We consider a square-integrable semimartingale and investigate the convex order relations between its discrete, continuous, and predictable quadratic variation. As the main result, we show that if the semimartingale has conditionally independent increments and symmetric jump measure, then its discrete realized variance dominates its quadratic variation in increasing convex order. The result has immediate applications to the pricing of options on realized variance. For a class of models including independently time-changed Lévy models and Sato processes with symmetric jumps our results show that options on variance are typically underpriced if quadratic variation is substituted for the discretely sampled realized variance.