Explaining Smiles

Abstract

Implied volatilities from equity options invariably show patterns that are inconsistent with lognormality. Time variation in volatility is an obvious, and widely explored, potential explanation. One approach has been to assume an explicit model of the volatility dynamics, typically a GARCH process. Another has been to compute an implied volatility smile/skew from option prices and then to model its behavior. This article presents a comparison of three such approaches using a large set of DAX option prices to imply out the model parameters. The “ad hoc Black-Scholes” technique of Dumas, Fleming, and Whaley fits an empirical volatility smile, and the model of Heston and Nandi produces GARCH volatilities in a closed-form framework. The third model is a very flexible GARCH variant: an EGARCH model with disturbances drawn from a GED process. This last model allows an asymmetric volatility response to positive and negative return shocks, plus additional skewness and kurtosis beyond that of standard GARCH, coming from the non-normal distribution of return shocks. In both in-sample and out-of-sample analysis, the EGARCH-GED model produced the best fit to observed option prices.