If you examine the structure of listed index options prices through the prism of the implied tree model, you observe the local volatility surface of the under- lying index.

In the same way as fixed income investors analyze the yield curve in terms of forward rates, so index options investors should analyze the volatility smile in terms of local volatilities.

In this report we explain the local volatility surface, give examples of its applications, and propose sev- eral heuristic rules of thumb for understanding the relation between local and implied volatilities. In essence, the model allows the extraction of the fair local volatility of an index at all future times and market levels, as implied by current options prices. We use these local volatilities in markets with a pro- nounced smile to measure options market senti- ment, to compute the evolution of standard options implied volatilities, to calculate the index exposure of standard index options, and to value and hedge exotic options. In markets with significant smiles, all of our results show large discrepancies from the results of the standard Black-Scholes approach.

Investors who buy or sell standard index options for the exposure they provide, as well as market partici- pants interested in the fair price of exotic index options, should find interest in the deviations we predict from the Black-Scholes results.