The article explores how to price barrier options using a modified version of the Black-Scholes model. By using a binomial framework, the researchers show how to calculate prices for different types of barrier options through backward induction or closed formulas. They also analyze how these prices in discrete time converge to continuous-time values. The study addresses numerical challenges with quadratic interpolation and delves into American barrier options, providing formulas for pricing them. Additionally, the researchers apply their approach to contracts with local and partial barrier checks.