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Semi-analytical pricing of barrier options in the time-dependent $\lambda$-SABR model

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  • Andrey Itkin
  • Dmitry Muravey

Abstract

We extend the approach of Carr, Itkin and Muravey, 2021 for getting semi-analytical prices of barrier options for the time-dependent Heston model with time-dependent barriers by applying it to the so-called $\lambda$-SABR stochastic volatility model. In doing so we modify the general integral transform method (see Itkin, Lipton, Muravey, Generalized integral transforms in mathematical finance, World Scientific, 2021) and deliver solution of this problem in the form of Fourier-Bessel series. The weights of this series solve a linear mixed Volterra-Fredholm equation (LMVF) of the second kind also derived in the paper. Numerical examples illustrate speed and accuracy of our method which are comparable with those of the finite-difference approach at small maturities and outperform them at high maturities even by using a simplistic implementation of the RBF method for solving the LMVF.

Suggested Citation

  • Andrey Itkin & Dmitry Muravey, 2021. "Semi-analytical pricing of barrier options in the time-dependent $\lambda$-SABR model," Papers 2109.02134, arXiv.org.
    • Handle: RePEc:arx:papers:2109.02134
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    References listed on IDEAS

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    1. 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.
    2. Peter Carr & Jian Sun, 2007. "A new approach for option pricing under stochastic volatility," Review of Derivatives Research, Springer, vol. 10(2), pages 87-150, May.
    3. Andrey Itkin & Alexander Lipton & Dmitry Muravey, 2021. "Generalized Integral Transforms in Mathematical Finance," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 12147, December.
    4. Weston Barger & Matthew Lorig, 2016. "Approximate pricing of European and Barrier claims in a local-stochastic volatility setting," Papers 1610.05728, arXiv.org, revised Apr 2017.
    5. Peter Carr & Andrey Itkin & Dmitry Muravey, 2020. "Semi-closed form prices of barrier options in the time-dependent CEV and CIR models," Papers 2005.05459, arXiv.org.
    6. 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.
    7. 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.
    8. Andrey Itkin, 2015. "HIGH ORDER SPLITTING METHODS FOR FORWARD PDEs AND PIDEs," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-24.
    9. Weston Barger & Matthew Lorig, 2017. "Approximate pricing of European and Barrier claims in a local-stochastic volatility setting," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(02n03), pages 1-31, June.
    10. Nawdha Thakoor & Désiré Yannick Tangman & Muddun Bhuruth, 2019. "A Spectral Approach to Pricing of Arbitrage-Free SABR Discrete Barrier Options," Computational Economics, Springer;Society for Computational Economics, vol. 54(3), pages 1085-1111, October.
    11. Huaiqing Zhang & Yu Chen & Xin Nie, 2014. "Solving the Linear Integral Equations Based on Radial Basis Function Interpolation," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-8, June.
    12. Alexander Lipton, 2001. "Mathematical Methods for Foreign Exchange:A Financial Engineer's Approach," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4694, February.
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    Cited by:

    1. P. Carr & A. Itkin & D. Muravey, 2022. "Semi-analytical pricing of barrier options in the time-dependent Heston model," Papers 2202.06177, arXiv.org.
    2. Alexander Lipton & Artur Sepp, 2022. "Toward an efficient hybrid method for pricing barrier options on assets with stochastic volatility," Papers 2202.07849, arXiv.org.

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