Barrier option

The barrier option (English barrier option ) is a special type of option , that is a conditional forward transaction . It is one of the exotic options . The main difference to the standard option is that a barrier option is activated or deactivated by the occurrence of certain events. The main forms are the knock-out option, the knock-in option and the digital barrier option.

Knock-out option

A knock-out option is an option that expires if the underlying asset reaches a certain pre-set barrier or, depending on whether it is an upper or lower barrier, exceeds or falls below it.

If it is an upper barrier, the reaching or exceeding of which triggers the decline in value, one speaks of an up-and-out option (from English up and out ). At a lower barrier whose reaching or falling below triggers the loss of value, it is called a down-and-out option (of English down and out ).

Example: A barrier purchase option with a starting value , exercise price , a term of years and a barrier has the following payout profile: ${\displaystyle S_{0}=100}$ ${\displaystyle K=90}$ ${\displaystyle T=2}$ ${\displaystyle B=120}$

• If the price reaches the 120 mark or rises above it at some point during the course of the year, nothing will be paid out, regardless of the closing value. ${\displaystyle S_{t}}$
• If the closing value is below the exercise price of 90, the option also expires (regardless of whether the barrier has been reached in the meantime). ${\displaystyle S_{2}}$
• The difference between the closing price and the exercise price is only paid out if the closing price exceeds the exercise price (for example ), but the mark of 120 has never been reached (here for example ). ${\displaystyle S_{2}=112}$ ${\displaystyle 112-90=22}$

See also : "Leverage Certificates (also: Turbo or Knock-out Certificates, Mini-Futures)" in the Certificate (Economy) article

Knock-in option

The opposite is true for the knock-in option: it only remains valid if the barrier is reached at least once over time (or - depending on whether it is an upper or lower barrier - is exceeded or fallen below), otherwise it will expire.

If it is an upper barrier, the reaching or exceeding of which triggers the retention of value, one speaks of an up-and-in option (from English up and in ). At a lower barrier whose reaching or falling below triggers to maintain the value, it is called a down-and-in option (of English down and in ).

Example: A barrier purchase option with , Strike and Knock-In amount results in the following payout structure: ${\displaystyle S_{0}=100}$ ${\displaystyle K=110}$ ${\displaystyle B=130}$

• If the price never reaches the 130 mark during the entire term, nothing will be paid out, even if the closing price is above the strike price.
• Nothing is paid out if the closing price is below 110, regardless of whether the 130 mark has ever been reached.
• If the price exceeds the mark of 130 in the course of time and is still above 110 on the closing day (around 124), the difference between the closing price and the exercise price is paid out (here: 124-110 = 14).

A down-and-in call option corresponds to a bet on the following performance: In order not to expire, the underlying must first fall below the barrier and then be above the strike price again at the end.

Digital barrier option

In the case of digital barrier options, in contrast to the derivatives above, no strike is agreed that must ultimately be exceeded or fallen below. Here a predetermined nominal value is simply paid out if the barrier has been reached during the term.

Example: A digital barrier option with starting value , face value and barrier pays 50 on the closing day if the price has been below 80 at some point. The closing price does not matter. With such an option, an investor who is also invested in the underlying can hedge against strong price losses. ${\displaystyle S_{0}=100}$ ${\displaystyle N=50}$ ${\displaystyle B=80}$

Other variants

The option types mentioned above are by far the most common and liquid barrier options. However, there are a number of other, more complicated instruments that are also based on the barrier concept:

• Knock-out option with a premium : This is a knock-out option, but with an additional agreement: If the option expires because the barrier has been reached, a previously agreed premium is paid as compensation. This option is often not considered a separate type of barrier option, as it can be represented as a linear combination of knock-out option and digital barrier option.
• Dynamic Barrier Option : Here the barrier condition is modified insofar as the knock-out or knock-in level can change over time. For example, it can be agreed that the base value may not fall below 90 in the first year, not below 80 in the second, or that the price may not fall below the functional value at any time . This is particularly advantageous for long terms, as the expected exponential increase in the price can be taken into account. ${\displaystyle S_{t}}$ ${\displaystyle t\in [0,T]}$ ${\displaystyle B(t)=B_{0}e^{rt}}$
• Tunnel option : This represents the broadest possible generalization of the barrier option: Here the option expires if the price falls below a (possibly dynamic) lower limit or rises above an upper limit. So that the option does not expire, the underlying may only move within a certain bandwidth (the "tunnel"). This instrument can be seen as a bet on the volatility of the underlying asset: Shares that fluctuate strongly are more likely to break one of the boundaries than stable values.
• Parisian option : With this variant, the barrier condition is only triggered if it has been met for a specified period of time, for example if the price is below the barrier for a month with a down-and-in option. Depending on whether the duration has to be fallen short of in one piece or cumulatively, a distinction is made between the pure Parisian option and the cumulative Parisian option, also known as the Parasian option .

Evaluation of barrier options

The calculation of the fair option price is often not an easy task with barrier options, as is generally the case with path-dependent derivatives (the payout not only depends on the closing price, but also takes the entire price trend into account). Information about the probability distribution of the extreme values ​​of the course must be known here explicitly . In the Black-Scholes model which is for example the case, just as in binomial models . With more complicated capital market models, however, often only a discretization (i.e. an approximation using a time-discrete model) or a Monte Carlo simulation helps .

Individual evidence

1. ↑ Barrier option. In: Equity forecast. Gerhard Merk , accessed on December 12, 2019 .