng Liu, 2019. "Drawdown and Drawup for Fractional Brownian Motion with Trend," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1581-1612, September.
    16. Zhenyu Cui & Duy Nguyen, 2018. "Magnitude and Speed of Consecutive Market Crashes in a Diffusion Model," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 117-135, March.
    17. 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.
    18. 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.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:0912.4533. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.