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Primary Article

Explaining Smiles

GARCH Option Pricing with Conditional Leptokurtosis and Skewness

Thorsten Lehnert
The Journal of Derivatives Spring 2003, 10 (3) 27-39; DOI: https://doi.org/10.3905/jod.2003.319199
Thorsten Lehnert
An assistant professor of finance at the Limburg Institute of Financial Economics at Maastricht University in the Netherlands.
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  • For correspondence: t.lehnert@berfin.unimaas.nl
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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.

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The Journal of Derivatives
Vol. 10, Issue 3
Spring 2003
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Explaining Smiles
Thorsten Lehnert
The Journal of Derivatives Feb 2003, 10 (3) 27-39; DOI: 10.3905/jod.2003.319199

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Explaining Smiles
Thorsten Lehnert
The Journal of Derivatives Feb 2003, 10 (3) 27-39; DOI: 10.3905/jod.2003.319199
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