IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v90y2014icp85-92.html
   My bibliography  Save this article

The randomly stopped geometric Brownian motion

Author

Listed:
  • Balanzario, Eugenio P.
  • Ramírez, Rosalva Mendoza
  • Ortiz, Jorge Sánchez

Abstract

In this short note we compute the probability density function of the random variable XT, where Xt is a geometric Brownian motion, and where T is a random variable independent of Xt and has either a Gamma distribution or it is uniformly distributed. In the last section of the note, the distribution obtained for XT is fitted to the data consisting in the academic production of a set of mathematicians.

Suggested Citation

  • Balanzario, Eugenio P. & Ramírez, Rosalva Mendoza & Ortiz, Jorge Sánchez, 2014. "The randomly stopped geometric Brownian motion," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 85-92.
    • Handle: RePEc:eee:stapro:v:90:y:2014:i:c:p:85-92
    DOI: 10.1016/j.spl.2014.03.013
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715214001060
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spl.2014.03.013?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    3. Reed, William J., 2001. "The Pareto, Zipf and other power laws," Economics Letters, Elsevier, vol. 74(1), pages 15-19, December.
    4. Hélyette Geman & Dilip B. Madan & Marc Yor, 2001. "Time Changes for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 79-96, January.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Johnston, Josh & Andersen, Tim, 2022. "Random processes with high variance produce scale free networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).

    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. Robert J. Elliott & Carlton-James U. Osakwe, 2006. "Option Pricing for Pure Jump Processes with Markov Switching Compensators," Finance and Stochastics, Springer, vol. 10(2), pages 250-275, April.
    2. Sun, Qi & Xu, Weidong, 2015. "Pricing foreign equity option with stochastic volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 437(C), pages 89-100.
    3. Wendong Zheng & Chi Hung Yuen & Yue Kuen Kwok, 2016. "Recursive Algorithms For Pricing Discrete Variance Options And Volatility Swaps Under Time-Changed Lévy Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(02), pages 1-29, March.
    4. Geman, Helyette, 2002. "Pure jump Levy processes for asset price modelling," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1297-1316, July.
    5. Kao, Lie-Jane & Wu, Po-Cheng & Lee, Cheng-Few, 2012. "Time-changed GARCH versus the GARJI model for prediction of extreme news events: An empirical study," International Review of Economics & Finance, Elsevier, vol. 21(1), pages 115-129.
    6. Ewald, Christian-Oliver & Nawar, Roy & Siu, Tak Kuen, 2013. "Minimal variance hedging of natural gas derivatives in exponential Lévy models: Theory and empirical performance," Energy Economics, Elsevier, vol. 36(C), pages 97-107.
    7. Yanhui Mi, 2016. "A modified stochastic volatility model based on Gamma Ornstein–Uhlenbeck process and option pricing," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(02), pages 1-16, June.
    8. Buchmann, Boris & Kaehler, Benjamin & Maller, Ross & Szimayer, Alexander, 2017. "Multivariate subordination using generalised Gamma convolutions with applications to Variance Gamma processes and option pricing," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2208-2242.
    9. Dilip B. Madan & Wim Schoutens & King Wang, 2017. "Measuring And Monitoring The Efficiency Of Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(08), pages 1-32, December.
    10. Yongxin Yang & Yu Zheng & Timothy M. Hospedales, 2016. "Gated Neural Networks for Option Pricing: Rationality by Design," Papers 1609.07472, arXiv.org, revised Mar 2020.
    11. Fu, Qi & So, Jacky Yuk-Chow & Li, Xiaotong, 2024. "Stable paretian distribution, return generating processes and habit formation—The implication for equity premium puzzle," The North American Journal of Economics and Finance, Elsevier, vol. 70(C).
    12. Kirkby, J. Lars & Nguyen, Duy, 2021. "Equity-linked Guaranteed Minimum Death Benefits with dollar cost averaging," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 408-428.
    13. Calvet, Laurent E. & Fisher, Adlai J., 2008. "Multifrequency jump-diffusions: An equilibrium approach," Journal of Mathematical Economics, Elsevier, vol. 44(2), pages 207-226, January.
    14. Lorenzo Torricelli, 2012. "Valuation of asset and volatility derivatives using decoupled time-changed L\'evy processes," Papers 1210.5479, arXiv.org, revised Jan 2015.
    15. Oliver X. Li & Weiping Li, 2015. "Hedging jump risk, expected returns and risk premia in jump-diffusion economies," Quantitative Finance, Taylor & Francis Journals, vol. 15(5), pages 873-888, May.
    16. Cl'ement M'enass'e & Peter Tankov, 2015. "Asymptotic indifference pricing in exponential L\'evy models," Papers 1502.03359, arXiv.org, revised Feb 2015.
    17. Dilip B. Madan & King Wang, 2022. "Two sided efficient frontiers at multiple time horizons," Annals of Finance, Springer, vol. 18(3), pages 327-353, September.
    18. Ron Chan & Simon Hubbert, 2014. "Options pricing under the one-dimensional jump-diffusion model using the radial basis function interpolation scheme," Review of Derivatives Research, Springer, vol. 17(2), pages 161-189, July.
    19. Buckley, Winston & Long, Hongwei & Marshall, Mario, 2016. "Numerical approximations of optimal portfolios in mispriced asymmetric Lévy markets," European Journal of Operational Research, Elsevier, vol. 252(2), pages 676-686.
    20. Pérez-Abreu, Victor & Stelzer, Robert, 2014. "Infinitely divisible multivariate and matrix Gamma distributions," Journal of Multivariate Analysis, Elsevier, vol. 130(C), pages 155-175.

    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:eee:stapro:v:90:y:2014:i:c:p:85-92. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    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.