European barrier options have a vanilla payoff at expiry plus they also have a single European barrier. For a European knock-out (EKO) barrier option, if spot at maturity is beyond the barrier level, the contract expires worthless despite being in-the-money. The payoff at maturity of a long European knock-out call with 1.3000 strike and 1.4500 European knock-out barrier is shown in Exhibit 22.1. European barriers must be positioned in-the-money, otherwise they have no impact.
European knock-in (EKI) barrier options have a vanilla payoff at expiry only if spot at maturity is beyond the barrier level. The payoff at maturity of a long European knock-in call with 1.3000 strike and 1.4500 European knock-in barrier is shown in Exhibit 22.2.
In the European knock-out barrier case, the curtailing of the payoff beyond the barrier can significantly reduce the cost of the European barrier option compared to the equivalent European vanilla option. This is shown in Exhibit 22.3.