Types
Barrier options are path-dependent exotics that are similar in some ways to ordinary options. You can call or put in American, Bermudan, or European exercise style. But they become activated (or extinguished) only if the underlying reaches a predetermined level (the barrier).
"In" options start their lives worthless and only become active in the event that a predetermined knock-in barrier price is breached. "Out" options start their lives active and become null and void in the event that a certain knock-out barrier price is breached.
If the option expires inactive, then it may be worthless, or there may be a cash rebate paid out as a fraction of the premium.
The four main types of barrier options are:
- Up-and-out: spot price starts below the barrier level and has to move up for the option to be knocked out.
- Down-and-out: spot price starts above the barrier level and has to move down for the option to become null and void.
- Up-and-in: spot price starts below the barrier level and has to move up for the option to become activated.
- Down-and-in: spot price starts above the barrier level and has to move down for the option to become activated.
For example, a European call option may be written on an underlying with spot price of $100 and a knockout barrier of $120. This option behaves in every way like a vanilla European call, except if the spot price ever moves above $120, the option "knocks out" and the contract is null and void. Note that the option does not reactivate if the spot price falls below $120 again. Once it is out, it's out for good.
In-out parity is the barrier option's answer to put-call parity. If we combine one "in" option and one "out" barrier option with the same strikes and expirations, we get the price of a vanilla option: . A simple arbitrage argument—simultaneously holding the "in" and the "out" option guarantees that exactly one of the two will pay off identically to a standard European option while the other will be worthless. The argument only works for European options without rebate.
Read more about this topic: Barrier Option
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