Option Pricing Under Skewness and Kurtosis Using a Cornish–Fisher Expansion

• DOI: http://doi.org/10.1002/fut.21787
• Published date: 01 December 2016
• Date: 01 December 2016
• Author: Sofiane Aboura,Didier Maillard

Option Pricing Under Skewness

and Kurtosis Using a

Cornish–Fisher Expansion

Sofiane Aboura*and Didier Maillard

This paper revisits the pricing of options, in a context of financial stress, when the underlying

asset’s returns displays skewness and excess kurtosis. For that purpose, we use a Cornish–

Fisher transformation for valuing option contracts with an exact formula allowing for heavy-

tails. An application to the S&P 500 stock index option contracts is carried out during both

stress (October 2008) and calm (May 2015) periods. It provides evidence about the capability

of the Cornish–Fisher model to fairly price options during a period of stress without violating

the skewness–kurtosis boundaries given its large domain of validity.©2016 Wiley Periodicals,

Inc. Jrl Fut Mark 36:1194–1209, 2016

1. INTRODUCTION

The question of whether option pricing models have the ability to derive fair contract prices

and risk measures during stress conditions may not yet be solved. The financial literature on

option theory has documented many well-known pervasive features that affect pricing, which

are not taken into account in the standard Black–Scholes–Merton framework. Indeed, there

is an important body of literature attempting to overcome the limitations of the Gaussian

assumption. The objective of this article is to set up an exact formula that allows for skew-

ness and excess kurtosis within a large domain of validity using a non-parametric approach.

The non-parametric techniques have proven to be very successful in option pricing notably

because they make no assumptions about the probability distributions. Consequently, they

provide flexible specifications that allow to approximate any distribution with a given degree

of accuracy since they have not a finite set of parameters contrary to parametric techniques.

Semi-nonparametric approaches lie in between as they have both a finite-dimensional vec-

tor of parameters and an infinite-dimensional function. The practical application of semi-

nonparametric price density functions requires the truncation of its polynomial expansion.

Sofiane Aboura is Professeur at the Université de Paris XIII, Sorbonne Paris Cit´

e, CEPN (UMR-CNRS

7234), 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France. Didier Maillard is Professeur at the

Conservatoire National des Arts et Métiers, LIRSA Laboratory & Senior Advisor, Amundi, France, 292 rue

Saint-Martin 75141 Paris cedex 03. We wish to thank the Editor Prof. Robert Webb and the anonymous

referee for their invaluable comments that improved the quality of the paper. We thank Essam N’Zoulou

(Amundi) for having provided and organized the data according to our research needs. We also thank the

participants who attended to the Dauphine-Amundi Chair of Asset Management Workshopand to the CEPN

seminar for their useful remarks. The usual disclaimer applies.

JEL Classification: C02, G11, G12, G21

*Correspondence author,Universit ´

e de Paris XIII, Sorbonne Paris Cit´

e, CEPN (UMR-CNRS 7234), 99 avenue

Jean-Baptiste Clément, 93430 Villetaneuse, France. Tel: +33+149403323, Fax: +33+149403334, e-mail:

sofiane.aboura@univ-paris13.fr

Received December 2014; Accepted March 2016

The Journal of Futures Markets, Vol. 36, No.12, 1194–1209 (2016)

©2016 Wiley Periodicals, Inc.

Published online 19 May 2016 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21787

Pricing Options with a Cornish-Fisher Expansion 1195

Indeed, most density functions can be expressed as a possibly infinite expansion multiplier

of the normal density, as shown by Charlier (1905) and Edgeworth (1896, 1907).

Since then, much effort has been made to approximate the exact distribution with semi-

nonparametric option pricing formulae. Jarrow and Rudd (1982) model the distribution of

stock price with a Edgeworth series expansion. Corrado and Su (1996a)1model the distribu-

tion of stock log prices with a Gram–Charlier series expansion, while Corrado and Su (1996b)

performed the same type of study with an Edgeworth expansion2. Notice that Gram–Charlier

is itself a particular density expansion of the Edgeworth expansions class. These methods

focus on the skewness and kurtosis deviation from normality for stock returns. For clarity, it

should be noted that, to correct the bias of the Black–Scholes (1973) model, Corrado and Su

(1996a) sum up the Black–Scholes formula with the adjustment terms accounting for non-

normal skewness and kurtosis by truncating the expansion after the fourth moment. Under

risk-neutral probability, they apply the Gram–Charlier density function to derive European

call price formula. The main advantage of Gram–Charlier and Edgeworth expansions is that

they allow for additional flexibility over a normal density because they introduce a skewness

and kurtosis parameter in the distribution. A main interest relies on the connection between

the expansion parameters and the volatility smile. In clear, the volatility parameter controls

for the at-the-money volatility level in the smile. The skewness parameter controls for the

slope of the smile while the kurtosis parameter controls for the curvature of the smile. Many

applications on contingent claims have been done. More generally,several papers provide an

extension of the Edgeworth/Gram–Charlier density expansions (Jurczenko, Maillet, & Ne-

grea, 2004; Ki, Choi, Chang, & Lee, 2005; Rubinstein, 1998, etc.); other papers provide an

empirical studies on option pricing based on Edgeworth/Gram–Charlier density expansions

(An´

e, 1999; Collin-Dufresne & Goldstein, 2002; Flamouris & Giamouridis, 2002; Navatte

& Villa, 2000; Tanaka, Yamada, & Watanabe, 2010, etc.).

However, these approaches have noticeable drawbacks. Since Gram–Charlier expan-

sions are polynomial approximations, they have the important drawback of yielding negative

values for a probability (Barton & Dennis, 1952; Jondeau & Rockinger, 2001; Niguez &

Perote, 2014, etc.). Actually, it is not guaranteed to be positive, and therefore may violate

the domain of validity of the probability distribution. This arises from the fact that the ex-

pansions are usually truncated after the fourth power, which may imply negative densities

over some interval of their domain of variation (Leon, Mencia, & Sentana, 2009), thereby

probabilities can be negative for such expansions. In addition, their sum does not necessarily

equal unity. This is an undesirable outcome because this situation might occur when the

financial markets are in distress, which means that these nearly Gaussian distributions may

fail when they are needed most.

Some papers try to overcome this shortcoming. First, Corrado (2007) expresses Eu-

ropean option prices in terms of the untruncated Gram–Charlier series expansion with an

embedded or a hidden no-arbitrage martingale restriction. In the same vein, Schlogl (2013)

presents an option pricing formula in terms of the untruncated Gram–Charlier series arguing

that here is no theoretical reason to truncate the series after the arbitrary fourth moment.

However, he did not display the economic rationale to have more than 4 moments as op-

posed to the economic literature. In both papers, some empirical tests are carried out, but

no empirical evidence has been provided. Second, the requirement of guaranteeing positive

1The formula given in Corrado and Su (1996) contains an error in the Hermite polynomial corrected by Brown and

Robinson (2002).

2See Barton and Dennis (1952) or Stuart and Ord (1987) for discussion on the distinction between Edgeworth and

Gram–Charlier expansions.