Volatility and jump risk in option returns

• Author: Biao Guo,Hai Lin
• Date: 01 November 2020
• DOI: http://doi.org/10.1002/fut.22107
• Published date: 01 November 2020

J Futures Markets. 2020;40:1767–1792. wileyonlinelibrary.com/journal/fut © 2020 Wiley Periodicals, Inc.

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1767

Received: 4 February 2020

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Accepted: 6 February 2020

DOI: 10.1002/fut.22107

RESEARCH ARTICLE

Volatility and jump risk in option returns

Biao Guo

1

|Hai Lin

2

1

School of Finance, Renmin University of

China, Beijing, China

2

School of Economics and Finance,

Victoria University of Wellington,

Wellington, New Zealand

Correspondence

Hai Lin, School of Economics and

Finance, Victoria University of

Wellington, Wellington 6140,

New Zealand.

Email: hai.lin@vuw.ac.nz

Funding information

National Natural Science Foundation of

China, Grant/Award Number: 71871190

Abstract

We examine the importance of volatility and jump risk in the time‐series

prediction of S&P 500 index option returns. The empirical analysis pro-

vides a different result between call and put option returns. Both volatility

and jump risk are important predictors of put option returns. In contrast,

only volatility risk is consistently significant in the prediction of call op-

tion returns over the sample period. The empirical results support the

theory that there is option risk premium associated with volatility and

jump risk, and reflect the asymmetry property of S&P 500 index

distribution.

KEYWORDS

aggregate volatility, jump, option return, prediction

JEL CLASSIFICATION

G12; G14

1|INTRODUCTION

Theimportanceoftime‐varying volatility and jump risk in option pricing has been extensively documented in

the literature. For example, D. S. Bates (2000), Duffie, Pan, and Singleton (2000), Pan (2002), Eraker (2004),

and Santa‐Clara and Yan (2010)proposeno‐arbitrage option pricing mo dels to combine them into the standard

Black–Scholes–Merton (BSM) option pricing model. Naik and Lee (1990), Broadie, Detemple, Ghysels, and

Torrés (2000),andBranger,Schlag,andSchneider(2008) investigate the general equilibrium of options market

with stochastic volatility or (and) jump. Recently, Christoffersen, Feunou, and Jeon (2015), Yang and

Kanniainen (2017), and H.‐L. Chang, Chang, Cheng, Peng, and Tseng (2019) introduce models allowing the

underlying asset with volatility and jumps for option pricing. These models provide a rich option pricing

framework in which both stochastic volatility and jump risk are considered. However, whether these two risks

have predictive powers on option returns in the time series remains an unexplored question.

In this paper, we provide a comprehensive empirical study of the importance of volatility and jump risk in

the time‐series predictability of option returns. In particular, we investigate whether there is any difference

between the results of the call and put option returns. Theoretically speaking, call option returns reflect the

expectation of right‐hand side distribution of underlying asset value, while put option returns reflect the ex-

pectation of left‐hand side distribution. Any difference between call and put option indicates the asymmetry

property of the underlying asset value distribution, which is of great importance for asset pricing and risk

management.

Following numerous literatures on the predictability of asset returns,

1

we run several tests. We run in‐sample

regression to test the significance of aggregate volatility and jump risk predictors on predicting the call and put

option returns. We follow the in‐sample test by out‐of‐sample evaluation that investigates the statistical sig-

nificance and economic value of out‐of‐sample forecast. Finally, we run several robustness tests to make sure our

results are stable.

How to measure the aggregate volatility and jump risk is a critical question for our empirical analysis. We use variance risk

premium (VRP) as a measure of aggregate volatility risk following Bollerslev, Tauchen, and Zhou (2009), and follow Cremers,

Halling, and Weinbaum (2015) to construct two aggregate jump risk variables that they use in their robustness check. The two

jump risk predictors are SMIRK,

2

the slope of the implied volatility proposed by Yan (2011), and TAIL

3

proposed by Du and

Kapadia (2012). Since VRP is also affected by jump risk (Bollerslev & Todorov, 2011), we run the regression of VRP on SMIRK

and TAIL and use the residuals (VRP‐SMIRK and VRP‐TAIL) as the alternative measures of aggregate volatility risk. We use

these five variables in the empirical analysis and test their performance on the prediction of option returns. We also include the

variables that are used in the stock return forecast literature and compare our results with the forecast using Coval and Shumay

(CS, 2001). Coval and Shumway (2001) show that expected option return could be determined by the capital asset pricing

model (CAPM) model under Geometric Brownian Motion (GBM) assumption. We are interested in whether there is additional

improvement by aggregate volatility and jump risk predictors once the other predictors are controlled.

We employ both the standard predictive regression models and combination methods. The predictive regression model is a

good way of testing the predictability when the number of predictors is limited. However, it loses power when the number of

predictors is large. To address this concern, we follow Rapach et al. (2010) to use the combination forecast, and Lin et al. (2018)

to use the regressed combination forecast. They show that these approaches are better ways of extracting useful information

from a large set of predictors and provide a better forecast result than the predictive regression model that puts all variables into

the multiple regression. We use two popular combination forecast, the mean combination (MC) forecast and J. M. Bates and

Granger's (1969)weighted‐mean combination (WC) forecast. These two forecasts are then regressed to obtain the regressed

mean combination (RMC) forecast and the regressed weighted‐mean combination (RWC) forecast, respectively.

Our empirical result shows that aggregate volatility risk is important for the prediction of both call and put option

returns. The results are both statistically and economically significant. For example, using VRP as the predictor could

generate an in‐sample R

2

of 10.81% for the 30‐day call option returns, and 14.16% for 30‐day put option returns. The out‐

of‐sample R

2

’s and utility gains are 9.54% and 5.38%, respectively, for 30‐day call option returns, and 6.31% and 7.15%,

respectively, for 30‐day put option returns. Using the residuals of VRP provide similar results. The finding is also robust

across different option series and forecast horizons. The results by grouped mean are also significant. These findings

show the importance of volatility risk for both calls and puts.

In contrast, the results of jump risk show a different pattern between calls and puts. The jump risk is not significant

in the prediction of call option returns. On the other hand, it is highly significant for the put option return. The different

performances of jump risk predictors on the forecast of call and put option returns reflect the market's expectation

about the asymmetry distribution of S&P 500 index. The forecast using CS (2001) is not significant for either calls or

puts, which provides the evidence of rejection of the GBM assumption of S&P 500 index. The results show there is

option risk premium associated with volatility and jump risk. Call and put options share similarities on the aggregate

volatility risk component, while they are different on the jump risk component.

The combination forecast shows that aggregate volatility and risk predictors significantly improve the forecast performance

once they are used in the combination forecast. For example, if only the predictors from stock return forecast literature are

used, the out‐of‐sample R

2

’softhe30‐day call option and put option returns are only 3.74% and 2.84%, respectively, when MC

forecast is used. However, they increase to 7.24% and 5.70% when the aggregate volatility and jump risk predictors are also

utilized. Regressed combination forecast improves the forecast performance furthermore.

We also run several robustness tests. We evaluate the performance of other aggregate volatility risk predictors, including

Volatility Index (VIX) that is a measure of the implied volatility of S&P 500 index options, Bernische Kantonal‐Musikverband

1

For example, Fama and Schwert (1977), Fama and French (1988), Campbell and Shiller (1988), Kothari and Shanken (1997), Pontiff and Schall (1998), Ang and Bekaert (2007), Rapach, Strauss, and

Zhou (2010), Dangl and Halling (2012), Rapach, Strauss, and Zhou (2013), Pettenuzzo, Timmermann, and Valkanov (2014), and Huang, Jiang, Tu, and Zhou (2015) for predicting stock returns; Keim

and Stambaugh (1986), Fama and French (1989), Greenwood and Hanson (2013), Lin, Wang, and Wu (2014), and Lin, Wu, and Zhou (2018) for predicting corporate bonds; and Fama and Bliss (1987),

Campbell and Shiller (1991), Cochrane and Piazzesi (2005), Ludvigson and Ng (2009), Thornton and Valente (2012), Goh, Jiang, Tu, and Zhou (2012), Sarno, Schneider, and Wagner (2016), Gargano,

Pettenuzzo, and Timmermann (2017), and Fulop, Li, and Wan (2017) for predicting Treasury bonds.

2

SMIRK is defined as the difference in the implied volatilities of put and call options.

3

TAIL is defined as the difference between the realized volatility and the annualized integrated volatility.

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GUO AND LIN