IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2510.15423.html
   My bibliography  Save this paper

On the short-time behaviour of up-and-in barrier options using Malliavin calculus

Author

Listed:
  • `Oscar Bur'es

Abstract

In this paper we study the short-maturity asymptotics of up-and-in barrier options under a broad class of stochastic volatility models. Our approach uses Malliavin calculus techniques, typically used for linear stochastic partial differential equations, to analyse the law of the supremum of the log-price process. We derive a concentration inequality and explicit bounds on the density of the supremum in terms of the time to maturity. These results yield an upper bound on the asymptotic decay rate of up-and-in barrier option prices as maturity vanishes. We further demonstrate the applicability of our framework to the rough Bergomi model and validate the theoretical results with numerical experiments.

Suggested Citation

  • `Oscar Bur'es, 2025. "On the short-time behaviour of up-and-in barrier options using Malliavin calculus," Papers 2510.15423, arXiv.org.
    • Handle: RePEc:arx:papers:2510.15423
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2510.15423
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    2. Takashi Kato & Akihiko Takahashi & Toshihiro Yamada, 2013. "An Asymptotic Expansion Formula for Up-and-Out Barrier Option Price under Stochastic Volatility Model," Papers 1302.3306, arXiv.org.
    3. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    4. Florit, Carme & Nualart, David, 1995. "A local criterion for smoothness of densities and application to the supremum of the Brownian sheet," Statistics & Probability Letters, Elsevier, vol. 22(1), pages 25-31, January.
    5. Elisa Alòs & Jorge León & Josep Vives, 2007. "On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility," Finance and Stochastics, Springer, vol. 11(4), pages 571-589, October.
    6. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    7. Takashi Kato & Akihiko Takahashi & Toshihiro Yamada, 2013. "An Asymptotic Expansion Formula for Up-and-Out Barrier Option Price under Stochastic Volatility Model," CIRJE F-Series CIRJE-F-873, CIRJE, Faculty of Economics, University of Tokyo.
    8. Haydyn Brown & David Hobson & L. C. G. Rogers, 2001. "Robust Hedging of Barrier Options," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 285-314, July.
    9. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
    10. Fannu Hu & Charles Knessl, 2010. "Asymptotics of Barrier Option Pricing Under the CEV Process," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(3), pages 261-300.
    11. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    12. Masaaki Fukasawa, 2017. "Short-time at-the-money skew and rough fractional volatility," Quantitative Finance, Taylor & Francis Journals, vol. 17(2), pages 189-198, February.
    13. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    14. Hossein Jafari & Ã’Scar Burã‰S & Josep Vives & Yiqiang Q. Zhao, 2025. "Option Price Asymptotics Under A Stochastic Volatility Lã‰Vy Model With Infinite Activity Jumps," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 28(01n02), pages 1-29, March.
    15. Zvan, R. & Vetzal, K. R. & Forsyth, P. A., 2000. "PDE methods for pricing barrier options," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1563-1590, October.
    16. Elisa Alòs & Kenichiro Shiraya, 2019. "Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach," Finance and Stochastics, Springer, vol. 23(2), pages 423-447, April.
    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. Giulia Di Nunno & Kk{e}stutis Kubilius & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2023. "From constant to rough: A survey of continuous volatility modeling," Papers 2309.01033, arXiv.org, revised Aug 2025.
    2. Elisa Al`os & `Oscar Bur'es & Rafael de Santiago & Josep Vives, 2025. "Volatility Modeling with Rough Paths: A Signature-Based Alternative to Classical Expansions," Papers 2507.23392, arXiv.org, revised Aug 2025.
    3. Elisa Alòs & Jorge A. León, 2021. "An Intuitive Introduction to Fractional and Rough Volatilities," Mathematics, MDPI, vol. 9(9), pages 1-22, April.
    4. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2022. "The rough Hawkes Heston stochastic volatility model," Papers 2210.12393, arXiv.org.
    5. repec:hal:wpaper:hal-03827332 is not listed on IDEAS
    6. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Finance and Stochastics, Springer, vol. 26(4), pages 733-769, October.
    7. Aït-Sahalia, Yacine & Li, Chenxu & Li, Chen Xu, 2021. "Closed-form implied volatility surfaces for stochastic volatility models with jumps," Journal of Econometrics, Elsevier, vol. 222(1), pages 364-392.
    8. R. Merino & J. Pospíšil & T. Sobotka & J. Vives, 2018. "Decomposition Formula For Jump Diffusion Models," Journal of Enterprising Culture (JEC), World Scientific Publishing Co. Pte. Ltd., vol. 21(08), pages 1-36, December.
    9. Raul Merino & Jan Posp'iv{s}il & Tom'av{s} Sobotka & Josep Vives, 2019. "Decomposition formula for jump diffusion models," Papers 1906.06930, arXiv.org.
    10. M.E. Mancino & S. Scotti & G. Toscano, 2020. "Is the Variance Swap Rate Affine in the Spot Variance? Evidence from S&P500 Data," Applied Mathematical Finance, Taylor & Francis Journals, vol. 27(4), pages 288-316, July.
    11. Elisa Alòs & Yan Yang, 2014. "A closed-form option pricing approximation formula for a fractional Heston model," Economics Working Papers 1446, Department of Economics and Business, Universitat Pompeu Fabra.
    12. Li, Hongshan & Huang, Zhongyi, 2020. "An iterative splitting method for pricing European options under the Heston model☆," Applied Mathematics and Computation, Elsevier, vol. 387(C).
    13. Hongshan Li & Zhongyi Huang, 2020. "An iterative splitting method for pricing European options under the Heston model," Papers 2003.12934, arXiv.org.
    14. Jan Matas & Jan Pospíšil, 2023. "Robustness and sensitivity analyses of rough Volterra stochastic volatility models," Annals of Finance, Springer, vol. 19(4), pages 523-543, December.
    15. Hideharu Funahashi, 2017. "Pricing derivatives with fractional volatility," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 1-28, March.
    16. Yicun Li & Yuanyang Teng, 2022. "Estimation of the Hurst Parameter in Spot Volatility," Mathematics, MDPI, vol. 10(10), pages 1-26, May.
    17. Archil Gulisashvili, 2017. "Large deviation principle for Volterra type fractional stochastic volatility models," Papers 1710.10711, arXiv.org, revised Aug 2018.
    18. Alessandro Bondi & Sergio Pulido & Simone Scotti, 2024. "The rough Hawkes Heston stochastic volatility model," Post-Print hal-03827332, HAL.
    19. Jacquier, Antoine & Pannier, Alexandre, 2022. "Large and moderate deviations for stochastic Volterra systems," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 142-187.
    20. Siow Woon Jeng & Adem Kilicman, 2020. "Series Expansion and Fourth-Order Global Padé Approximation for a Rough Heston Solution," Mathematics, MDPI, vol. 8(11), pages 1-26, November.
    21. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Working Papers hal-02946146, HAL.

    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:2510.15423. 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.