4/2 rough and smooth

We propose a novel 4/2 rough and smooth stochastic volatility model by combining the rough Heston (rough 1/2) and smooth 3/2 models through a convex specification. This parsimonious two-factor model admits semi-closed-form pricing formulas for equity and volatility index (VIX) derivatives, while capturing key stylized facts documented in empirical studies. The model flexibly generates elasticity of variance estimates consistent with empirical findings from equity markets and produces realistic variance distributions. Although the rough 1/2 component carries a small weight, our numerical experiments confirm a degree of roughness comparable with that obtained with the rough Heston model. Empirical analysis using S&P 500 and VIX option data shows that the model outperforms benchmark specifications both in- and out-of-sample. We further provide insights into how rough volatility modeling influences the estimation of risk-neutral return moments and variance risk premia.