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Digital options

Key concepts
A digital option, also known as a binary option or an all-or-nothing option, either pays out zero or a pre-set amount.
Unlike standard options therefore its payout is discontinuous: if the strike price is reached, a pre-determined amount is payable no matter by how much the option is in the money.
The payout is expressed either as a multiple of the premium a payout of 1:2.5 would mean that the option would pay 2.5 times the premium invested or as a percentage of premium to payout, in this example the premium would be 40%.
The conditions that determine a digital option's payout are many:
for example, a spot rate trading or not trading at a certain level, a range maintained or broken, a level trading only after another level trades or if that level has never traded.
Like barrier options, digital options are difficult to value and hedge because, around the one particular spot price, small moves in the underlying can have very large effects on the value of the option because of the discontinuity in payout around that point.
Digital options can be combined to create products that enable the holders to express extremely precise views on market movements.
They are often embedded in fixed-income instruments to create innovative structured assets.
And basic digital puts and calls are a fundamental building block for many structured products including installment options, range and timer options and structured forwards.
The three basic structures are the European digital option, the American digital option and the range digital option.
The other products are variations on these three themes.

American Digital Option
An option that pays a fixed amount at expiry providing that the underlying has traded above (call) or below (put) a pre-determined level at any time over the life of the option. If the underlying fails to achieve this strike level then the option pays nothing. Also known as one-touch options (as it only needs the underlying to reach the strike price/trade within the one-sided range once within the life of the option for there to be a payout) and no-touch options (as if there is no-touch there is no payout).

Asset-or-nothing digital option
Not a true digital option, these are options that pay out the value of the underlying at maturity if spot trades above (call) or below (put) the strike levels. A slight variation is the digital gap option that pays out an amount defined by the underlying asset price minus a constant providing that the underlying has traded above or below the strike. The payout profile is very similar to that of standard knock-in options whose knock-in level is in-the-money.

Digital barrier option
A digital option whose payout depends not only upon the digital strike level but also on whether the spot price has reached a knock-out or knock-in barrier level. A knock-out digital call {put} pays a fixed amount at expiry providing that the underlying is trading above {below} the strike at maturity and that it has never hit a predetermined knock-out strike level at any time over the life of the option. There are two basic types: i. in one the strike is set at-the-money and the knock-out is set out-of-the-money relative to the strike. If the underlying moves away from the strike to the knock-out the option is extinguished. If it does not hit the knock-out and trades above the strike at expiry (European) or has ever traded above it (American) then the option pays out. This is known as an out-of-the-money knock-out digital option. ii. in the other, the strike and knock-out are set at the same level which is below the current spot price. As long as the underlying never trades through the knock-out level the option pays out a set amount. If it hits the knock-out at any time it is extinguished. This second variety is essentially a range option with only one boundary. This is known as an in-the-money knock-out digital option and is (like the American digital) known as a no-touch option though in this case because the underlying must not touch the knock-out level. A knock-in digital option is an American-style digital whose payout is not made at expiry but is automatically triggered as soon as the underlying asset price hits the barrier levels. Some varieties require that the barrier has been hit more than once.

European/At-maturity digital option
An option that pays a fixed amount at expiry providing that the underlying is trading above (call) or below (put) a pre-determined level at expiry. If the underlying is trading outside the level, then the option pays nothing even if it has previously traded through the barrier. For example, a digital cap {floor} pays a fixed amount if the interest rate moves above {below} the strike. These are useful where rates are expected to move just far enough to trigger the option but not much further and so where a conventional cap or floor would represent the purchase of unwanted protection.

Mandarin cylinder/collar
A collar that is restructured using an overlay of digital options. The underlying collar remains in place but the holder buys a range binary option whose range matches that of the collar. As long as the underlying trades within the boundaries of the collar the range binary option produces an additional payout. If spot trades outside the range, the holder loses the premium paid for the option. In the example illustrated, the holder has bought a GBP put struck at 1.45, has sold a GBP call struck at 1.54 and has bought a GBP 1.45/1.54 range binary option. If the barriers are not hit, then at maturity the holder receives 0.0620 GBP plus the value of the cylinder.

Range [digital] option
A digital option with more than one boundary condition. The basic version comes in European and American varieties. The European-style range option, often known simply as a range option, pays a fixed amount providing that spot is trading within a predetermined range at expiry. If spot is trading outside the range then the option pays out nothing. The American-style or knock-out range option only pays a fixed amount at expiry providing that spot has never traded outside the range during the life of the option. Options can be structured that pay out only if both boundaries have been touched (limit digital option); if either boundary has been touched (double-one-touch option); if neither boundary has been touched (double-no-touch option); or if the underlying either stays inside a predetermined range, or hits both extremes of the range. The latter is sometimes known as a boundary [digital] option and is the combination of a range option with a limit digital option. All these options take views on trading ranges and thus volatility rather than any directional trend. For example, the boundary binary option takes the view that volatility will either be low, or high against no strong directional trend. They are embedded in structured assets to create various types of accrual or range note.

Timing/timer option
An option whose payout is based on the amount of time the underlying has traded within a pre-set range or above or below a specified barrier. The resurrecting/standard timing option accrues a fixed proportion of a total amount for every period (typically one day) that the underlying continues to trade within a predetermined range or band over the life of the option. The buyer specifies the range and pays a premium upfront. The option's maximum payout is specified in advance as a multiple of the premium. Then, for every period in which the underlying trades within the range, a portion of that maximum payout is locked in. No daily payment is made for the time that the underlying trades outside the range. The final payout is calculated on a pro rata basis and is payable on expiry. Also known as corridor options, memory options or accrual options. The extinguishing/knock-out timing option is a standard timer option but with the additional condition that if the underlying trades outside the range during any given period then the option will stop accruing any further payment from that point on the option is knocked-out. The total amount accrued prior to knock-out is payable at expiry.

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