Barrier options are the first path-dependent contracts most desks price daily: the payoff depends on whether the underlying touches a level before expiry, not only on . That single gate makes closed-form Black–Scholes prices unavailable and forces either a lattice with killed nodes, a PDE with altered boundary conditions, or Monte Carlo with a careful treatment of monitoring frequency. This note classifies the contract zoo, proves in–out parity, prices knock-outs on binomial trees and finite-difference grids, and applies the Brownian-bridge survival probability to remove the systematic bias of discrete path simulation against continuous monitoring.
Learning objectives. By the end of this note you should be able to:
- classify single-barrier options (in/out × up/down) and explain the economic motivation for each;
- state and prove in–out parity and use it to price knock-ins from knock-outs;
- implement knock-out pricing on a CRR tree with correct barrier alignment;
- write FDM boundary conditions for each barrier type;
- explain discrete-monitoring bias in Monte Carlo and apply the bridge correction;
- distinguish contractual discrete monitoring from numerical discretisation error.
In this lesson
- The contract taxonomy
- In–out parity
- Pricing on binomial trees
- Pricing by finite differences
- Monte Carlo and discrete monitoring
Before you start
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