IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2107.12439.html
   My bibliography  Save this paper

Proof of non-convergence of the short-maturity expansion for the SABR model

Author

Listed:
  • Alan L. Lewis
  • Dan Pirjol

Abstract

We study the convergence properties of the short maturity expansion of option prices in the uncorrelated log-normal ($\beta=1$) SABR model. In this model the option time-value can be represented as an integral of the form $V(T) = \int_{0}^\infty e^{-\frac{u^2}{2T}} g(u) du$ with $g(u)$ a "payoff function" which is given by an integral over the McKean kernel $G(s,t)$. We study the analyticity properties of the function $g(u)$ in the complex $u$-plane and show that it is holomorphic in the strip $|\Im(u) | 0$). In a certain limit which can be defined either as the large volatility limit $\sigma_0\to \infty$ at fixed $\omega=1$, or the small vol-of-vol limit $\omega\to 0$ limit at fixed $\omega\sigma_0$, the short maturity $T$-expansion for the implied volatility has a finite convergence radius $T_c = \frac{1.32}{\omega\sigma_0}$.

Suggested Citation

  • Alan L. Lewis & Dan Pirjol, 2021. "Proof of non-convergence of the short-maturity expansion for the SABR model," Papers 2107.12439, arXiv.org, revised Jul 2021.
  • Handle: RePEc:arx:papers:2107.12439
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2107.12439
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Nick Deguillaume & Riccardo Rebonato & Andrey Pogudin, 2013. "The nature of the dependence of the magnitude of rate moves on the rates levels: a universal relationship," Quantitative Finance, Taylor & Francis Journals, vol. 13(3), pages 351-367, February.
    2. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    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. Christoffersen, Peter & Heston, Steven & Jacobs, Kris, 2010. "Option Anomalies and the Pricing Kernel," Working Papers 11-17, University of Pennsylvania, Wharton School, Weiss Center.
    2. Fisher, Travis & Pulido, Sergio & Ruf, Johannes, 2019. "Financial models with defaultable numéraires," LSE Research Online Documents on Economics 84973, London School of Economics and Political Science, LSE Library.
    3. Nicolas Langrené & Geoffrey Lee & Zili Zhu, 2016. "Switching To Nonaffine Stochastic Volatility: A Closed-Form Expansion For The Inverse Gamma Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(05), pages 1-37, August.
    4. Peter Carr & Travis Fisher & Johannes Ruf, 2014. "On the hedging of options on exploding exchange rates," Finance and Stochastics, Springer, vol. 18(1), pages 115-144, January.
    5. Keller-Ressel, Martin, 2015. "Simple examples of pure-jump strict local martingales," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4142-4153.
    6. Manley, Bruce & Niquidet, Kurt, 2010. "What is the relevance of option pricing for forest valuation in New Zealand?," Forest Policy and Economics, Elsevier, vol. 12(4), pages 299-307, April.
    7. Slanina, František, 2010. "A contribution to the systematics of stochastic volatility models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(16), pages 3230-3239.
    8. Bertram During & Christian Hendricks & James Miles, 2016. "Sparse grid high-order ADI scheme for option pricing in stochastic volatility models," Papers 1611.01379, arXiv.org.
    9. Björn Lutz, 2010. "Pricing of Derivatives on Mean-Reverting Assets," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-02909-7, October.
    10. Archil Gulisashvili, 2022. "Multivariate Stochastic Volatility Models and Large Deviation Principles," Papers 2203.09015, arXiv.org, revised Nov 2022.
    11. Fazlollah Soleymani & Andrey Itkin, 2019. "Pricing foreign exchange options under stochastic volatility and interest rates using an RBF--FD method," Papers 1903.00937, arXiv.org.
    12. Antonio Mele, 2003. "Fundamental Properties of Bond Prices in Models of the Short-Term Rate," The Review of Financial Studies, Society for Financial Studies, vol. 16(3), pages 679-716, July.
    13. Fergusson, Kevin, 2020. "Less-Expensive Valuation And Reserving Of Long-Dated Variable Annuities When Interest Rates And Mortality Rates Are Stochastic," ASTIN Bulletin, Cambridge University Press, vol. 50(2), pages 381-417, May.
    14. Jim Gatheral & Radoš Radoičić, 2019. "Rational Approximation Of The Rough Heston Solution," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-19, May.
    15. Andrey Itkin, 2023. "The ATM implied skew in the ADO-Heston model," Papers 2309.15044, arXiv.org.
    16. Falko Baustian & Katev{r}ina Filipov'a & Jan Posp'iv{s}il, 2019. "Solution of option pricing equations using orthogonal polynomial expansion," Papers 1912.06533, arXiv.org, revised Jun 2020.
    17. Travis Fisher & Sergio Pulido & Johannes Ruf, 2019. "Financial Models with Defaultable Numéraires," Post-Print hal-01240736, HAL.
    18. Minqiang Li, 2015. "Derivatives Pricing on Integrated Diffusion Processes: A General Perturbation Approach," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 35(6), pages 582-595, June.
    19. Martin Keller-Ressel, 2008. "Moment Explosions and Long-Term Behavior of Affine Stochastic Volatility Models," Papers 0802.1823, arXiv.org, revised Oct 2008.
    20. Viatcheslav Gorovoi & Vadim Linetsky, 2004. "Black's Model of Interest Rates as Options, Eigenfunction Expansions and Japanese Interest Rates," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 49-78, January.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:2107.12439. 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.