Computing exponential moments of the discrete maximum of a Lévy process and lookback options

Liming Feng*, Vadim Linetsky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

38 Scopus citations


We present a fast and accurate method to compute exponential moments of the discretely observed maximum of a Lévy process. The method involves a sequential evaluation of Hilbert transforms of expressions involving the characteristic function of the (Esscher-transformed) Lévy process. It can be discretized with exponentially decaying errors of the form O(exp (-aMb)) for some a,b > 0, where M is the number of discrete points used to compute the Hilbert transform. The discrete approximation can be efficiently implemented using the Toeplitz matrix-vector multiplication algorithm based on the fast Fourier transform, with total computational cost of O(N M log (M)), where N is the number of observations of the maximum. The method is applied to the valuation of European-style discretely monitored floating strike, fixed strike, forward start and partial lookback options (both newly written and seasoned) in exponential Lévy models.

Original languageEnglish (US)
Pages (from-to)501-529
Number of pages29
JournalFinance and Stochastics
Issue number4
StatePublished - Jul 2009


  • Discrete lookback options
  • Discrete maximum
  • Esscher transform
  • Exponential moments
  • Fourier transform
  • Hilbert transform
  • Lévy processes
  • Sinc expansion

ASJC Scopus subject areas

  • Statistics and Probability
  • Finance
  • Statistics, Probability and Uncertainty


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