Step options

Vadim Linetsky*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

56 Scopus citations


Motivated by risk management problems with barrier options, we propose a flexible modification of the standard knock-out and knock-in provisions and introduce a family of path-dependent options: step options. They are parametrized by a finite knock-out (knock-in) rate, ρ. For a down- and-out step option, its payoff at expiration is defined as the payoff of an otherwise identical vanilla option discounted by the knock-out factor exp(-ρτ-B) or max(1 - ρ τ-B, 0), where τ-B is the total time during the contract life that the underlying price was lower than a prespecified barrier level (occupation time). We derive closed-form pricing formulas for step options with any knock-out rate in the range [0, ∞). For any finite knock-out rate both the step option's value and delta are continuous functions of the underlying price at the barrier. As a result, they can be continuously hedged by trading the underlying asset and borrowing. Their risk management properties make step options attractive "no-regrets" alternatives to standard barrier options. As a by-product, we derive a dynamic almost-replicating trading strategy for standard barrier options by considering a replicating strategy for a step option with high but finite knock-out rate. Finally, a general class of derivatives contingent on occupation times is considered and closed-form pricing formulas are derived.

Original languageEnglish (US)
Pages (from-to)55-96
Number of pages42
JournalMathematical Finance
Issue number1
StatePublished - Jan 1999


  • Barrier options
  • Feynman-Kac formula
  • Laplace transform
  • Occupation time
  • Path-dependent options

ASJC Scopus subject areas

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

Fingerprint Dive into the research topics of 'Step options'. Together they form a unique fingerprint.

Cite this