S&P 500 Index Option Surface Drivers and their Real World and Risk Neutral Covariations

Abstract

Observing first that the daily option surface may be summarized by the level of the spot price and the four parameters of the Sato process based on the variance gamma process, a time series is constructed for this five dimensional set of factors driving the surface of S&P 500 index option prices. Next we show that parameter movements can be hedged at zero cost and a delta hedged position then earns in its theta compensation for the exposure to the fifteen second order squared and cross product terms. The coefficients of compensation form an estimate for the risk neutral quadratic covariation between the five surface drivers. A constrained optimization forcing positive semidefinite coefficients is then employed to estimate risk neutral covariations. Simultaneously statistical covariations are estimated and it is shown that the two sets of covariations are quite different. We then ask if quadratic variations in all directions risk neutrally exceed their statistical counterparts. This is not the case and quadratic variations of the stock price unaccompanied by other movements of the surface have a lower risk neutral expectation. All other directions have a higher risk neutral quadratic variation from their statistical counterparts.