Variance Swap: Trade Pure Volatility Without Delta Hedging

Key Takeaways

• A variance swap pays (realized variance minus strike variance) times the variance notional; a doubling of realized volatility from 16% to 32% produces a payoff proportional to 1,024 minus 256, not a linear gain.
• Variance is volatility squared: a swap struck at 16% vol is struck at 256 variance points, confusing these two units is the most common beginner error in variance swap pricing.
• Selling variance has been profitable on average because implied volatility consistently embeds a risk premium above realized, but the 2008, 2018 (Volmageddon), and 2020 drawdowns wiped out years of accumulated premium in weeks.
• Variance swaps can be replicated by a static portfolio of vanilla options weighted by 1/K², which is exactly the calculation behind the Cboe VIX methodology.

What It Is

A variance swap settles at maturity on the difference between realized variance, measured from daily log returns over the life of the contract, and a preset strike variance. The payoff is multiplied by the variance notional.

Payoff = variance notional * (realized variance - strike variance)

Realized variance is typically annualized using the convention (252 / N) times the sum of squared daily log returns, where N is the number of trading days. The strike is set at inception so the contract has zero fair value at trade date.

Variance swaps became widely traded after 2000 as dealers developed replication techniques using vanilla options. They are a core part of how volatility is priced and hedged on major equity indices.

The Intuition

Options give you exposure to volatility, but also to direction, time decay and skew. Cleaning volatility out of an options position requires constant delta and gamma hedging. A variance swap strips all of that away. The only thing that matters at expiry is how much the underlying moved, not which way.

That purity is why hedge funds, structured product desks and pension funds use variance swaps. They can bet on calm markets, hedge a tail-risk portfolio, or lock in today's implied volatility without running a dynamic delta hedge.

How It Works

The variance notional is usually quoted in dollars per variance point, where a variance point equals one squared volatility percentage. A swap with $100,000 vega notional at a 20 percent strike has a variance notional of $100,000 / (2 * 20) = $2,500 per variance point.

Pricing relies on a theoretical result: the fair strike of a variance swap can be replicated with a static portfolio of out-of-the-money puts and calls, weighted by the inverse of the squared strike, plus a rolling position in the underlying. This insight, developed in the Goldman Sachs 1999 note and the later JPMorgan 2001 note, is why variance swaps can be priced and hedged on any liquid equity index.

fair variance strike = (2 / T) * integral of option prices weighted by 1 / K^2

Because the payoff is in variance, not volatility, the swap has a convex exposure to moves in realized volatility. A doubling of realized vol more than doubles the payoff.

Worked Example

An investor enters a 3-month variance swap on the S&P 500 with a variance strike of 16 percent volatility squared (256 variance points) and a vega notional of $50,000, corresponding to a variance notional of $1,562 per point.

Over the three months, the index is more volatile than expected. Realized volatility comes in at 22 percent, or 484 variance points.

Payoff: $1,562 * (484 - 256) = $1,562 * 228 = $356,136.

If realized vol had instead come in below the strike at 12 percent (144 variance points), the payoff would have been $1,562 * (144 - 256) = -$174,944. Notice the asymmetry: the receiver of variance gains more in a volatile environment than they lose in a quiet one, because variance is convex in volatility.

Common Mistakes

1. Confusing variance with volatility. Variance is volatility squared. A variance swap struck at 16 percent vol is struck at 256 variance points, not 16. Mispricing by a factor of vol is one of the classic pitfalls for newcomers.

2. Ignoring path dependency on a short gamma position. Short variance is similar, but not identical, to short gamma on a delta-hedged options book. When a gap happens overnight, variance accrues that cannot be hedged away, and realized variance can spike past any dynamic hedge.

3. Treating VIX and variance strikes as the same thing. The VIX is a volatility index derived from a variance-swap-style calculation on the S&P 500, but quoted in volatility units. A 20 VIX is not a 20 variance strike. The conversion and the rolling 30-day window on VIX matter when hedging.

4. Underestimating jump risk. Variance swaps have uncapped downside for the seller. A single-day 10 percent drop can add hundreds of variance points to the realized number. Single-stock variance swaps became rare for this reason, and most post-crisis dealer books focus on indices with vol caps built in.

5. Relying on implied vol as a predictor. The fair strike is set at expected risk-neutral realized variance, which includes a volatility risk premium. Selling variance consistently below that premium has been profitable on average but has suffered large drawdowns in 2008, 2018 (the Volmageddon episode), and 2020. Premium is not free money.

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 amount agreed at the start. The long side profits if the market is more volatile than expected; the short side profits if the market stays calm.

Q: How does a variance swap affect investment decisions? Variance swaps let volatility traders take a pure view on future market turbulence without running a dynamic delta hedge. Pension funds use them to hedge tail risks in an equity portfolio; structured-product desks use them to offset the volatility exposure accumulated from selling autocallable notes.

Q: What is a real-world example of a variance swap? An investor enters a 3-month S&P 500 variance swap at a strike of 256 variance points (equivalent to 16% vol) with $1,562 variance notional. Realized volatility comes in at 22% (484 variance points). The payoff is $1,562 times (484 minus 256), or $356,136, a significant gain from higher-than-expected turbulence.

Q: How can investors use variance swaps to hedge a portfolio? An investor holding a portfolio of equity index options that are short gamma can buy a variance swap to offset realized-variance risk without managing the complex greeks of the options book. The variance swap directly hedges the P&L sensitivity to large daily moves in the underlying.

Q: How is a variance swap different from buying a VIX futures contract? VIX futures track a forward-starting 30-day expected variance and are marked daily. A variance swap locks in a specific realized-variance payoff over a defined window from trade date to maturity. The two instruments capture different aspects of the volatility market and can diverge significantly across term-structure shifts.

Sources

1. Demeterfi, Derman, Kamal, Zou. "More Than You Ever Wanted To Know About Volatility Swaps." Goldman Sachs Quantitative Strategies Research Notes, 1999. https://emanuelderman.com/wp-content/uploads/1999/02/gs-volatility_swaps.pdf
2. Allen, Einchcomb, Granger. "Just What You Need to Know About Variance Swaps." JPMorgan Equity Derivatives Strategy, 2006. http://docs.sbossu.com/bossu-strasser-guichard-varswap.pdf
3. Carr, P. and Lee, R. "Realized Volatility and Variance: Options via Swaps." https://www.math.uchicago.edu/~rl/OVSwithAppendices.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.