The Price of Higher Order Catastrophe Insurance: The Case of VIX Options

We develop an equilibrium pricing model aimed at explaining observed characteristics in equity returns, VIX futures and VIX options data. To derive our model we first specify a general framework based on affine jump-diffusive state-dynamics and representative agent endowed with Duffie-Epstein recursive utility. This allows us to derive moments of equity returns under the objective and risk-neutral measures, and subsequently semi-closed form solutions to prices of equity options, VIX futures, and VIX options. We calibrate this model to fit the salient features of the data, including moments of consumption and equity returns, variance premium, and various features of VIX derivatives data. The model matches the extremely right-skewed volatility smiles seen in VIX options, a downward-sloping term structure of implied Black'76 volatilities, large negative rates of return on VIX futures, and large VIX option risk premia. It also matches other characteristics of VIX options data, including time-variation in the shape of implied volatilities.