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Efficient inverse $Z$-transform and pricing barrier and lookback options with discrete monitoring

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  • Svetlana Boyarchenko
  • Sergei Levendorskiu{i}

Abstract

We prove simple general formulas for expectations of functions of a random walk and its running extremum. Under additional conditions, we derive analytical formulas using the inverse $Z$-transform, the Fourier/Laplace inversion and Wiener-Hopf factorization, and discuss efficient numerical methods for realization of these formulas. As applications, the cumulative probability distribution function of the process and its running maximum and the price of the option to exchange the power of a stock for its maximum are calculated. The most efficient numerical methods use a new efficient numerical realization of the inverse $Z$-transform, the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-10 and better in several dozen of milliseconds, and E-14 - in a fraction of a isecond.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:2207.02858
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    References listed on IDEAS

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    1. Svetlana Boyarchenko & Sergei Levendorskii, 2005. "American options: the EPV pricing model," Annals of Finance, Springer, vol. 1(3), pages 267-292, August.
    2. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    3. 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.
    4. Kirkby, J. Lars & Nguyen, Duy & Cui, Zhenyu, 2017. "A unified approach to Bermudan and barrier options under stochastic volatility models with jumps," Journal of Economic Dynamics and Control, Elsevier, vol. 80(C), pages 75-100.
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    6. 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.
    7. S. I. Boyarchenko & S. Z. Levendorskii, 2002. "Pricing of perpetual Bermudan options," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 432-442.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Gianluca Fusai & I. Abrahams & Carlo Sgarra, 2006. "An exact analytical solution for discrete barrier options," Finance and Stochastics, Springer, vol. 10(1), pages 1-26, January.
    13. 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.
    14. Svetlana I Boyarchenko & Sergei Z Levendorskii, 2002. "Non-Gaussian Merton-Black-Scholes Theory," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4955, January.
    15. 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.
    16. 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.
    17. Svetlana Boyarchenko & Sergei Levendorskii, 2007. "Irreversible Decisions under Uncertainty," Studies in Economic Theory, Springer, number 978-3-540-73746-9, March.
    18. 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.
    19. 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.
    20. 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.
    21. 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.
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    Cited by:

    1. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2022. "Efficient evaluation of double-barrier options and joint cpdf of a L\'evy process and its two extrema," Papers 2211.07765, arXiv.org.
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