Convexity

(a) Prove that if $f$ is convex and $Y$ is a mean-preserving spread of $X$ ($Y=X+\varepsilon$ with $\mathbb E[\varepsilon\mid X]=0$), then $\mathbb E f(Y)\ge\mathbb E f(X)$. (b) Apply this to a long straddle on Brent: why does a rise in geopolitical uncertainty with no change in expected price raise the straddle's value? (c) Why does the same argument imply that selling options into a geopolitical event is a short-convexity bet whose losses are concentrated in exactly the states you care about?