Volatility pays convex things and bills concave ones — one slider for curvature, one for shocks, and the gap computes itself.
The machine under this page is Jensen's inequality: for a convex payoff f, E[f(X)] ≥ f(E[X]) — the average of the outcomes beats the outcome of the average — with equality exactly when f is affine on the convex hull of the support of X — here, affine across the whole stretch between −σ and +σ. It is old and unglamorous: J.L.W.V. Jensen, “Sur les fonctions convexes et les inégalités entre les valeurs moyennes”, Acta Mathematica 30 (1906), 175–193, read to the Danish Mathematical Society in January 1905. The modern reading is the sign: a thing that loses from volatility is fragile, a thing indifferent to it is robust, a thing that gains from it is antifragile — a triad of curvature against shocks, paraphrasing Nassim Nicholas Taleb, Antifragile: Things That Gain from Disorder (Random House, 2012), and made mathematical in Taleb & Douady, “Mathematical definition, mapping, and detection of (anti)fragility”, Quantitative Finance 13(11) (2013), 1677–1689.
Why a two-point world? Because when X sits at ±σ with equal odds, E[f(X)] is literally the midpoint of a chord, and Jensen collapses into exact arithmetic — no Monte Carlo, no simulation fog, nothing left to converge. The gap even has a closed form here, (cosh(kσ)−1)/k; the page computes it straight from f, and the two agree. At k=1, σ=1 the gap is cosh(1)−1 ≈ 0.543081, while the small-volatility estimate ½f″(0)σ² = kσ²/2 guesses 0.5 — good in the small, honest about drifting in the large. One line of salt: real payoffs — options, engineering margins, dose–response curves — carry curvature too, but mapping them is analysis of structure, not advice. Next: the two averages →