Volatility, Variance & Gamma Swaps
Derivatives-based Portfolio Solutions
VOLATILITY, VARIANCE & GAMMA SWAPS GIVE PURE VOL EXPOSUREAs spot moves away from the strike of an option the gamma decreases, and it becomes more difficult to profit via delta hedging. Second-generation volatility products, such as volatility swaps, variance swaps and gamma swaps, were created to give volatility exposure for all levels of spot and also to avoid the overhead and cost of delta hedging. While volatility and variance swaps have been traded since 1993, they became more popular post-1998, when Russia defaulted on its debts and Long-Term Capital Management (LTCM) collapsed. The naming of volatility swaps, variance swaps and gamma swaps is misleading, as they are in fact forwards. This is because their payoff is at maturity, whereas swaps have intermediate payments. **Volatility swaps.**Volatility swaps were the first product to be traded significantly and became increasingly popular in the late 1990s until interest migrated to variance swaps. Following the collapse of the single-stock variance market in the credit crunch, they are having a renaissance due to demand from dispersion traders. A theoretical drawback of volatility swaps is the fact that they require a volatility of volatility (vol of vol) model for pricing, as options need to be bought and sold during the life of the contract (which leads to higher trading costs). However, in practice, the vol of vol risk is small and volatility swaps trade roughly in line with ATM forward (ATMf) implied volatility.
**Variance swaps.**The difficulty in hedging volatility swaps drove liquidity towards the variance swap market, particularly during the 2002 equity collapse. As variance swaps can be replicated by delta hedging a static portfolio of options, it is not necessary to buy or sell options during the life of the contract. The problem with this replication is that it assumes options of all strikes can be bought, but in reality very OTM options are either not listed or not liquid. Selling a variance swap and only hedging with the available roughly ATM options leaves the vendor short tail risk. As the payout is on variance, which is volatility squared, the amount can be very significant. For this reason, liquidity on single-stock variance disappeared in the credit crunch.
**Gamma swaps.**Dispersion traders profit from overpriced index-implied volatility by going long single-stock variance and short index variance. The portfolio of variance swaps is not static; hence, rebalancing trading costs are incurred. Investment banks attempted to create a liquid gamma swap market, as dispersion can be implemented via a static portfolio of gamma swaps (and, hence, it could better hedge the exposure of their books from selling structured products). However, liquidity never really took off due to limited interest from other market participants.
VOLATILITY SWAP ≤ GAMMA SWAP ≤ VARIANCE SWAP Variance and gamma swaps are normally quoted as the square root of variance to allow easier comparison with the options market. However, typically variance swaps trade in line with the 30 delta put (if skew is downward sloping as normal). The square root of the variance strike is always above volatility swaps (and ATMf implied as volatility swaps ≈ ATMf implied). This is due to the fact a variance swap payout is convex (hence, will always be greater than or equal to volatility swap payout of identical vega, which is explained later in the section). Only for the unrealistic case of no vol of vol (ie, future volatility is constant and known) will the price of a volatility swap and variance swap (and gamma swap) be the same. The fair price of a gamma swap is between volatility swaps and variance swaps. 1. VOLATILITY SWAPSThe payout of a volatility swap is simply the notional, multiplied by the difference between the realised volatility and the fixed swap volatility agreed at the time of trading. As can be seen from the payoff formula below, the profit and loss is completely path independent as it is solelybased on the realised volatility. Volatility swaps were previously illiquid, but are now more popular with dispersion traders, given the single stock variance market no longer exists postcredit crunch. Unless packaged as a dispersion, volatility swaps rarely trade. As dispersion is short index volatility, long single stock volatility, single stock volatility swaps tend to be bid only (and index volatility swaps offered only). Volatility swap payoff(σF – σS) × volatility notional where: σF = future volatility (that occurs over the life of contract) σS = swap rate volatility (fixed at the start of contract) Volatility notional = Vega = notional amount paid (or received) per volatility point 2. VARIANCE SWAPSVariance swaps are identical to volatility swaps except their payout is based on variance (volatility squared) rather than volatility. Variance swaps are long skew (more exposure to downside put options than upside calls) and convexity (more exposure to OTM options than ATM). One-year variance swaps are the most frequently traded. Variance swap payoff (σ _{F}^{2} - σ_{S}^{2}) × variance notionalwhere: Variance notional = notional amount paid (or received) per variance point NB: Variance notional = Vega / (2 × σ _{S}) where σ_{S} = current variance swap priceVARIANCE SWAPS CAPS ARE EFFECTIVELY SHORT OPTION ON VAR Variance swaps on single stocks and emerging market indices are normally capped at 2.5 times the strike, in order to prevent the payout from rising towards infinity in a crisis or bankruptcy. A cap on a variance swap can be modelled as a vanilla variance swap less an option on variance whose strike is equal to the cap. More details can be found in the section Options on Variance. Capped variance should be hedged with OTM calls, not OTM putsThe presence of a cap on a variance swap means that if it is to be hedged by only one option it should be a slightly OTM call, not an OTM (approx delta 30) put. This is to ensure the option is so far OTM when the cap is hit that the hedge disappears. If this is not done, then if a trader is long a capped variance swap he would hedge by going short an OTM put. If markets fall with high volatility hitting the cap, the trader would be naked short a (now close to ATM) put. Correctly hedging the cap is more important than hedging the skew position. S&P500 variance market is increasing in liquidity, while SX5E has become less liquidThe payout of volatility swaps and variance swaps of the same vega is similar for small payouts, but for large payouts the difference becomes very significant due to the quadratic (ie, squared) nature of variance. The losses suffered in the credit crunch from the sale of variance swaps, particularly single stock variance (which, like single stock volatility swaps now, was typically bid), have weighed on their subsequent liquidity. Now variance swaps only trade for indices (usually without cap, but sometimes with). The popularity of VIX futures has raised awareness of variance swaps, which has helped S&P500 variance swaps become more liquid than they were before the credit crunch. S&P500 variance swaps now trade with a bid-offer spread of c30bp and sizes of approximately US$5mn vega can regularly trade every day. However, SX5E variance swap liquidity is now a fraction of its pre-credit-crunch levels, with bid-offer spreads now approx. 80bp compared with approx. 30bp previously. CORRIDOR VARIANCE SWAPS ARE NOT LIQUIDAs volatility and spot are correlated, volatility buyers would typically only want exposure to volatility levels for low values of spot. Conversely, volatility sellers would only want exposure for high values of spot. To satisfy this demand, corridor variance swaps were created. These only have exposure when spot is between spot values A and B. If A is zero, then it is a down variance swap. If B is infinity, it is an up variance swap. There is only a swap payment on those days the spot is in the required range, so if spot is never in the range there is no payment. Because of this, a down variance swap and up variance swap with the same spot barrier is simply a vanilla variance swap. The liquidity of corridor variance swaps was always far lower than for variance swaps and, since the credit crunch, they are rarely traded. Corridor variance swap payoff(σ _{F when in corridor}^{2} - σ_{S}^{2}) × variance notional × percentage of days spot is within corridorwhere: σ _{F when in corridor} = future volatility (of returns P_{i}/P_{i-1} which occur when B_{L}< P_{i-1} ≤ B_{H})B _{L} and B_{H}, are the lower and higher barriers, where B_{L} could be 0 and B_{H} could be infinity.GAMMA SWAPSThe payout of gamma swaps is identical to that of a variance swap, except the daily P&L is weighted by spot (pricen) divided by the initial spot (price0). If spot range trades after the position is initiated, the payouts of a gamma swap are virtually identical to the payout of a variance swap. Should spot decline, the payout of a gamma swap decreases. Conversely, if spot increases, the payout of a gamma swap increases. This spot-weighting of a variance swap payout has the following attractive features: - Spot weighting of variance swap payout makes it unnecessary to have a cap, even for single stocks (if a company goes bankrupt with spot dropping close to zero with very high volatility, multiplying the payout by spot automatically prevents an excessive payout).
- If a dispersion trade uses gamma swaps, the amount of gamma swaps needed does not change over time (hence, the trade is ‘fire and forget’, as the constituents do not have to be rebalanced as they would if variance swaps were used).
- A gamma swap can be replicated by a static portfolio of options (although a different static portfolio to variance swaps), which reduces hedging costs. Hence, no volatility of volatility model is needed (unlike volatility swaps).
Gamma swap market has never had significant liquidityA number of investment banks attempted to kick start a liquid gamma swap market, partly to satisfy potential demand from dispersion traders and partly to get rid of some of the exposure from selling structured products (if the product has less volatility exposure if prices fall, then a gamma swap better matches the change in the vega profile when spot moves). While the replication of the product is as trivial as for variance swaps, it was difficult to convince other market participants to switch to the new product and liquidity stayed with variance swaps (although after the credit crunch, single-stock variance liquidity moved to the volatility swap market). If the gamma swap market ever gains liquidity, long skew trades could be put on with a long variance-short gamma swap position (as this would be long downside volatility and short upside volatility, as a gamma swap payout decreases/increases with spot). Gamma swap payoff(σ _{G}^{2} - σ_{S}^{2}) × variance notionalwhere: σ _{G}^{2}= future spot weighted (i.e. multiplied by price_{n / } price_{0 } ) varianceσ _{S}^{2} = swap rate variance (fixed at the start of contract) |

### Vol, Var & Gamma

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