Barrier Option Mechanics — Knock-In, Knock-Out, and No-Arbitrage Relationships

Barrier options are the simplest path-dependent derivatives: a European option that activates (knock-in) or deactivates (knock-out) if the stock touches a barrier level during the option's life. They are the building block of structured products (autocallables, capital-guaranteed notes) and are actively traded in FX and equity markets.

This lesson catalogues the four standard barrier types, establishes in-out parity (the model-free relationship that makes knock-ins and knock-outs interchangeable), and computes closed-form prices for continuously monitored barriers under GBM — the starting point for all subsequent barrier pricing theory.

• define all four standard barrier types (DI, DO, UI, UO) and state their relationships;
• apply in-out parity to reduce a knock-in to a vanilla minus a knock-out;
• compute the Black-Scholes price of a down-and-out call using the reflection formula;
• state what "rebate" means and how it affects the barrier price.

1. 1.1 Barrier option types
2. 1.2 In-out parity
3. 1.3 Closed-form prices under GBM (the reflection principle)
4. 1.4 Model sensitivity