Barrier option

In investment, a barrier option is an exotic option on an underlying asset whose existence depends upon the underlying asset's price reaching pre-set barrier level: the derivative either springs into existence or, if the option already exists, it is extinguished.

A barrier option has a lower premium than a similar option without a barrier. Barrier options were created to provide the hedge of an option at a lower premium than a conventional option. For example, if an investor believes that the price of a particular common stock, now trading at $100 per share, will increase within the next 6 months, but will not reach $150 per share, you could buy the option with a barrier level of $150 at a lower premium than a conventional option on the same common stock.

There are disputes over the tax treatment of these options in the United States.

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:

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. Also note that once it's in, it's in 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.

Barrier events

A barrier event occurs when the underlying crosses the barrier level. While it seems straightforward to define a barrier event as "underlying trades at or above a given level," in reality it's not so simple. What if the underlying only trades at the level for a single trade? How big would that trade have to be? Would it have to be on an exchange or could it be between private parties? When barrier options were first introduced to options markets, many banks had legal trouble resulting from a mismatched understanding with their counterparties regarding exactly what constituted a barrier event.

Variations

Barrier options are sometimes accompanied by a rebate, which is a payoff to the option holder in case of a barrier event. Rebates can either be paid at the time of the event or at expiration.

Barrier options can have either American, Bermudan or European exercise style.

Valuation

The valuation of barrier options can be tricky, because unlike other simpler options they are path-dependent – that is, the value of the option at any time depends not just on the underlying at that point, but also on the path taken by the underlying (since, if it has crossed the barrier, a barrier event has occurred). Although the classical Black–Scholes approach does not directly apply, several more complex methods can be used:

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