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Semi-analytical pricing of barrier options in the time-dependent Heston model

Author

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  • P. Carr
  • A. Itkin
  • D. Muravey

Abstract

We develop the general integral transforms (GIT) method for pricing barrier options in the time-dependent Heston model (also with a time-dependent barrier) where the option price is represented in a semi-analytical form as a two-dimensional integral. This integral depends on yet unknown function $\Phi(t,v)$ which is the gradient of the solution at the moving boundary $S = L(t)$ and solves a linear mixed Volterra-Fredholm equation of the second kind also derived in the paper. Thus, we generalize the one-dimensional GIT method, developed in (Itkin, Lipton, Muravey, Generalized integral transforms in mathematical finance, WS, 2021) and the corresponding papers, to the two-dimensional case. In other words, we show that the GIT method can be extended to stochastic volatility models (two drivers with inhomogeneous correlation). As such, this 2D approach naturally inherits all advantages of the corresponding 1D methods, in particular, their speed and accuracy. This result is new and has various applications not just in finance but also in physics. Numerical examples illustrate high speed and accuracy of the method as compared with the finite-difference approach.

Suggested Citation

  • P. Carr & A. Itkin & D. Muravey, 2022. "Semi-analytical pricing of barrier options in the time-dependent Heston model," Papers 2202.06177, arXiv.org.
    • Handle: RePEc:arx:papers:2202.06177
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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.
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    3. Long Teng, 2021. "The Heston Model with Time-Dependent Correlation Driven by Isospectral Flows," Mathematics, MDPI, vol. 9(9), pages 1-8, April.
    4. Alexander Lipton & Andrey Gal & Andris Lasis, 2014. "Pricing of vanilla and first-generation exotic options in the local stochastic volatility framework: survey and new results," Quantitative Finance, Taylor & Francis Journals, vol. 14(11), pages 1899-1922, November.
    5. 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.
    6. Luca De Gennaro Aquino & Carole Bernard, 2019. "Semi-analytical prices for lookback and barrier options under the Heston model," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 715-741, December.
    7. 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.
    8. 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.
    9. 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.
    10. Belomestny, Denis & Schoenmakers, John, 2016. "Statistical inference for time-changed Lévy processes via Mellin transform approach," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 2092-2122.
    11. 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.
    12. Andrey Itkin & Dmitry Muravey, 2020. "Semi-closed form prices of barrier options in the Hull-White model," Papers 2004.09591, arXiv.org, revised Sep 2020.
    13. 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.
    14. 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.
    15. 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.
    16. Andrey Itkin & Dmitry Muravey, 2021. "Semi-analytical pricing of barrier options in the time-dependent $\lambda$-SABR model," Papers 2109.02134, arXiv.org.
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    Cited by:

    1. Andrey Itkin, 2023. "The ATM implied skew in the ADO-Heston model," Papers 2309.15044, 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.
    3. Andrey Itkin & Dmitry Muravey, 2023. "American options in time-dependent one-factor models: Semi-analytic pricing, numerical methods and ML support," Papers 2307.13870, arXiv.org.
    4. Andrey Itkin, 2023. "Semi-analytic pricing of American options in time-dependent jump-diffusion models with exponential jumps," Papers 2308.08760, arXiv.org, revised Feb 2024.

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