And the extent to which the expectation of the function of the random variable exceeds the function of the expectation of the random variable depends on the variable’s variability (or variance), as can be seen eg by a Taylor expansion around the expectation.

That’s the reason why linear (or affine) financial derivatives (such as forwards) can be priced without using volatility as an input, while products with convexity (such as options) require volatility as an input.

(Side note: I think Delta One desks should rename to Gamma Zero…)