Pricing Barrier Options with Flexible Monitoring Periods

This paper will derive explicit unified pricing formulas for eight types of inside and outside barrier options, respectively. The monitoring periods of these options start at an arbitrary date and end at another arbitrary date before maturity. The eight types of barrier options are up-and-in, up-and-out, down-and-in and down-and-out call (or put) options. Sections 3, 4 and 5 assume that the underlying assets pay no dividends. In contrast, Section 6 will derive a unified pricing formula for the barrier options when their underlying assets pay dividends continuously at a rate proportional to their prices.