Index option returns and systemic equity risk

In an environment characterized by stochastic variances and correlations, we demonstrate through construction of the equilibrium index option value from constituent components, that the generalized PDE identifies the stochastic elements differentially affecting index option prices relative to prices of aggregated constituent stock options. A unified treatment of the generalized partial differential system for index and constituent stock options in Theorem 1 illustrates that nonlinear interactive terms emanating from stochastic correlation affect index option price and return essentially different from contributions to the aggregated risks of the constituent stock options. Our study contributes to the growing evidence of priced correlation risk in markets for index and constituent stock options.

Theorem 1 illustrates the pricing differential, while Proposition 1 illustrates that the pricing differential produces a quantifiable metric of the measure of the nonlinear interactive terms. The quantifiable metric is constructed from the difference between the model free implied variance of the index and a weighted aggregate of the model free implied variances of the constituent stocks. Proposition 2 identifies that index variance risk premium includes additional significant contributions from the nonlinear interactive risks not present in the aggregated returns of the constituent stocks. The nonlinear interactive risks produce a wedge between the instantaneous expected excess index and aggregated stock option returns.