IDEAS home Printed from https://ideas.repec.org/a/taf/oaefxx/v5y2017i1p1384125.html
   My bibliography  Save this article

Bismut–Elworthy–Li formula for subordinated Brownian motion applied to hedging financial derivatives

Author

Listed:
  • M. Kateregga
  • S. Mataramvura
  • D. Taylor
  • Xibin Zhang

Abstract

The objective of the paper is to extend the results in Fournié, Lasry, Lions, Lebuchoux, and Touzi (1999), Cass and Fritz (2007) for continuous processes to jump processes based on the Bismut–Elworthy–Li (BEL) formula in Elworthy and Li (1994). We construct a jump process using a subordinated Brownian motion where the subordinator is an inverse α$ \alpha $-stable process (Lt)t≥0$ (L_t)_{t \ge 0} $ with (0,1]$ (0,\, 1] $. The results are derived using Malliavin integration by parts formula. We derive representation formulas for computing financial Greeks and show that in the event when Lt≡t$ L_t \equiv t $, we retrieve the results in Fournié et al. (1999). The purpose is to by-pass the derivative of an (irregular) pay-off function in a jump-type market by introducing a weight term in form of an integral with respect to subordinated Brownian motion. Using MonteCarlo techniques, we estimate financial Greeks for a digital option and show that the BEL formula still performs better for a discontinuous pay-off in a jump asset model setting and that the finite-difference methods are better for continuous pay-offs in a similar setting. In summary, the motivation and contribution of this paper demonstrates that the Malliavin integration by parts representation formula holds for subordinated Brownian motion and, this representation is useful in developing simple and tractable hedging strategies (the Greeks) in jump-type derivatives market as opposed to more complex jump models.

Suggested Citation

  • M. Kateregga & S. Mataramvura & D. Taylor & Xibin Zhang, 2017. "Bismut–Elworthy–Li formula for subordinated Brownian motion applied to hedging financial derivatives," Cogent Economics & Finance, Taylor & Francis Journals, vol. 5(1), pages 1384125-138, January.
  • Handle: RePEc:taf:oaefxx:v:5:y:2017:i:1:p:1384125
    DOI: 10.1080/23322039.2017.1384125
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1080/23322039.2017.1384125
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1080/23322039.2017.1384125?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. Michael Kateregga & Sure Mataramvura & David Taylor, 2017. "Parameter estimation for stable distributions with application to commodity futures log returns," Papers 1706.09756, arXiv.org.
    2. Reiichiro Kawai & Arturo Kohatsu-Higa, 2010. "Computation of Greeks and Multidimensional Density Estimation for Asset Price Models with Time-Changed Brownian Motion," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(4), pages 301-321.
    3. Hans-Peter Bermin, 2000. "Hedging lookback and partial lookback options using Malliavin calculus," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(2), pages 75-100.
    4. T. R. Cass & P. K. Friz, 2006. "The Bismut-Elworthy-Li formula for jump-diffusions and applications to Monte Carlo pricing in finance," Papers math/0604311, arXiv.org, revised May 2007.
    5. M. Kateregga & S. Mataramvura & D. Taylor, 2017. "Parameter estimation for stable distributions with application to commodity futures log-returns," Cogent Economics & Finance, Taylor & Francis Journals, vol. 5(1), pages 1318813-131, 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. Bielak, Łukasz & Grzesiek, Aleksandra & Janczura, Joanna & Wyłomańska, Agnieszka, 2021. "Market risk factors analysis for an international mining company. Multi-dimensional, heavy-tailed-based modelling," Resources Policy, Elsevier, vol. 74(C).
    2. Taurai Muvunza, 2020. "An $\alpha$-Stable Approach to Modelling Highly Speculative Assets and Cryptocurrencies," Papers 2002.09881, arXiv.org, revised Jul 2023.
    3. Abootaleb Shirvani & Svetlozar T. Rachev & Frank J. Fabozzi, 2019. "Multiple Subordinated Modeling of Asset Returns," Papers 1907.12600, arXiv.org.
    4. Muroi, Yoshifumi & Suda, Shintaro, 2017. "Computation of Greeks in jump-diffusion models using discrete Malliavin calculus," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 140(C), pages 69-93.
    5. Guillaume Bernis & Emmanuel Gobet & Arturo Kohatsu‐Higa, 2003. "Monte Carlo Evaluation of Greeks for Multidimensional Barrier and Lookback Options," Mathematical Finance, Wiley Blackwell, vol. 13(1), pages 99-113, January.
    6. Reiichiro Kawai, 2009. "Sensitivity Analysis And Density Estimation For The Hobson-Rogers Stochastic Volatility Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(03), pages 283-295.
    7. Gian Luca Tassinari & Michele Leonardo Bianchi, 2014. "Calibrating The Smile With Multivariate Time-Changed Brownian Motion And The Esscher Transform," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(04), pages 1-34.
    8. Reiichiro Kawai, 2012. "Likelihood ratio gradient estimation for Meixner distribution and Lévy processes," Computational Statistics, Springer, vol. 27(4), pages 739-755, December.
    9. Reiichiro Kawai, 2013. "Local Asymptotic Normality Property for Ornstein–Uhlenbeck Processes with Jumps Under Discrete Sampling," Journal of Theoretical Probability, Springer, vol. 26(4), pages 932-967, December.
    10. Lee, Hangsuck, 2003. "Pricing equity-indexed annuities with path-dependent options," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 677-690, December.
    11. Yuji Hishida & Kenji Yasutomi, 2009. "Asymptotic behavior of prices of path dependent options," Papers 0911.5579, arXiv.org.
    12. Kim, Geonwoo & Jeon, Junkee, 2018. "Closed-form solutions for valuing partial lookback options with random initiation," Finance Research Letters, Elsevier, vol. 24(C), pages 321-327.

    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:taf:oaefxx:v:5:y:2017:i:1:p:1384125. 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: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/OAEF20 .

    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.