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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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    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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