Asymptotic Distribution Expansions in Option Pricing

Abstract

The previous article compares different methods for estimating the empirical probability distribution for asset returns. This article looks at methods for approximating an implied risk neutral density. Constraining it to be lognormal is common practice, of course, but does not give a very good fit to actual option prices. A unique feature of this article is to examine American options, for which the implied densities must produce prices that satisfy certain bounds, without being pinned down to exact values. Some of the same procedures as described by Jensen and Poulsen, like Hermite approximation, have been explored, but Giamouridis and Tamvakis find that these techniques and several others, including a mixture of lognormals, do not work as well as an Edgeworth Series Expansion (ESE). The ESE technique relaxes the constraints on skewness and kurtosis imposed by the lognormal, without a proliferation of parameters to estimate. The ESE model is then compared in terms of performance to methods examined in 12 other studies, and is found to behave comparably, with densities that are close to a median structure across models.