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

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. The pronounced implied volatility smile produces artificially inflated fitted parameters, resulting in unrealistically high VIX option implied volatilities. To remedy these shortcomings, jumps are introduced. The resulting model is able to better fit short-term index option implied volatilities while producing more realistic VIX option implied volatilities, without a loss in tractability.

Key Words:

• Stochastic volatility plus jumps model
• 3/2 model
• VIX derivatives

Notes

1 The method is applicable to all conditionally Gaussian stochastic volatility models for which the Laplace transform of realized variance is known explicitly.