Pricing Options With Curved Boundaries

Naoto Kunitomo*, Masayuki Ikeda

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

176 Citations (Scopus)


This paper provides a general valuation method for the European options whose payoff is restricted by curved boundaries contractually set on the underlying asset price process when it follows the geometric Brownian motion. Our result is based on the generalization of the Levy formula on the Brownian motion by T. W. Anderson in sequential analysis. We give the explicit probability formula that the geometric Brownian motion reaches in an interval at the maturity date without hitting either the lower or the upper curved boundaries. Although the general pricing formulae for options with boundaries are expressed as infinite series in the general case, our numerical study suggests that the convergence of the series is rapid. Our results include the formulae for options with a lower boundary by Merton (1973), for path‐dependent options by Goldman, Sossin, and Gatto (1979), and for some corporate securities as special cases.

Original languageEnglish
Pages (from-to)275-298
Number of pages24
JournalMathematical Finance
Issue number4
Publication statusPublished - 1992
Externally publishedYes


  • curved boundaries
  • geometric Brownian motion
  • options

ASJC Scopus subject areas

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


Dive into the research topics of 'Pricing Options With Curved Boundaries'. Together they form a unique fingerprint.

Cite this