Why is the Index Smile So Steep?

Empirical evidence shows that the implied volatility smiles for index options are significantly steeper than those for individual options. We propose a model setup where we start from the joint dynamics of the stocks and where the index value is a weighted sum of individual stock prices. Then the differences between the index smile and the smiles for individual stocks are entirely determined by the dependence structure among the stocks. We illustrate our idea in a jump-diffusion framework where both the diffusion and the jumps are decomposed into common and idiosyncratic components. Empirical data for options on the German stock index DAX and on Deutsche Bank are used to show that the model can explain the stylized facts on implied volatility smiles.