Valuing step barrier options and their icicled variations

This paper intends to investigate an interesting class of barrier options, called step barrier options, whose barrier levels are a piecewise constant function of time. These options, while having transparent, simple, and flexible payoff structures, allow for explicit pricing formulas under the Black-Scholes model, and thus can be easily embedded into equity-linked products to enhance the yield or reduce the downside risk. Moreover, the class can be further generalized by attaching vertical branches of barriers to the horizontal one as in Lee and Ko (2018). Using the actuarial method of Esscher transform and the factorization formula, we derive the option pricing formulas under a more general framework with vertical branches attached to horizontal barriers. We explore the formulas through numerical examples, demonstrating their applicability to equity-linked investment with the step barrier option embedded.