A smiling bear in the equity options market and the cross‐section of stock returns

• Author: Haehean Park,Hyeongsop Shim,Baeho Kim
• DOI: http://doi.org/10.1002/fut.22000
• Published date: 01 November 2019
• Date: 01 November 2019

Received: 20 January 2019

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Accepted: 20 January 2019

DOI: 10.1002/fut.22000

RESEARCH ARTICLE

A smiling bear in the equity options market and the

cross‐section of stock returns

Haehean Park

1

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Baeho Kim

2

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Hyeongsop Shim

3

1

Southwestern University of Finance and

Economics (SWUFE), Chengdu, Sichuan,

China

2

Korea University Business School, Seoul,

Republic of Korea

3

College of Business and Economics,

Gachon University, Seongnam‐si,

Republic of Korea

Correspondence

Baeho Kim, Korea University Business

School, Anam‐dong, Sungbuk‐gu,

Seoul 02841, Republic of Korea.

Email: baehokim@korea.ac.kr

Funding information

National Research Foundation of Korea,

Grant/Award Number:

2015S1A5A8014515

Abstract

We propose a measure for the convexity of an option‐implied volatility curve,

IV convexity, as a forward‐looking measure of risk‐neutral tail‐risk contribution

to the perceived variance of underlying equity returns. Using equity options

data for individual US‐listed stocks during 2000–2013, we find that the average

realized return differential between the lowest and highest IV convexity quintile

portfolios exceeds 1% per month, which is both economically and statistically

significant on a risk‐adjusted basis. Our empirical findings indicate the

contribution of informed options trading to price discovery in terms of the

realization of tail‐risk aversion in the stock market.

KEYWORDS

convexity, equity options, implied volatility, predictability, stock returns

JEL CLASSIFICATION

G12, G13, G14

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INTRODUCTION

The nonnormality of stock returns has been well documented in literature as a natural extension of the traditional

mean‐variance approach to portfolio optimization.

1

In general, a rational investor’s utility is a function of higher moments,

as the investor tends to have an aversion to negative skewness and high excess kurtosis of her asset return (see Dittmar,

2002; Guidolin & Timmermann, 2008; Kimball, 1990; Scott & Horvath, 1980).

2

While considerable research has examined

whether the higher moment preference is priced in the realized returns (e.g., Chung, Johnson, & Schill, 2006; Dittmar,

2002; Harvey & Siddique, 2000; Kraus & Litzenberger, 1976; D. R. Smith, 2007), the higher moment pricing effect is also

embedded in equity option prices in a forward‐looking manner. The shape of an option‐implied volatility curve reveals the

ex ante higher moment implications beyond the standard mean‐variance framework, as the curve expresses the degree of

abnormality in the option‐implied distribution of the underlying stock return by inverting the standard Black and Scholes

(1973) option‐pricing model. Although empirical research abounds in examining the risk‐neutral skewness of stock returns

implied by equity option prices, the option‐implied kurtosis has received less attention.

3

Our study attempts to fill this gap.

This paper explores asset‐pricing implication of the ex ante higher order moments extracted from the shape of

option‐implied volatility curve. We find from numerical analyses that slope and convexity of the option‐implied volatility

J Futures Markets. 2019;39:1360–1382.wileyonlinelibrary.com/journal/fut1360

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© 2019 Wiley Periodicals, Inc.

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The mean‐variance approach is consistent with the maximization of expected utility if either (a)the investors’utility functions are quadratic, or (b) the assets’return distributions are jointly normal.

Arrow (1971) claims that a quadratic utility function may be unrealistic for practical purposes, as it exhibits increasing absolute risk aversion, consistent with investors who reduce the dollar amount

invested in risky assets as their initial wealth increases.

2

Decreasing absolute prudence implies the kurtosis aversion according to Kimball (1990).

3

There are some notable exceptions from this trend (refer to Bali, Hu, & Murray, 2019; Chang, Christoffersen, & Jacobs, 2013).

curve convey distinct information about risk‐neutral skewness and excess kurtosis of the underlying return distributions,

respectively. Based on our findings, we propose a method of decomposing the shape of option‐implied volatility curves into

the slope and convexity components (IV slope and IV convexity hereafter) and examine whether the risk‐neutral kurtosis,

proxied by IV convexity, predicts the cross‐section of underlying stock returns, even after the skewness effect is controlled.

4

Using equity options data for both individual US‐listed stocks and the Standard & Poor’s 500 (S&P500) Index

during 2000–2013, we study the ability of IV convexity to predict the cross‐section of future equity returns across quintile

portfolios ranked by the convexity of the option‐implied volatility curve. We find a significantly negative relationship

between IV convexity and subsequent stock returns. The average return differential between the lowest and highest

IV convexity quintile portfolios exceeds 1% per month, which is both economically and statistically significant on a

risk‐adjusted basis. The results are robust across various definitions of the IV convexity measure. Both time‐series and

cross‐sectional tests show that existing risk factors and firm characteristics do not subsume the additional return on the

zero‐cost portfolio formed on IV convexity. The predictive power of IV convexity is significant for both the systematic and

idiosyncratic components, and the results are robust even after controlling for the slope of the option‐implied volatility

curve and other known predictors based on stock characteristics.

Where does the predictive power of IV convexity come from? Our finding is consistent with earlier studies demonstrating

the informational leading role of the options market to the stock market. Extensive research demonstrates that equity option

markets provide informed traders with opportunities to capitalize on their informational advantage from several benefits of

options trading relative to stock trading. The benefits include (a) reduced trading costs (Cox & Rubinstein, 1985), (b) the lack

of restrictions on short selling (Diamond & Verrecchia, 1987), and (c) greater leverage effect and built‐in downside protection

(Manaster & Rendleman, 1982). In this context, Chakravarty, Gulen, and Mayhew (2004) document that the option market’s

contribution to price discovery is approximately 17% on average.

Why do we observe the negative relationship between IV convexity and the return of underlying stock in the

subsequent month? We suggest that the convexity of implied volatility curves reveals the implication of risk‐neutral

tail‐risk contribution to the perceived volatility of underlying return distributions. As the estimated volatility shapes the

probability distribution of investors’holding period return, expected hedging errors owing to uncertain volatility

estimation create nonnegligible risk exposure. This leads to the aversion against excess kurtosis of the equity return

distributions under the risk‐neutral probability measure.

Given that excessive tail‐risk is certainly unfavorable to rational equity investors, the fat‐tailed risk‐neutral

distribution incorporates ex ante estimates of market risk premia regarding future rare events. In turn, a heavier tail in

the risk‐neutral distribution is associated with a higher required return (or, equivalently, a larger discount rate), and the

informed option trading related to IV convexity predicts a negative short‐term return on the underlying stock returns as

a result of price discovery driven by an asymmetric information transmission from options traders to stock investors.

Our empirical findings indicate the contribution of informed options trading to price discovery in terms of the realization of

tail‐risk aversion in the stock market. Consistent with the above arguments, the predictive power of IV convexity becomes more

pronounced for the firms with stronger information asymmetry and more severe short‐sale constraints, especially during

economic contraction periods. The cross‐sectional predictability of IV convexity disappears as the forecasting horizon increases.

A strand of literature examines the relationship between option‐implied higher order moments and the cross‐section of

stock returns. Conrad, Dittmar, and Ghysels (2013) find a negative relationship between quarterly averages of daily

risk‐neutralskewnessestimatesandrealizedquarterly stock returns under the no‐arbitrage assumption between the options

market and the underlying stock market. Contrarily, Stilger, Kostakis, and Poon (2016) report a positive relationship between

the risk‐neutral skewness of individual stock return’s distribution and future realized monthly stock returns using the

risk‐neutral skewness estimates extracted on the last trading day of each month to examine the temporary mispricing effect

from the intermarket information asymmetry. Unlike our study, they mainly focus on the risk‐neutral skewness rather than

kurtosis.Amaya,Christoffersen,Jacobs,andVasquez(2015) find a very strong negative relationship between realized

skewness and next week’s stock returns using high‐frequency intraday stock market data. They also find a positive

relationship between realized kurtosis and next week’s stock returns, but the evidence is not always robust and statistically

significant. In contrast, Bollerslev, Li, and Zhao (2018) find that the realized kurtosis negatively predicts the future stock

returns after controlling for individual stocks’relative good minus bad volatility.

Our study extends the recent literature that shows an increased interest in intermarket information

transmission, leading to a proliferation of studies into the potential lead–lag relationship between options and

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Hereafter, we use kurtosis and excess kurtosis interchangeably, despite their conceptual differences.

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