Volatility Metrics Intermediate The additive dispersion quantity

Variance σ²

The market quotes volatility because humans read it; it models variance because variance is the one that adds.

Quick answer: Variance is the square of volatility — the average of the squared deviations of returns from their mean — and it is the quantity that adds across independent time periods and independent positions, which volatility does not.

In simple words

Volatility is easy to read: 13% means something a human can picture. But volatility has a hidden inconvenience — it does not add. If one month has a volatility of 12% and the next has 16%, the two months together do not have a volatility of 28%, or of 14%, or of any number you can get by combining the two directly. Variance fixes this. Variance is simply volatility squared, and unlike volatility, variances of independent periods add together. So the market shows you volatility because you can read it, but every model underneath computes in variance, because variance is the quantity that behaves.

Here is the fact that makes variance matter, stated once and precisely: for two independent periods a and b, σ²(a+b) = σ²(a) + σ²(b), but σ(a+b) ≠ σ(a) + σ(b). Variances add; volatilities do not. Almost everything strange about volatility arithmetic — the square root of time, forward volatility, the existence of variance swaps — is a consequence of this one line. Once you internalise that variance is the additive quantity and volatility is merely its square root, the rest stops being mysterious.

Not to be confused with: Standard deviation, which is variance's own square root and is quoted in the same units as the data. Variance is in squared units — %², which no trader can intuit — so it is almost never quoted, only computed. The two carry identical information; the difference is that variance adds and standard deviation does not, and that single distinction is the whole reason both exist.

Variance is convex in volatility

Doubling volatility quadruples variance

Variance plotted against volatility, showing the quadratic curve.

Because variance is volatility squared, the relationship is a parabola, not a line: raising volatility from 12% to 24% lifts variance from 1.44 to 5.76 — a quadrupling. This convexity is exactly why a variance swap pays far more in a crisis than a volatility swap on the same notional, and why sellers of variance are structurally short the tail.

Professional explanation

Why the market quotes volatility but models variance

There are two numbers and each is good at one job. Volatility is in the units of returns — a percentage — so a human can look at 13% and picture a range. Variance is in squared units — 13% squared is 1.69 %², which pictures nothing. So volatility wins for communication. But the moment you need to combine dispersion across time or across positions, volatility fails, because it is not additive, and variance wins, because it is. A pricing model that needs the total dispersion over the life of an option sums the variance of each independent sub-period and takes the square root only at the very end to report a volatility. The division of labour is clean: variance is the currency of the calculation, volatility is the currency of the quote, and a good trader keeps a mental note of which one they are holding at any moment.

This one fact generates the square-root-of-time rule

If returns are independent from one day to the next, then the variance of n days is n times the variance of one day, because variances of independent periods add. Take the square root of both sides and the volatility of n days is √n times the daily volatility. That is the entire derivation of the square-root-of-time rule — the √252 that annualises a daily volatility, the √(days ÷ 365) inside every expected move, the √T in Black–Scholes. None of it is a separate convention to memorise. It is the additivity of variance, seen through the square root that converts variance back into volatility. Whenever you see a √t anywhere in options, you are looking at variance quietly adding underneath.

Convexity, and why a variance swap is long the tail

Because variance is the square of volatility, equal steps in volatility are unequal steps in variance. Moving volatility from 12% to 24% doubles the volatility but takes variance from 0.12² = 0.0144 to 0.24² = 0.0576 — a factor of four. This convexity has a direct financial consequence. A volatility swap pays out linearly in realised volatility; a variance swap pays out in realised variance, which is convex. On the same notional, in a calm market the two pay similarly, but in a crisis, when volatility spikes, the variance swap pays vastly more because its payoff is squared. A long variance-swap position is therefore structurally long the tail — it profits disproportionately from extreme moves — and a short position is short the tail, which is precisely why sellers of variance swaps are the ones who suffer catastrophically when volatility explodes.

Why variance can be replicated model-free, and why VIX-style indices do exactly that

Here is the deep and genuinely surprising fact: the realised variance of an underlying can be replicated by a static portfolio of options across all strikes, weighted by one over the strike squared, without assuming any particular model of how volatility behaves. Variance is special in this way — volatility is not replicable model-free, but variance is, because a strip of options with 1/K² weighting has a payoff whose sensitivity to the underlying reproduces the mathematics of accumulated variance. This is not a curiosity; it is the foundation of the entire VIX family. India VIX and the Cboe VIX are not computed by inverting Black–Scholes on one option — they are computed as the fair strike of a variance swap, a weighted sum across the whole option chain, which is why they are called model-free. Every time you read a VIX number you are reading the market's price for variance, dressed up as a volatility by a final square root.

Formula

Variance as the square of volatility

σ² = ( Σ(x_i − x̄)² ÷ (n − 1) )

Variance is the mean of the squared deviations from the mean, in squared units. Its defining property is additivity: for independent periods, variances add while volatilities do not. Take the square root of variance to recover the volatility that gets quoted.

σ²The variance — the average squared deviation of returns from their mean, in squared units (%² or fraction²).
x_iThe i-th return observation (log return of day i).
x̄The mean of the returns, often set to zero over short windows in finance.
nThe number of observations.
ΣSummation over all observations from i = 1 to n.

The additivity property that makes variance special

σ²(a + b) = σ²(a) + σ²(b) for independent a, b

Variances of independent periods or positions add; volatilities do not — σ(a+b) is not σ(a) + σ(b). This single identity generates the square-root-of-time rule, forward volatility and the replicability of variance swaps.

How to work with variance

To combine independent periods, always convert to variance first. Square each period's volatility to get its variance — you cannot add volatilities directly.
Scale each period's variance to a common time unit if needed, since variance scales linearly with time: a 30-day variance is 30 times a 1-day variance under independence.
Add the variances of the independent periods together. This is the step that volatility cannot do.
Divide the total variance by the total time to get the variance per unit time over the whole span.
Take the square root at the very end to convert the combined variance back into a volatility you can quote.
Sanity-check the convexity: if one period's volatility is much larger than the other's, its squared contribution dominates the total variance far more than its volatility suggests.

Practical example

NIFTY worked example

Two independent 30-day windows, each with an annualised volatility of 13%, combine to a 60-day volatility of exactly 13% — and the arithmetic shows why. Each window contributes a variance of 0.13² × (30 ÷ 365) = 0.0169 × 0.0822 = 0.001389. Two of them add to 0.002778 of variance over 60 days. Divide by the time span, 60 ÷ 365 = 0.1644, to get 0.002778 ÷ 0.1644 = 0.0169, and the square root is √0.0169 = 0.13 — back to 13%, as it must be when both periods are identical. Now change the second window to 20%. Variance one is 0.12² and variance two is 0.20², and adding them: (0.0144 + 0.0400) ÷ 2 = 0.0272, whose square root is 0.1649, about 16.5%. Note what happened: the combined volatility of 16.5% is not the average of 12 and 20, which would be 16 — it leans toward the larger figure, because variance adds and the bigger number's square dominates.

BANKNIFTY worked example

BANKNIFTY shows convexity biting in a real position. Suppose a desk is short a variance swap on BANKNIFTY struck at a 16% volatility — meaning it collects if realised volatility comes in below 16% and pays if above. If realised volatility doubles to 32% in a banking crisis, a naive trader expects to lose roughly twice the notional. They do not. Variance goes from 0.16² = 0.0256 to 0.32² = 0.1024 — a factor of four, not two — so the variance-swap loss is quadruple, not double. This is why the 2008 and 2020 blow-ups among variance sellers were so much larger than the volatility spike alone suggested: the payoff is squared, and the square of a crisis is a catastrophe. A volatility swap on the same notional would have lost only double. The convexity that looks like a footnote in calm markets is the entire risk in a crash.

Lot sizes used above (NIFTY 75, BANKNIFTY 30) are those in force at the time of writing; NSE revises them periodically. Figures exclude brokerage, STT, exchange charges, stamp duty and GST. Examples are teaching scenarios built on round numbers — they are not historical quotes, not backtests and not trade calls.

Risk note. The replicability and convexity of variance are routinely used to sell variance swaps and short-volatility structures as sources of steady premium, on the reasoning that realised variance usually comes in below implied. It usually does — and then, occasionally, it comes in at four or nine times the strike, because variance is the square of a volatility that can triple overnight. The convexity that makes a short-variance position bleed slowly in calm markets is the same convexity that destroys it in a crisis. A payoff that is squared in the underlying risk is not a premium-harvesting strategy; it is selling insurance on the tail.

Advantages & limitations

What it is good for

It is additive across independent periods and independent positions, which volatility is not — the single property that makes it the natural quantity for every model.
It generates the square-root-of-time rule directly, so the √t seen throughout options pricing is a consequence of variance's additivity rather than a separate convention.
It is the quantity that can be replicated model-free from a strip of options, which is what makes VIX-style indices and variance swaps possible at all.
Its convexity is a feature for anyone wanting exposure to extreme moves: a long-variance position profits disproportionately from a crisis, making it a purer tail hedge than a volatility position.
It decomposes cleanly. The variance of a portfolio splits into the variances of its parts plus their covariances, which is the entire basis of portfolio risk mathematics.

Where it breaks down

It is in squared units — %² — which no human can intuit, so it is useless for communication and must be converted to volatility before quoting.
Its additivity holds only when the periods or positions are independent; when returns are autocorrelated or positions are correlated, cross terms appear and simple addition understates or overstates the total.
Its convexity, which is an advantage for a buyer, is a severe hazard for a seller, because the squared payoff turns a large volatility move into a far larger loss.
It gives disproportionate weight to the largest observations, since squaring a big return dominates the sum — meaning a variance figure can be driven almost entirely by a handful of extreme days.
The model-free replication assumes a continuum of strikes with liquidity at every one; in reality the far wings are illiquid, so a real variance swap is imperfectly hedged exactly where the tail lives.
It shares volatility's blindness to direction and to the order of returns, because squaring destroys the sign — a variance figure cannot distinguish a rally from a crash of the same magnitude.

Common mistakes

Adding volatilities of two periods directly. Volatility does not add — you must square each to variance, add the variances, then take the square root. Adding 12% and 16% to get 28% is simply wrong.
Assuming the combined volatility of two periods is the average of their volatilities. Because variance adds and is convex, the combined figure leans toward the larger volatility — 12% and 20% combine to about 16.5%, not 16%.
Selling variance swaps as a steady-premium strategy without respecting the convexity. A doubling of realised volatility quadruples the variance payoff, which is how short-variance books blew up in 2008 and 2020.
Confusing a variance swap with a volatility swap on the same notional. They pay similarly in calm markets and wildly differently in a crisis, because one payoff is squared and the other is linear.
Reading a variance figure as representative of the whole period. Because squaring amplifies the biggest days, a variance can be dominated by two or three sessions and tell you almost nothing about the other fifty.
Applying variance additivity to correlated positions. The clean σ²(a+b) = σ²(a) + σ²(b) holds only under independence; correlated positions have covariance terms that the simple sum ignores.
Thinking the VIX inverts Black–Scholes like a single option's implied volatility. It does not — it is the fair strike of a variance swap, a model-free weighted sum across the chain, which is a different construction entirely.

Professional usage

Variance is the working currency of a volatility-trading desk even though the screen shows volatility. When a desk prices the total dispersion of an option over its life, it sums the variance of each sub-period — including the extra variance contributed by scheduled events like an RBI decision or Union Budget — and converts to a volatility only to quote it. Forward volatility, the volatility implied for a future window between two expiries, is computed by subtracting the near variance from the far variance and taking the square root, because only variance can be subtracted this way. And the entire variance-swap market exists because variance, unlike volatility, can be replicated from a static strip of options, letting a desk trade realised dispersion directly and hedge it with the chain.

Risk management is variance mathematics from top to bottom. Portfolio variance is the sum of each position's variance plus twice every pairwise covariance, and this decomposition is how a risk desk attributes total risk to individual books and to the correlations between them. Value-at-risk models propagate variances and covariances rather than volatilities precisely because variances add and volatilities do not. When a risk manager stresses a correlation assumption, they are altering the covariance terms in a variance sum, and watching the total portfolio variance — and therefore the capital required against it — move in response.

Key takeaways

Variance is volatility squared — the average squared deviation of returns — and it is the quantity that adds across independent periods and positions, where volatility does not.
The single identity σ²(a+b) = σ²(a) + σ²(b) for independent a and b generates the square-root-of-time rule, forward volatility and the existence of variance swaps.
Variance is convex in volatility: doubling volatility from 12% to 24% quadruples variance from 1.44 to 5.76, which is why variance swaps pay far more in a crisis than volatility swaps.
Variance can be replicated model-free from a strip of options weighted by 1/K², which is exactly how VIX-style indices, including India VIX, are constructed.
The market quotes volatility because humans read it and models variance because variance is the one that behaves — keep track of which you are holding.

Variance is the number the market never shows you and always uses. Learn to reach for it the instant you need to combine dispersion across time or across positions, because it is the only version that adds, and reach back for volatility only at the end, to quote. The deeper lesson is in the convexity: variance is a squared quantity, and a squared quantity turns a bad day into a disaster and a crisis into a catastrophe. Anyone selling you exposure to variance as a source of calm premium is selling you the square of a risk that can triple overnight, and the market has closed that trade violently at least twice in living memory.

Frequently asked questions

What is variance in simple terms?

Variance is the square of volatility — the average of the squared distances of returns from their mean. It is in squared units, so no one quotes it, but it is the quantity every model computes with, because unlike volatility it adds across independent periods and positions.

What is the difference between variance and volatility?

Volatility is the standard deviation of returns and variance is its square. They carry the same information, but variance adds across independent periods and volatility does not — so the market quotes volatility because humans read it and models compute in variance because variance behaves.

Why do variances add but volatilities don't?

Because for independent periods the squared dispersions accumulate linearly: σ²(a+b) = σ²(a) + σ²(b). Taking the square root breaks the additivity, since the square root of a sum is not the sum of the square roots. That is why you must convert to variance before combining periods.

Why does the market quote volatility if it models variance?

Because volatility is in the units of returns — a percentage a human can picture — while variance is in squared units that mean nothing intuitively. Volatility wins for communication; variance wins for calculation. Models compute in variance and take a square root only to report a volatility.

How does variance generate the square-root-of-time rule?

If daily returns are independent, the variance of n days is n times the daily variance, because variances add. Taking the square root gives volatility scaling with √n. Every √t in options pricing — √252, √(days/365), √T in Black–Scholes — traces back to this additivity of variance.

Why is variance convex in volatility?

Because variance is volatility squared, so the graph of variance against volatility is a parabola. Equal steps in volatility are unequal steps in variance: going from 12% to 24% doubles volatility but quadruples variance from 0.0144 to 0.0576. This convexity is central to how variance swaps behave in a crisis.

Does doubling volatility double variance?

No — it quadruples it. Because variance is the square of volatility, doubling volatility from 12% to 24% takes variance from 1.44 to 5.76, a factor of four. This is why a variance swap pays roughly four times as much when volatility doubles, not twice.

What is a variance swap?

A variance swap is a contract that pays the difference between realised variance and a strike variance agreed upfront. Because its payoff is in variance — squared volatility — it is convex, paying far more in a crisis than a volatility swap on the same notional. It can be replicated from a strip of options.

How is a variance swap different from a volatility swap?

A volatility swap pays linearly in realised volatility; a variance swap pays in realised variance, which is squared and therefore convex. On the same notional they pay similarly in calm markets but wildly differently in a crisis, when the variance swap's squared payoff explodes.

Why can variance be replicated model-free but volatility cannot?

Because the payoff of accumulated variance can be reproduced by a static portfolio of options across all strikes weighted by 1/K², without assuming any model of how volatility evolves. Volatility, the square root, has no such clean replication. This is why VIX-style indices are built on variance, not volatility.

How is India VIX related to variance?

India VIX is computed as the fair strike of a variance swap on NIFTY — a model-free weighted sum across the near-dated option chain — and then converted to a volatility by a final square root. It does not invert Black–Scholes on a single option; it prices variance and reports volatility.

Why did variance swap sellers blow up in 2008 and 2020?

Because variance is the square of volatility, so when volatility tripled the variance payoff rose ninefold, far beyond what the collected premium could cover. Short-variance positions that bled slowly in calm markets lost catastrophically when the squared payoff hit its tail. Convexity that is a footnote in calm is the whole risk in a crash.

Why is variance in squared units?

Because it is computed by squaring the deviations of returns from their mean, so a 13% return contributes 0.0169 or 1.69 %². Those squared units have no intuitive meaning, which is exactly why the square root is taken to recover volatility before anyone quotes a figure.

How do you combine the variances of two periods?

Convert each period's volatility to variance by squaring, scale to a common time unit if needed, add the variances, then take the square root of the total to get a combined volatility. Two 30-day windows at 13% combine to 60 days at 13%; 12% and 20% combine to about 16.5%.

Why does combining 12% and 20% volatility give 16.5%, not 16%?

Because variance adds and is convex, so the larger volatility's square dominates. The variances 0.0144 and 0.0400 average to 0.0272, whose square root is 16.5% — pulled above the simple 16% average toward the bigger figure. Averaging volatilities directly is always wrong.

What is forward volatility and how does variance produce it?

Forward volatility is the volatility implied for a future window between two expiries. It is found by subtracting the near-term variance from the far-term variance and taking the square root, because only variance can be subtracted across periods this way. Volatilities cannot be netted directly.

Does variance tell you about direction?

No. Squaring the deviations destroys their sign, so variance treats a rally and a crash of equal size identically. Like volatility, it measures dispersion, not direction, and cannot distinguish an up-move from a down-move of the same magnitude.

Why does variance give so much weight to big days?

Because it squares each return, so a day twice as large contributes four times as much to the sum. A handful of extreme sessions can dominate a whole period's variance, which means a variance figure is often a summary of its worst few days rather than a description of the typical one.

Is portfolio variance just the sum of position variances?

Only if the positions are independent. In general, portfolio variance is the sum of each position's variance plus twice every pairwise covariance, so correlations matter enormously. The clean additivity of variance holds strictly under independence, and covariance terms capture the rest.

Why do risk models use variance instead of volatility?

Because variances and covariances add, letting a portfolio's total risk be decomposed into the contributions of its parts and their correlations. Value-at-risk and margin models propagate variances precisely because volatilities cannot be summed, then convert to a volatility or a loss figure at the end.

Is a variance figure enough to describe a period's risk?

No. Because squaring amplifies the largest moves, a variance can be driven almost entirely by two or three days and say little about the rest. It also shares volatility's blindness to skew, to fat tails beyond the squared term, and to the order in which returns arrived.

Voice search & related questions

Natural-language questions people ask about variance.

What is variance?

Variance is volatility squared — the average of the squared distances of returns from their mean. Nobody quotes it because its units are squared and unreadable, but every model computes in it, because variance is the one quantity that adds across independent periods and positions.

Why can't I just add two volatilities together?

Because volatility does not add — only variance does. If you have 12% one month and 16% the next, square each to a variance, add the variances, then take the square root to get the combined volatility. Adding 12 and 16 to get 28 is one of the most common volatility mistakes there is.

Why does doubling volatility quadruple variance?

Because variance is the square of volatility, and squaring turns a doubling into a quadrupling: 0.24² is four times 0.12². That convexity is why a variance swap pays roughly four times more when volatility doubles, and why selling variance is so dangerous in a crisis.

What makes a variance swap risky to sell?

Its payoff is squared, so a volatility spike hits far harder than it looks. If volatility triples, the variance payoff rises ninefold, and the premium you collected never covers that. Short-variance books blew up in 2008 and 2020 for exactly this reason — the convexity is the whole risk. None of this is investment advice.

How does the VIX use variance?

The VIX and India VIX are the fair strike of a variance swap — a model-free weighted sum across the whole option chain — converted to a volatility by a final square root. They price variance because variance, unlike volatility, can be replicated from a strip of options. The volatility figure is just the readable wrapper.

Why do models love variance so much?

Because it adds, and volatility doesn't. Across time, across positions, across a whole portfolio, variances and covariances sum cleanly, which makes every risk and pricing calculation tractable. The models live in variance and only step out to volatility when they need to show a human a number.

Is variance just a more complicated way of saying volatility?

They hold the same information, but they are not interchangeable — variance adds and volatility doesn't, and variance is convex where volatility is linear. That difference is why both exist: volatility to be read, variance to be computed. Knowing which one you are holding at any moment prevents most volatility arithmetic errors.

Sources & references

Last reviewed 10 July 2026. Educational content only — not investment advice.