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

Svetlana Boyarchenko () and Sergei Levendorskiĭ ()
Additional contact information
Svetlana Boyarchenko: The University of Texas at Austin
Sergei Levendorskiĭ: Calico Science Consulting

Finance and Stochastics, 2025, vol. 29, issue 2, No 4, 443-468

Abstract: 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.

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; 60-08; 42A38; 42B10; 44A10; 65R10; 65G51; 91G20; 91G60 (search for similar items in EconPapers)
JEL-codes: C02 C65 C67 G13 (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s00780-025-00556-6 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:finsto:v:29:y:2025:i:2:d:10.1007_s00780-025-00556-6

Ordering information: This journal article can be ordered from
http://www.springer. ... ance/journal/780/PS2

DOI: 10.1007/s00780-025-00556-6

Access Statistics for this article

Finance and Stochastics is currently edited by M. Schweizer

More articles in Finance and Stochastics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().