Wednesday 10 Jun 2026. The Nifty options expiring on 30 Jun had 20 days left, and the at the money one priced a volatility of 15.03 percent. Add up every out of the money option of that expiry, each weighted by one over its strike squared, and the variance they carry is a volatility of 16.06 percent. That is the fair strike of a variance swap. Then the days passed. From the 10 Jun close to the 30 Jun close Nifty realised 12.77 percent over 13 sessions, and the buyer of variance at the fair strike lost money. I found no variance swap listed on NSE or BSE. The arithmetic is still the cleanest way to see what an option strip sells. Every figure comes from NSE end of day files, dated, so nothing here is live and nothing is a view on the next fortnight.
3By the end you can
- Compute a variance swap payoff from a strike, a variance notional and a realised volatility.
- Explain why the strip is weighted by one over the strike squared, and read the units it holds.
- Say why a variance swap is convex in volatility, and why a volatility swap has no static replication.
The idea, plainly
What does a variance swap pay?
A variance swap is a contract on realised variance. The two sides agree a strike variance and a notional. At the end the buyer receives the notional times the gap between realised variance and the strike. Variance is volatility squared, so both sides quote it in volatility points, and the gap is in points squared. If realised volatility ends above the strike volatility the buyer is paid. If it ends below, the seller is.
Payoff = N x (RV − K)
N = variance notional
RV = realised variance
K = strike variance
N = vega notional
/ (2 x strike vol)
Read 2 more paragraphs, 74 words
Traders think in vega, so the size is usually quoted as a vega notional: the rupees gained for one volatility point near the strike. The variance notional follows from it.
Realised variance is the sum of squared daily log returns, scaled to a year, with the average return taken as zero. The contract names which closes count and how many days make a year. This module uses Nifty closes and the zero mean convention.
Why can options replicate it?
A delta hedged option earns, day by day, about half its rupee gamma times the gap between realised and implied variance. Rupee gamma is gamma times the index squared. For one option it peaks at the money and fades away from it. A variance swap pays the same per unit of variance wherever the index trades, so the hedge needs a flat rupee gamma. Weighting each option by one over its strike squared gives that: puts below the forward and calls above it, with units falling as the strike squared rises. The strip is bought once and held. Only a small futures position is rebalanced as the index moves.
A static strip plus a moving futures hedge
Where does the daily result come from?
Hold an option and hedge its delta every day. Over one day the result is close to half the option's rupee gamma times the gap between the squared return that happened and the squared return its volatility implied. Add the days and you have a sum of gaps. The holder is paid when moves exceed what was priced, and the writer when they fall short. A variance swap pays the same gap with one difference: the rupee gamma is held flat, so the pay per squared return does not depend on where the index stands that day.
The maths: the result of a hedged option in one day
Day result =
(1/2) x Gamma x S^2
x (r^2 − vol^2 x dt)
Here r is the day's log return, vol the implied volatility, and dt one day as a fraction of a year. The first factor is the rupee gamma. For one option it changes with S, which is why one option cannot stand for a variance swap. A strip with units proportional to dK over K squared makes the sum of the rupee gammas the same at every S inside the strip, so each squared return earns the same. The delta hedging module shows the result on a simulated path.
The maths: from a log to a strip
For a path with no jumps, Ito's lemma ties the variance accumulated over the life of the contract to a log.
Variance x T =
2 x (sum of dS / S)
− 2 x ln(S_T / S_0)
The first term is a futures position of one over S units, resized as S moves. The second is minus twice a log contract. A log contract is itself a strip, because any payoff of S can be built from options:
ln(S_T / K0) =
(S_T − K0) / K0
− sum of puts / K^2
− sum of calls / K^2
The sums run over strikes below and above K0, each option weighted by the strike gap dK over K squared. Collect the constants and the fair variance is the sum of option prices, each times dK over K squared, times two over T, less a small term for the forward sitting off K0. On a real chain the integral becomes a sum over listed strikes, so the answer depends on which strikes trade and on the gaps between them.
What is the fair strike?
It is the price of the strip, read as a variance. India VIX is the same sum for a 30 day horizon, blended across two expiries. The India VIX rebuild module walks through it line by line. Here the question is what the number means as a contract.
What changes the realised number?
Three choices. Close to close returns miss what happens inside a session. The zero mean convention ignores drift, which over a month is small against the swing. And the day count is a convention: scaling by trading days a year gives one figure and scaling by calendar days gives another. A contract spells these out. When two quotes of realised variance disagree, I look at those three first.
The term this section turns on: Realised variance
Worked example: the 30 Jun 2026 expiry
What strike does the strip give?
The parity forward was 23,249.60 and K0, the strike just below it, 23,200. Add the settlement price of each out of the money option times its strike gap over the strike squared, and the sum is 0.000707. Double it, divide by the time to expiry and take off the small forward term: the fair variance is 0.02581 a year, a strike volatility of 16.06 percent. India VIX, which blends the 30 Jun expiry with the next to a 30 day horizon, closed the day at 15.63. The at the money option priced 15.03 percent, so the strip priced 1.03 points more. The extra is the left wing. Far puts cost little each, but they carry large weights.
| Step | 30 Jun 2026 |
|---|---|
| Days left | 20 |
| Forward | 23,249.60 |
| K0 | 23,200 |
| The sum | 0.000707 |
| Variance | 0.02581 |
| Strike vol, % | 16.06 |
| ATM vol, % | 15.03 |
What does the strip hold?
Take a vega notional of Rs 1,00,000 a volatility point. That is an illustrative contract size, not a market quote. At the strike above, the variance notional is Rs 3,112 for each point squared. For that size the strip holds the following, from the formula in the fold. Gap is the strike gap dK, units are units of the index, and price is rupees a unit.
| Option | Gap | Units | Lots | Price |
|---|---|---|---|---|
| 18,000 put | 1,000 | 3,506 | 53.9 | 1.65 |
| 19,000 put | 750 | 2,360 | 36.3 | 3.15 |
| 22,700 put | 50 | 110 | 1.7 | 133.70 |
| 24,500 call | 50 | 95 | 1.5 | 22.90 |
| 28,000 call | 750 | 1,087 | 16.7 | 1.05 |
Low strikes weigh more than high ones
Read one more paragraph, 119 words
The 18,000 put settled at 1.65 a unit, yet the strip holds 3,506 units of it, 53.9 lots of 65 units, because its gap to the next strike is wide and its strike is small. The 22,700 put near the money costs 133.70 a unit and the strip holds 110 units. Whole lots cannot be split, and nothing lists beyond the last strike, so a real hedge truncates the strip. The weight per unit alone also climbs as the strike falls. Set to one at K0, it is 1.09 at 22,200, 1.22 at 21,000, 1.35 at 20,000 and 1.66 at 18,000 on the put side, and 0.92 at 24,200 and 0.86 at 25,000 on the call side.
What did it pay?
From the 10 Jun close to the 30 Jun close there were 13 returns. Square each, add them, scale to a year of 252 days and the realised variance is 0.01632, a realised volatility of 12.77 percent. In points squared the strike was 258.1 and the realised figure 163.2. The gap, 94.9 points squared on each unit of variance notional, cost the buyer 2.95 lakh on the notional above, and paid the seller the same. A flat payoff of Rs 1,00,000 a point would have lost 3.29 lakh for the same gap of volatility points, which is the convexity at work in the next section.
The strip priced the most, and Nifty delivered the least
This is one window. It does not say whether the gap between a strip's price and what follows is usual. The variance risk premium module asks that question with more than one window.
Why is a variance swap not a volatility swap?
A volatility swap pays in volatility points. A variance swap pays in points squared, so it pays more per point when volatility rises and less when it falls. At the vega notional above, realised volatility four points above the strike pays 4.50 lakh on the variance swap and 4.00 lakh on a flat vega payoff. Four points below loses 3.50 lakh against 4.00 lakh. The variance buyer gains in both tails. That convexity has a price, so the fair strike of a volatility swap sits below the square root of the fair variance. A volatility swap cannot be replicated with one static strip, since a square root of a sum of options is not a sum of options.
The payoff curves upward, not along a line
Where does the replication break?
- The strip stops. The last listed strike is the end of the sum. In this chain 9.8 percent of the sum sits five to ten percent from K0 and 5.4 percent more than ten percent away, 15.2 percent together, so what lies beyond the last strike is not small.
- Prices are settlements. Some are stale, far below the strikes beside them. The strip above keeps every strike that traded that day and leaves out the two that did not.
- The hedge is daily at best. A gap day of the index is hedged less well than an ordinary one, and the squared return of a gap is large.
- The contract decides the rest. Which closes count, the day count, a cap on the payout and the treatment of the forward all move the fair strike.
Contracts often cap the payout at a multiple of the strike. A cap removes the extreme tail of the sum, so it lowers the fair strike by the value of the options it cuts off. A swap that is not capped hands the seller an unlimited loss in a crash, which is one reason an uncapped quote is a different product from a capped one.
Read one more paragraph, 64 words
What I can say is that variance swaps trade over the counter in global markets, and that India's version of the idea is India VIX, a strip read as a volatility. A short strangle held to expiry is short variance in a rough way, but it depends on where the index ends, not only on how it travelled. The variance swap isolates the travel.
What does the strip say to an option writer?
A writer of index options collects premium and is short gamma. The strip shows where a swap would put the cover for that exposure: more units in the low strike puts than at the money, at a price that is small each and large together. The module on hedging an option selling book works the same exposure with listed instruments. Nothing here is a hedge for a real book. It is a way to see what the premium is paying for.
Two terms to hold on to from 16.14
Appendix: the discrete sum and the units
On a real chain, with the gap dK taken as half the distance between neighbouring strikes and the whole step at the two ends, the fair variance and the units at each strike are:
Fair variance =
(2 / T) x sum of
dK / K^2 x e^(rT) x Q
− (1/T) x (F/K0 − 1)^2
Units at K =
N x (2 / T) x dK / K^2
Q is the settlement price of the out of the money option at K, and the price at K0 is the average of the put and the call. N is the variance notional with volatility written as a decimal, so a notional quoted in points squared is multiplied by 10000 first, as the table above does. T is 0.0548 years here. The forward term is small when K0 sits close to the forward. This sum is the same one the India VIX white paper sets out, which is why the two numbers share a method and differ only in horizon and quotes.
The common mistake
The claim: India VIX is the market's forecast of Nifty's volatility, so a variance swap struck there is a fair bet.
A price is a forecast plus a charge for risk, and a strip adds its own weights. Here the strip priced 16.06 percent, 1.03 points over the at the money option, and Nifty delivered 12.77 percent. One window cannot show whether the charge is usual. It can show that the strike is not a forecast. The seller of variance collected in a quiet window like this one and can give much of it back on a single gap day, and the buyer's loss is limited to the strike while the seller's is not. I read the strike as a price for covering the left tail, and I check the window before I read anything else.
Key takeaways
- A variance swap pays realised variance less a strike, in points squared.
- A strip of options weighted by one over strike squared prices it.
- Far puts carry many units but cost little each.
- On the example window the strip priced more than Nifty delivered.
- Variance is convex in volatility; a volatility swap has no static strip.
- A strike is a price including risk charge, not a forecast.
Where this section sits in module 16.14
Before you trade. Derivatives trading carries a high risk of loss. SEBI's study "Profitability of Individual Traders in the Equity Derivatives Segment (FY25 to FY26)" (August 2026) found that 87.7% of individual traders in equity F&O made net losses in FY26, with aggregate net losses of Rs 91,685 crore. Read the SEBI study