Variance
The squared foundation of all volatility math — why the option market lives in variance space even when it quotes in vol.
Definition
Variance, in the context of financial markets and options pricing, is the average of the squared deviations of an asset's returns from their mean. It is the square of volatility (σ²) and serves as the fundamental building block for every volatility calculation — implied, realised, or forward. While traders speak in volatility (percent per year), option pricing models — including Black-Scholes — operate in variance space internally. Understanding variance clarifies why volatility cannot simply be added across time periods while variance can, and why the forward volatility formula uses squared terms rather than raw vol figures.
Why it matters
For F&O traders, variance matters in three concrete ways. First, it explains the additivity of vol across time: if Nifty has a daily variance of 0.0001 (equiv. 1 percent daily vol squared), then two-day variance is 0.0002 — you can add variances, not vols. Second, variance is the payoff underlying a variance swap — an OTC instrument common among institutional desks globally where the buyer receives realised variance and pays implied variance. The profit is directly proportional to how much actual daily squared returns exceed what was implied at trade entry. Third, understanding variance helps in decomposing an option portfolio's sensitivity: vomma (the rate of change of vega with vol) is more naturally understood as a second-order variance sensitivity. For retail traders on NSE, the most practical application is internalising that daily moves need to be squared before averaging — a week with moves of +2%, −1%, +3% is not simply "6% volatile" but has a realised variance that weights the outlier day more heavily.
Formula
Realised variance over N daily log-returns ri: σ² = (1 ÷ N) × ∑(ri − r̄)², where r̄ is the mean return. For annualisation, multiply by 252 (trading days per year). Annualised volatility = √(σ² × 252). Implied variance at any expiry is simply the square of the annualised implied volatility: IV². In the forward vol context: σfwd² = (σ2² × T2 − σ1² × T1) ÷ (T2 − T1).
Example
Suppose Nifty logs daily returns of +1.5%, −0.5%, +2.0%, −1.0%, +0.5% over five sessions. Mean = 0.5%. Deviations: +1.0%, −1.0%, +1.5%, −1.5%, 0.0%. Squared deviations: 0.0001, 0.0001, 0.000225, 0.000225, 0. Average squared deviation = 0.000110. Annualised variance = 0.000110 × 252 = 0.02772. Annualised volatility = √0.02772 ≈ 16.6 percent. Notice that the two larger moves (+2%, −1%) dominate the variance — a property that makes variance a better risk measure than average absolute move. All figures are illustrative.
See realised vs implied on TradePulse
TradePulse tracks both realised and implied volatility on Nifty so you can judge whether variance is being over- or under-priced by the market.