Variance Swap Deep Dive: Replication, VIX, and Vol Premium

Key Takeaways

• A variance swap payoff equals variance notional times (realized variance minus strike variance); the 1/K² option replication strip from the 1999 Demeterfi-Derman paper proves the fair strike can be computed from listed vanilla option prices across all strikes.
• The fair variance strike consistently sits above ATM implied vol because the 1/K² weighting puts more weight on low-strike puts, the skew premium is structural, not a pricing error.
• The Cboe VIX uses exactly the variance swap replication formula on a rolling 30-day window, which is why VIX squared equals the 30-day fair variance strike times 10,000.
• Variance caps (set at 2.5x the strike) on post-2018 contracts limit the short side's loss but turn the long into something resembling a straddle in tail scenarios, making it materially different from pure variance exposure.

What It Is

A variance swap exchanges at maturity the realized variance of an underlying over the swap's life against a fixed variance strike, times a notional called the variance vega notional. The payoff to the long side is:

Payoff = N_var * (sigma_realized^2 - K_var^2)

Where sigma_realized^2 is the annualized realized variance (typically computed from log returns), K_var is the variance strike set at inception, and N_var is the variance notional. Trade tickets often quote the notional in vega terms, where N_var = N_vega / (2 * K_vol), so a 100,000 vega notional at a 20 vol strike corresponds to a variance notional of 2,500.

Variance swaps trade primarily on equity indices (SPX, Euro Stoxx 50, Nikkei), with smaller markets on single names and commodities. They cleared principally OTC under ISDA until the 2010s and are now offered cleared on some venues.

The Intuition

A vanilla option is a position on both spot and volatility. A delta-hedged option strips out most of the spot exposure, leaving a payoff driven by the difference between realized and implied volatility, weighted by gamma. The problem: that weighting is not uniform. A delta-hedged option captures variance most where spot is near the strike and less elsewhere. If spot drifts far from the strike, the P&L stalls.

The insight of the 1999 Demeterfi, Derman, Kamal, and Zou paper is that a weighted strip of vanilla options with weights proportional to 1/K^2 has a gamma profile that is locally constant in log-space, which means a dynamic delta-hedge of that strip captures pure realized variance, independent of where spot goes. That strip is the replication portfolio. The variance swap is the packaged, standardized version of that hedge.

How It Works

The fair variance strike is the expected future realized variance under the risk-neutral measure. The Demeterfi-Derman-Kamal-Zou formula expresses this as an integral over vanilla option prices:

K_var^2 = (2/T) * [integral over K of put(K)/K^2 for K < F
                 + integral over K of call(K)/K^2 for K > F]
        + (forward adjustment term)

Where F is the forward price at expiry. In practice, dealers discretize the integrals with the traded option strikes on the listed market. The Cboe VIX methodology uses exactly this formula, with a 30-day constant-maturity target, which is why the VIX is often described as the square root of the 30-day fair variance strike times 100.

Realized variance is computed at settlement, typically as:

sigma_realized^2 = (A/N) * sum of [ln(S_i / S_(i-1))]^2

Where A is the annualization factor (252), N is the number of sampling days, and the sum is over the log returns. Some termsheets include a variance cap at 2.5 times or 3 times the strike to limit downside for the short side after episodes like the VIX spike of February 2018 (Volmageddon).

The replication strip explains many features of variance swap behavior. The 1/K^2 weighting puts more weight on low strikes, which is why variance swap strikes sit above at-the-money implied volatility and why the steepness of the skew drives the gap, an effect known as the vol-of-vol premium.

Worked Example

A trader enters a three-month variance swap on the SPX at a variance strike of 400 (corresponding to a volatility strike of 20) with a vega notional of 100,000 USD per vol point. The variance notional is 100,000 divided by twice the vol strike, or 2,500.

Over the next three months, the SPX realizes a volatility of 24. Realized variance is 576. Payoff to the long side is 2,500 times (576 minus 400), or 440,000 USD.

If realized vol had come in at 16 instead, realized variance would be 256 and the payoff would be 2,500 times (256 minus 400), or minus 360,000 USD. Notice the convex payoff: the upside from a 4-point vol beat (20 to 24) is larger in absolute terms than the downside from a 4-point vol miss (20 to 16). That convexity is exactly the gap between variance and volatility swaps and is one reason sell-side desks often prefer to quote volatility swaps while pricing them off variance replication.

Common Mistakes

• Confusing variance swaps with volatility swaps. Volatility swaps pay on realized volatility; variance swaps pay on its square. The two have different convexity and different replicability. Volatility swaps are not perfectly replicable from vanilla options.
• Ignoring the vol-of-vol premium. The variance strike typically sits above ATM implied vol because of the skew weighting. Shorting variance because it looks rich versus ATM implied is a common beginner mistake that ignores the structural premium.
• Missing the cap in the termsheet. Capped variance swaps limit the short side's loss but turn the long payoff into something resembling a straddle. In tail scenarios the cap binds, and the contract behaves nothing like pure variance exposure.
• Forgetting discrete-sampling adjustments. The replication formula assumes continuous log-return observation. Real contracts sample daily. That gap is usually small but matters for very short-dated contracts and during fast-moving regimes.
• Treating a variance swap as a VIX substitute. VIX futures track a forward-starting 30-day variance expectation, while a traded variance swap is on realized variance over a specific window. They price differently, especially across term-structure shifts.

Frequently Asked Questions

Q: What is a variance swap in simple terms? A variance swap is a contract where one side pays the actual squared daily volatility of an asset over a set period and the other side pays a fixed strike agreed at inception. The long side profits when the market is more volatile than expected; the short side profits in calm markets. The key insight is that variance (not volatility) is what is actually traded.

Q: How does a variance swap affect investment decisions? Variance swaps provide the cleanest available exposure to future market turbulence, free from directional and skew contamination. Pension funds use them to hedge portfolio tail risk; structured-product desks use them to flatten volatility books they accumulate from selling complex structured notes.

Q: What is a real-world example of a variance swap deep dive? A trader enters a 3-month SPX variance swap at a strike of 400 (20% vol equivalent) with a variance notional of $2,500. Realized vol comes in at 24% (576 variance points). Payoff: $2,500 times (576 minus 400) = $440,000. The long side gained more from the 4-point upside surprise than it would have lost from a 4-point downside miss, because variance is convex in vol.

Q: How can investors use variance swaps to improve a volatility hedging strategy? By buying variance swaps as a tail-risk hedge, investors gain a payoff that accelerates as realized volatility rises, the convexity of variance means a doubling of realized vol more than doubles the payout. This makes variance swaps more efficient than linear volatility instruments for extreme-scenario protection.

Q: How is a variance swap different from a volatility swap? A variance swap pays on realized variance (vol squared). A volatility swap pays on realized volatility directly. Because variance is convex in volatility, these two products have different premium levels and different convexity profiles. Variance swaps can be replicated with options; volatility swaps cannot be replicated perfectly, which is why dealers tend to quote volatility swaps but hedge them using variance replication.

Sources

1. Demeterfi, K., Derman, E., Kamal, M., Zou, J. "More Than You Ever Wanted to Know About Volatility Swaps." Goldman Sachs Quantitative Strategies Research Notes, March 1999. https://emanuelderman.com/wp-content/uploads/1999/02/gs-volatility_swaps.pdf
2. Cboe Exchange. "Volatility Index Mathematics Methodology." https://cdn.cboe.com/resources/indices/Cboe_Volatility_Index_Mathematics_Methodology.pdf
3. Gatheral, J. "Variance Swaps." First Baruch Volatility Workshop, Session 5. https://mfe.baruch.cuny.edu/wp-content/uploads/2015/06/VW5.pdf
4. Martin, I. "Simple Variance Swaps." London School of Economics. https://personal.lse.ac.uk/martiniw/simple%20variance%20swaps%20latest.pdf

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.