Polynomial Variation, VIX Decomposition, and Tail Risk Premium

We identify the VIX index is innately a risk-neutrally forward-looking measure of the polynomial (not quadratic) variation of market returns. Correspondingly, the VIX risk premium, squared VIX minuses the realized VIX (a physically conditional measure of the polynomial variation), compensates jointly for the risk of stochastic volatility and that of jump and tail. The VIX index consequently can be decomposed into four fundamentally different components: the realized variance (RV), the variance risk premium (VRP*), the realized tail (RT), and the tail risk premium (TRP), respectively. The empirical results reveal that VRP*, RT, and TRP (except RV) help predict future market returns.