IDEAS home Printed from https://ideas.repec.org/a/wsi/ijtafx/v20y2017i03ns0219024917500133.html
   My bibliography  Save this article

Functional Analytic (Ir-)Regularity Properties Of Sabr-Type Processes

Author

Listed:
  • LEIF DÖRING

    (Department of Mathematics, University of Mannheim, Germany)

  • BLANKA HORVATH

    (#x2020;Department of Mathematics, Imperial College London, UK)

  • JOSEF TEICHMANN

    (#x2021;Department of Mathematics, ETH Zürich, Switzerland)

Abstract

The stochastic alpha, beta, rho (SABR) model is a benchmark stochastic volatility model in interest rate markets, which has received much attention in the past decade. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods. As markets moved to historically low rates, this expansion appeared to yield inconsistent prices. Since the model is deeply embedded in the markets, alternative pricing methods for SABR have been addressed in numerous approaches in recent years. All standard option pricing methods make certain regularity assumptions on the underlying model, but for SABR, these are rarely satisfied. We examine here regularity properties of the model from this perspective with a view to a number of (asymptotic and numerical) option pricing methods. In particular, we highlight delicate degeneracies of the SABR model (and related processes) at the origin, which deem the currently used popular heat kernel methods and all related methods from (sub-) Riemannian geometry ill-suited for SABR-type processes, when interest rates are near zero. We describe a more general semigroup framework, which permits to derive a suitable geometry for SABR-type processes (in certain parameter regimes) via symmetric Dirichlet forms. Furthermore, we derive regularity properties (Feller properties and strong continuity properties) necessary for the applicability of popular numerical schemes to SABR-semigroups and identify suitable Banach and Hilbert spaces for these. Finally, we comment on the short-time and large time asymptotic behavior of SABR-type processes beyond the heat-kernel framework.

Suggested Citation

  • Leif Döring & Blanka Horvath & Josef Teichmann, 2017. "Functional Analytic (Ir-)Regularity Properties Of Sabr-Type Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(03), pages 1-48, May.
    • Handle: RePEc:wsi:ijtafx:v:20:y:2017:i:03:n:s0219024917500133
    DOI: 10.1142/S0219024917500133
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219024917500133
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0219024917500133?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. Martin Forde & Andrey Pogudin, 2013. "The Large-Maturity Smile For The Sabr And Cev-Heston Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(08), pages 1-20.
    2. S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 1-12, January.
    3. Philipp Doersek & Josef Teichmann, 2010. "A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential Equations," Papers 1011.2651, arXiv.org.
    4. Jim Gatheral & Tai-Ho Wang, 2012. "The Heat-Kernel Most-Likely-Path Approximation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(01), pages 1-18.
    5. Jim Gatheral & Tai-Ho Wang, 2012. "The Heat-Kernel Most-Likely-Path Approximation," World Scientific Book Chapters, in: Matheus R Grasselli & Lane P Hughston (ed.), Finance at Fields, chapter 17, pages 389-406, World Scientific Publishing Co. Pte. Ltd..
    6. David Hobson, 2010. "Comparison results for stochastic volatility models via coupling," Finance and Stochastics, Springer, vol. 14(1), pages 129-152, January.
    7. Archil Gulisashvili, 2015. "Left-Wing Asymptotics Of The Implied Volatility In The Presence Of Atoms," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(02), pages 1-25.
    8. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    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. Blanka Horvath & Oleg Reichmann, 2018. "Dirichlet Forms and Finite Element Methods for the SABR Model," Papers 1801.02719, arXiv.org.

    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. Blanka Horvath & Oleg Reichmann, 2018. "Dirichlet Forms and Finite Element Methods for the SABR Model," Papers 1801.02719, arXiv.org.
    2. Jacquier, Antoine & Roome, Patrick, 2016. "Large-maturity regimes of the Heston forward smile," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1087-1123.
    3. Vimal Raval & Antoine Jacquier, 2021. "The Log Moment formula for implied volatility," Papers 2101.08145, arXiv.org.
    4. Gaoyue Guo & Antoine Jacquier & Claude Martini & Leo Neufcourt, 2012. "Generalised arbitrage-free SVI volatility surfaces," Papers 1210.7111, arXiv.org, revised May 2016.
    5. Stefano De Marco & Peter Friz, 2013. "Varadhan's formula, conditioned diffusions, and local volatilities," Papers 1311.1545, arXiv.org, revised Jun 2016.
    6. Michael R. Tehranchi, 2015. "Uniform bounds for Black--Scholes implied volatility," Papers 1512.06812, arXiv.org, revised Aug 2016.
    7. Francesco Caravenna & Jacopo Corbetta, 2015. "The asymptotic smile of a multiscaling stochastic volatility model," Papers 1501.03387, arXiv.org, revised Jul 2017.
    8. Christoffersen, Peter & Jacobs, Kris & Chang, Bo Young, 2013. "Forecasting with Option-Implied Information," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 2, chapter 0, pages 581-656, Elsevier.
    9. Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
    10. Sergey Badikov & Mark H. A. Davis & Antoine Jacquier, 2018. "Perturbation analysis of sub/super hedging problems," Papers 1806.03543, arXiv.org, revised May 2021.
    11. Caravenna, Francesco & Corbetta, Jacopo, 2018. "The asymptotic smile of a multiscaling stochastic volatility model," Stochastic Processes and their Applications, Elsevier, vol. 128(3), pages 1034-1071.
    12. Archil Gulisashvili & Peter Tankov, 2014. "Implied volatility of basket options at extreme strikes," Papers 1406.0394, arXiv.org.
    13. Aleksandar Mijatovi'c & Peter Tankov, 2012. "A new look at short-term implied volatility in asset price models with jumps," Papers 1207.0843, arXiv.org, revised Jul 2012.
    14. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2017. "Shapes of implied volatility with positive mass at zero," Working Papers 2017-77, Center for Research in Economics and Statistics.
    15. A. Gulisashvili, 2009. "Asymptotic Formulas with Error Estimates for Call Pricing Functions and the Implied Volatility at Extreme Strikes," Papers 0906.0394, arXiv.org.
    16. Philip Stahl, 2022. "Asymptotic extrapolation of model-free implied variance: exploring structural underestimation in the VIX Index," Review of Derivatives Research, Springer, vol. 25(3), pages 315-339, October.
    17. Lingjiong Zhu, 2015. "Short maturity options for Azéma–Yor martingales," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(04), pages 1-32, December.
    18. Antoine Jacquier & Patrick Roome, 2013. "The Small-Maturity Heston Forward Smile," Papers 1303.4268, arXiv.org, revised Aug 2013.
    19. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2013. "Shapes of implied volatility with positive mass at zero," Papers 1310.1020, arXiv.org, revised May 2017.
    20. Louis-Pierre Arguin & Nien-Lin Liu & Tai-Ho Wang, 2018. "Most-Likely-Path In Asian Option Pricing Under Local Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(05), pages 1-32, August.

    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:wsi:ijtafx:v:20:y:2017:i:03:n:s0219024917500133. 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: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/ijtaf/ijtaf.shtml .

    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.