Basket Option

ORE scopes the product as follows:

The supported underlying types are Equity, Fx or Commodity resulting in corresponding trade types and trade data container names EquityBasketOption / EquityBasketOptionData FxBasketOption / FxBasketOptionData CommodityBasketOption / CommodityBasketOptionData

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

ORE represents basket options as traditional trades or as scripted trades. Each variation has its own payoff script. The catalogue maps them as follows: the vanilla form runs the VanillaBasketOption script, the Asian form the AsianBasketOption script, the average strike form the AverageStrikeBasketOption script, and the lookback forms the LookbackCallBasketOption and LookbackPutBasketOption scripts.

A basket option is a bet on several assets at once, through one option. The assets are combined with weights into one basket value. A vanilla basket call pays when the basket is above the strike at expiry. The other forms soften the payoff. The Asian form averages the basket over time. The average strike form uses the basket's own average as the strike. The lookback forms use the best or worst basket value over the observation period.

The vanilla form is the plain option on the basket. ORE describes its input as follows:

relevant are the long/short flag, the call/put flag, the payoff type (must be set to Vanilla to identify the payoff), and the exercise date (exactly one date must be given). A Premiums node can be added to represent deterministic option premia to be paid by the option holder.

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

Currency is the pay currency. Notional is the quantity for equity and commodity underlyings, and the foreign amount for an FX underlying. Strike is the strike of the option, expressed as a basket price. Underlyings lists the basket constituents, each with its Type, Name and Weight. OptionData carries LongShort, OptionType, PayoffType, one ExerciseDate and optionally a Premiums node. Settlement is the settlement date; it defaults to the exercise date.

The Asian form adds ObservationDates on which the basket value is observed for the average. Its payoff type is Asian. The averaging style is selectable:

PayoffType2 can be optionally set to Arithmetic or Geometric and defaults to Arithmetic if not given.

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

The average strike form uses the same observation dates. Its payoff type is AverageStrike, and it carries no strike: the strike is the average basket value itself. The lookback call and put forms observe the basket on the observation dates and track its lowest, or highest, value there. Their payoff types are LookbackCall and LookbackPut.

All forms share the currency machinery. The PayCcy of the scripted form defines the payment currency. ORE describes the quanto case as follows:

If CCY1 or the currency of the underlying (for EQ and COMM underlyings), this will result in a quanto payoff.

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

In the scripted representation each variation is a scripted trade with its own data node, named after its script, such as VanillaBasketOptionData or AsianBasketOptionData. The nodes carry Expiry, Settlement, the PutCall and LongShort flags, Notional, Strike where the form has one, the index Underlyings list and the matching Weights list, in the same order, and PayCcy. The observation-based forms add an ObservationDates schedule.

The basket value on a date is the weighted sum of its constituents:

basketPrice = sum over u of Underlyings[u](date) x Weights[u].

The scripts implement the payoffs on top of it. The vanilla payoff is the positive part of the signed distance from the strike, capped at zero:

Payoff = max(PutCall x (basketPrice - Strike), 0).

The option value is the long-short signed notional times the payoff, settled at expiry. The Asian script averages the basket over the observation dates and applies the same payoff against the strike. The average strike script compares the basket at expiry with the time average of the basket. The lookback call script pays the basket at expiry minus its minimum over the observation dates. The lookback put script pays the maximum over the observation dates minus the basket at expiry. The lookback forms carry their direction in the payoff itself, so they take no call-put flag.

• The prices of the basket constituents.
• The correlation between the constituents. It drives the volatility of the basket: lower correlation diversifies it.
• The weights, which set the exposure to each constituent.
• The volatility of each constituent.
• The strike, for the forms that carry one.
• The FX rates, when the pay currency differs from the constituent currencies and the payoff is quanto.

The observation schedule is the heart of the averaging forms. Each observation date adds a sample to the Asian average or to the lookback running minimum or maximum. The averaging shortens the volatility the payoff sees as samples accumulate. At expiry the final basket is fixed and the payoff settles on the settlement date.

A basket option buys exposure to several assets in one trade, with one payoff profile. It costs less than a strip of single-asset options because the basket is less volatile than its parts. The averaging forms further reduce the cost and smooth the payoff. In ORE Studio a customer books basket options to value them and run sensitivities on the ORE engine.

ORE's catalogue shows a vanilla basket call on the S&P 500 and the Euro Stoxx 50, struck at 5000, in its traditional representation:

<Trade id="VanillaBasketOption#1">
  <TradeType>EquityBasketOption</TradeType>
  <Envelope>
    <CounterParty>CPTY_A</CounterParty>
    <NettingSetId>CPTY_A</NettingSetId>
    <AdditionalFields/>
  </Envelope>
  <EquityBasketOptionData>
    <Currency>USD</Currency>
    <Notional>1</Notional>
    <Strike>5000</Strike>
    <Underlyings>
      <Underlying>
        <Type>Equity</Type>
        <Name>RIC:.SPX</Name>
        <Weight>1.0</Weight>
      </Underlying>
      <Underlying>
        <Type>Equity</Type>
        <Name>RIC:.STOXX50E</Name>
        <Currency>EUR</Currency>
        <Weight>1.0</Weight>
      </Underlying>
    </Underlyings>
    <OptionData>
      <LongShort>Long</LongShort>
      <OptionType>Call</OptionType>
      <PayoffType>Vanilla</PayoffType>
      <ExerciseDates>
        <ExerciseDate>2020-02-15</ExerciseDate>
      </ExerciseDates>
      <Premiums> ... </Premiums>
    </OptionData>
    <Settlement>2020-02-20</Settlement>
  </EquityBasketOptionData>
</Trade>

Source: ORE User Guide, Products catalogue, basketoption.tex, the vanilla basket option example trade. The catalogue shows the Asian, average strike and lookback variants with the same basket.