Why is the Index Smile So Steep? *

Abstract

Empirical evidence shows that the implied volatility smiles for index options are signi- ficantly 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. 1. Introduction and Motivation There is growing empirical evidence that the implied volatility smile for index op- tions is different from that for options on individual stocks. While implied volatility is in most cases a downward sloping function of the strike price, index smiles are usually significantly steeper than the corresponding curves for individual stocks, which are sometimes even flat. See, for example, Bakshi et al. (2003) for options on the S&P 100 index, and Bollen and Whaley (2001) for the S&P 500, who find that the slope of the function relating the implied volatility of options to their moneyness is around −0.308 for the typical stock in their sample, while for the index the curve is much steeper with a slope of −0.886 (the computational details behind these numbers are explained in Section 4). Two questions arise in this context. The first is how to explain in general a downward sloping volatility smile, and the second is how to explain the differences between the implied volatility functions for indices and individual stocks. One hypothesis that has been put forward in the literature to explain downward sloping smiles is the so-called leverage effect. Interpreting the stock as a call option on the assets of the firm, a drop in firm value causes a drop in the stock price, which � Earlier versions of this paper were presented at Cass Business School in London, the Brown Bag Seminar in Finance at Goethe University, the 9th Symposium on Finance, Banking, and Insurance in Karlsruhe, the annual meetings of the Swiss Finance Association in Zurich, the Money, Macro, and Finance Conference in Cambridge, and the annual meetings of the European Investment Review in Geneva. The authors would like to thank the seminar and conference participants as well as an anonymous referee and Raman Uppal, the editor in charge of the paper, for useful comments and suggestions.