Skewness Versus Kurtosis: Implications for Pricing and Hedging Options

• Published date: 01 December 2017
• Author: Yuen Jung Park,Geul Lee,Sol Kim
• DOI: http://doi.org/10.1111/ajfs.12200
• Date: 01 December 2017

Skewness Versus Kurtosis: Implications for

Pricing and Hedging Options*

Sol Kim

College of Business, Hankuk University of Foreign Studies, Republic of Korea

Geul Lee

NH Finance Research Institute, Republic of Korea

Yuen Jung Park**

Department of Finance, Hallym University, Republic of Korea

Received 15 January 2016; Accepted 16 August 2017

Abstract

We examine the relative influence of the skewness and kurtosis of option-implied risk-neutral

density on pricing and hedging performance in the S&P 500 Index options market. We find

that skewness exerts a greater impact on pricing and hedging errors than kurtosis. The model

that considers skewness shows better performance for pricing and hedging options than the

model that only considers kurtosis. Our results are consistent, even when the underlying

return is extremely high or low, as well as for options on individual stocks. Overall, risk-neu-

tral skewness is more important than risk-neutral kurtosis for pricing and hedging options.

Keywords Volatility smiles; Options pricing; Risk-neutral distribution; Skewness; Kurtosis

JEL Classification: G13

1. Introduction

The Black and Scholes (1973, BS) option pricing model simplifies option pricing by

imposing a rather strict normality assumption on the underlying return density.

Contradictorily, however, implied volatility curves (IVCs) in most options markets

are nonlinear, that is, implied volatility levels vary across strike prices, which is

*We are very grateful to Kwangwoo Park (editor) and two anonymous referees for valuable

comments. The authors are responsible for any errors. This work was supported by the Min-

istry of Education of the Republic of Korea and the National Research Foundation of Korea

(NRF-2015S1A5A2A01009743) and by Hankuk University of Foreign Studies Research Fund

of 2017.

**Corresponding author: Yuen Jung Park, Department of Finance, Hallym University, 1 Hal-

lymdaehak-gil, Chuncheon, Gangwondo 24252, Korea. Tel: +82-33-248-1855, Fax: +82-33-

248-1804, email: yjpark@hallym.ac.kr.

Asia-Pacific Journal of Financial Studies (2017) 46, 903–933 doi:10.1111/ajfs.12200

©2017 Korean Securities Association 903

commonly known as a volatility smile or skew. The phenomenon can be an issue

when the BS model is employed to price or hedge options across different strike

prices, because the implied volatility that varies across moneyness clashes with the

constant volatility assumption, thereby incurring hedging and pricing errors (Fig-

ure 1).

A common interpretation of a nonlinear IVC is that it implies a skewed and

leptokurtic risk-neutral density (RND). The nonnormality can be attributed to a

few factors, including nonnormality in physical density due to return asymmetry

and jumps in return, as well as the structure of the pricing kernel, which reflects an

asymmetric risk preference. Although identification and characterization of the

specific mechanism that determines the shape of the RND are of academic interest,

what also matters in financial markets is whether hedging and pricing errors can be

addressed by utilizing the concept of nonnormal RND and, if so, by how much.

The popularity of simple adjusted option pricing methods, for example, an ad hoc

BS model, that try to incorporate a volatility smile and skew into option prici ng

and hedging suggests that the effective reconciliation of option pricing models with

nonlinear IVCs is closely related to investors’ interest in options markets.

This study sheds light on how useful it is to consider the shape of the implied

RND to address hedging and pricing errors that cannot be avoided with the BS

model. Specifically, we separately consider the two major properties of the nonnor-

mal density, that is, skewness and kurtosis, and compare the significance of the two

implied moments in pricing and hedging. Roughly speaking, the skewness of the

implied RND reflects the “slope” of IVCs, that is, the degree to which out-of-the-

money (OTM) calls and in-the-money (ITM) puts are overpriced compared to

OTM puts and ITM calls, while kurtosis reveals the “curvature” of IVCs, that is,

the degree to which OTM and ITM option prices are more inflated than at-the-

money (ATM) option prices in terms of implied volatility. Hence, by evaluating the

relevance of the two moments of the implied RND with hedging and pricing errors,

we assess which property of the nonlinear IVC is more important when adjusting

the option price to reduce the errors. To the best of our knowledge, this is the first

study to examine the relative importance of the skewness and kurtosis of the

implied RND for pricing and hedging options.

We compare the significance of implied moments in hedging and pricing Stan-

dard & Poor’s (S&P 500) Index options. The implied moments are estimated using

the parametric method of Corrado and Su (1996, CS) and the nonparametric

method of Bakshi et al. (2003, BKM). After the implied moment estimation, we

evaluate the relative importance of the implied moments on hedging and pricing in

two ways. First, we regress the hedging and pricing errors on skewness and kurtosis

to determine how closely the moments are related to the magnitude of the errors.

Second, we compare the hedging and pricing performance with and without consid-

ering higher moments to investigate the extent to which hedging and pricing per-

formance can be enhanced by considering implied skewness and kurtosis.

S. Kim et al.

904 ©2017 Korean Securities Association