The Fine Structure of Variance: Pricing VIX Derivatives in Consistent and Log-VIX Models

Abstract

We analyze pricing models for VIX derivatives which account for the theoretical link to stock options, taking Log-VIX models as a benchmark. We focus on up to three risk factors to model variance risk. To assess the performance of the models, we do not only look at the pricing errors, but also at the level and dynamics of the VIX' risk-neutral moments which vary considerably over time. We find that both model classes, consistent- and Log-VIX models, can reproduce the empirical patterns if three risk factors are included. In both approaches, a stochastic central tendency is of first order importance to capture the term structure of VIX futures prices, i.e. the first moment of the risk-neutral distribution. A stochastic vol-of-vol then helps to match the prices of VIX options, i.e. the higher order moments. Finally, variance jumps add the finishing touches to the model performance. All in all, consistency comes at notable costs in-sample, while out-of-sample performances are close. We find that the main difference between both model classes is the ability to capture the second moment of the VIX risk-neutral distribution.

Keywords: Consistent pricing of VIX derivatives, Log-VIX model, volatility derivatives, VIX

JEL Classification: G13