Double Digital Option

ORE defines the product as follows:

A double digital option is a binary option that pays out a fixed amount if the two underlyings (FX spots, Equity or Commodity prices, interest rates) are simultaneously in the money w.r.t. given strikes and option types at the option expiry.

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

ORE notes the range of supported underlying types:

Equity, Commodity and IR underlyings are also supported in arbitrary combinations.

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

A double digital option is an all-or-nothing bet on two markets at once. The buyer names a level, or a range, for each market. The payout happens only when both markets are on the right side of their levels on the same day. One missed market kills the payout. The trade suits a view on two markets together, and on how they move together, more than on either alone.

The data node DoubleDigitalOptionData is the trade data container for the DoubleDigitalOption trade type. Expiry is the expiry date of the option and Settlement the payout settlement date. BinaryPayout is the amount paid when the option is in the money.

Each underlying has its own strike node. BinaryLevel1 is the strike for underlying 1, and BinaryLevel2 the strike for underlying 2. For a call or put option the level is the strike; for a collar option it is the lower bound. BinaryLevelUpper1 and BinaryLevelUpper2 are the optional upper bounds, used by collar options only. The meaning of a level follows the underlying type. For an FX underlying it is the number of units of CCY2 per unit of CCY1; for equity it is the price in the equity currency; for commodity the price in the commodity currency; for an interest rate it is the rate in decimal form. Allowable level values are non-negative numbers.

Type1 and Type2 name the option type that applies to each underlying: Call, Put or Collar. ORE defines the in-the-money condition for a call and put as follows:

Underlying 1 is considered to be in the money if the spot is above (Call) / below (Put) the BinaryLevel1 resp. between (Collar) the BinaryLevel1 and BinaryLevelUpper1 at the expiry.

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

Underlying 2 is in the money under the same rule with its own level and upper bound. Position takes Long or Short. Underlying1 and Underlying2 name the two underlyings; each takes Equity, Commodity, FX or IR as its type. The optional Underlying3 and Underlying4 turn each side into a spread: when Underlying3 is defined, the first underlying is treated as the fixing of Underlying1 minus the fixing of Underlying3. The same applies to Underlying2 with Underlying4. Each spread side must match the type of the underlying it modifies. PayCcy is the currency in which the binary payout is paid.

The payout is a fixed amount or nothing. The option pays the binary payout when both underlying values satisfy their in-the-money conditions at expiry, and pays zero otherwise. Each underlying condition is a digital in its own right: the spot above the level for a call, below it for a put, or inside the lower and upper bounds for a collar. The joint condition is the product of the two digitals. The position flag decides whether the holder receives or pays the payout. When a spread side is present, the underlying value is the difference of two fixings instead of a single fixing.

• The spot levels of both underlyings at expiry. They decide the two digital conditions.
• The volatility of each underlying. It sets the probability of each condition.
• The correlation between the two underlyings. The joint condition fires only when both are in the money, so its probability depends on the dependence between them.
• The strike levels and the upper bounds.
• The binary payout amount.
• The interest rates that discount the payout.

The trade has no cashflows before expiry. Both underlyings diffuse until the expiry date. At expiry each spot is compared with its levels and type. When both conditions hold, the fixed payout is paid on the settlement date. The value of the trade decays toward the discounted probability that both digitals fire together.

A double digital option packages two market views into one binary payout. It costs less than two single digitals because it demands both conditions at once. The structure suits investors who expect two markets to land in specific regions at the same date, and who want a known payout if they are right. In ORE Studio a customer books double digital options to value them and run sensitivities on the ORE engine.

ORE's catalogue shows a double digital on EUR-USD and JPY-USD. The trade pays 12000000 USD when EUR-USD is above 1.1 and JPY-USD sits between 0.006 and 0.008 on the expiry date:

<DoubleDigitalOptionData>
  <Expiry>2021-09-01</Expiry>
  <Settlement>2021-09-03</Settlement>
  <BinaryPayout>12000000</BinaryPayout>
  <BinaryLevel1>1.1</BinaryLevel1>
  <BinaryLevel2>0.006</BinaryLevel2>
  <BinaryLevelUpper2>0.008</BinaryLevelUpper2>
  <Type1>Call</Type1>
  <Type2>Collar</Type2>
  <Position>Long</Position>
  <Underlying1>
    <Type>FX</Type>
    <Name>ECB-EUR-USD</Name>
  </Underlying1>
  <Underlying2>
    <Type>FX</Type>
    <Name>ECB-JPY-USD</Name>
  </Underlying2>
  <PayCcy>USD</PayCcy>
</DoubleDigitalOptionData>

Source: ORE User Guide, Products catalogue, doubledigitaloption.tex, listing Double Digital Option data. The catalogue also walks through a term sheet variant of the same structure with a put on the second underlying.