Every option you have met so far is a volatility bet with strings attached. A straddle's vega is biggest at its strike and fades as the stock walks away, so you can be right about volatility and still lose on where the price went. A variance swap cuts the strings: it pays on the volatility the index actually delivers, wherever the price goes. It is also, near enough, the thing the VIX measures.
Variance, not volatility
Volatility is the typical size of a move, scaled to a year. Square it and you get variance. Traders keep it in vol-point units so the numbers stay readable: 20 vol is 400 variance points, 30 vol is 900. Variance is the natural unit because it adds up over time: a year's variance is the sum of its days' variances, and volatilities do not add.
The swap
A variance swap has no option in it. The two sides agree a variance strike, quoted as a vol, say 20, and nothing changes hands. At the end the buyer receives the realized volatility of the index over the swap's life, squared, minus the strike squared, times a notional per variance point.
Take $250 per variance point. Near the strike one vol point is worth about $10,000 (250 × 2 × 20), and desks quote the size that way, as vega notional. Here is the payoff next to a straight $10,000 per vol point, which is what a volatility swap pays.
| Realized vol | Variance swap | $10,000 per vol point |
|---|---|---|
| 10 | -$75,000 | -$100,000 |
| 20 | $0 | $0 |
| 30 | $125,000 | $100,000 |
| 40 | $300,000 | $200,000 |
Convexity: big moves pay more
Ten points above the strike pays $125,000; ten points below costs only $75,000. The payoff bends upward because it is built on squares, and that bend is called convexity. For the buyer the worse the storm, the faster the payout grows. For the seller a calm year pays modestly and a 40-vol year costs $300,000, half as much again as the straight bet. That is why sellers want a strike above where they expect realized to land, and why single-stock variance swaps usually carry a cap.
Building variance out of options
Nobody can deliver realized variance, so how does the dealer who sells the swap hedge it? With options. Think back to Module 6: a delta-hedged option earns half its gamma times the squared move each day and pays theta for it. If a position's gamma, counted in dollars per 1% move, were the same at every price, its hedged profit would be realized variance minus the variance it was priced at, wherever the index went. Such a position exists: out-of-the-money puts and calls at every strike, each held in proportion to one over its strike squared. On a $100 index an $80 put gets 1.56 times the at-the-money weight and a $120 call 0.69.
Puts from $80 to $100 and calls from $100 to $120, each weighted by one over its strike squared (the $80 put at 1.56, the $120 call at 0.69). The expiry line is a bowl: roughly the square of the move. Drag volatility up and watch the whole position gain value while the stock stays put.
At expiry the strip pays roughly the square of the move: 5 points either way pays $5, 10 points about $15. Before expiry its value bends upward as volatility rises, while a single at-the-money straddle's value rises in a straight line. Curved in the move and curved in volatility: that is variance made of options.
The same strip, valued at every level of implied volatility with the index at $100. The line curves upward: doubling the vol from 20 to 40 roughly triples the value. A single at-the-money straddle would be close to a straight line on this chart. Drag days to see the curve hold its shape.
The VIX is a variance strike
Price that strip on the S&P 500 with 30 days to run, turn its cost into a vol, and you have the VIX: the recipe sketched in Module 12. The VIX is the fair strike of a 30-day index variance swap. It is not the at-the-money vol; it is the price of variance.
Why skew lifts the variance strike
If every strike traded at one implied vol, the strip would price variance at exactly that vol. Index options do not. The skew puts the downside strikes on higher vols, and the one-over-strike-squared weights lean toward exactly those strikes. On a made-up 30-day smile with 15 at the money, the 25-delta put near 17.7 and the 25-delta call near 13.1, the strip prices variance at about 17.5, two and a half points over the at-the-money vol. The puts supply almost three quarters of the value, against about half on a flat smile.
A made-up 30-day index smile: 15 at the money, rising steeply below it. The strip weights the low strikes most, so the higher their vols, the higher the variance strike. Drag skew to 0 and curvature to 0 and the variance strike falls back to the at-the-money 15.
That gap is why the VIX usually prints above SPY's at-the-money IV and widens when the skew steepens, and why selling variance means selling the crash puts that carry most of its price.
The maths: the swap payoff and the VIX formula
The buyer's payoff is N × (σR² - K²): σR is realized vol, K the variance strike, both in vol points, and N the notional per variance point. Vega notional is N × 2K: 250 × 2 × 20 = $10,000. Realized 30 pays 250 × (900 - 400) = $125,000; realized 10 pays 250 × (100 - 400) = -$75,000.
The VIX recipe for one expiration is σ² = (2 / T) × Σ (ΔK / K²) × e^(RT) × Q(K) - (1 / T) × (F / K0 - 1)². Q(K) is the mid of the out-of-the-money option at strike K (puts below K0, calls above, the average of both at K0), ΔK the strike spacing around K, F the forward from put-call parity, K0 the first strike at or below F, R the interest rate and T the time in years. The last term corrects for K0 not sitting on the forward. Two expirations either side of 30 days are blended, and the VIX is 100 times the square root.
One strike on the made-up smile: the 30-day $90 put at 24% vol is worth about $0.18. With strikes 5 apart it adds (2 / 0.0822) × (5 / 8,100) × 0.178 ≈ 0.0027 to σ², about 27 of the roughly 306 variance points (17.5 squared) the whole strip sums to.
Your turn
Open the SPY chain on the monthly expiration. Read the at-the-money IV, then the IV of the put whose delta is nearest minus 0.25. Subtract the 25-delta call's IV to get the skew, and add the two wings' mids to price the strangle the strip leans on.