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Efficient evaluation of expectations of functions of a Lévy process and its extremum

Author

Listed:
  • Svetlana Boyarchenko

    (The University of Texas at Austin)

  • Sergei Levendorskiĭ

    (Calico Science Consulting)

Abstract

We prove a simple general formula for the expectation of a function of a Lévy process and its running extremum. Under additional conditions, we derive analytical formulas using the Fourier/Laplace inversion and Wiener–Hopf factorisation, and discuss efficient numerical methods for the realisation of these formulas. As applications, the cumulative probability distribution function of the current value of the process and of the value of the supremum process, both evaluated at some moment in the past, and the price of the option to exchange the supremum of the stock price for a power of the price are calculated. The most efficient numerical methods use the sinh-acceleration technique and simplified trapezoid rule. The program in MATLAB running on a Mac with moderate characteristics achieves the precision E-7 and better in several milliseconds, and E-14 in a fraction of a second.

Suggested Citation

  • Svetlana Boyarchenko & Sergei Levendorskiĭ, 2025. "Efficient evaluation of expectations of functions of a Lévy process and its extremum," Finance and Stochastics, Springer, vol. 29(2), pages 443-468, April.
    • Handle: RePEc:spr:finsto:v:29:y:2025:i:2:d:10.1007_s00780-025-00556-6
    DOI: 10.1007/s00780-025-00556-6
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    References listed on IDEAS

    as
    1. Svetlana Boyarchenko & Sergei Levendorskii, 2005. "American options: the EPV pricing model," Annals of Finance, Springer, vol. 1(3), pages 267-292, August.
    2. Lingfei Li & Vadim Linetsky, 2015. "Discretely monitored first passage problems and barrier options: an eigenfunction expansion approach," Finance and Stochastics, Springer, vol. 19(4), pages 941-977, October.
    3. Svetlana Boyarchenko & Sergei Levendorskiĭ, 2020. "Static and semistatic hedging as contrarian or conformist bets," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 921-960, July.
    4. S. Z. Levendorskiǐ, 2004. "Pricing Of The American Put Under Lévy Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(03), pages 303-335.
    5. Fusai, Gianluca & Germano, Guido & Marazzina, Daniele, 2016. "Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options," European Journal of Operational Research, Elsevier, vol. 251(1), pages 124-134.
    6. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2022. "Efficient inverse $Z$-transform and pricing barrier and lookback options with discrete monitoring," Papers 2207.02858, arXiv.org, revised Jul 2022.
    7. Svetlana Boyarchenko & Sergei Levendorskiĭ, 2013. "Efficient Laplace Inversion, Wiener-Hopf Factorization And Pricing Lookbacks," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(03), pages 1-40.
    8. Svetlana Boyarchenko & Sergei Levendorskii, 2023. "Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in L\'evy models," Papers 2312.03915, arXiv.org.
    9. Svetlana I. Boyarchenko & Sergei Z. Levendorskiǐ, 2000. "Option Pricing For Truncated Lévy Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 3(03), pages 549-552.
    10. Mitya Boyarchenko & Marco De Innocentis & Sergei Levendorskiĭ, 2011. "Prices Of Barrier And First-Touch Digital Options In Lévy-Driven Models, Near Barrier," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(07), pages 1045-1090.
    11. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    12. Sergei Levendorskiĭ, 2012. "Efficient Pricing And Reliable Calibration In The Heston Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(07), pages 1-44.
    13. Svetlana Boyarchenko & Sergei Levendorskiĭ, 2019. "Sinh-Acceleration: Efficient Evaluation Of Probability Distributions, Option Pricing, And Monte Carlo Simulations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-49, May.
    14. Gareth G. Haslip & Vladimir K. Kaishev, 2014. "Lookback option pricing using the Fourier transform B-spline method," Quantitative Finance, Taylor & Francis Journals, vol. 14(5), pages 789-803, May.
    15. Mitya Boyarchenko & Svetlana Boyarchenko, 2011. "Double Barrier Options In Regime-Switching Hyper-Exponential Jump-Diffusion Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 14(07), pages 1005-1043.
    16. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
    17. Svetlana Boyarchenko & Sergei Levendorskii, 2007. "Irreversible Decisions under Uncertainty," Studies in Economic Theory, Springer, number 978-3-540-73746-9, April.
    18. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2023. "Efficient inverse $Z$-transform: sufficient conditions," Papers 2305.10725, arXiv.org.
    19. Mitya Boyarchenko & Sergei Levendorskiĭ, 2009. "Prices And Sensitivities Of Barrier And First-Touch Digital Options In Lévy-Driven Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(08), pages 1125-1170.
    20. Svetlana Boyarchenko & Sergei Levendorskii, 2023. "Efficient evaluation of joint pdf of a L\'evy process, its extremum, and hitting time of the extremum," Papers 2312.05222, arXiv.org.
    21. Liming Feng & Vadim Linetsky, 2009. "Computing exponential moments of the discrete maximum of a Lévy process and lookback options," Finance and Stochastics, Springer, vol. 13(4), pages 501-529, September.
    22. Sergei Levendorskiĭ, 2014. "METHOD OF PAIRED CONTOURS AND PRICING BARRIER OPTIONS AND CDSs OF LONG MATURITIES," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(05), pages 1-58.
    23. Leif Andersen & Alexander Lipton, 2013. "Asymptotics For Exponential Lévy Processes And Their Volatility Smile: Survey And New Results," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(01), pages 1-98.
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    More about this item

    Keywords

    Lévy process; Extrema of a Lévy process; Lookback options; Barrier options; Wiener–Hopf factorisation; Fourier transform; Laplace transform; Gaver–Wynn rho algorithm; Sinh-acceleration;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • C67 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Input-Output Models
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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