Options on realized variance and convex orders

Realized variance option and options on quadratic variation normalized to unit expectation are analysed for the property of monotonicity in maturity for call options at a fixed strike. When this condition holds the risk-neutral densities are said to be increasing in the convex order. For Lévy processes, such prices decrease with maturity. A time series analysis of squared log returns on the S&P 500 index also reveals such a decrease. If options are priced to a slightly increasing level of acceptability, then the resulting risk-neutral densities can be increasing in the convex order. Calibrated stochastic volatility models yield possibilities in both directions. Finally, we consider modeling strategies guaranteeing an increase in convex order for the normalized quadratic variation. These strategies model instantaneous variance as a normalized exponential of a Lévy process. Simulation studies suggest that other transformations may also deliver an increase in the convex order.