FX Digital Barrier Option

Table of Contents

An FX digital barrier option couples a digital payout with a barrier condition. It pays a fixed amount if the rate both touches a level and settles on the right side of a strike. ORE models it with the trade type FxDigitalBarrierOption. This note records the domain grounding, as ORE documents it in its product catalogue.

1. Identifiers

What a trader books it as, and what to grep for. The codes are the trade type catalogue's, which is what the database enforces.

Field Value
Full name FX Digital Barrier Option
Short name FX Digital Barrier Option
ORE Studio product code FxDigitalBarrierOption
Asset class fx
Family Option
ORE trade data node FxDigitalBarrierOptionData
ORE documentation fx_digitalbarrieroption.tex

2. Summary

An FX digital barrier option combines a cash-or-nothing digital underlying with a continuously monitored single barrier. At expiry it pays either the digital payoff or nothing, depending on whether the rate reached the barrier. The four main barrier types apply. A knock-in pays if the barrier is breached. A knock-out pays if it is not. The strike test at expiry decides the digital payout.

3. Detail

3.1. What it is

ORE defines the product as follows:

This product has a continuously monitored single barrier with an Cash-or-Nothing digital underlying option.

Source: ORE User Guide, Products catalogue, fx_digitalbarrieroption.tex.

ORE states the payoff condition as follows:

At expiry a digital FX Barrier option pays either the payoff of a digital Cash-or-Nothing underlying option depending upon an FX spot rate reaching a pre-set barrier level.

Source: ORE User Guide, Products catalogue, fx_digitalbarrieroption.tex.

ORE also notes that the option can be of the same four main types as a regular barrier FX option: up-and-out, down-and-out, up-and-in, and down-and-in. A knock-in or one-touch pays the underlying option payoff if the barrier is breached, and no payout otherwise. A knock-out or no-touch pays nothing if the barrier is breached, and the underlying option payoff otherwise.

3.2. In plain terms

An FX digital barrier option is a digital option with a door. The door opens or closes when the rate touches a level. On the paying side, the rate must also end past the strike. Both conditions decide the fixed payout.

3.3. How it works in ORE

The FxDigitalBarrierOptionData node includes one OptionData node and one BarrierData node. This is a single-barrier instrument, so the barrier data holds one level. The level is quoted as the amount in DomesticCurrency per one unit of ForeignCurrency. The barrier Type is one of the four main types. StrictComparison chooses the boundary of the barrier check. The Strike test at expiry uses the same currency terms. Call pays when the rate at expiry is above the strike. Put pays when it is below. PayoffAmount and PayoffCurrency set the payout. A StartDate before today triggers a check of past fixings for a breach, with the Calendar and FXIndex naming the fixing source. FXIndexDailyLows and FXIndexDailyHighs track intraday barrier breaches.

3.4. Mathematical notes

The value is the probability that the barrier condition and the strike condition both resolve in favour of the buyer, discounted. The barrier path probability and the terminal moneyness interact. The payout is fixed, so volatility acts through the joint probability, not through a payoff tail.

3.5. What moves its value (static sensitivities)

  • The FX spot rate. Its distance from the barrier and the strike sets the state.
  • The volatility of the pair. It drives both the touch probability and the terminal moneyness.
  • The interest rates in both currencies.
  • The barrier level, the strike, and the fixed payout.
  • The payoff currency and the fixing source.

The sensitivities change sign near the barrier. A knock-in gains as spot approaches the barrier. A knock-out loses. A digital near the strike is sensitive to the terminal rate.

3.6. How the profile ages (dynamic sensitivities)

The barrier is monitored continuously through the option's life. A knock-out can end at any moment. A knock-in waits for its breach. At expiry the digital strike test decides the final payout. The payout settles in the payoff currency and the trade stops.

3.7. Why a customer would want it

A customer who wants a fixed payout on a rate range view buys a digital barrier option. The barrier shapes the payout at a lower premium than a plain digital. In ORE Studio a customer books FX digital barrier options to value them and run sensitivities on the ORE engine.

3.8. Example

ORE's catalogue shows a long European digital call with a down-and-in barrier: 

<FxDigitalBarrierOptionData>
  <OptionData>
    <LongShort>Long</LongShort>
    <OptionType>Call</OptionType>
    <Style>European</Style>
    <ExerciseDates>
      <ExerciseDate>2021-12-14</ExerciseDate>
    </ExerciseDates>
    ...
  </OptionData>
  <BarrierData>
    <Type>DownAndIn</Type>
    <Levels>
      <Level>1.18</Level>
    </Levels>
  </BarrierData>
  <StartDate>2019-01-25</StartDate>
  <Calendar>TARGET</Calendar>
  <FXIndex>FX-ECB-EUR-USD</FXIndex>
  <Strike>1.1</Strike>
  <PayoffAmount>100000</PayoffAmount>
  <PayoffCurrency>USD</PayoffCurrency>
  <ForeignCurrency>EUR</ForeignCurrency>
  <DomesticCurrency>USD</DomesticCurrency>
</FxDigitalBarrierOptionData>

Source: ORE User Guide, Products catalogue, fx_digitalbarrieroption.tex, listing FX Digital Barrier Option data (optional elements omitted).

4. See also

Emacs 29.3 (Org mode 9.6.15)