Jump variance risk: Evidence from option valuation and stock returns

• Author: Yen‐Cheng Chang,Kevin Tseng,Po‐Hsiang Peng,Hung‐Wen Cheng,Hsuan‐Ling Chang
• Date: 01 July 2019
• DOI: http://doi.org/10.1002/fut.22009
• Published date: 01 July 2019

Received: 1 June 2018

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Revised: 17 March 2019

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Accepted: 18 March 2019

DOI: 10.1002/fut.22009

RESEARCH ARTICLE

Jump variance risk: Evidence from option valuation and

stock returns

Hsuan‐Ling Chang

1

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Yen‐Cheng Chang

1,2

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Hung‐Wen Cheng

3

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Po‐Hsiang Peng

4

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Kevin Tseng

5

1

Department of Finance, National Taiwan

University, Taipei, Taiwan

2

Center for Research in Econometric

Theory and Applications, National

Taiwan University, Taipei, Taiwan

3

Department of Financial Engineering

and Actuarial Mathematics, Soochow

University, Taipei, Taiwan

4

Institute of Statistics, National Tsing Hua

University, Hsinchu, Taiwan

5

School of Business, University of Kansas,

Lawrence, Kansas

Correspondence

Yen‐Cheng Chang, National Taiwan

University, No. 1, Sec. 4, Roosevelt Rd.,

10617 Taipei, Taiwan.

Email: yenchengchang@ntu.edu.tw

Funding information

Ministry of Education of Taiwan (R.O.C.),

Grant/Award Number: 108L900202;

Ministry of Science and Technology of

Taiwan (R.O.C.), Grant/Award Numbers:

107‐3017‐F‐002‐004, 106‐2410‐H‐002‐

043‐MY2

Abstract

We study jump variance risk by jointly examining both stock and option

markets. We develop a GARCH option pricing model with jump variance

dynamics and a nonmonotonic pricing kernel featuring jump variance risk

premium. The model yields a closed‐form option pricing formula and improves

in fitting index options from 1996 to 2015. The model‐implied jump variance

risk premium has predictive power for future market returns. In the cross‐

section, heterogeneity in exposures to jump variance risk leads to a 6%

difference in risk‐adjusted returns annually.

KEYWORDS

jump variance risk, nonmonotonic pricing kernel, option valuation, return predictability

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INTRODUCTION

There is now overwhelming evidence that aggregate market return has time‐varying volatility and exhibits volatility

clustering.

1

These features lead to additional uncertainty for investors about price risk and thus command a risk

premium. Consistent with this view, many studies show that the difference between realized and risk‐neutral

volatilities, or variance risk premium, is negative and also time‐varying (see, e.g., Bakshi & Kapadia, 2003; Bali &

Hovakimian, 2009; Carr & Wu, 2009). In addition, there is also evidence that jumps in aggregate market returns are

time‐varying. For example, Bates (1991) finds that out‐of‐the‐money puts become unusually expensive compared to out‐

of‐the‐money calls before market crashes. His results show that expected jump probabilities are time‐varying and

exhibit clustering patterns similar to total return variation.

2

To the extent that variation in jumps is a source of

uncertainty for investors, it should manifest as a nontrivial component in variance risk premium.

In this paper, we use an option pricing framework to explore the role of jumps in variance risk premium and its asset

pricing implications. Options provide a desirable instrument to study how different kinds of risks that are ex ante perceived

J Futures Markets. 2019;39:890–915.wileyonlinelibrary.com/journal/fut890

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© 2019 Wiley Periodicals, Inc.

1

For a review, see Andersen, Bollerslev, Christoffersen, and Diebold (2013).

2

For more recent evidence on dynamic jump clustering, see Aït‐Shahalia, Cacho‐Diaz, and Laeven (2015), Maheu and McCurdy (2004).

by investors and priced in the economy. The tremendous growth in options markets is a testament to its flexibility in

tailoring to different investor risk preferences (Coval & Shumway, 2001). Specifically, option valuation requires one to specify

the return and variance dynamics of the underlying asset, including jumps. Option valuation also necessitates a pricing

kernelthatinformsusofthepricesofriskfactorsandprovides a connection between the physical measure and risk‐neutral

measure that is essential for pricing. Estimating the model will in turn allow us to derive variance risk premium due to

jumps and explore its pricing implications in both the time‐series and cross‐section of stock returns. If jump variance risk is a

legitimate concern for investors, then we should observe its effect in both option valuation and asset pricing tests.

Our option pricing approach follows the generalized Heston and Nandi (2000) GARCH option pricing framework.

The discrete‐time GARCH framework is an attractive setup because of its emphasis on volatility dynamics and relative

ease of estimation (Christoffersen, Jacobs, & Ornthanalai, 2012). In our model, the underlying asset’s return process

features both normal and jump innovations. These innovations have dynamic variances governed by Heston and

Nandi’s (2000) GARCH‐type processes with jumps (i.e., we allow for jumps in both normal and jump variances).

Importantly, our pricing kernel specification includes multiple state variables. The classical pricing kernels by

Rubinstein (1976) and Brennan (1979) are monotonically decreasing in a single‐state variable (market return),

reflecting decreasing marginal utility of wealth. However, Rosenberg and Engle (2002) find that the empirical pricing

kernel is typically nonmonotonic or U‐shaped in returns.

3

In light of this stylized fact, Christoffersen, Heston, and

Jacobs (2013) introduce a variance risk component into the option pricing kernel that enables the model to capture

various option pricing anomalies. In this case, the pricing kernel projected on market return yields a nonlinear U‐shape

as observed in data. We follow this methodology and incorporate separately risk premia associated with both jump and

continuous variance dynamics, in addition to market price risk.

We derive a closed‐form option pricing solution and estimate model parameters using index options. Specifically, we use

joint maximum likelihood estimator (MLE) with both daily S&P500 index returns and index options from 1996 to 2015.

Combining both returns and options in model estimation avoids overfitting concerns and ensures that return data are properly

accounted for in parameter estimation. This idea is first pointed out by Bates (1996) and recent studies emphasize the

importance of joint estimation.

4

Our model has significantly higher total log‐likelihood and lower mean‐squared error (MSE)

compared to other benchmark models with monotonic pricing kernel or without jump variance dynamics. Our full model

yields an MSE 10% lower than the standard GARCH with only return jumps and a pricing kernel with a single‐state variable. In

particular, the contribution of incorporating jump variance risk in the pricing kernel is larger than that of continuous variance

risk. This pattern in likelihoods gives a preview of the prominent role of jump variance risk premium in our asset pricing tests.

From estimation, we obtain the model‐implied jump variance risk premium and continuous jump variance risk

premium. The two components are not very correlated (correlation coefficient 0.28), with the jump variance risk

premium process showing larger absolute magnitude and variation. In addition, the estimated jumps in market returns

are consistent with stylized evidence that returns are asymmetrically distributed. On the daily frequency, the smallest

(largest) jump in market return is –8% (7.2%) with the majority of them (55%) being negative.

We explore further asset pricing implications of jump variance risk in expected stock returns to underscore the

importance of incorporating this state variable in the pricing kernel. These “out‐of‐sample”tests also help mitigate model

overfitting concerns in addition to providing further support for the validity of jump variance risk. First, individual stocks

with returns that covary positively with jump variance risk premium tend to have high returns when jump variance is

low.

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Due to leverage effect, high market return tends to coincide with low variance (French, Schwert, & Stambaugh,

1987; Glosten, Jagannathan, & Runkle, 1993). Therefore, this type of stock has higher systematic risk and should be

compensated with higher expected returns in the cross‐section. We test this hypothesis by first estimating a “jump

variance beta”for each stock in our sample and subsequently sort firmsinto jump variance beta decile portfolios. We then

construct a long–short portfolio withlong positions in the high jump variance beta portfolio and short positions in the low

jump variance beta portfolio. Indeed, we find that this long–shortstrategyyieldsupto6%annualizedrisk‐adjusted return.

Second, we explore time‐series market return predictability of jump variance risk premium. There are now a number

of papers that find that variance risk premium helps predict future market returns (e.g, Bollerslev, Tauchen, & Zhou,

2009; Drechsler & Yaron, 2011; Li & Zinna, 2018). These studies argue that variations in variance risk premium capture

time‐varying market attitudes toward uncertainty. In times of high uncertainty, investors’risk aversion will lead to high

3

For a review of the “pricing kernel puzzle,”Jackwerth (2004) and Cuesdeanu and Jackwerth (2018) survey this literature and find largely consistent findings using alternative methodologies and

datasets.

4

For example, Christoffersen et al. (2012), Christoffersen et al. (2013), and Ornthanalai (2014).

5

By construction, jump variance risk premium is low when jump variance under the physical measure is high.

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