A Valuation Formula for Chained Options with  n -Barriers

Won Choi,Doobae Jun,Hyejin Ku,Gaetano Luciano

doi:10.1155/2022/9563019

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.

Highlights

Barrier options are a widely used class of path-dependent financial derivatives because they are flexible and less expensive than vanilla options
Merton [2], Rubinstein and Reiner [3], and Rich [4] provided a mathematical framework and derived closed-form pricing formulas for various types of single-barrier options. ese studies assume that the underlying asset price is monitored with respect to a single constant barrier for the entire life of the option
All the papers described above are concerned with barrier options, where the monitoring of the barrier starts at a predetermined date. ere is another class of barrier options with two barriers. e option is a chained option in which another barrier option is activated when a primary barrier is hit. is option has become popular in over-thecounter equity and foreign exchange derivative markets

Summary

Introduction

Barrier options are a widely used class of path-dependent financial derivatives because they are flexible and less expensive than vanilla options. A down-and-in chained call option (DICu) is a downand-in call option activated at a time when the underlying asset price hits an upper-barrier level.

Results

Conclusion