Pricing Options With Curved Boundaries

Naoto Kunitomo, Masayuki Ikeda

Research output: Contribution to journalArticle

134 Citations (Scopus)

Abstract

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
Volume2
Issue number4
DOIs
Publication statusPublished - 1992
Externally publishedYes

Fingerprint

Curved Boundary
Brownian movement
Option Pricing
maturity
pricing
assets
Geometric Brownian Motion
Costs
Sequential Analysis
European Options
Infinite series
Date
Valuation
Pricing
Brownian motion
Numerical Study
Interval
Series
sequential analysis
Option pricing

Keywords

  • curved boundaries
  • geometric Brownian motion
  • options

ASJC Scopus subject areas

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

Cite this

Pricing Options With Curved Boundaries. / Kunitomo, Naoto; Ikeda, Masayuki.

In: Mathematical Finance, Vol. 2, No. 4, 1992, p. 275-298.

Research output: Contribution to journalArticle

Kunitomo, Naoto ; Ikeda, Masayuki. / Pricing Options With Curved Boundaries. In: Mathematical Finance. 1992 ; Vol. 2, No. 4. pp. 275-298.
@article{19b3d73801314408a39deaa56a4fb1a8,
title = "Pricing Options With Curved Boundaries",
abstract = "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.",
keywords = "curved boundaries, geometric Brownian motion, options",
author = "Naoto Kunitomo and Masayuki Ikeda",
year = "1992",
doi = "10.1111/j.1467-9965.1992.tb00033.x",
language = "English",
volume = "2",
pages = "275--298",
journal = "Mathematical Finance",
issn = "0960-1627",
publisher = "Wiley-Blackwell",
number = "4",

}

TY - JOUR

T1 - Pricing Options With Curved Boundaries

AU - Kunitomo, Naoto

AU - Ikeda, Masayuki

PY - 1992

Y1 - 1992

N2 - 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.

AB - 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.

KW - curved boundaries

KW - geometric Brownian motion

KW - options

UR - http://www.scopus.com/inward/record.url?scp=84986779671&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84986779671&partnerID=8YFLogxK

U2 - 10.1111/j.1467-9965.1992.tb00033.x

DO - 10.1111/j.1467-9965.1992.tb00033.x

M3 - Article

VL - 2

SP - 275

EP - 298

JO - Mathematical Finance

JF - Mathematical Finance

SN - 0960-1627

IS - 4

ER -