IDEAS home Printed from https://ideas.repec.org/a/bpj/mcmeap/v18y2012i4p353-377n4.html
   My bibliography  Save this article

Improving the Monte Carlo estimation of boundary crossing probabilities by control variables

Author

Listed:
  • Pötzelberger Klaus

    (Institute for Statistics and Mathematics, Augasse 2-6, A-1090 Vienna, Austria)

Abstract

We propose an efficient Monte Carlo approach to compute boundary crossing probabilities (BCP) for Brownian motion and a large class of diffusion processes, the method of adaptive control variables. For the Brownian motion the boundary b (or the boundaries in case of two-sided boundary crossing probabilities) is approximated by a piecewise linear boundary , which is linear on m intervals. Monte Carlo estimators of the corresponding BCP are based on an m-dimensional Gaussian distribution. Let N denote the number of (univariate) Gaussian variables used. The mean squared error for the boundary is of order , leading to a mean squared error for the boundary b of order with , if the difference of the (exact) BCP's for b and is . Typically, for infinite-dimensional Monte Carlo methods, the convergence rate is less than the finite-dimensional .

Suggested Citation

  • Pötzelberger Klaus, 2012. "Improving the Monte Carlo estimation of boundary crossing probabilities by control variables," Monte Carlo Methods and Applications, De Gruyter, vol. 18(4), pages 353-377, December.
    • Handle: RePEc:bpj:mcmeap:v:18:y:2012:i:4:p:353-377:n:4
    DOI: 10.1515/mcma-2012-0013
    as

    Download full text from publisher

    File URL: https://doi.org/10.1515/mcma-2012-0013
    Download Restriction: For access to full text, subscription to the journal or payment for the individual article is required.
    File URL: https://libkey.io/10.1515/mcma-2012-0013?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. K. Borovkov & Alexander Novikov, 2004. "Explicit Bounds for Approximation Rates for Boundary Crossing Probabilities for the Wiener Process," Research Paper Series 115, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Sheldon Lin, X., 1998. "Double barrier hitting time distributions with applications to exotic options," Insurance: Mathematics and Economics, Elsevier, vol. 23(1), pages 45-58, October.
    3. G. O. Roberts & C. F. Shortland, 1997. "Pricing Barrier Options with Time–Dependent Coefficients," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 83-93, January.
    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. Bekiros, Stelios & Kouloumpou, Dimitra, 2019. "On the pricing of exotic options: A new closed-form valuation approach," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 153-162.
    2. Fernández Lexuri & Hieber Peter & Scherer Matthias, 2013. "Double-barrier first-passage times of jump-diffusion processes," Monte Carlo Methods and Applications, De Gruyter, vol. 19(2), pages 107-141, July.
    3. K. Borovkov & Alexander Novikov, 2004. "Explicit Bounds for Approximation Rates for Boundary Crossing Probabilities for the Wiener Process," Research Paper Series 115, Quantitative Finance Research Centre, University of Technology, Sydney.
    4. Tung-Lung Wu, 2020. "A Note on Erdös and Kac’s Identity: Boundary Crossing Probabilities of Brownian Motion Over Constant Boundaries," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 161-171, March.
    5. Sbuelz, A., 2000. "Hedging Double Barriers with Singles," Discussion Paper 2000-112, Tilburg University, Center for Economic Research.
    6. Marco Avellaneda & Robert Buff, 1999. "Combinatorial implications of nonlinear uncertain volatility models: the case of barrier options," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(1), pages 1-18.
    7. Lee, Hangsuck & Ko, Bangwon & Lee, Minha, 2023. "The pricing and static hedging of multi-step double barrier options," Finance Research Letters, Elsevier, vol. 55(PA).
    8. Gregor Dorfleitner & Paul Schneider & Kurt Hawlitschek & Arne Buch, 2008. "Pricing options with Green's functions when volatility, interest rate and barriers depend on time," Quantitative Finance, Taylor & Francis Journals, vol. 8(2), pages 119-133.
    9. Sbuelz, A., 2000. "Hedging Double Barriers with Singles," Other publications TiSEM e810e3ab-1936-457e-a3ae-7, Tilburg University, School of Economics and Management.
    10. Vaibhav Srivastava & Samuel F. Feng & Jonathan D. Cohen & Naomi Ehrich Leonard & Amitai Shenhav, 2015. "A martingale analysis of first passage times of time-dependent Wiener diffusion models," Papers 1508.03373, arXiv.org, revised Sep 2016.
    11. repec:dau:papers:123456789/5374 is not listed on IDEAS
    12. Lin, X. Sheldon & Wu, Panpan & Wang, Xiao, 2016. "Move-based hedging of variable annuities: A semi-analytic approach," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 40-49.
    13. Pingjin Deng & Xiufang Li, 2017. "Barrier Options Pricing With Joint Distribution Of Gaussian Process And Its Maximum," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(06), pages 1-18, September.
    14. Dimitrina S. Dimitrova & Zvetan G. Ignatov & Vladimir K. Kaishev, 2017. "On the First Crossing of Two Boundaries by an Order Statistics Risk Process," Risks, MDPI, vol. 5(3), pages 1-14, August.
    15. Rolf Poulsen, 2006. "Barrier options and their static hedges: simple derivations and extensions," Quantitative Finance, Taylor & Francis Journals, vol. 6(4), pages 327-335.
    16. Zhang, Xiaoyuan & Zhang, Tianqi, 2022. "Barrier option pricing under a Markov Regime switching diffusion model," The Quarterly Review of Economics and Finance, Elsevier, vol. 86(C), pages 273-280.
    17. Gabriela Pesce & Florencia Verónica Pedroni & Etelvina Chávez & María de la Paz Moral & María Andrea Rivero, 2021. "Exotic options: conceptualization and evolution in the literature from a systematic review," Lecturas de Economía, Universidad de Antioquia, Departamento de Economía, issue 95, pages 231-275, July-Dece.
    18. Lee, Hangsuck & Jeong, Himchan & Lee, Minha, 2022. "Multi-step double barrier options," Finance Research Letters, Elsevier, vol. 47(PA).
    19. Péter Farkas, 2013. "Counting Process Generated by Boundary-crossing Events. Theory and Statistical Applications," CEU Working Papers 2013_4, Department of Economics, Central European University.
    20. Roy Cerqueti, 2022. "A new concept of reliability system and applications in finance," Annals of Operations Research, Springer, vol. 312(1), pages 45-64, May.
    21. Youngchul Han & Geonwoo Kim, 2016. "Efficient Lattice Method for Valuing of Options with Barrier in a Regime Switching Model," Discrete Dynamics in Nature and Society, Hindawi, vol. 2016, pages 1-14, October.

    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:bpj:mcmeap:v:18:y:2012:i:4:p:353-377:n:4. 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: Peter Golla (email available below). General contact details of provider: https://www.degruyter.com .

    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.