Aggregate Jump and Volatility Risk in the Cross‐Section of Stock Returns

• Date: 01 April 2015
• DOI: http://doi.org/10.1111/jofi.12220
• Published date: 01 April 2015
• Author: MARTIJN CREMERS,MICHAEL HALLING,DAVID WEINBAUM

THE JOURNAL OF FINANCE •VOL. LXX, NO. 2 •APRIL 2015

Aggregate Jump and Volatility Risk

in the Cross-Section of Stock Returns

MARTIJN CREMERS, MICHAEL HALLING, and DAVID WEINBAUM∗

ABSTRACT

We examine the pricing of both aggregate jump and volatility risk in the cross-section

of stock returns by constructing investable option trading strategies that load on one

factor but are orthogonal to the other. Both aggregate jump and volatility risk help

explain variation in expected returns. Consistent with theory,stocks with high sensi-

tivities to jump and volatility risk have low expected returns. Both can be measured

separately and are important economically, with a two-standard-deviation increase

in jump (volatility) factor loadings associated with a 3.5% to 5.1% (2.7% to 2.9%) drop

in expected annual stock returns.

AGGREGATE STOCK MARKET volatility varies over time. This has important im-

plications for asset prices in the cross-section and is the subject of much re-

cent research (e.g., Ang et al. (2006)).1There is also evidence that aggregate

jump risk is time-varying. For example, Bates (1991) shows that out-of-the-

money puts became unusually expensive during the year preceding the crash of

October 1987. His analysis reveals significant time variation in the conditional

expectations of jumps in aggregate stock market returns. Santa-Clara and Yan

(2010) use option prices to calibrate a model in which both the volatility of

the diffusion shocks and the intensity of the jumps are allowed to change over

time. They likewise find substantial time variation in the jump intensity pro-

cess, with aggregate implied jump probabilities ranging from 0% to over 99%.

∗Cremers is at Mendoza College of Business, University of Notre Dame. Halling is at

Stockholm School of Economics, University of Utah. Weinbaum is at Whitman School of Manage-

ment, Syracuse University. The authors thank Gurdip Bakshi, Turan Bali, Hank Bessembinder,

Oleg Bondarenko, Nicole Branger, Fousseni Chabi-Yo (WFA discussant), Joseph Chen, Magnus

Dahlquist, James Doran, Wayne Ferson, Fangjian Fu, Kris Jacobs, Chris Jones, Nikunj Kapadia,

Christian Schlag, Grigory Vilkov (EFA discussant), Shu Yan, Yildiray Yildirim, Hao Zhou, and

seminar participants at Boston University, ESMT Berlin, Imperial College London, Stockholm

School of Economics, the 2013 IFSID and Bank of Canada conference on tail risk and derivatives,

the 21st Annual Conference on Financial Economics and Accounting (CFEA) at the University of

Maryland, the 12th Symposium on Finance, Banking and Insurance, the 2012 WFA Meetings, and

the 2012 EFA Meetings for helpful comments and discussions. The authors are grateful to two

anonymous referees, an anonymous Associate Editor, and Campbell Harvey, the Editor,for helpful

suggestions that greatly improved the paper. The authors are responsible for any errors.

1Considerable research examines the time-series relation between aggregate stock market

volatility and expected market returns. See, for example, Bali (2008), Campbell and Hentschel

(1992), and Glosten, Jagannathan, and Runkle (1993).

DOI: 10.1111/jofi.12220

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578 The Journal of Finance R

While they examine the time-series relation between systematic jump risk and

expected stock market returns, the question of how aggregate jump risk affects

the cross-section of expected returns has received less attention.

The main objective of this paper is to provide a comprehensive empirical

investigation of the pricing of time-varying jump and volatility risk in the cross-

section of expected stock returns. In particular, we consider whether aggregate

jump and volatility constitute separately priced risk factors. Several papers

argue that aggregate volatility may be a priced factor in part because assets

with high sensitivities to volatility risk hedge against the risk of significant

market declines (e.g., Bakshi and Kapadia (2003), Ang et al. (2006)). This

argument suggests that jump and volatility risk may be similar. In addition,

as markets tend to be more volatile in times of extreme returns, separating

jump and volatility risk is an empirical challenge. In this paper, we show that

they are in fact different: they can be measured separately using option returns

and they are both important economically. Economic theory provides several

reasons why aggregate jump and volatility risk should constitute priced risk

factors. The importance of these risks is now a fundamental premise of the

option pricing literature (see, e.g., the reduced-form models in Bates (2000),

Pan (2002), and Santa-Clara and Yan (2010)). General equilibrium models can

be used to shed light on the economic mechanisms that drive jump and volatility

risk premia. Naik and Lee (1990) introduce jumps into general equilibrium

models, Pham and Touzi (1996) introduce stochastic volatility, and Branger,

Schlag, and Schneider (2007) examine the equilibrium with both jumps and

stochastic volatility.

While these models use standard preferences, Bates (2008) considers in-

vestors who are both risk and crash averse. Jump and diffusive risks are both

priced even in the absence of crash aversion, but introducing crash aversion

allows for greater divergence between the two risk premia. An important fea-

ture of the model in Bates (2008) is a representative investor who treats jump

and diffusive risks differently,which formalizes the intuition that investors can

treat extreme events differently than they treat more common and frequent

ones.2

These models provide a rich framework in which both volatility and jump

risk are separately priced. Investors seeking to hedge against changes in invest-

ment opportunities will find assets that covary positively with market volatility

attractive, and thus require lower expected returns. Separately, investors who

seek to insure themselves against tail events such as the recent financial cri-

sis, that is, more extreme events that go beyond business cycle fluctuations in

investment opportunities, will find stocks with a positive loading on jump risk

attractive and thus require lower expected returns.

To examine the cross-sectional pricing of aggregate jump and volatility risk,

we construct investable option trading strategies that load on one factor but

2Liu, Pan, and Wang (2005) examine the equilibrium when stock market jumps can occur and

investors are both risk averse and averse to model uncertainty with respect to jumps; they obtain

similar pricing implications for jump and diffusive risk.

Aggregate Jump and Volatility Risk 579

are orthogonal to the other. Because traded S&P 500 futures options are highly

liquid, their prices encode market participants’ ex ante assessment of ex-

pected aggregate jump and volatility risk. These prices should therefore contain

forward-looking information that we expect to be highly relevant for our anal-

ysis. The ex ante jump risk perceived by investors may be quite different from

ex post realized jumps in prices because even high-probability jumps may fail

to materialize in sample (Santa-Clara and Yan (2010)). Therefore, employing

options alleviates the “Peso problem” in measuring jump risk from observed

stock returns.

A straddle involves the simultaneous purchase of a call and a put option.

Coval and Shumway (2001) motivate the use of delta-neutral straddles for

studying the effect of stochastic volatility by their high sensitivity to volatility—

they have large vegas—and their insensitivity to market returns. However, this

only holds for small diffusive shocks. In a world with jumps, straddle returns

are subject to hedging error due to the positive gamma of the options: if the

underlying asset experiences a large move in any direction, the straddle will not

remain delta neutral and will earn a positive return. This implies that straddle

returns are affected by both volatility and jump risk. More importantly, this

observation suggests alternative trading strategies that allow us to focus on

each risk separately.

A strategy constructed to be market (i.e., delta) neutral and gamma neutral

but vega positive is essentially insulated from jump risk and thus only subject

to volatility risk. Similarly, a strategy that is market neutral and vega neutral

but gamma positive is ideal to study the effects of jump risk. We show that

both strategies can be constructed by setting up long/short strategies involving

market-neutral straddles. Our resulting jump risk factor-mimicking portfolio

(JUMP) is a market-neutral, vega-neutral, and gamma-positive strategy in-

volving two at-the-money straddles with different maturities. Similarly, we

construct the volatility risk factor-mimicking portfolio (VOL) by combining two

at-the-money straddles with different maturities into a position that is market

neutral, gamma neutral, and vega positive. The JUMP and VOL strategies are

directly tradable strategies that are constructed to load on one factor while

being orthogonal to the other. Empirically, we find that the returns on the two

strategies are essentially uncorrelated.

Our approach to finding a premium for bearing volatility and jump risk

closely follows Ang, Chen, and Xing (2006). Specifically, we estimate jump and

volatility risk factor loadings at the individual stock level using daily returns,

we sort stocks on the realized factor loadings estimated over a given time pe-

riod, and we investigate whether stocks with higher volatility and jump betas

have lower average returns contemporaneously (i.e., over the same period).

This approach considers both requirements that must be met for any factor to

be priced in the cross-section of stock returns. First, there must be a contempo-

raneous pattern between factor loadings and average returns. Therefore, our

analysis focuses on uncovering contemporaneous relations between volatility

and jump risk loadings and average stock returns. Second, the pattern should

be robust to controls for various stock characteristics and other factors known