Smiling twice: The Heston++ model

We recommend the addition of a deterministic displacement to multi-factor affine models to calibrate and hedge SPX and VIX derivatives jointly. The proposed model, labeled Heston++, calibrates both markets with an average relative error (on quoted implied volatilities over two years of data) of 2%, and a maximum relative error of 4%, without additional computational costs with respect to traditional affine benchmarks. Hedging performance on both markets is also drastically improved. The displacement can be interpreted as a volatility push-up reflecting expectations about a (risk-neutral) lower bound on forward VIX dynamics. Our empirical results document substantial correlation between the dynamics of the displacement and the variance risk premium, and still provide strong support for the presence of both price/volatility negatively correlated co-jumps and idiosyncratic jumps in the volatility dynamics, even when the displacement is added.