The S&P 500 Index as a Sato Process Travelling at the Speed of the VIX.

The logarithm of SPX is modeled as a Sato process running at a speed proportional to the current level of the VIX. When the logarithm of the VIX is an exponential compound Poisson process with drift one may obtain exact expressions for the prices of equity options taken at an independent exponential maturity. The parameters for the Lévy process are calibrated from VIX options while the parameters for the Sato process driving the stock may be calibrated from market option prices taken at an independent exponential maturity. Results confirm that both the option surface and the VIX time changed Sato process volatilities, skews and term volatility spreads are responsive to the VIX level and the VIX option surface.