Variance vs Volatility: What's the Difference?

Why squaring matters for vol products — and why variance is not just a technicality.

Ask most traders what volatility is and they will say standard deviation of returns. Ask them what variance is and they will say volatility squared. Both answers are correct, and the distinction between variance vs volatility looks purely notational until you start building products around it. Variance is additive over time; volatility is not. Variance swaps pay on realized variance, not realized volatility, because variance is what replicates cleanly from an option strip. VIX is quoted in volatility units but computed from a variance calculation. The difference between σ and σ² shapes the entire market for realized volatility, and it is worth understanding precisely.

Definitions: volatility, variance, and the relationship between them

Start from daily log returns r_t = ln(S_t/S_t−1). The realized variance over N days is the average squared deviation:

σ² = (1/N) · Σ r_t²

(under the convention that the mean is approximately zero for short horizons, which is standard in vol markets). Realized volatility — the number you see quoted — is simply the square root: σ = √σ². The options pricing calculator takes volatility as its input, not variance, because that is the market convention. But under the hood, every option model is working with σ² in the exponent of the lognormal density.

The notation is consistent across the field:

• Volatility σ is expressed as an annualized percentage — e.g. 20%.
• Variance σ² is expressed in squared-percentage units — e.g. 0.04 (= 0.20²).
• Converting between them is a squaring or square-root operation; there is no additive adjustment.

Why variance is additive and volatility is not

This is the point that matters most for anything involving multiple time periods. If two non-overlapping periods have variances σ₁² and σ₂², the combined variance is simply their sum:

σ_total² = σ₁² + σ₂²

That follows directly from the independence of returns across non-overlapping windows — variances of independent random variables add. Volatilities do not add. The combined volatility is:

σ_total = √(σ₁² + σ₂²)

A worked example makes the gap concrete. Suppose the first 30 days of a month run at 16% annualized volatility and the final 21 days run at 32% annualized volatility. Converting to variance on a per-day basis (annualized variance ÷ 252):

• Period 1 daily variance: (0.16)²/252 ≈ 0.0001016
• Period 2 daily variance: (0.32)²/252 ≈ 0.0004063
• Combined daily variance over 51 days: you average, weighted by days — (30 × 0.0001016 + 21 × 0.0004063)/51 ≈ 0.0002271
• Combined annualized volatility: √(0.0002271 × 252) ≈ 23.9%

If you had simply averaged the two vol numbers — (16% + 32%)/2 = 24% — you would get close but for the wrong reason. Blend unequal weights and the error compounds. Only working through variance and then taking the square root is correct.

The practical implication: variance scales linearly with time, so a T-day variance is just T times the daily variance. Volatility scales with √T, which is why 1-month at-the-money options are priced using σ·√(30/252), not σ·(30/252).

Variance swaps and the replication argument

A variance swap pays the difference between realized variance and a strike variance, multiplied by a notional:

Payoff = Notional · (σ_realized² − K_var)

Note that the payoff is linear in realized variance, not in realized volatility. That is not an arbitrary contract choice — it is a consequence of replication. A static strip of options across all strikes, held to expiration, delivers a payoff equal to realized variance. The replication works because the option strip captures all of the quadratic variation of the price path. Volatility swaps, which pay on √(realized variance), cannot be replicated statically; they require dynamic hedging and involve a convexity adjustment. Pricing a vol swap from a variance swap quote always requires subtracting a Jensen's inequality term, because E[√X] < √E[X] for any non-degenerate random variable X.

The convexity this creates matters for buyers of variance. A long variance swap benefits disproportionately from large moves — a day with a 3% return contributes nine times as much variance as a day with a 1% return. If volatility spikes during a market dislocation, a variance buyer profits on a convex payoff function, not a linear one. This is why variance swaps were the preferred instrument for tail hedges before the basis risk from the 2008 crisis (and subsequent ISDA documentation changes) compressed the market.

VIX: quoted in volatility, built from variance

The VIX is the canonical example of this split. The index is expressed as an annualized volatility — 20 VIX means the market implies 20% annualized vol over the next 30 days. But the calculation that produces that number is a variance calculation. The VIX methodology takes a weighted sum of out-of-the-money call and put prices across a continuum of strikes — approximating the same option strip used to replicate a variance swap — to produce an expected variance. That variance is then annualized and square-rooted to yield the volatility number users see on their screens. The details of how the VIX is calculated are worth reviewing if you trade any VIX-linked instrument, because changes in the volatility of volatility affect the variance estimate asymmetrically in ways a quoted vol number conceals.

Practical consequences for vol traders

The variance-vs-volatility distinction creates several concrete differences in how products behave:

• P&L attribution. A delta-hedged option book generates daily P&L proportional to gamma times the squared move — i.e., realized variance. A single large move contributes quadratically, not linearly. Traders tracking "daily theta vs gamma" are implicitly working in variance units.
• Vol-of-vol exposure. A variance swap has a fixed payoff function regardless of how volatility evolves day to day. A vol swap or VIX option has payoff that depends on realized vol, which means it depends on vol-of-vol. The two instruments have very different tail characteristics.
• Strike setting. Variance swap strikes are quoted in variance units (e.g. 400 variance points = 20 vol points squared). Converting back and forth is straightforward, but confusion between the two has produced real pricing errors in structured products.
• Correlation products. Dispersion trades — long single-stock variance, short index variance — exploit the fact that index variance equals the variance-weighted sum of single-stock variances plus a correlation term. That decomposition only works cleanly in variance space; in vol space it requires approximations.

A note on which number to use when

The rule of thumb is simple: use volatility when communicating with humans, use variance when doing mathematics. Option prices, implied vol surfaces, and risk reports all live in vol space because percentages are more intuitive than squared percentages. But whenever you are combining periods, summing exposures across assets, computing expected payoffs, or checking replication arguments, convert to variance first, do the arithmetic, and convert back. The square root is cheap; the error from skipping it is not.

The same principle applies when reading model documentation or term sheets. A contract that references "realized variance" and one that references "realized volatility" are economically different instruments. The former has a static replication; the latter does not. If the term sheet is ambiguous, the question to ask is whether the payoff formula involves σ or σ² — that single character determines the hedging strategy, the fair value, and the convexity profile of the position.