Exotic Variance and Volatility Derivatives

ORE defines the family as follows:

These are vanilla variance/volatility swaps and options on an underlying with some additional payoff features, such as knock-in/knock-out barrier/s, conditions for the accrual of variance, basket underlyings, optionality on the payoff, etc.

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

ORE states the input representation:

Exotic variance/volatility swaps and options are represented as scripted trades, refer to the scripted trade documentation in ore/Docs/ScriptedTrade for an introduction.

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

All swaps have an optional cap/floor feature.

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

A variance swap is a bet on how much a market moves. The buyer pays a fixed strike and receives the realised variance of the market. The exotic family dresses that bet up. A barrier can switch the swap on or off. A corridor can make the bet count only on the days the market sits inside a window. The window can depend on another market. The variance can accrue only under one-sided conditions, or on a pair of markets, or on whole baskets. One form weighs the variance by the level of the market. Every variation can cap or floor the payout.

The variations of the family, with the script each one prices, are:

• Variance Option, script VarianceOption.
• Variance Swap with KI/KO Barrier, script KIKOVarianceSwap.
• Dual European Binary Option with Volatility and Spot KO Barrier, script DualEuroBinaryOptionDoubleKO.
• Corridor Variance Swap, script CorridorVarianceSwap.
• Indexed Corridor Variance Swap, script IndexedCorridorVarianceSwap.
• Corridor Variance Swap with KI/KO Barrier, script KIKOCorridorVarianceSwap.
• Conditional Variance Swap 01, script ConditionalVarianceSwap01.
• Conditional Variance Swap 02, script ConditionalVarianceSwap02.
• Pairwise Variance Swap, script PairwiseVarianceSwap.
• Variance Dispersion Swap, script VarianceDispersionSwap.
• Corridor Variance Dispersion Swap, script CorridorVarianceDispersionSwap.
• KO Corridor Variance Dispersion Swap, script KOCorridorVarianceDispersionSwap.
• Pairwise Geometric Variance Dispersion Swap, script PairwiseGeometricVarianceDispersionSwap.
• Gamma Swap, script GammaSwap.
• Basket Variance Swap, script BasketVarianceSwap.

The trade data nodes live under the ScriptedTrade trade type and carry typed fields. The common input pattern is the variance swap core: LongShort, Strike, Notional, Underlying, the daily ValuationSchedule that defines the variance observation period, a SquaredPayoff flag that selects variance or volatility, Cap and Floor, SettlementDate and PayCcy. The PayCcy payment currency: for FX, the underlying name has the form FX-SOURCE-CCY1-CCY2 and PayCcy should be CCY2. Choosing CCY1, or the underlying currency for equity and commodity underlyings, produces a quanto payoff.

ORE describes the individual variations as follows. The variance option is:

An option on the realised variance/volatility of an underlying.

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

Its node adds PutCall, PremiumAmount and PremiumDate, VarianceReference and the ValuationSchedule. The option carries a call or put on the realised variance or volatility against the strike, with the premium paid on the premium date.

The variance swap with a barrier is:

A capped and/or floored variance swap with a knock-in/knock-out feature.

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

ORE notes a reuse of the trade type:

Note that this trade type is also used for regular capped/floored variance swaps without barrier, by setting the BarrierType to UpIn and BarrierLevel to zero.

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

The dual European binary pairs a volatility knock-out with a spot knock-out. Its node carries a fixed SettlementAmount, a SpotBarrierLevel, a VolBarrierLevel with a VolBarrierDate, and a terminal-only flag.

The corridor variance swap is:

A capped and/or floored variance swap where variance is accrued only when the underlying trades within a pre-specified window defined by an upper and lower bound.

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

Its node adds UpperBarrierLevel and LowerBarrierLevel, a CountBothObservations flag and an AccrualAdjustment. The corridor with a barrier adds a knock-in or knock-out type and level of its own. The indexed corridor moves the window to a second index:

Also called Cross Corridor Variance Swap. The payoff depends on the variance of one equity index (the Underlying) for the days another equity index (the CorridorIndex) is within a corridor.

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

The conditional variance swaps accrue variance in a single-barrier window. ORE describes the 01 variant:

ConditionalVarianceSwap01 is a type of conditional variance swap that accrues variance contributions only when the underlying trades within a specified single barrier window - (differing from the CorridorVarianceSwap that has a double barrier window.)

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

In the 01 variant the variance amount that scales the notional comes directly from the Strike parameter. The node supports caps, floors, accrual adjustments and the count-both-observations choice. ORE describes the 02 variant:

ConditionalVarianceSwap02 accrues variance only when the underlying trades within the single barrier window, but differs from the 01 variant by including an additional VarianceReference parameter. This parameter acts as a separate variance strike used to scale the notional amount, distinct from the Strike parameter used in the payoff calculation.

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

The pairwise variance swap is:

A capped and/or floored variance swap on a basket of two underlyings, with an optional lag in the variance accrual.

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

Its node carries one underlying strike and notional per underlying plus a basket notional and basket strike, and a lagged valuation schedule with an accrual lag and a payoff limit.

The variance dispersion swap is:

A capped and/or floored variance swap that pays on the difference in the realised volatilities between two baskets of underlyings.

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

The corridor dispersion form restricts the accrual to a corridor, and the knock-out form adds a double-barrier knock-out:

This instrument is a variance dispersion swap with a double-barrier knock-out and a corridor feature.

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

The pairwise geometric form is documented by its node alone, with two baskets, per-pair and basket strikes, basket and variance weights, a lag, a cap multiplier, a cap and a floor.

The gamma swap is:

A vanilla variance swap with a weight function on the underlying.

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

The basket variance swap is:

A capped and/or floored variance swap on a basket of underlyings.

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

Its trade types are FxBasketVarianceSwap, EquityBasketVarianceSwap and CommodityBasketVarianceSwap, with a weight per underlying in the basket.

ORE computes the realised variance from the daily underlying prices on the valuation schedule. The daily log return is the natural log of the ratio of one price to the previous price. The realised variance is the sum of the squared daily log returns, scaled by the number of observation days and annualised to 252 days per year. The realised volatility is the square root of the realised variance.

The variance option converts the vega notionals. The notional of the trade is the vega notional. ORE derives the variance notional from the vega notional and the variance reference: with the squared-payoff flag set, the payout is the converted notional times the positive difference between the realised variance and the squared strike, in the call or put direction. With the flag clear, the payout is the vega notional times the positive difference between the realised volatility and the strike. The strike is quoted as an absolute volatility level in decimal form, so a 20 percent volatility strike is 0.20. When the option was struck in variance terms, the square root of the variance is entered as the strike. The variance reference converts between the two notions, so the variance amount used to scale the notional can differ from the strike that determines the payoff.

The corridor forms accrue only the observations inside the window, between the upper and the lower barrier level. The conditional forms accrue only inside a single-barrier window. The indexed corridor accrues the variance of one index only on the days a second index sits in its corridor. The count-both-observations choice and the accrual adjustment tune the accrual. The dispersion forms pay on the volatility difference between two baskets. Caps and floors bound the payouts, and the barrier forms condition them on their knock events.

• The realised path of the underlying, or of each basket member. It drives the realised variance.
• The implied volatility of the underlying, which sets the expected variance and the option value.
• The strike, the variance reference and the notional.
• The corridor levels and the accrual conditions.
• The barrier types and levels of the barrier forms.
• The correlation between the basket members and between the baskets, in the dispersion and basket forms.
• The caps and floors.
• The interest rates that discount the payouts.

Each date of the valuation schedule adds a daily return to the realised variance, subject to the accrual conditions of the variation. Barrier forms watch for their knock events over the observation period. The corridor forms accrue only inside their windows. The averaging of the variance ends with the schedule, and the strike comparison, the cap or floor, and the settlement follow on the settlement date.

A variance or volatility swap trades the size of market moves directly, without a view on direction. The exotic forms refine that exposure: corridors and conditions shape when the variance counts, barriers switch the trade on or off, and the dispersion forms trade the spread between baskets. In ORE Studio a customer books exotic variance and volatility derivatives to value them and run sensitivities on the ORE engine.

ORE's catalogue shows a long call on the realised variance of the S&P 500. The trade has a vega notional of 138000 USD, a strike of 0.19 and a variance reference of 0.19, and observes the index daily from 2020-11-26 to 2021-09-18:

<Trade id="EQ_VarianceOption">
  <TradeType>ScriptedTrade</TradeType>
  <Envelope>
    .....
  </Envelope>
  <VarianceOptionData>
    <LongShort type="longShort">Long</LongShort>
    <PutCall type="optionType">Call</PutCall>
    <PremiumAmount type="number">0</PremiumAmount>
    <PremiumDate type="event">2020-11-26</PremiumDate>
    <Notional type="number">138000</Notional>
    <VarianceReference type="number">0.19</VarianceReference>
    <Strike type="number">0.19</Strike>
    <Underlying type="index">EQ-RIC:.SPX</Underlying>
    <ValuationSchedule type="event">
      <ScheduleData>
        <Rules>
          <StartDate>2020-11-26</StartDate>
          <EndDate>2021-09-18</EndDate>
          <Tenor>1D</Tenor>
          <Convention>Following</Convention>
          <TermConvention>Following</TermConvention>
          <Calendar>USA</Calendar>
          <Rule>Forward</Rule>
        </Rules>
      </ScheduleData>
    </ValuationSchedule>
    <SquaredPayoff type="bool">true</SquaredPayoff>
    <SettlementDate type="event">2021-09-22</SettlementDate>
    <PayCcy type="currency">USD</PayCcy>
  </VarianceOptionData>
</Trade>

The squared-payoff flag is true, so the trade is a variance option. The premium is zero and the premium date equals the first valuation date. ORE prices the trade with the VarianceOption script, shown in full in the catalogue.

Source: ORE User Guide, Products catalogue, var_and_vol_derivatives.tex, the example variance option trade in the section Variance Option.