Consistent Modeling of VIX and Equity Derivatives Using a 3/2 plus Jumps Model

TL;DR

This paper shows that a pure-diffusion 3/2 model can effectively capture VIX implied volatility skew, but adding jumps improves short-term index option smile modeling without sacrificing tractability.

Contribution

It demonstrates that a pure-diffusion 3/2 model suffices for VIX skew, and introducing jumps enhances short-term option smile modeling while maintaining tractability.

Findings

Pure-diffusion 3/2 model captures VIX skew.

Adding jumps improves short-term index option smile.

Model remains tractable for pricing derivatives.

Abstract

The paper demonstrates that a pure-diffusion 3/2 model is able to capture the observed upward-sloping implied volatility skew in VIX options. This observation contradicts a common perception in the literature that jumps are required for the consistent modelling of equity and VIX derivatives. The pure-diffusion model, however, struggles to reproduce the smile in the implied volatilities of short-term index options. One remedy to this problem is to augment the model by introducing jumps in the index. The resulting 3/2 plus jumps model turns out to be as tractable as its pure-diffusion counterpart when it comes to pricing equity, realized variance and VIX derivatives, but accurately captures the smile in implied volatilities of short-term index options.