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<p>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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年3月4日

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  • Svetlana Boyarchenko,
  • Sergei Levendorskiĭ

摘要

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.