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Pricing Options With Curved Boundaries 1
Author(s) -
Kunitomo Naoto,
Ikeda Masayuki
Publication year - 1992
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/j.1467-9965.1992.tb00033.x
Subject(s) - geometric brownian motion , brownian motion , mathematics , mathematical economics , boundary (topology) , valuation (finance) , stochastic game , valuation of options , lévy process , interval (graph theory) , mathematical analysis , diffusion process , finance , economics , econometrics , combinatorics , economy , statistics , service (business)
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.