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Key Takeaways
- A variance swap’s buyer receives variance notional times realized variance minus the agreed variance strike.
- A quoted 20-volatility-point strike corresponds to 400 variance points.
- At $50,000 vega notional, that strike gives $1,250 per variance point.
- Realized volatility of 26 produces a $345,000 payment in the worked contract; realized volatility of 14 produces a $255,000 payment by the buyer.
- The worked comparison with a volatility swap uses the same strike and matched vega notional.
What is a variance swap?
A variance swap is a forward contract on realized variance. It pays the buyer the difference between realized variance over the contract period and the agreed variance strike, multiplied by the variance notional, so the buyer gains when realized volatility finishes above the strike and pays when it finishes below. At a fair entry strike, the contract starts with zero value.
The unit conversion is central. A quote of 20 volatility points corresponds to 400 variance points. Realized volatility of 26 corresponds to 676. The resulting difference is 276 variance points.
Variance swap payoff: vega notional and variance notional
Divide vega notional by twice the volatility strike to obtain variance notional. At $50,000 vega notional and strike 20, the result is $1,250 per variance point.
The 276-point difference therefore pays the buyer $345,000. At realized volatility 14, the difference is 196 minus 400, and the buyer pays $255,000.
Formula
Buyer payoff = N_var × (RV − K²); N_var = N_vega / (2K). Worked daily RV = 10,000 × (252 / N) × Σ[ln(P_i / P_(i−1))]².- K is the strike in volatility points; K² is the variance strike.
- N_var is dollars per variance point and N_vega is dollars per volatility point.
- The worked observation convention uses N daily log returns and 252-day annualization.
- Apply the contract’s agreed observation prices, dates and cap to its settlement.
Worked Example
Compare a three-month variance swap with strike 20 volatility points and $50,000 vega notional against a volatility swap with the same 20 strike and $50,000 per-point notional. Use these identical strikes for the payoff comparison.
- Variance strike = 20² = 400 points; variance notional = $50,000 / 40 = $1,250 per point
- At realized volatility 26: variance payoff = $1,250 × (676 − 400) = $345,000
- At realized volatility 26: volatility payoff = $50,000 × (26 − 20) = $300,000
- At realized volatility 14: variance payoff = $1,250 × (196 − 400) = −$255,000
- At realized volatility 14: volatility payoff = $50,000 × (14 − 20) = −$300,000
- At realized volatility 20: both payoffs equal zero
At the same strike and matched vega notional, variance payoff minus volatility payoff equals $1,250 × (realized volatility - 20)². The difference is $45,000 at either 14 or 26, and zero at 20. Price each contract’s fair entry strike separately.
Variance swap vs volatility swap payoff
Use the same strike of 20 and $50,000 vega notional for the worked comparison. A volatility swap pays $300,000 at realized volatility 26 and loses $300,000 at 14. The variance payoff is $45,000 higher in both cases. Both pay zero at 20.
The difference equals variance notional × (realized volatility − strike)². This identity compares the payoff shapes at identical strikes. Each instrument’s fair entry strike is priced separately.
Variance swap observation terms and caps
The worked daily convention squares each log return, sums the values, annualizes by 252 divided by the observation count, then multiplies by 10,000 for variance points. The agreed contract specifies observation prices, dates and any cap.
For a worked cap of 50 volatility points, the buyer’s maximum receipt is $1,250 times 2,500 minus 400, or $2,625,000. Read the cap in the same units as the strike.
How variance swaps relate to options and the VIX
For continuous price paths and a complete strike range, replication combines inverse-squared-strike option weights with dynamic underlying trading. This connects the option surface with the fair variance price. VIX uses related option-strip mathematics for its 30-day implied-volatility measure.
Cboe’s S&P 500 variance futures provide a listed contract with a $1 multiplier per variance point. For listed-option trades, compare implied and realized volatility, inspect the selected strikes and use the options profit calculator to calculate each structure’s payoff.
Limitations of variance swaps
- Size the short side for convexity. The payoff grows with the square of realized volatility, which helps the buyer in a volatility spike and makes the seller’s losses accelerate, limited only by a cap when the contract includes one.
- Hedge jumps separately when you replicate with options. Option replication assumes continuous price paths and a complete strike range, so price jumps and missing far strikes leave the option strip an imperfect hedge.
- Read the term sheet before you compare two contracts. Settlement depends on the agreed observation prices, dates, annualization convention and cap, so two contracts with the same strike can settle differently.
- Price each contract’s fair strike separately. The payoff identity against a volatility swap compares shapes at identical strikes, not fair entry levels.
- Trade it as a view on the size of moves. The contract pays on realized variance alone, and a large move in either direction raises realized variance.
Bottom line
A variance swap trades realized variance against a fixed strike, sized in dollars per variance point. Convert vega notional by dividing by twice the strike, account for the convexity that makes large moves pay more than a volatility swap at the same strike, and read the observation terms and cap before trading it.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- Demeterfi, Derman, Kamal and Zou: More Than You Ever Wanted To Know About Volatility Swaps (Goldman Sachs Quantitative Strategies Research Notes, 1999)
- Cboe: Selected Broad-Based Index, Equity and ETF Volatility Indices Methodology
- Cboe Futures Exchange: S&P 500 Variance Futures FAQ
- Cboe Futures Exchange: Summary Product Specifications Chart for S&P 500 Variance Futures
- Bollerslev and Zhou: Expected Stock Returns and Variance Risk Premia (Federal Reserve FEDS 2007-11)
How entries are sourced, calculated and updated: editorial methodology.