Discretely monitored first passage problems and barrier options: an eigenfunction expansion approach

Lingfei Li*, Vadim Linetsky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


This paper develops an eigenfunction expansion approach to solve discretely monitored first passage time problems for a rich class of Markov processes, including diffusions and subordinate diffusions with jumps, whose transition or Feynman–Kac semigroups possess eigenfunction expansions in L2$L^{2}$-spaces. Many processes important in finance are in this class, including OU, CIR, (JD)CEV diffusions and their subordinate versions with jumps. The method represents the solution to a discretely monitored first passage problem in the form of an eigenfunction expansion with expansion coefficients satisfying an explicitly given recursion. A range of financial applications is given, drawn from across equity, credit, commodity, and interest rate markets. Numerical examples demonstrate that even in the case of frequent barrier monitoring, such as daily, approximating discrete first passage time problems with continuous solutions may result in unacceptably large errors in financial applications. This highlights the relevance of the method to financial applications.

Original languageEnglish (US)
Pages (from-to)941-977
Number of pages37
JournalFinance and Stochastics
Issue number4
StatePublished - Oct 1 2015


  • Barrier options
  • Diffusions
  • Eigenfunction expansions
  • First passage times
  • Subordination

ASJC Scopus subject areas

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


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