Jensen's inequality and convexity in pricing

State Jensen's inequality and describe why it matters in finance.

For convex $f$, $E[f(X)]\ge f(E[X])$ (reversed for concave $f$). Option payoffs are convex, so an option is worth more than its payoff evaluated at the expected price, and that gap (time value) grows with volatility. It also explains convexity adjustments in rates (e.g. futures versus forwards, CMS) and why expected log return lies below the log of expected return.