Variance & Vol Swaps

You already know the difference between realized and implied volatility, and you know your way around the Greeks. So here’s the natural next question: if you have a view on volatility — you think the market will be wilder, or calmer, than the options market is pricing — how do you actually bet on it? The obvious answer (“buy a straddle”) turns out to be a surprisingly clumsy way to trade vol, for reasons we’ll dismantle in a second. The clean answer is a contract that pays you the difference between the volatility that actually happens and a number you agree today — no delta, no gamma, no babysitting. Meet the variance swap and its cousin the vol swap.

Why a swap to trade volatility?

The problem with options. Say you want to bet purely on how much the market moves, regardless of direction. Your first instinct is options — buy a straddle (a call plus a put), which gains from big moves either way. Reasonable, but messy. The moment the underlying ticks away from the strike, your straddle sprouts a delta: it starts caring which way the market goes, not just how far. To keep it a clean volatility bet you must delta-hedge continuously — trade the underlying over and over to neutralize that drift. That’s expensive, operationally annoying, and your final profit ends up path-dependent: it depends on when the moves happened and at what spot level, not simply on how volatile the period was. Two months with identical realized volatility can hand a delta-hedged straddle wildly different P&L. As a vol instrument, it leaks.

The fix: trade vol with a forward contract. A variance swap (and its simpler relative the vol swap) is an over-the-counter (OTC) forward contract whose payoff is tied directly to realized volatility, with no delta to manage. You don’t hedge anything. At expiry you compare the volatility that actually occurred against a strike fixed at the start, and cash settles the difference. It’s the purest vol bet on the menu — exactly the “clean shot” the straddle can’t give you.

A vol swap is the intuitive one. Its payoff is linear in realized volatility:

$\text{payoff}_{\text{vol}} = N_{\text{vega}}\left(\sigma_{\text{realized}} - K_{\text{vol}}\right)$

Here $\sigma_{\text{realized}}$ is the volatility the underlying actually printed over the life of the swap (in vol points, e.g. 18 means 18%), $K_{\text{vol}}$ is the strike volatility agreed at inception, and $N_{\text{vega}}$ is the vega notional — the dollars you make per one vol point that realized beats the strike. Long the swap, realized comes in 3 points above strike, you pocket $3 \times N_{\text{vega}}$.

A variance swap does the same thing but on variance — volatility squared:

$\text{payoff}_{\text{var}} = N_{\text{var}}\left(\sigma_{\text{realized}}^2 - K^2\right)$

where $K$ is the strike vol, so the strike variance is $K^2$, and $N_{\text{var}}$ is the variance notional — the dollars per one variance point ($\text{vol}^2$ unit) of difference. That little square is the entire personality of the product, as we’ll see.

Because traders think in vol points but the contract pays in variance points, dealers quote a variance swap’s size in vega notional anyway and convert. The standard bridge between the two notionals is:

$N_{\text{vega}} = N_{\text{var}} \times 2 K_{\text{vol}}$

The factor $2K_{\text{vol}}$ is just the slope of $\sigma^2$ at the strike (the derivative of $\sigma^2$ is $2\sigma$). It means: near the strike, a variance swap behaves like a vol swap with that vega notional. Far from the strike, the square takes over — and that’s where the magic is.

Worked settlement: run the numbers

Let’s settle a real ticket. We strike a variance swap at 20 (so strike vol $K_{\text{vol}} = 20$ and strike variance $K^2 = 400$), and we size it so the vega notional is roughly $100,000 per vol point. Using the conversion:

$N_{\text{var}} = \frac{N_{\text{vega}}}{2 K_{\text{vol}}} = \frac{100{,}000}{2 \times 20} = 2{,}500 \text{ per variance point}$

So the variance notional is $2,500 per variance point. Now two scenarios. The payoff to the long (the side that wants volatility) is $N_{\text{var}}\left(\sigma_{\text{realized}}^2 - K^2\right)$. We’ll also compute what a vol swap struck at the same level with $100,000 vega notional would have paid, $N_{\text{vega}}(\sigma_{\text{realized}} - K_{\text{vol}})$, to see the difference.

Stare at that table, because it contains the entire lesson. In both scenarios realized moved 5 vol points away from the strike. The vol swap, being linear, pays a symmetric $500,000 either way. The variance swap does not: it pays more on the up-move ($562,500 vs $500,000) and loses less on the down-move (−$437,500 vs −$500,000).

That asymmetry — bigger gain when vol rises, smaller loss when vol falls, for the same move size — is not a quirk of these numbers. It is convexity, and it is the reason variance swaps exist as their own product instead of everyone just trading vol swaps. Hold that thought; the next section is entirely about it.

Long variance is long convexity

Here’s the core idea, stated plainly. The variance swap’s payoff is built on $\sigma_{\text{realized}}^2$, and a square is a convex function: its graph bends upward. So if you plot the swap’s payoff against realized vol (not variance), the variance swap traces a curve, while the vol swap traces a straight line. They’re tangent at the strike — near 20 they behave almost identically — but as you move away in either direction, the curve pulls above the line.

Above the strike, “above the line” means bigger gains. Below the strike, “above the line” means smaller losses. Either way the long-variance holder is better off than the long-vol-swap holder for the same realized vol. That free upgrade is convexity, and in trader slang being long it is being long “vol of vol” — you benefit from volatility being uncertain and jumpy, because the square rewards the extremes.

The interactive below is the whole concept in one picture. Toggle between the linear vol-swap payoff, the convex variance-swap payoff, and both overlaid, and watch the curve sit above the line everywhere except the single tangent point at the strike.

Now, nothing in finance is free. Because that convexity is valuable — you’d always rather hold the curve than the line — the long has to pay for it. The way the market charges you is in the strike: a variance swap’s strike (the “variance swap rate,” often quoted as its square root, the fair vol) trades slightly above at-the-money implied volatility. That gap is the convexity premium (you’ll also hear it tied to the “variance risk premium”). So if 30-day ATM implied is 18, the fair strike on a 30-day variance swap might be 19 or 20 — the extra point or two is what you hand over for the upward-bending payoff. A vol swap, lacking that convexity, strikes closer to the plain ATM implied.

Replication: a variance swap is a strip of options

Now for the result that makes variance swaps genuinely beautiful, and ties this whole topic together. A variance swap can be replicated — its payoff manufactured — by a static portfolio of options plus a little dynamic trading in the underlying. “Static” means you buy the options once at the start and don’t rebalance them; the only ongoing trade is a delta-hedge in futures that follows a simple rule. Crucially, the replication is model-free: you don’t need to assume any volatility, any Black-Scholes, anything. The fair variance strike falls straight out of the prices of options you can see in the market.

What’s in the portfolio? You hold out-of-the-money options across the entire range of strikes — OTM puts below the current price, OTM calls above it — and you weight each one by $1/K^2$, inversely proportional to the square of its strike. So low-strike puts get heavy weights, high-strike calls get light ones, in a smooth $1/K^2$ taper.

Why $1/K^2$? Keep it intuitive: a single option’s sensitivity to the underlying isn’t constant across price levels, and the $1/K^2$ weighting is precisely the recipe that makes the whole portfolio’s aggregate payoff come out proportional to realized variance — the sum of squared returns — rather than to the price level or to plain volatility. Stack options with exactly that taper and their combined gamma exposure is flat in log-price, which is the mathematical fingerprint of a variance payoff. The dynamic futures position just sweeps up the leftover directional drift so what remains is pure variance. You don’t need to memorize the derivation; the takeaway is that a basket of vanilla options, correctly weighted, is a variance bet.

Risks & the cap

The same square that makes long variance so attractive makes short variance genuinely dangerous. Realized variance is the sum of squared daily returns, so a single catastrophic day doesn’t just add to the total — it gets squared into it. One −20% session contributes $20^2 = 400$ to the sum where a −2% day contributes only $4$: a hundred times the impact for a tenfold-bigger move. The distribution of realized variance therefore has a fat right tail — most of the time it’s quiet, and occasionally it explodes. For the long, that tail is a jackpot. For the short (often a dealer who sold the swap to harvest the variance risk premium), it’s a potential wipeout.

To stay in business, dealers usually sell capped variance swaps, where realized variance is clipped at a ceiling — a common one is 2.5× the strike variance (which, since you take the square root to get vol, corresponds to roughly $\sqrt{2.5} \approx 1.58\times$ the strike vol). The cap bounds the short’s worst-case loss, turning an unbounded tail into a merely-very-bad one.

History is blunt about why this matters. In 2008, short-variance and short-vol positions were savaged as realized vol on the S&P 500 ran far above anything implied beforehand. In February 2018 — the episode nicknamed “Volmageddon” — a spike in volatility detonated short-vol products (most infamously the inverse-VIX ETN, XIV, which lost almost its entire value in a day) and torched anyone short variance without a cap. The lesson practitioners keep relearning: being short variance is collecting nickels in front of a steamroller, and the steamroller’s size is squared.

Putting it together

A variance swap and a vol swap let you bet on realized volatility directly — an OTC forward that pays the gap between what actually happened and a strike fixed today, with no delta to hedge and none of the path-dependence that makes a straddle a leaky vol bet. The vol swap is linear in vol, $N_{\text{vega}}(\sigma_{\text{realized}} - K_{\text{vol}})$; the variance swap is linear in variance, $N_{\text{var}}(\sigma_{\text{realized}}^2 - K^2)$, with the two notionals bridged by $N_{\text{vega}} = N_{\text{var}} \times 2K_{\text{vol}}$. That square makes the variance payoff convex in vol: for the same-size move it pays more on the upside and loses less on the downside, so long variance is long convexity (long “vol of vol”) — and you pay for it via a strike that sits above ATM implied. The deepest result is that a variance swap is a static, $1/K^2$-weighted strip of OTM options plus a futures hedge — model-free, and exactly how the VIX is computed. Finally, because realized variance squares the daily returns, it carries a fat right tail that has bankrupted short-variance traders (2008, Volmageddon 2018), so dealers sell capped swaps. Get all that, and you can trade volatility as cleanly as you trade anything else — and you’re ready to see the VIX for what it really is.

Next — the VIX itself: we’ll take the option-strip replication you just met and walk the actual CBOE calculation strike by strike, turning a wall of SPX option quotes into the single number everyone calls the fear gauge.