Variance Swap

A variance swap is a derivative contract whose cash settlement depends on the difference between an underlying asset’s annualized realized variance and a fixed variance strike. A long-variance buyer receives money when realized variance finishes above the strike and pays when it finishes below the strike, subject to the contract’s notional, cap, and calculation terms.

Variance is volatility squared. That makes a variance swap more sensitive to very large price moves than a contract with a payoff linear in realized volatility.

Key Takeaways

• A variance swap references the magnitude of returns, not whether the underlying asset rises or falls.
• The contract must define observation dates, price sources, log returns, annualization, treatment of disruptions, strike, notional convention, cap, and settlement date.
• Variance units are not percentage-return units. A volatility level of 20 points corresponds to 400 variance points because (20^2=400).
• A trade quoted in vega notional must be converted to variance notional under the contract convention before a dollar payoff can be calculated.
• Long variance gains when realized variance exceeds the strike; short variance has the opposite payoff and can face severe losses after large moves.
• A variance swap can reduce some directional exposure, but it retains path, jump, volatility-surface, liquidity, collateral, model, and counterparty risks.

Contract Terms That Determine the Payoff

Without these terms, “long volatility” does not describe the actual cash exposure precisely enough.

Payoff Formula

For a contract stated directly with currency notional per variance point, the long-variance payoff is:

where:

• (N_{\text{var}}) is currency per variance point;
• (\sigma^2_{\text{realized}}) is the contract-calculated annualized realized variance; and
• (K_{\text{var}}) is the fixed variance strike in the same units.

The short-variance payoff is the negative of this amount. Some market documents quote trade size as vega notional and convert it to variance units using the strike and product convention. Analysts must use the confirmation’s conversion rather than substituting a generic “notional amount.”

Variance Notional Versus Vega Notional

The two sizing terms answer different questions:

Under a common convention, variance notional is set from vega notional as:

where (K_{\text{vol}}) is the volatility strike in volatility points and (K_{\text{var}}=K_{\text{vol}}^2). This conversion makes a small move in realized volatility near the strike produce approximately the stated vega-notional exposure because:

Near the strike, (\sigma+K_{\text{vol}}) is approximately (2K_{\text{vol}}). Farther from the strike, the approximation diverges because the payoff is convex in volatility.

Vega-Notional Example

Assume a variance swap has a 20-point volatility strike and USD 100,000 vega notional. Under the convention above:

If realized volatility is 21 points, the variance difference is (21^2-20^2=41) points, so the payoff is USD 102,500. That is close to, but not exactly, the USD 100,000 suggested by one point of vega notional.

If realized volatility is 25 points, the variance difference is 225 points and the payoff is USD 562,500. Multiplying the five-volatility-point difference by USD 100,000 would give USD 500,000 and would understate the contractual variance payoff by USD 62,500.

This conversion is common, not universal. Forward-start dates, accrued observations, caps, and platform-specific conventions can change the sizing formula.

How Realized Variance Is Calculated

Suppose the contract observes prices (S_0,S_1,\ldots,S_n). A log return is:

One simplified zero-mean daily convention annualizes the sum of squared log returns as:

where (A) is the annualization factor. If the result is expressed in decimal variance, multiplying by 10,000 converts it to volatility-points-squared convention. For example, decimal variance of 0.04 corresponds to volatility of 20%, or 20 volatility points, and therefore 400 variance points.

This formula is educational, not universal. A contract can use a different denominator, assume zero mean, subtract a sample mean, include an expected-observation adjustment, or prescribe special handling for missing observations and market disruptions. Small methodology differences can materially change settlement after a volatile period.

Observation-Path Example

Consider a simplified 252-observation year in which every daily log return has a magnitude of 1%. Because signs disappear when returns are squared:

That equals 252 variance points after multiplying decimal variance by 10,000, or approximately 15.87 volatility points after taking the square root and multiplying by 100.

Now replace one 1% observation with a 5% observation. The annualized decimal variance becomes:

The result is 276 variance points, or approximately 16.61 volatility points. One larger move adds 24 variance points even though the other 251 observations do not change. This simplified example uses stated log returns and no cap, disruption, or mean adjustment.

Measurement Choices That Change Settlement

A data vendor’s historical-volatility field is not settlement evidence unless it reproduces the contract’s exact schedule and formula. The reviewer should retain the source prices, adjustment records, observation count, and calculation workbook or system output.

Worked Example: Correct Variance Units

Assume a one-year long variance swap has:

• variance strike quoted as 20 volatility points squared, so (K_{\text{var}}=20^2=400) variance points
• realized volatility of 25 points, so realized variance is (25^2=625) variance points
• variance notional of USD 100 per variance point

The settlement is:

The long-variance buyer receives USD 22,500. If realized volatility were 15 points, realized variance would be 225 and the payoff would be:

The negative amount means the long-variance buyer pays USD 17,500 to the seller. These examples ignore collateral, discounting, fees, early termination, and any cap.

Why Large Moves Matter Disproportionately

Because returns are squared, one extreme observation can contribute more to realized variance than many ordinary observations.

For example, before applying annualization:

• a 1% return contributes roughly (0.01^2=0.0001)
• a 5% return contributes roughly (0.05^2=0.0025)

The 5% move is five times as large, but its squared contribution is about 25 times as large. Positive and negative returns both increase variance because both become positive when squared.

This convexity benefits a long-variance position during large moves but creates substantial tail exposure for a short-variance seller. A contractual cap can limit settlement, but the cap level and its interaction with the notional must be read carefully.

How a Variance Cap Changes the Payoff

A cap may limit recognized realized volatility, recognized variance, or the final payment. Those are not interchangeable structures.

Suppose the USD 2,500 variance-notional example has a contractually specified realized-volatility cap of 50 points. The maximum recognized realized variance is then (50^2=2{,}500) variance points. With a 400-point variance strike, the maximum simplified long payoff is:

If observed realized volatility reaches 65 points, this particular contract would still use 50 points for settlement. A contract that instead caps the dollar payment could produce a different maximum. The cap reduces the short side’s contractual tail exposure but also limits the long side precisely when extreme realized variance is greatest.

Variance Strike and Option Prices

The strike is fixed when the swap is entered. It is not simply the implied volatility of one at-the-money option. Dealers can infer a fair variance level from a broad strip of option prices across strikes and then incorporate market conventions, hedging costs, liquidity, supply and demand, and counterparty terms.

The link to an option strip explains why variance swaps are sensitive to the volatility surface, including downside skew. Far out-of-the-money options can matter to the theoretical replication of variance because large tail moves contribute heavily to squared returns.

In actual markets, replication is imperfect. Available strikes are finite, trading is discrete, transaction costs exist, jumps occur, and contract adjustments may differ from listed-option terms.

Interim Value Before Settlement

The final payoff is not known until the observation period ends, but the swap has a market value before then. Its valuation combines:

• accrued realized variance: squared returns already observed under the contract;
• implied remaining variance: the market’s priced expectation for the unobserved period;
• discounting and funding: timing and collateral effects on the expected payment;
• cap and tail value: the probability and consequence of reaching the cap; and
• counterparty and closeout terms: the value that can actually be realized on termination.

Observed squared returns cannot be undone by later calm markets, although their weight in the final annualized calculation depends on the specified denominator. After a large early move, a short-variance position can therefore face a substantial mark-to-market loss and collateral call even if the final settlement date is months away.

The quoted variance strike at inception is the fixed leg, not a guarantee that future realized variance will equal it. The difference between option-implied variance and subsequently realized variance is sometimes discussed as a variance risk premium, but that historical or expected relationship is not assured for a particular trade.

Variance Swap Versus Volatility Swap

If both strikes are expressed at 20 volatility points, the unscaled differences illustrate the payoff shapes:

The numbers cannot be compared as dollar payoffs until appropriate volatility and variance notionals are applied.

Why Market Participants Use Variance Swaps

Volatility exposure. A participant can take a position on realized variance without choosing a single option strike.

Portfolio hedging. Long variance can offset some losses during turbulent markets, but hedge effectiveness depends on the portfolio, horizon, cap, and volatility response.

Realized-versus-strike trading. A trader can take a view on whether future realized variance will finish above or below the contractual strike.

Relative-value analysis. Positions can compare maturities, indexes, option-implied variance, or related volatility products. Apparent price differences are not riskless because replication, funding, liquidity, and contract terms differ.

Risks and Limitations

• Tail and jump risk: Large single-day moves can dominate settlement, especially for the short side.
• Measurement risk: Observation time, holidays, missing prices, annualization, and return convention affect realized variance.
• Cap risk: A cap changes the long buyer’s tail payoff and the seller’s maximum contractual exposure.
• Notional-convention risk: Confusing vega notional with variance notional can create a large sizing or P/L error.
• Volatility-surface risk: Changes in skew and tail-option prices affect mark-to-market value before settlement.
• Path risk: The ending asset price does not determine realized variance; the sequence of returns does.
• Counterparty risk: An OTC gain remains a claim on the counterparty, subject to collateral and closeout terms.
• Collateral and liquidity risk: Adverse marks can create margin calls, and bespoke positions may be difficult to unwind.
• Model risk: Valuation before maturity requires assumptions about future variance, the option surface, rates, and disruptions.
• Basis risk: A variance swap on an index may not offset volatility in a concentrated or differently weighted portfolio.
• Leverage: Dollar exposure can be large relative to initial cash or collateral.

U.S. Regulatory Context

U.S. classification depends partly on the underlying. A variance swap on a single security or narrow-based security index can fall within the SEC’s security-based-swap framework, while a variance swap on a broad-based equity index generally falls within the CFTC swap framework. Requirements can vary with product terms, counterparties, and current rules.

How to Evaluate a Variance Swap

1. Confirm the underlying, observation period, price source, currency, and settlement date.
2. Reproduce the realized-variance formula, including return type, mean treatment, annualization, and disruption rules.
3. Determine whether the strike is in decimal variance, variance points, or a volatility quote that must be squared.
4. Identify whether size is variance notional or vega notional and verify the conversion.
5. Review cap, floor, early termination, adjustment, collateral, and calculation-agent provisions.
6. Stress isolated jumps, repeated volatility, missing observations, and a changing volatility skew.
7. Compare the contract exposure with the portfolio risk being hedged rather than relying on the label “long volatility.”
8. Reconcile interim marks to accrued observations, remaining implied variance, cap treatment, discounting, and collateral records.

Official and Primary Sources

• CFTC-filed equity index variance swap specifications describe realized-variance and strike legs, quotation conventions, vega amounts, caps, and settlement.
• CFTC: Swaps Report Data Dictionary identifies variance swaps as broad-based equity products driven by the magnitude rather than direction of change.
• Cboe: S&P 500 Variance Futures specifications provide an exchange example of annualized squared log-return methodology and variance-unit conventions.
• SEC: Security-Based Swap Markets provides the SEC’s current framework and resources for security-based swaps.
• ISDA: Index and Share Variance Swap Annexes identifies standardized documentation for index and share variance swaps.

Related Terms

• Volatility Swap: A contract with payoff linear in realized volatility rather than variance.
• Volatility: A measure of return dispersion whose square is variance.
• Implied Volatility: Volatility backed out from option prices, which helps inform volatility and variance markets.
• Vega: Option-value sensitivity to implied volatility and a term also used in variance-swap quoting conventions.
• Volatility Surface: Implied volatilities across strikes and maturities that help inform variance pricing and interim valuation.
• Standard Deviation: The square root of variance.
• Counterparty Risk: The risk that the other swap party cannot perform.

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This article is general financial education, not personalized investment, trading, or legal advice. Variance swaps are complex leveraged derivatives whose calculation and governing documentation determine the actual exposure.