Pricing discretely monitored barrier options and defaultable bonds in Lévy process models: A fast Hilbert transform approach

Liming Feng*, Vadim Linetsky

*Corresponding author for this work

Research output: Contribution to journalArticle

90 Scopus citations

Abstract

This paper presents a novel method to price discretely monitored single- and double-barrier options in Lévy process-based models. The method involves a sequential evaluation of Hilbert transforms of the product of the Fourier transform of the value function at the previous barrier monitoring date and the characteristic function of the (Esscher transformed) Lévy process. A discrete approximation with exponentially decaying errors is developed based on the Whittaker cardinal series (Sinc expansion) in Hardy spaces of functions analytic in a strip. An efficient computational algorithm is developed based on the fast Hilbert transform that, in turn, relies on the FFT-based Toeplitz matrix-vector multiplication. Our method also provides a natural framework for credit risk applications, where the firm value follows an exponential Lévy process and default occurs at the first time the firm value is below the default barrier on one of a discrete set of monitoring dates.

Original languageEnglish (US)
Pages (from-to)337-384
Number of pages48
JournalMathematical Finance
Volume18
Issue number3
DOIs
StatePublished - Jul 1 2008

Keywords

  • Credit risk
  • Defaultable bonds
  • Discrete barrier options
  • Esscher transform
  • First passage time problems
  • Fourier transform
  • Hilbert transform
  • Lévy processes
  • Sinc expansion
  • Whittaker cardinal series

ASJC Scopus subject areas

  • Accounting
  • Finance
  • Social Sciences (miscellaneous)
  • Economics and Econometrics
  • Applied Mathematics

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