Convex order of discrete, continuous and predictable quadratic variation & applications to options on variance

TL;DR

This paper establishes convex order relations between different types of quadratic variation for certain semimartingales and demonstrates implications for pricing variance options, showing underpricing when using quadratic variation as a proxy.

Contribution

It proves that for semimartingales with conditionally independent, symmetric jumps, realized variance dominates quadratic variation in convex order, impacting option pricing models.

Findings

Realized variance dominates quadratic variation in convex order under specified conditions.

Options on realized variance tend to be underpriced when quadratic variation is used as a substitute.

Results apply to models including time-changed Levy and Sato processes with symmetric jumps.

Abstract

We consider a square-integrable semimartingale and investigate the convex order relations between its discrete, continuous and predictable quadratic variation. As the main results, 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 results have immediate applications to the pricing of options on realized variance. For a class of models including time-changed Levy 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.