A Valuation Formula for Chained Options with n -Barriers

Abstract

This study examines chained options that are connected in the sense that another barrier option becomes active continuously after the underlying asset price crosses a primary barrier. These barrier options have several advantages. First, they preserve the merit of regular barrier options, but demand far lower option premiums, which appeal to option traders. Second, they reduce the higher risk of loss of double barrier options, making option strategies more profitable in certain cases. Third, they have closed-form pricing formulas, unlike double-barrier options, and, thus, avoid the complexity of option pricing. Therefore, they help to enlarge the range of trader’s choice according to a variety of demand of buyers. The values of chained options are compared to those of similar single- and double-barrier options. This study extends the chained option with two barriers to a generalized chained option with $n$ -barriers. In addition, this paper proves the closed formulas of generalized chained options with n-barriers using mathematical induction.