Mathematics · Core
Variance
Variance is the expected squared deviation from the mean, Var(X) = E[(X − μ)²]. It is the second moment that becomes volatility after a square root and a convention.
Definition
Var(X) = E[X²] − (E[X])². For independent variables, variances add. Annualization √t assumes independent increments. Fat tails and jumps make variance a poor complete description — it still exists; it just does not capture the shape.
Why it matters
Markowitz, Black–Scholes, and VaR all lean on it. Using variance as ‘risk’ for a short-option book is a category error.
Case
Two strategies with the same variance, one Gaussian and one with a 10-sigma jump every decade, are not the same risk. 2008 taught that in credit.
How to read it
Report variance and a tail statistic. If they disagree, believe the tail for sizing.
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