Short‐Term Market Risks Implied by Weekly Options

• Author: VIKTOR TODOROV,TORBEN G. ANDERSEN,NICOLA FUSARI
• Date: 01 June 2017
• DOI: http://doi.org/10.1111/jofi.12486
• Published date: 01 June 2017

THE JOURNAL OF FINANCE •VOL. LXXII, NO. 3 •JUNE 2017

Short-Term Market Risks Implied by Weekly

Options

TORBEN G. ANDERSEN, NICOLA FUSARI, and VIKTOR TODOROV∗

ABSTRACT

We study short-maturity (“weekly”) S&P 500 index options, which provide a direct

way to analyze volatility and jump risks. Unlike longer-dated options, they are largely

insensitive to the risk of intertemporal shifts in the economic environment. Adopting

a novel seminonparametric approach, we uncover variation in the negative jump

tail risk, which is not spanned by market volatility and helps predict future equity

returns. As such, our approach allows for easy identification of periods of heightened

concerns about negative tail events that are not always “signaled” by the level of

market volatility and elude standard asset pricing models.

RECENT YEARS HAVE WITNESSED A RAPID INCREASE in the trading of short-dated

options. For instance, S&P 500 option contracts with one week or less to matu-

rity have seen their share of trading at the Chicago Board of Options Exchange

(CBOE) rise steadily from about 10% in early 2010 to almost 30% in mid-2015.

Furthermore, the volume in shorter-dated options is disproportionately skewed

toward out-of-the-money (OTM) options relative to the pattern for longer-dated

options. Similar developments are observed in many other index option mar-

kets and for options on individual names. This process has been facilitated

by the introduction of a new option category, featuring sequential issuance of

contracts expiring one week apart, the so-called weekly options, or “weeklies.”

∗Torben G. Andersen and ViktorTodorov are at the Kellogg School of Management, Northwest-

ern University. Nicola Fusari is at the Carey Business School, The Johns Hopkins University. We

are thankful to Ken Singleton (the Editor) and anonymous referees, as well as seminar partici-

pants at the Booth School of Business, University of Chicago, the Kellogg School of Management,

Northwestern University, Stanford University, Johns Hopkins University, the Second Interna-

tional Workshop in Financial Econometrics, Salvador, Brazil, October 2015, the European Econo-

metric Society Meeting in Milan, December 2015, the Empirical Finance Workshop at ESSEC

Business School, Office of Financial Research, Washington, DC, Boston University and theMarket

Microstructure and High Frequency Data Conference at the Stevanovich Center, University of

Chicago, May 2016. We also thank Ian Dew-Becker,Anna Cieslak, Ravi Jagannathan, Roger Lee,

Jeff Russell, Fabio Trojani, and Dacheng Xiu for helpful comments. YupengWang provided skillful

research assistance. Finally,we are very grateful to John Angelos and Mike Warsh from the CBOE

for detailed explanations regarding the institutional organization of trading in weekly options, and

to Luca Benzoni and Ivana Ruffini from the Federal Reserve Bank of Chicago for facilitating this

information exchange. Andersen gratefully acknowledges support from CREATES, Center for Re-

search in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research

Foundation. The work is partially supported by NSF Grant SES-1530748. We also acknowledge

financial support from the CME Group and access to data from the CME DataMine system.

DOI: 10.1111/jofi.12486

1335

1336 The Journal of Finance R

The emergence of active trading in the weeklies represents a step toward

market completion. This topic has a long history, with Ross (1976)emphasiz-

ing the enhanced spanning and the potential for efficiency gains from options

trading, and Breeden and Litzenberger (1978) stressing the ability to replicate

a wide range of payoffs through a static option portfolio.1

Nevertheless, we face the question of why these particular contracts have

been so successful. The primary distinguishing feature of these securities, rela-

tive to longer-dated options, is the intimate link between the pricing of options

close to expiry and the state of the underlying asset return process.2When

tenor is short, the expected volatility and jump intensity do not vary much over

the remaining life of the option. This implies, in particular, that the relative

prices of deep OTM options are largely independent of the level of diffusive

volatility, and instead reflect the characteristics of the risk-neutral jump pro-

cess. Likewise, the pricing of short-dated at-the-money (ATM) options depends

primarily on current spot volatility. Hence, the weeklies improve market par-

ticipants’ ability to acquire or lay off exposure to diffusive and jump price risks.

Such arguments fail in the case of longer-dated instruments for which the ex-

pected variation in the future volatility and jump intensity cannot be ignored

in valuation. In fact, realistic models for their joint dynamics invariably in-

volve complex interactions, rendering semi-closed-form option pricing feasible

only under strong parametric assumptions, for example, the specifications re-

side within the affine jump-diffusion model class of Duffie, Pan, and Singleton

(2000).

Reversing the above reasoning, we infer that prices for actively traded short-

dated options simplify the task of identifying the concurrent spot volatility

and pertinent features of the risk-neutral jump process, subject to minimal as-

sumptions on the return generating process. Specifically, short-maturity ATM

options help pin down spot volatility, while the relative prices of deep OTM op-

tions assist in determining the intensity and distribution of jumps. The emer-

gence of the weekly options moves this observation from the realm of theory

to the domain of practical empirical work. The requisite quotes for short-dated

options are now available on a daily basis. The goal of the current paper is to

capitalize on the new opportunities afforded by the trading of weeklies to ex-

plore the characteristics of the risk-neutral distribution of equity-index returns

as implied directly by the option data, largely avoiding reliance on parametric

restrictions.

The current paper is, as far as we know, the first to explore the information

content of weekly options for the underlying risk-neutral return dynamics in

a systematic way. In part, this reflects the very recent emergence of near-

continuous trading in short-dated options. Therefore, we first carefully review

1The latter statement is more formally explored in Green and Jarrow (1987)andNachman

(1988). For further developments, see, for example, Bakshi and Madan (2000) and Pan and Liu

(2003), among many others.

2A similar type of connection is present for short-maturity bond prices. Collin-Dufresne, Gold-

stein, and Jones (2008) explore this relation to identify the state vector driving the short-rate

dynamics in a model-free way.

Short-Term Market Risks Implied by Weekly Options 1337

the basic features of our weekly option sample and provide detailed descriptions

of our filtering procedures, imposed to control for excessive noise or errors.

In order to exploit information from all available short-dated options, we de-

velop a new asymptotic option pricing approximation that is operative across

all strikes, and not just for ATM and deep OTM options. It exploits the fact

that, over short intervals of time, volatility and jump intensity do not change

much in expectation. Using our new approximation, we proceed seminonpara-

metrically, that is, we impose only weak parametric restrictions on the jump

distribution, while remaining silent about the dynamics of the volatility and

jump intensity. The approach allows us to infer the spot characteristics for

the risk-neutral return distribution exclusively from short-dated options. In

particular, we generate separate estimates of the current volatility and jump

intensity as well as the jump size distribution at the end of each trading

day.

Our approach bears superficial resemblance to calibration procedures com-

monly applied in approximating option-implied volatility surfaces. The differ-

ences are critical and fundamental, however. Weexplicitly impose no-arbitrage

constraints in estimation and synthesize the option price information into con-

sistent estimates for spot volatility and key jump characteristics, amenable to

formal econometric analysis. From the perspective of summarizing the state

of the local risk-neutral distribution in a continuous-time setting, the spot

volatility and jump characteristics constitute sufficient statistics. They fully

characterize the local behavior of the underlying semimartingale representing

the risk-neutral asset price process. Furthermore, our ability to extract con-

sistent point estimates and generate suitable confidence regions for the state

vector over time sets the stage for analysis of the dynamic properties of the

system. In contrast, standard calibration delivers smoothed risk-neutral den-

sities for specific horizons, but does not enforce any form of internal dynamic

consistency and provides no guidance for the extraction of spot volatility, jump

intensities or jump size distributions. We label the most general version of our

methodology “structural calibration” to emphasize the fact that it generates

valid asymptotic inference for the key components of the state vector govern-

ing the evolution of the risk-neutral distribution at the point in time when we

observe the option prices.3

Our seminonparametric procedure enables far more general modeling of the

time variation in jump risk than in the prior literature. In particular, standard

option pricing models allow no time variation in the jump distribution and

only limited variation in the jump intensity. Relative to recent nonparametric

approaches focused on tail estimation, we offer a far more comprehensive anal-

ysis. For example, Bollerslev and Todorov (2014) and Bollerslev, Todorov, and

Xu (2015) are strictly concerned with the tails and rely exclusively on very deep

OTM contracts, obtained from regular option samples. In contrast, we estimate

3See Jarrow and Kwok (2015) for an alternative method of disentangling model misspecification

from estimation error through the imposition of minimal restrictions on the calibration of the option

surface.