Reconciling Smiles for Index and Stock Options

The fact that the implied volatility smiles for equity indices are strongly downward sloping, while the typical individual stock exhibits either a flat or even an upward sloping smile is sometimes considered puzzling. We show that this effect can easily be generated in a simple and parsimonious two-factor stochastic volatility model in the spirit of Bates (2000). From a theoretical perspective our paper adds to the literature by proving a ’diversification’ result with respect to idiosyncratic stochastic volatility components of individual stocks, so that in the limit for infinitely many stocks, the prices of index options are solely based on the common volatility component. This result holds without restrictive assumptions on the structure of the correlations between stock returns and volatility changes. We employ Monte Carlo simulation to show that our model can reproduce the empirical observations quite well for an index of 30 stocks similar to the Dow Jones Industrial Average.