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

The logarithm of the S&P 500 Index is modelled as a Sato process running at a speed proportional to the current level of the VIX. When the VIX is itself modelled as the exponential of a compound Poisson process with drift, we show that exact expressions are available for the prices of equity options, taken at an independent exponential maturity. The parameters for the compound Poisson process are calibrated from VIX options whereas the parameters for the Sato process driving the stock may be inferred from market option prices. Results confirm that both the S&P 500 index option surface and the parameters of the VIX time-changed Sato process have volatilities, skews and term volatility spreads that are responsive to the VIX level and the VIX option surface.