All Risks Matter

Abstract

We derive the total variance risk premium for an index in the stochastic environment of Driessen, Maenhout and Vilkov (2009) and correct the previous authors omission of certain components which contribute significantly to index option expected returns. This study provides a mathematically complete decomposition of an index's total variance risk premium, and a mathematically complete description of the dynamic asset pricing for the system consisting of the index and the index's component stocks. Previous authors studying this exact stochastic structure, have neglected important elements which contribute to the index's total variance risk premium. We illustrate that an index's total variance risk premium is due not only to changes in index component's variance and changes in component's return correlations, but is also due to important interactions between component's variances, changes in component's variances; correlations among component's returns and changes in correlations. We identify the roles of the total risk components within an option pricing framework and provide empirical verification of the statistical significance of the previously omitted adapting correlations and interactive risks. Furthermore we establish the generalized Black-Scholes-Merton partial differential system in the presence of nontrivial and stochastic correlations. The unified treatment of the partial differential system identifies interactive risks affecting index option price which have been ignored before. We also introduce the quantified systemic risk indicator. We show that all risks are economically significant determinants of option prices.

Keywords: stochastic correlation, stochastic volatility, equity index option

JEL Classification: G11, G12, G13, G14