The structure of listed index options prices, examined through the prism of the implied tree model, reveals the local volatility surface of the underlying 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. This article explains the concept of local volatility and its use. It also presents three heuristic rules that relate local and implied volatilities. Local volatilities are used in markets with a pronounced smile to measure market sentiment, to compute the evolution of implied volatilities through time, to calculate the index exposure of standard index options, and to value and hedge exotic options. In markets with significant smiles, all of the results show large discrepancies from those of the standard Black–Scholes approach.