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Barrier Option

A barrier option is a path dependent option that has one of two features:
A knockout feature causes the option to immediately terminate if the underlier reaches a specified barrier level, or
A knock-in feature causes the option to become effective only if the underlier first reaches a specified barrier level.
Premiums are paid in advance. Due to the contingent nature of the option, premiums tend to be lower than for a corresponding vanilla option.

Consider a knock-in call option with a strike price of EUR 100 and a knock-in barrier at EUR 110. Suppose the option was purchased when the underlier was at EUR 90. If the option expired with the underlier at EUR 103, but the underlier never reached the barrier level of EUR 110 during the life of the option, the option would expire worthless. On the other hand, if the underlier first rose to the EUR 110 barrier, this would cause the option to knock-in. It would then be worth EUR 3 when it expired with the underlier at EUR103. This is illustrated in Exhibit 1:

Example: Up-And-In Barrier Call Option
Exhibit 1

An up-and-in barrier call option expires worthless unless the underlier value hits the barrier at some time during the life of the option.

The particular option in this example is known as an “up-and-in” option because the underlier must first go “up” to the barrier before the option knocks “in.”
In all, there are eight flavors of barrier options comprising European puts or calls having barriers that are:
up-and-in,
down-and-in,
up-and-out, or
down-and-out.

Of the eight, four either knock-in or knockout when they are in-the-money. These are called reverse barrier options. They can pose significant hedging challenges for the issuer.
Alternative structures include multiple barriers or barriers incorporated into other types of derivatives. For example, binary options can be structured with barriers.
Merton priced a down-and-out call option in his seminal (1973) paper. The classic paper providing analytic pricing formulas for barriers is Reiner and Rubinstein (1991). See Haug (1997) for an alternative treatment of the same formulas. A shortcoming of analytic formulas is their use of a single implied volatility. Because barrier options are path-dependent, it is desirable to model a term structure of implied volatilities. See Taleb (1996).

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