Jointly Modeling of VIX and SPX Options at a Single and Common Maturity with Risk Management Applications

Abstract

A double gamma model is proposed for the VIX. The VIX is modeled as gamma distributed with a mean and variance that respond to a gamma distributed realized variance over the preceeding month. Conditionally on VIX and the realized variance, the logarithm of the stock is variance gamma distributed with affine conditional drift and quadratic variation. The joint density for the triple, realized variance, VIX and the SPX is in closed form. Calibrating the model jointly to SPX and VIX options a risk management application illustrates a hedge for realized volatility options.