Variance and Volatility Swaps

A variance swap is a contract that pays the difference between the realised variance of an asset over a period and a fixed variance level agreed at the start. A volatility swap does the same with volatility (the square root of variance). Unlike options, these swaps give exposure to realised volatility without any need to delta hedge and without depending on where the price ends up. They are mostly traded over the counter between banks and institutions, especially on equity indices.

How a variance swap works#

At maturity, the payoff to the buyer (long variance) is:

payoff = variance notional × (σ²_realised - K²_var)

• σ_realised: annualised realised volatility over the period, computed from daily log returns.
• K_var: the strike, quoted in volatility points.
• Variance notional: usually set from a vega notional: variance notional = vega notional / (2 × K_var).

Volatility swaps#

A volatility swap pays vega notional × (σ_realised minus K_vol). The payoff is linear in volatility, which is easier to understand, but volatility swaps are harder to hedge and price. Because of convexity, the fair volatility swap strike is slightly below the variance swap strike for the same period.

How variance swaps are priced#

A key result, developed in work by Demeterfi, Derman, Kamal and Zou at Goldman Sachs (1999) and others, is that variance can be replicated with a portfolio of out of the money options across all strikes, weighted by 1/K², plus a delta hedge. The fair variance strike can be read from the option prices:

K²_var ≈ (2 / T) × Σ (ΔK / K²) × e^(rT) × Q(K)

where Q(K) is the price of the out of the money option at each strike. This is essentially the same calculation Cboe uses for the VIX, which is why the VIX squared approximates the 30 day S&P 500 variance swap rate. See The VIX.

Because the replication puts weight on far out of the money puts, variance swap strikes on equity indices sit above at the money implied volatility, reflecting the skew. See Volatility Smile and Skew.

Uses#

• Pure volatility views: profit from realised volatility without delta hedging. See Volatility Trading.
• Volatility risk premium harvesting: selling variance has historically earned a premium on average. See Theta Harvesting.
• Hedging: long variance protects against volatile, crashing markets.
• Relative value: comparing variance swaps with options, across indices, or across maturities. See Dispersion and Correlation Trading.

Risks#

• Short variance tail risk: in a crash, realised variance can explode. Many dealers and funds selling variance suffered heavy losses in 2008, after which the market shrank and caps became standard.
• Caps: most single stock variance swaps, and many index ones, now cap realised variance at a multiple of the strike (often 2.5 times the strike in volatility terms), limiting losses for sellers and gains for buyers.
• Jumps: replication assumes continuous prices; large gaps break the hedge for dealers. See Fat Tails.
• Liquidity and counterparty risk: over the counter contracts depend on the counterparty. See Market, Credit and Counterparty Risk.
• Realised volatility calculation details: missing days, market closures and dividend adjustments are defined in the contract.

Listed alternatives#

VIX futures give exposure to forward implied volatility, not realised volatility. Some exchanges have listed variance futures at times, but liquidity has been limited compared with the over the counter market.

Frequently asked questions#

What is a variance swap?#

A contract that pays the difference between the realised variance of an asset over a period and a fixed variance strike, multiplied by a notional.

What is the difference between a variance swap and a volatility swap?#

A variance swap pays on volatility squared, giving convex payoffs; a volatility swap pays linearly on volatility and is harder to hedge.

How is the VIX related to variance swaps?#

The VIX is calculated with a method closely related to variance swap replication, so VIX squared approximates the fair 30 day variance swap rate on the S&P 500.

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