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Slow Diffusion of Information and Price Momentum inStocks: Evidence from Options Markets
Zhuo Chen∗ Andrea Lu†
November 10, 2014
Abstract
This paper investigates the source of price momentum in the stock market usinginformation from options markets. We provide direct evidence of the gradual infor-mation diffusion model in Hong and Stein (1999): momentum profits are larger forstocks whose information diffuses slowly into the stock market. We exploit the optionsmarkets to identify stocks with slow information diffusion speed. As informed traderstrade options to realize the information that has not been fully incorporated in thestock price, we are able to enhance the momentum strategy by selecting winner/loserstocks with high growth/large drop in call option implied volatility. Our empiricalstrategy generates a risk-adjusted alpha of 1.8% per month over the 1996–2011 period,during which the simple momentum strategy fails to perform. The results are robust tothe impact of earnings announcement, transaction costs, industry concentration, andchoice of options’ moneyness and time-to-maturity. Finally, our finding is not driven byexisting stock- or option-related characteristics that are known the improve momentum.
JEL Classification: G10, G11, G12, G13Keywords: Momentum, Implied Volatility
∗PBC School of Finance, Tsinghua University. Email: chenzh@pbcsf.tsinghua.edu.cn. Tel: +86-10-62781370.†Department of Finance, The University of Melbourne. Email: andrea.lu@unimelb.edu.au. Tel: +61-
3-83443326. For helpful comments and discussions, the authors thank Torben Andersen, Snehal Banerjee,Zhi Da, Stephen Figlewski, Kathleen Hagerty, Ravi Jagannathan, Robert Korajczyk, Chayawat Ornthanalai(discussant), Todd Pulvino, Costis Skiadas, Viktor Todorov, and participants at the 3rd International Con-ference on Futures and Derivative Markets, the 2014 NFA Annual Conference, the 2014 PanAgora CrowellMemorial Prize Presentation, the 2013 FMA Doctoral Student Consortium, the 2013 Chicago QuantitativeAlliance Fall Conference, the 2013 PKU-Tsinghua-Stanford Conference in Quantitative Finance, and theKellogg finance baglunch. All errors remain ours
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1 Introduction
The diffusion of information plays a crucial role in explaining price momentum. Researchers
attempt to understand momentum from investors’ process and reaction to firm-specific in-
formation, and how such information is conveyed into stock price. Among them, Hong and
Stein (1999) propose a model that shows how slow diffusion of information and interaction
of two types of investors, newswatchers and momentum traders, can explain price underre-
action in the short run and overreaction in the median run.1 A direct prediction of their
model is that momentum should be stronger for stocks with slower information diffusion
speed. In this paper, we provide empirical support for their theoretical prediction by iden-
tifying stocks’ information diffusion speed using options markets. We show that momentum
profit concentrates in stocks with slow information diffusion speed. An enhanced momentum
strategy that is constructed within such stocks perform well, even during periods when the
simple momentum strategy fails to perform.
Although the correct identification of information diffusion speed is important in explaining
momentum, in reality it is easier said than done. Hong, Lim and Stein (2000) use size and
analyst coverage to classify stocks into slow and fast diffusion groups. They find momentum
effect is stronger for the slow diffusion group characterized by small size and low analyst
coverage. However, size and analyst coverage are static firm-specific characteristics that do
not change much over time, while information diffusion speed could be information-specific
and time-varying. For example, the manager of a company tends to have a piece of positive
information to be perceived by investors fast, but may try to delay the diffusion of another
piece of negative information (Kothari, Shu, and Wysocki (2009)). Therefore, our goal is to
1In their model setup, newswatchers trade stocks only based on fundamental information while momentumtraders make their investment decisions solely based on forecasts from historical prices. As information slowlydiffuses into the market, the stock price first experiences underreaction as newswatchers adjust to the newinformation, and then experiences overreaction as momentum traders attempt to profit from the previousprice movement.
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identify individual stocks’ information diffusion speed and construct the momentum portfolio
using stocks with continued information diffusion in the holding period.
We take advantage of the options markets to dynamically refine our momentum portfolio
selection according to stocks’ information diffusion speed. Options markets provide an effec-
tive channel for price discovery and information diffusion (Manaster and Rendleman (1982)).
Previous researchers find that informed traders may prefer options markets to the stock mar-
ket for various reasons, such as embedded leverage of options (Black (1975), Frazzini and
Pedersen (2012)), investors’ short sale constraints (Figlewski and Webb (1993)), transaction
costs (Cox and Rubinstein (1985)), and so forth. Thus, options prices may contain material
information that has not been fully reflected in stock prices. When a piece of information
starts to diffuse into the market, option traders have informational advantage to distinguish
whether the informational content has conveyed into the stock price. Billings and Jennings
(2011) find that an increase in uncertainty-adjusted option prices prior to earnings announce-
ments is positively related to the sensitivity between the stock market reaction and earnings
announcements. Their finding indicates that option traders prefer options of those stocks
with slower information diffusion speed regarding earnings announcements. We generalize
their argument through out the course of information diffusion. When the information dif-
fusion speed is slow, upon discovering more information to continue releasing in the stock
market, they will realize their superior information in the options markets, causing options
prices to change. Therefore, within those winner/loser stocks that have started their infor-
mation diffusion process, trades in options markets allow us to identify those stocks with
slower information diffusion speed and thus further price adjustment. Specifically, for winner
stocks, if we also observe prices of call options increase or prices of put options decrease, it
indicates that informed option traders believe that not all relevant information has released
and there will be further price appreciation. The same logic applies to loser stocks: informed
option traders can either sell call options or buy put options if they think that the negative
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information associated with those loser stocks has not been fully incorporated in the stock
prices.
Based on the logic above, we use implied volatility growth of options to identify stocks’
information diffusion speed and construct the enhanced momentum portfolio. A large
growth/decline in the call option implied volatility reflects informed option traders’ buy/sell
position and their belief that positive/negative information will continue to convey into stock
price. Thus to enhance the stock selection based on information diffusion speed, conditional
on winner stocks, we long those with the largest growth in call option implied volatility. Simi-
larly, conditional on loser stocks, we short those with the largest decline in call option implied
volatility. Our enhanced momentum strategy generates a risk-adjusted alpha of 1.78% per
month over the 1996–2011 period, while a simple momentum strategy fails to perform dur-
ing the same period. Moreover, Fama-MacBeth regression shows that the return spread is
attributed to the interaction of the momentum effect and the correct identification of in-
formation diffusion speed with implied volatility growth. Similar improvement can be done
using implied volatility growth of put options, i.e., longing winner stocks with the largest
decline in put option implied volatility and shorting loser stocks with the largest increase in
put option implied volatility. However, the size of the improvement is smaller.
Our results are robust to a battery of alternatives. First, although it is well-known that
options are more actively traded before earnings announcements, we find that the informa-
tiveness of implied volatility growth about stocks’ information diffusion is not limited to
earnings announcements. Similar results can be found even if we exclude stocks that have
earnings announcements in the holding month. Second, the profitability of the enhanced
momentum strategy is robust in consideration of transaction costs. Third, we show that the
good performance is not driven by industry momentum (Moskowitz and Grinblatt (1999)) or
several stock-level characteristics (Bandarchuk and Hilscher (2012)). Fourth, the portfolio
return remains significant if we use implied volatility growth of options that have time-to-
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maturity equal to the holding period (two, three, or six months), while the magnitude of
return is smaller. This result indicates that besides the correct identification of information
diffusion speed, repeatedly updating the identification also contributes to the improvement.
Finally, our results also hold if we use out-of-the-money options, control for the persistency
of implied volatility, implement independent double sorting, and construct a value-weighted
portfolio.
Our paper adds to the momentum literature.2 Although numerous papers have shown the
existence of price momentum across asset classes, sample periods, and geographic markets,
our paper, to the best of our knowledge, is the first to explain momentum profit using in-
formation from individual stock options. In addition, our paper is the first to document the
poor performance of the simple momentum strategy for the most recent 1996–2011 period
(the OptionMetrics data range). However, by incorporating information from options mar-
kets, we are able to generate statistically significant raw/risk-adjusted returns for momentum
strategy in the same period. Although many researchers find that momentum is stronger
when uncertainty is higher, our finding is different from this uncertainty explanation. The
high momentum profit is not due to mechanically finer sorting on more volatile stocks, but
it is attributed to the selection of stocks with slow information diffusion speed.
Our paper is also related to the growing research that studies the implication of options
markets on equity returns. Our empirical approach of using growth of individual stock option
implied volatility is similar to An et al. (2014), who find that the returns of stocks with a
large first difference of call option implied volatility tend to rise in the next month, whereas
2Papers documenting momentum effect include Jagadeesh and Titman (1993), (2001), (2012), Carhart(1997), Novy-Marx (2012), and Israel and Moskowitz (2013). Several papers also find that momentum effectexists in international markets (Asness et al. (2013), Asness (2011), Chui et al. (2010)). Many papersimplement double sorting on firm-level characteristics to strengthen momentum profit (Hong et al (2000),Lee and Swaminathan (2000), Zhang (2006), Da et al. (2012), Hillert et al. (2012)). Bandarchuk andHilscher (2012) provide a comprehensive summary of all the characteristic-based double-sorting strategiesthat are actually thinner sorting on more extreme returns. Papers that try to explain momentum effectinclude Daniel, Hirshleifer, and Subrahmanyam (1998), Johnson (2002), and Sagi and Seasholes (2007)
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a large first difference in put option implied volatility predicts lower future returns. Our
paper is different from An et al.’s (2014) as our focus is on the role played by option implied
volatility growth on identifying stocks’ information diffusion speed and how it interacts with
the momentum effect. Fama-MacBeth regression also confirms that the outperformance of
our double-sorted momentum strategy is not a simple linear summation of the momentum
effect and the option effect. Besides An et al. (2014), many others have studied the predictive
relationship between options markets and equity returns. Xing et al. (2010) find that
volatility smirk defined as the difference between implied volatility of out-of-the-money put
option and implied volatility of at-the-money call option has strong predictive power for
future returns. Bali and Hovakimian (2009) find that realized volatility and implied volatility
spread predicts lower future returns, whereas call and put implied volatility spread predicts
higher future returns. Cremers and Weinbaum (2010) find that deviation from put-call parity
predicts future stock returns.3
The remainder of the paper is organized as follows. In Section 2 we describe the data.
Section 3 presents the identification of slow information diffusion stocks and main empirical
findings. We conduct a battery of robustness tests in Section 4. Section 5 concludes.
3Other papers include Easley, O’Hara and Srinivas (1998), who find that option volume contains superiorinformation as informed traders may choose options markets over the stock market under some condition; Panand Poteshman (2006), who find that the put-call option volume ratio has information content about futurestock returns; Goyal and Saretto (2009), who find that stocks with a larger difference between historicalrealized volatility and at-the-money implied volatility have higher monthly returns; Johnson and So (2012),who show a negative relationship between option to stock trading volume ratio and future stock returns;Kehrle and Puhan (2013), who discover that option demand imbalance has predictive power for equityreturns; Yan (2011), who finds that expected stock return is a function of the slope of option implied volatilitysmile; and Hu (2014), who finds that the stock market order imbalance induced by option transactionshave strong cross-sectional predictive power for stock returns. There are also papers that study otherimplications of options markets on financial market or firm decisions: Cremers et al. (2008) study optionimplied volatility’s predictive power on corporate bond; Roll et al. (2009) find that options trading ispositively associated with firm values as well as information production; Stein and Stone (2012) studythe impact of option implied volatility on firm’s investment and hiring decisions; Vilkov and Xiao (2012)investigate the role of option implied volatility in predicting stocks extreme returns; Baltussen et al. (2012)find cross-sectional return prediction captured by higher second-order uncertainty using individual optionvol-of-vol data; Cao et al. (2005) study the relationship between option volume and takeover announcementday returns; and Roll et al. (2010) suggest that the option/stock trading volume ratio relates positivelywith the post-announcement absolute returns. Acharya and Johnson (2007) also find evidence of significantincremental information revelation in the credit default swap market.
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2 Data
2.1 Data overview
Our data primarily come from two sources. The stock data are downloaded from the
CRSP. We include all common stocks that are traded on NYSE, Amex, and Nasdaq. To
ensure liquidity, we exclude stocks with market capitalization that are below the 10% NYSE
cutoff or have price less than five dollars at the end of formation month. Data of implied
volatility are from OptionMetrics’ Implied Volatility Surface. Our sample includes implied
volatility of options written on stocks that also have data in CRSP. Because OptionMetrics
data start in 1996, our final sample is from January 1996 to December 2011.
[Insert Table 1]
Table 1 presents the year-by-year descriptive statistics, including average number of stocks,
average market size, and median market size. The average number of stocks in CRSP that
satisfies the liquidity requirement is 2,762, across all years from 1996 to 2011. Nevertheless,
the average number has gradually decreased from 3,564 in 1996 to 2,135 in 2011. Contrary
to the decrease in the number of stocks, market size has increased significantly: the average
size in 2011 is more than three and a half times than that in 1996. Not all stocks that pass
the liquidity filter have listed option contracts; ones that do are often large in size. The
CRSP-OptionMetrics-merged data set contains 1,536 stocks on average, covering more than
half of the stocks in the CRSP data set. The average market capitalization in the merged
data set is 6,741 million, which is about one and a half times larger than the average size of
stocks in the CRSP data set. The median market capitalization is much smaller than the
mean in both data sets. Note that the fraction of stocks with option contracts has expanded
over time: only 15.8% CRSP stocks do not have options in 2011, whereas this number is
73.8% in 1996. Stocks with option contracts are less subject to the liquidity requirement:
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almost half of all common stocks do not meet the liquidity requirement, whereas this rate
is only 13% for stocks with option contracts. Overall, stocks we study in this paper are
relatively large stocks that are typically considered to exhibit weaker momentum.
2.2 Data on option implied volatility
OptionMetrics provides a comprehensive coverage of option data for U.S. equities. The
Implied Volatility Surface covers almost 100% of equity options currently or historically
listed on U.S. exchanges. The Implied Volatility Surface provides the implied volatility of
standardized call and put options for each stock on a fixed grid in the time-to-expiration
and moneyness coordinates. It considers expirations of 30, 60, 91, 122, 152, 182, 273, 365,
547, and 730 calendar days. Moneyness is measured in deltas, taking values from 0.20 to
0.80 with an increment of 0.05 (negative deltas for put options). The implied volatility of
standardized option with a combination of expiration and moneyness is interpolated from
implied volatilities of available listed option contracts with various maturities and strikes
using a kernel smoothing technique. The implied volatility is computed using a pricing
algorithm that is based on the Cox-Ross-Rubinstein binomial tree. Specifically, the option
expiration period is dividend into N subperiods.4 In each subperiod, the underlying stock’s
price is assumed to move either up or down, and the size of the movement is determined by
the implied volatility and the length of the subperiod. The tree that contains all the possible
security price movements is then built according to the current security price. The price of
the option can be computed using backward computation. The model is run iteratively until
the theoretical price of the option converges to its current market price. 5 The market price
4The number of subperiods used is typically over 1,000.5One advantage of this approach is that it correctly adjusts for the potential early exercise or future
dividend payments. The security’s current dividend yield, defined as the most recently announced dividendpayment divided by the security’s most recent closing price, is assumed to remain constant over the remaininglife of the option. It is also assumed that the security pays dividends at specified predetermined times.When future dividend dates are not announced, OptionMetrics uses an extrapolation algorithm to projectex-dividend dates according to securities’ usual dividend payment frequency.
8
of the option used in the implied volatility computation is the settlement price. In case the
settlement price is not available, the last traded price, or the midpoint of the closing bid
and offer prices (in the order of availability), is taken as the price. The interpolated implied
volatility, with a given combination of delta and maturity, is calculated as a weighted average
of implied volatilities of all existing option contracts on that date, where the weights are
functions of options’ Vegas and measures of the distance between the option and the target
maturity and delta. An interpolated implied volatility is only included if there exists enough
underlying option price data on that date. Computation is done using only the current
available data without forward-looking bias.
We use implied volatilities of call and put options with a delta of 0.5 (-0.5 for put) and
time-to-maturity from one-month (30 days) to six-month (182 days). Options with a delta
of 0.5 have two advantages. First, they have high trading volume as they are closest to at-
the-money options. High liquidity means tighter bid-ask spreads and more accurate implied
volatility interpolation. Second, although being close to at-the-money, options with a delta
of 0.5 are slightly out-of-the-money so that the simple Put-Call Parity does not have to
hold. Empirical “deviation” of put-call parity leaves room for us to explore the difference
of information content between the call option implied volatility and put option implied
volatility. In the robustness test, we also use implied volatility of call and put options with
other moneyness and time-to-maturity.
The variable we use to identify stocks’ information diffusion speed is option implied volatil-
ity growth, denoted by ∆IV C and ∆IV P for call and put options, respectively. The implied
volatility growth is calculated over the last five trading days of each calendar month. Specif-
ically, the implied volatility growth in month t is defined as the implied volatility on the
last trading day of month t divided by the implied volatility five trading days earlier. For
example, suppose the last trading day falls on a Wednesday, the growth is computed using
the implied volatility of that day and the previous Wednesday. In case when the previous
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Wednesday is a public holiday, we use the implied volatility of the nearest previous trading
day.
Table 2 presents summary statistics of option implied volatility growth. Average and
standard deviation are computed from the monthly observations across all options and then
averaged across 12 months within a year. The average implied volatility growth for call and
put options with a delta of 0.5 and maturity of one month is 0.31% and 0.19%, respectively.
Call option has a slightly higher cross-sectional dispersion in implied volatility growth relative
to that of put option. This pattern also holds in longer maturity options. In addition, implied
volatility growth of options with longer maturity exhibits lower cross-sectional average and
standard deviation, consistent with the fact that long maturity options are less traded than
short maturity options.
[Insert Table 2]
3 Momentum strategy enhanced by options markets
information
3.1 Performance of the traditional momentum strategy for the
1996–2011 period
We first examine the performance of a simple momentum strategy for the 1996–2011 pe-
riod. Momentum portfolios are constructed following the standard procedure described by
Jegadeesh and Titman (1993). Specifically, we assign stocks into ten equal-weighted portfo-
lios according to their past J-month cumulative returns and then hold the winner portfolio
and short the loser portfolio for K months. We follow Jegadeesh and Titman (1993) to skip
one month between the formation month and the holding month to mitigate the influence of
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temporary price pressure due to high-frequency phenomena or bid-ask bounce. We construct
the momentum portfolio using each of the two sets of stocks: the first is all U.S. common
stocks traded on the three major exchanges that satisfy the liquidity requirement at the end
of formation month. The second group is a subset of the first that contains stocks with
listed option contracts. Table 3 presents monthly winner-minus-loser returns for various
combinations of formation and holding periods.
[Insert Table 3]
Surprisingly, the traditional momentum strategy is only marginally profitable for the 1996
to 2011 period.6 Panel A shows that the returns of various momentum portfolios are al-
most always insignificant. The return is only marginally significant when a combination
of (J = 6,K = 1) is used for portfolio formation and holding. In Panel B, we report the
results constructed over stocks with listed options: no return spread is significant while the
magnitudes are all below 1% per month.
This finding is in contradiction with common wisdom about momentum. A few reasons
could explain the disappearance of momentum profit. First, our sample period contains a
couple of crises that may lead to volatile momentum performance. Jegadeesh and Titman
(2012) find that the raw return of the momentum strategy is -36.50% in 2009 when the
market rebounded from the financial crisis. Daniel et al. (2013) also find that the momentum
strategy could have a sharp performance decline as the market rebounds. Second, stocks with
options are relatively large stocks that are associated with small momentum returns. Our
finding is consistent with Hong et al. (2000), who find that the profitability of momentum
strategies declines sharply with firm size.
6Some papers study momentum in the most recent two decades. Israel and Moskowitz (2013) find thatmomentum strategy is still profitable for the 1990–2011 period, but the monthly return is only about 70bps. Novy-Marx (2012) finds that momentum strategy is much stronger if investors skip six months insteadof one month after formation for the 1990–2011 period. Ours is the first paper that looks at the 1996–2011period as the OptionMetrics data begin in 1996.
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3.2 Identification of stocks’ information diffusion speed using im-
plied volatility growth
Hong and Stein (1999) propose a theoretical explanation for momentum. In their model,
two groups of investors, newswatchers and momentum traders, interact in the stock market.
Newswatchers trade stocks only based on information of future fundamentals, without relying
on current or past prices. On the other hand, momentum traders trade solely on their
price forecast formed using historical prices. As firm-specific information gradually diffuses
into the stock market, prices first experience underreaction as newswatchers slowly adjust
to such information. Prices then experience overreaction as momentum traders attempt
to profit from the underreaction caused by newswatchers. Slow diffusion of information
generates both underreaction and overreaction repeatedly, driving the price momentum.
One important prediction of their paper is that stocks with slower information diffusion
should exhibit more pronounced momentum.7 Their story also implies that different pieces
of news across stocks and time are heterogeneous in terms of information diffusion speed.
The price of one stock could reflect firm-related information faster than the price of another
stock. Alternatively, the price of one stock could reflect one piece of information faster than
the other piece of information at a different time. Therefore, momentum traders can benefit
if they construct the portfolio with stocks that have slow information diffusion speed, i.e.,
stocks whose prices have not fully incorporated the relevant information.
We identify the speed of information diffusion using options markets. Compared to stocks’
static characteristics such as size or analyst coverage, options provide a more timely and pre-
cise identification in several ways. First, options provide a number of advantages that attract
informed investors to realize their superior information in the options markets instead of the
stock market. For example, leverage constrained investors could use options to invest with
7Hong et al. (2000) test this prediction and find that momentum portfolios formed on small stocks orstocks with low analyst coverage deliver higher returns.
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“borrowed” money. When a stock has short sale constraint, investors can also use options
to express their negative view on the stock. Second, due to the benefit of options markets,
researchers have empirically found that options markets could be a more informed channel
for price discovery and information diffusion.8 When sophisticated informed investors have
positive private news on a company, they buy call options or sell put options if such infor-
mation has not been reflected in the stock price. Therefore, call price appreciation or put
price depreciation might convey informative content about informed traders’ view on the
information diffusion stage of individual stocks. Third, Billings and Jennings (2011) find
that pre earnings announcement option price change is positively related to option traders’
view on the sensitivity between the stock price reaction and the earnings announcements.
For example, a large increase in call option price (or a large drop in put option price) implies
informed option traders’ belief that the stock price will have a large increase to the potential
positive earnings announcement, which is equivalent to saying that positive information is
going to be diffused into the stock price.
While the study of Billings and Jennings (2011) focuses on earnings announcements, the
same logic could be applied to normal periods as well. We exploit the information of options
markets to improve the momentum strategy. Past cumulative returns (momentum effect)
of individual stocks could be used to detect stocks’ information diffusion, while possible
future information diffusion and time-varying diffusion speed cannot be identified by only
looking at such past returns. On the other hand, option prices reflect informed investors’
view on whether such information diffusion would continue being conveyed into stock prices.
If we observe positive past cumulative return paired with call option price appreciation (put
option price depreciation), it implies that option traders believe that the diffusion speed
associated with the current piece of information is slow in the sense that it has started and
will continue to diffuse in the stock market leading to further stock price increase. The same
8Papers include but not limited to Easley et al. (1998); Chakravarty et al. (2004); Cao, Chen, and Griffin(2005); Chern et al. (2008)
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applies to those loser stocks with negative past cumulative returns. Since option implied
volatility is a monotonic mapping of option price, we identify the sign and magnitude of
stocks’ information diffusion speed using option implied volatility growth. Notice that we
do not exclude the possibility that informed investors could also trade on the stock market.
What we assume here is that option traders are in general more sophisticated with better
understanding on whether information diffusion would continue into the stock price.
The identification of information diffusion speed helps to improve the price momentum in
the stock market. To construct the enhanced momentum portfolio, we first sort stocks into
ten groups based on their cumulative returns over the past six months. We fix the formation
period to keep the number of strategies tractable.9 The group of stocks with the highest
past cumulative returns is the winner portfolio, and the one with the lowest past cumulative
returns is the loser portfolio. These are stocks whose firm-specific good or bad information
has already started diffusing into the stock market. We skip one month post the formation
month. Instead of simply holding the winner portfolio (or shorting the loser portfolio), we
take positions on a subset of stocks in the winner and loser pools that are more likely to
experience continued information diffusion, as suggested by the options markets. Specifically,
at the beginning of each month during the holding period, we sort stocks within the winner
and loser pools into three groups, namely, slow, median, and fast, based on implied volatility
growth over the most recent trading week.10 They are named in this way because, given the
detection of information diffusion, the likelihood of continuation in information diffusion is
closely related to the diffusion speed. Stocks with slow (fast) information diffusion are more
(less) likely to experience further information diffusion. Using call options, stocks with slow
information diffusion are winners (or loser) stocks that option traders believe good (bad)
news will continue to diffuse into the stock market, and thus ones with large (small) call
9J =6 is also the formation period that delivers the highest return for the simple momentum strategy inthe relevant sample period.
10This is the last trading week of the previous month. In addition, to rule out the effect of extreme values,we winsorize the implied volatility growth at 1% and 99%.
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option implied volatility growth. Using put options, stocks with slow information diffusion
among winner (loser) stocks are the ones with small (large) growth in implied volatility. We
construct equal-weighted winner-minus-loser momentum portfolio with this double sorting
strategy by taking a long position in the refined winner stocks and a short position in the
refined loser stocks. We hold the portfolio for one month and re-rank stocks based on implied
volatility growth the next month during the holding period.11
3.3 Empirical results
Table 4 presents average monthly returns for the hedged winner-minus-loser portfolios (P10-
P1) with holding period K = 1, 3, 6 using call option information. DF , DM , and DS represent
portfolios constructed by selecting stocks with fast, median, and slow diffusion, respectively.
Although our sample contains large stocks that usually exhibit smaller momentum effect,
picking slow diffusion stocks within the winner and loser pools generates large momentum
profit. When the holding period is one month, the average excess return for the refined
momentum portfolio is 1.55% per month.12 We also compute the risk-adjusted alphas relative
to the CAPM, the Fama-French three-factor model, and a four-factor model of the Fama-
French three factors plus the short-term reversal (STR) factor.13 The alpha is 1.78% per
month after controlling for all four risk factors, which is around 21% per year. Large and
statistically significant returns remain for longer holding horizons. The risk-adjusted alpha
is 1.25% per month when the holding period is six months. Panel B of Table 4 assesses the
effect of selecting stocks with information diffusion continuation based on the call option
11We keep the momentum ranking constant over the holding months and use dependent double sorting.We also consider the case in which we implement independent double sorting strategy. Our results are robustto different portfolio construction procedures.
12The excess return for the momentum strategy using all NYSE/AMEX/NASDAQ stocks during the sameperiod is 1.12% per month, and the excess return for the momentum strategy using stocks that have listedoptions is 0.94% per month.
13Portfolios’ beta loadings and adjusted R2 under this four-factor model are presented in Table A.1. Betaloadings and adjusted R2 estimated under the CAPM or the Fama-French three-factor model are availableupon request.
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implied volatility growth. It reports the return difference of two winner/loser portfolios, one
is constructed within stocks with slow information diffusion and the other is constructed
within stocks with fast information diffusion. Taking the one-month holding period case
as an example, winner stocks with large call option implied volatility growth earn a higher
four-factor adjusted alpha of 55 bps per month than winner stocks with small call option
implied volatility growth. Similar result is found for loser stocks and for longer holding
horizons.
[Insert Table 4]
On the other hand, we do not observe the same pattern when sorting by put option implied
volatility growth as presented in Table 5. Momentum portfolio constructed using stocks with
slow information diffusion speed only generates a significant return when the holding period
is short (K = 1), but not for longer holding horizons (K = 3 or K = 6). The magnitude of
momentum profit is smaller than that generated by sorting on call implied volatility growth.
Furthermore, the differences between DS (slow diffusion group) and DF (fast diffusion group)
momentum portfolios are not significant as shown in the last row of Panel B of Table 5.
Empirical results indicate that the put option seems to be less informative for identifying
stocks’ information diffusion speed.
[Insert Table 5]
We provide empirical evidence on how identification of stocks’ future information diffusion
helps refine stock selection within the winner and loser stocks. The enhanced momentum
portfolio earns a risk-adjusted alpha over 20% per year by selecting winner/loser stocks with
large growth/drop in call option implied volatility. Previous researchers have found the
predictive power of options on stock returns. Our methodology is in line with theirs in terms
of private information on companies’ fundamentals being first perceived by option traders.
However, the outperfomance of our momentum strategy comes from the fact that we use
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information embedded in options to dynamically identify stocks’ information diffusion speed.
The return spread of our strategy is larger than that of a portfolio single-sorted on option
implied volatility growth (Table A.2).
We implement Fama-MacBeth regressions to examine whether the strong performance of
the double-sorted momentum portfolio comes from the interaction of the past cumulative
performance and the identification of slow information diffusion. The regression specification
is as following:
Ri,t+1 = β0+β1PastReturni,t+β2∆IVC(P )i,t +β3PastReturni,t×∆ ˆIV
C(P )
i,t +β4Xi,t+εi,t, (1)
where ∆ ˆIVC
i,t = ∆IV Ci,t ,∆ ˆIV
P
i,t = −∆IV Pi,t if PastReturni,t > median(PastReturni,t);
∆ ˆIVC
i,t = −∆IV Ci,t ,∆ ˆIV
P
i,t = ∆IV Pi,t if PastReturni,t < median(PastReturni,t). The specifi-
cation of ∆ ˆIVC(P )
i,t is consistent with our stock selection procedure in the portfolio construc-
tion. Xi,t consists of a list of control variables, including stock size, stock price, book-to-
market ratio, stock trading volume, number of analyst coverage, the maximum daily return,
market beta, Amihud illiquidity measure, realized volatility, idiosyncratic volatility, options’
open interest growth, and options’ trading volume change.14 Results are presented in Ta-
ble 6. We find that while call option implied volatility growth has a strong predictive power
on holding period return (Column (2)), past cumulative return does not (Column (3)), over
the sample period of 1996-2011. This finding is consistent with the predictive power of
options documented in previous studies and the weak performance of a simple momentum
strategy in the earlier section. However, the interaction of momentum and call option im-
plied volatility growth plays an important role, as the coefficient estimate on the cross term
14The open interest growth is calculated as the total open interest on the last trading day prior to theholding month divided by the value five trading days earlier. The option trading volume change is the totalvolume measured on the last trading day prior to the holding month minus the total volume five tradingdays earlier. We use the volume change instead of growth because of the presence of zero volume. Bothopen interest and volume are calculated using all call (put) options with maturities between 30 days and 365days. We exclude short maturity options to avoid the potential mechanical changes near expiration.
17
β3 is positive and statistically significant. Similar results are found when we use put options.
The last two columns show the results when only winner and loser stocks (extreme stocks)
are included in the sample. In case of call options, only the interaction term β3 is significant,
while neither past cumulative return or implied volatility growth has any predictive power
alone. Results of Fama-MacBeth regressions imply that it is indeed the interaction between
the momentum and implied volatility growth that contributes to the strong performance of
our portfolio.
To ensure that implied volatility growth is not related to other well-documented stock- or
option-specific characteristics that can improve momentum effect,15 we calculate the median
of several characteristics for stocks in the double-sorted portfolios. We consider ten charac-
teristics, including stock size, stock price, stock trading volume, stock analyst coverage, past
cumulative return, realized volatility, idiosyncratic volatility, maximum daily return, option
open interest growth, and option trading volume change. The median value of each char-
acteristic within each double-sorted portfolio is presented in Table 7. Instead of displaying
cells in terms of fast, medium, or slow (DF , DM , DS), which involves different rankings for
winner and loser stocks, we display cells according to the actual implied volatility growth
(small, medium, and large). The portfolios that we pick for the long and short sides as part
of the enhanced momentum portfolio are highlighted in bold. We see no obvious pattern in
those characteristics across volatility growth sorted portfolios, indicating that stock selec-
tion based on implied volatility growth is not equivalent to selecting stocks based on these
ten characteristics. In other words, by forming an enhanced momentum portfolio using im-
plied volatility growth, we are not simply implementing a narrower sorting on more extreme
winner or loser stocks based on these characteristics above.
[Insert Table 7]
15Bandarchuk and Hilscher (2012) provides a comprehensive summary of stock-level characteristics thatcan enhance momentum profit.
18
4 Robustness analysis
In this section, we present a number of robustness tests. We examine the earnings an-
nouncement effect, the impact of transaction cost, the industry concentration of the momen-
tum portfolio, and performance of portfolios that are refined using options with maturity
matched with holding horizon. Other tests include using out-of-the-money options, con-
structing value-weighted portfolios, using monthly implied volatility growth, and examining
the informational content of implied volatility growth.
4.1 Earnings announcement
In the previous section, we show that the hedged winner-minus-lower portfolio enhanced
with options markets information delivers a risk-adjusted alpha of 1.78% per month. The
outperformance is attributed to the effective identification of stocks’ information diffusion
speed using implied volatility growth. Meanwhile, it is well known that implied volatility
increases significantly before earnings announcements. We examine whether our finding is
caused by informational advantage of options traders around earnings announcements. We
construct the enhanced momentum portfolio using stocks without earnings announcements
in the holding month. Table 8 presents the monthly portfolio returns when the holding
period is one month.16 We find similar results for both unadjusted and risk-adjusted returns
compared to the case where all stocks are used. Results are also unaffected if we use the put
option implied volatility growth.
[Insert Table 8]
16Results with alternative holding periods are similar and available upon request.
19
4.2 Transaction cost
Momentum strategy is known to have high turnover. Such high turnover also applies to
our double sorting strategy. Taking the (J = 6,S = 1,K = 1) strategy as an example,
only 10% of the stocks do not need to rebalance each month. In this section, we assess the
profitability of the options improved momentum strategy after considering transaction costs.
Due to the lack of data on realized transaction costs, we take an alternative approach by
imposing a restriction on portfolio rebalancing. Specifically, each month, we rebalance x%
of the largest stocks that needs rebalancing. We find that the performance of the option
improved momentum strategy is robust to the imposed restriction. Table 9 presents the
results on the call option improved momentum portfolios.17 When 80% of the stocks are
allowed to rebalance, the risk-adjusted alpha is 1.57% per month with a t-statistic of 2.64.
When we only allow a turnover of 20%, the alpha is 0.86% with a t-statistic of 2.15. Similar
results are also found for value-weighted portfolios.
[Insert Table 9]
4.3 Industry concentration
It is possible that our portfolio construction procedure actually results in selecting stocks
with high industry concentration (Moskowitz and Grinblatt (1999)). To address this ques-
tion, we compute industry concentration of the winner and the loser portfolios over time,
and study its correlation with the portfolio return. We measure industry concentration using
the Herfindahl-Hirschman Index (HHI) as expressed in Equation 2.
HHIt = ΣNti=1s
2i,t (2)
17Results for the put options enhanced momentum portfolios are available upon request.
20
The HHI of a portfolio in a given month is computed as the sum of the squared stock share
of industry i, si,t, where si,t is the fraction of stocks that belong to industry i.18 As a result,
the HHI takes a positive value from zero to one with a larger number indicating higher
concentration.
Panel A of Figure. 1 plots the time series of the HHIs for the winner and loser portfolios
constructed using the call option-based benchmark strategy, in comparison to the HHIs of all
qualifying stocks. We see that the HHIs for both the winner and the loser portfolios exhibit
large time series variation, and such variation is more pronounced in the first half of the
sample. Although both portfolios are less “diversified” relative to the portfolio constructed
using all stocks, it is uncommon to see HHIs at extremely high levels: a vast majority
of the sample has an HHI that is less than 0.5. Moreover, the correlations between the
winner/loser portfolios’ HHIs and the return of the winner-minus-loser portfolio is 5.9% and
4.5%, respectively; the correlation between the HHIs of the winner/loser portfolio and the
returns of the corresponding winner/loser portfolio is 1.7% and 5.9%. Similar results are
found for portfolios sorted using put options implied volatility growth as shown in Panel B
of Figure 1.
[Insert Figure 1]
4.4 Lazy updating
In the benchmark strategy, stocks are sorted into portfolios according to past cumulative
return and option implied volatility growth. Options used are standardized call (put) op-
tions with 30-day-to-maturity and a delta of 0.5(-0.5). While the cumulative return rank hold
18We classify stocks into ten major industry groups based on the first two digits of their SIC code:agriculture, forestry, and fishing (0100–0999), mining (1000–1499), construction (1500–1799), manufacturing(2000–3999), transportation, communications, electric, gas, and sanitary service (4000–4999), wholesale trade(5000–5199), retail trade (5200–5999), finance, insurance, and real estate (6000–6799), service (7000–8999),and public administration (9100–9729).
21
through out the holding months, the option-based sorting is implemented for each holding
month. We can also match the holding horizon of the enhanced momentum portfolio with
the maturity of options. Specifically, for any holding horizon, we sort winner and loser stocks
only once according to implied volatility growth of options with the time-to-maturity that
matches with the holding horizon. We present the results in Tables 10 and 11. When call
option is used, raw excess return is 1.35% per month with a t-statistic of 2.04 for a two-month
holding horizon (K = 2); the raw return is not significant for K = 3 or K = 6. Risk-adjusted
alphas are all significant across different holding horizons. Two observations worth empha-
sizing. First, the lazy updating strategy yields lower momentum profit than the benchmark
monthly updating strategy. The risk-adjusted alpha is 1.26%/0.98% per month for the 3/6-
month holding horizon compared with 1.52%/1.25% under the benchmark strategy. Second,
the power of the lazy updating strategy comes from different sides of winner/loser stocks
compared to the timely updating strategy. Under the benchmark strategy, we find that the
marginal contribution of selecting stocks with slow information diffusion speed is similar for
winner and loser stocks. Under the lazy updating strategy, loser stocks with slow informa-
tion diffusion (DS) have a -0.66%/month lower return than the loser stocks identified to be
fast diffusion (DF ), whereas the return difference is only 0.1%/month for the winner stocks.
This finding suggests that the benefit of monthly updating concentrates more among winner
stocks.
[Insert Table 10 and Table 11]
4.5 Other robustness tests
In addition, we also consider a number of other robustness tests. To keep the presentation
succinct, we only present these tests in brief. Tables are included in the Appendix.
Out-of-the-money options: We also use out-of-the-money options to identify stocks’ infor-
22
mation diffusion speed. We classify options into slightly out-of-the-money group (delta =
0.35, 0.40, or 0.45) and far our-of-the-money group (delta = 0.20, 0.25, or 0.30). The implied
volatility for each group is calculated as the average value across options with three deltas.
Double sorting on slightly OTM call option implied volatility growth further improves the
momentum performance with a risk-adjusted alpha of 1.96% per month. On the other hand,
momentum enhanced by far out-of-the-money option implied volatility growth does not de-
liver a return spread as large as the benchmark strategy. Detailed results are presented in
Table A.3.
Value-weighted: In the benchmark test, stocks are equally weighted within each portfolio.
We also construct value-weighted momentum portfolios to check whether our results are
driven by small stocks. We find that the momentum profit is even larger for value-weighted
portfolios: the alphas are 1.73%, 1.96%, and 1.94% per month, respectively, for different
holding horizons. Again, momentum profits are insignificant when the put option is used.
Detailed results are presented in Table A.4.
Monthly implied volatility growth: In the benchmark test, the implied volatility growth
is computed over the last five trading days previous to the holding month. We also use
implied volatility growth over the last month and find similar results with weaker magnitude.
Detailed results can be found in Table A.5.
Information content of implied volatility growth: We use a vector-autoregression (VAR)
approach to examine whether our results are driven by the persistence of implied volatility.
Each month, we regress the log growth of call (put) implied volatility on day t on the log
growth rates of both call and put option implied volatilities on day t-1. Mean-reverting exists
in implied volatility growth: there is a negative relation between implied volatility growth
and its lagged value.19 Persistency is less a concern for our study because information of
option traders is not necessarily independent from day to day. However, as an extension, we
19Results on the VAR regression using the full sample are presented in the Appendix Table A.6.
23
decompose the implied volatility growth into two the persistency part and the innovation
part, and use the innovation part to construct double-sorted momentum portfolio.
log(∆IV Ct ) = βC1 log(∆IV C
t−1) + βC2 log(∆IV Pt−1) + εCt (3)
log(∆IV Pt ) = βP1 log(∆IV C
t−1) + βP2 log(∆IV Pt−1) + εPt (4)
We implement this decomposition through the VAR as specified in Equations 3 and 4. We
aggregate daily innovation terms, εCt and εPt , over the last week of each month to obtain
weekly cumulative innovation. We use this weekly innovation term to identify the information
diffusion speed of each stock. We find that winner (or loser) stocks that will continue to
win (or lose) are more effectively identified by the innovation component than by implied
volatility growth itself. The risk-adjusted alphas of the enhanced momentum portfolios are
1.85%, 1.70%, and 1.63% per month for the holding horizon K = 1, 3, 6. Stronger results
are also found when put options are used. Detailed results can be found in Table A.7.
5 Conclusion
To the best of our knowledge, this is the first paper that empirically investigates the
source of price momentum using information from options markets. We provide empirical
support for the slow diffusion model of Hong and Stein (1999). We construct a momentum
portfolio by selecting stocks with slower information diffusion speed using information from
options markets. We show that momentum profit is stronger in stocks with slow information
diffusion, even during periods when a simple momentum strategy fails to perform.
By double sorting winner and loser stocks using option implied volatility growth, we find
that among winner stocks, those with large growth in call option implied volatility continue
to experience strong price appreciation. For loser stocks, those with a sharp decline in call
24
option implied volatility exhibit strong continued price depreciation. By dynamically se-
lecting winner and loser stocks with continued information diffusion, we are able to earn a
risk-adjusted alpha of 1.78% per month for the enhanced momentum portfolio for the 1996–
2011 period. Moreover, the outperformance of our strategy is attributed to the interaction
of momentum effect based on past returns and the selection of slow information diffusion
stocks based on option information. Our results indicate that effective identification of infor-
mation diffusion speed is important in exploiting the underreaction of price to fundamental
information.
25
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30
Figure 1: Industry Concentration of the Enhanced Momentum Portfolios
This figure presents the industry concentration of the enhanced momentum portfolios. Panel A (PanelB) presents the industry concentration of the winner/loser side of the call (put) options implied volatilitygrowth enhanced momentum portfolio. Industry concentration is measured by the Herfindahl-Hirschmanindex (HHI) defined in Equation. 2 The enhanced momentum portfolios are constructed following theprocedure described in Section 3.2 where we fix six months for the past cumulative return calculation, skipone month, and hold the portfolio for one month. Options data used are the implied volatility growth of30-day to maturity at-the-money call (put) options. We exclude stocks with market capitalization less thanthe 10% NYSE cutoff or with price less than $5 at the end of the formation month to ensure liquidity.We also winsorize the data each month by excluding stocks that have implied volatility growth in the topand bottom 1%. The green line indicates the industry HHI for the enhanced winner portfolio; the red lineindicates the HHI for the enhanced loser portfolio; and, the blue line indicates the HHI calculated using allstocks that pass the filters.
Panel A: Industry concentration HHI: Call options enhanced momentum portfolios
Panel B: Industry concentration HHI: Put options enhanced momentum portfolios
31
Table 1: Sample characteristics
This table presents the descriptive statistics for the two sets of stocks used in this study. The first setincludes all common stocks traded on NYSE/NASDAQ/AMEX that have market capitalization greaterthan the 10% NYSE cutoff and a share price greater than $5 at the beginning of each month. Thesecond set contains a subset of the qualifying common stocks that also have option contract data availablefrom OptionMetrics. This table reports the average number of firms, the mean, and the median marketcapitalization from 1996 to 2011. Firm size is measured in millions of dollars.
CRSP common stocks CRSP-OptionMetrics mergedYear No. of firms Mean size Median size No. of firms Mean size Median size
1996 3,564 1,894.4 374.6 934 4,582.0 1,278.01997 3,565 2,413.0 460.3 1,124 5,094.5 1,148.31998 3,442 3,094.9 522.8 1,313 5,822.0 1,131.21999 3,468 3,758.9 499.6 1,477 6,858.8 1,074.02000 3,539 4,382.4 577.3 1,484 8,500.2 1,417.32001 2,991 4,290.1 604.1 1,497 7,432.9 1,359.32002 2,587 4,247.6 695.0 1,532 6,367.3 1,269.32003 2,505 4,206.0 733.2 1,526 6,195.6 1,318.62004 2,470 5,179.1 1,005.7 1,594 7,223.5 1,696.32005 2,443 5,681.9 1,184.3 1,650 7,567.8 1,806.42006 2,437 6,148.8 1,327.7 1,710 7,856.2 1,865.62007 2,400 6,902.7 1,441.0 1,775 8,353.3 2,001.52008 2,285 6,020.2 1,151.1 1,703 7,201.1 1,612.02009 2,177 4,631.8 877.1 1,686 5,385.3 1,188.42010 2,192 5,649.3 1,212.7 1,776 6,265.2 1,520.82011 2,135 6,735.4 1,530.6 1,797 7,148.0 1,768.9
Average 2,762 4,702.3 887.3 1,536 6,740.9 1,466.0
32
Tab
le2:
Sum
mar
ySta
tist
ics
ofIm
plied
Vol
atilit
yG
row
th
Th
ista
ble
pre
sents
mea
nan
dst
and
ard
dev
iati
on
of
impli
edvo
lati
lity
gro
wth
for
call
(pu
t)op
tion
sw
ith
ad
elta
of
0.5
(-0.5
)w
ith
diff
eren
tm
atu
riti
es(o
ne,
thre
e,an
dsi
xm
onth
s).
Th
egro
wth
isca
lcu
late
das
the
op
tion
imp
lied
vola
tili
tym
easu
red
onth
ela
sttr
ad
ing
day
of
agiv
enm
onth
div
ided
by
the
opti
onim
pli
edvol
atil
ity
mea
sure
dfi
vetr
ad
ing
day
sea
rlie
r.A
llst
ati
stic
sare
calc
ula
ted
over
op
tion
sof
stock
sth
at
hav
ein
form
atio
nav
aila
ble
from
CR
SP
.A
stock
isex
clu
ded
ifit
sm
ark
etca
pit
ali
zati
on
isle
ssth
an
the
10%
NY
SE
cuto
ffor
ith
as
ash
are
pri
cele
ssth
an$5
atth
eb
egin
nin
gof
each
mon
th.
All
nu
mb
ers
are
exp
ress
edin
per
centa
ge
term
s.
1-m
onth
3-m
onth
6-m
onth
Cal
lP
ut
Call
Pu
tC
all
Pu
tY
ear
Mea
nS
DM
ean
SD
Mea
nS
DM
ean
SD
Mea
nS
DM
ean
SD
1996
1.18
15.3
1-0
.05
14.0
01.1
010.5
10.2
59.8
70.3
16.6
2-0
.05
6.0
619
970.
3113
.00
-0.1
911.7
80.9
110.0
30.6
18.9
80.4
16.3
80.1
95.6
819
981.
4414
.69
0.8
612.0
41.7
810.2
81.5
59.2
81.0
76.9
80.8
56.3
619
99-0
.08
12.9
2-0
.01
10.8
40.2
110.3
20.3
58.0
5-0
.03
7.5
70.1
85.8
420
00-0
.12
14.3
4-0
.14
12.4
90.4
211.9
10.5
29.6
20.1
48.6
70.2
56.9
620
01-1
.43
11.4
3-0
.72
12.0
3-0
.96
8.6
1-0
.02
8.7
5-0
.84
5.8
4-0
.37
5.5
620
02-0
.73
12.0
1-0
.73
13.3
80.0
110.8
70.2
210.3
5-0
.11
7.3
8-0
.15
7.0
420
03-0
.32
11.2
8-0
.44
10.3
9-0
.58
8.2
3-0
.48
7.7
5-0
.63
5.4
8-0
.53
5.1
320
04-0
.52
12.2
20.3
914.7
4-0
.67
9.3
4-0
.06
9.6
5-0
.45
7.1
0-0
.16
5.8
720
050.
6620
.88
1.8
417.6
70.0
716.3
80.9
813.2
8-0
.14
13.1
30.4
29.5
620
060.
2232
.60
0.1
015.9
3-0
.14
21.4
6-0
.25
10.9
4-0
.22
13.0
4-0
.29
7.8
120
070.
9119
.89
0.7
016.1
11.1
418.5
80.8
711.6
20.6
311.3
00.5
59.1
720
08-2
.98
14.7
6-1
.36
14.2
7-1
.86
11.6
6-0
.77
10.8
0-1
.33
9.2
3-0
.63
7.3
620
092.
1015
.97
0.9
614.9
30.8
010.9
7-0
.33
9.8
20.1
88.4
0-0
.47
6.9
820
104.
7223
.00
1.0
021.0
03.0
015.6
70.2
315.7
02.2
612.8
40.1
210.6
820
11-0
.37
24.8
70.8
727.0
8-1
.26
19.7
5-0
.19
19.8
7-1
.26
13.9
5-0
.60
13.9
3
Ave
rage
0.31
16.8
20.1
914.9
20.2
512.7
80.2
210.9
00.0
08.9
9-0
.04
7.5
0
33
Table 3: Momentum Profits for the 1996-2011 Period
This table presents momentum strategy profits during the 1996–2011 period. We follow the portfolioconstruction strategy in Jegadeesh and Titman (1993): at the beginning of each month, stocks inNYSE/NASDAQ/AMEX are sorted into ten deciles according to their past J (= 3, 6, 9, 12) months’cumulative returns. We hold each equal-weighted portfolio for K (= 1, 3, 6, 12) months after skipping S =1 month upon portfolio formation. P10 indicates the winner portfolio; P1 indicates the loser portfolio; andP10-P1 indicates the hedged winner-minus-loser portfolio. We exclude stocks with market capitalizationless than the 10% NYSE cutoff or a share price less than $5 at the formation month to ensure liquidity.Panel A reports monthly portfolio returns and associated t-statistics (in parentheses) using all the stocks.Panel B reports monthly portfolio returns and associated t-statistics (in parentheses) for stocks that alsohave listed option contracts.
Panel A: All stocks Panel B: Stocks with optionsK=1 K=3 K=6 K=12 K=1 K=3 K=6 K=12
J=3 P1 0.56 0.49 0.45 0.67 0.48 0.31 0.33 0.58(0.78) (0.70) (0.64) (0.98) (0.64) (0.43) (0.46) (0.87)
P10 1.17 1.10 1.13 1.00 0.72 0.76 0.86 0.79(1.99) (1.81) (1.89) (1.67) (1.20) (1.26) (1.45) (1.35)
P10-P1 0.61 0.62 0.68 0.33 0.23 0.45 0.54 0.21(1.11) (1.23) (1.48) (1.00) (0.39) (0.87) (1.18) (0.65)
J=6 P1 0.34 0.37 0.46 0.72 0.20 0.27 0.42 0.70(0.47) (0.50) (0.63) (1.03) (0.26) (0.35) (0.57) (1.01)
P10 1.46 1.35 1.27 1.05 1.14 1.02 1.02 0.83(2.33) (2.15) (2.07) (1.71) (1.86) (1.66) (1.71) (1.40)
P10-P1 1.12 0.98 0.80 0.33 0.94 0.75 0.60 0.12(1.78) (1.62) (1.45) (0.82) (1.37) (1.19) (1.09) (0.30)
J=9 P1 0.48 0.49 0.61 0.87 0.45 0.44 0.59 0.89(0.66) (0.67) (0.85) (1.27) (0.57) (0.57) (0.80) (1.28)
P10 1.36 1.29 1.10 0.92 1.04 1.05 0.93 0.75(2.15) (2.03) (1.76) (1.48) (1.7) (1.72) (1.54) (1.26)
P10-P1 0.87 0.80 0.50 0.05 0.59 0.61 0.34 -0.13(1.36) (1.31) (0.92) (0.12) (0.86) (0.94) (0.60) (-0.30)
J=12 P1 0.53 0.61 0.76 0.99 0.41 0.55 0.73 0.98(0.75) (0.87) (1.09) (1.46) (0.54) (0.74) (1.01) (1.43)
P10 1.05 1.01 0.93 0.82 0.76 0.77 0.74 0.65(1.66) (1.58) (1.46) (1.31) (1.24) (1.25) (1.21) (1.08)
P10-P1 0.53 0.40 0.16 -0.17 0.34 0.21 0.01 -0.33(0.86) (0.70) (0.31) (-0.38) (0.51) (0.34) (0.02) (-0.72)
34
Tab
le4:
Mon
thly
Ret
urn
sfo
rP
ortf
olio
sB
ased
onM
omen
tum
and
Cal
lOpti
onIm
plied
Vol
atilit
yG
row
th:
Wee
kly
,D
epen
den
tSor
ting
Th
ista
ble
pre
sents
mon
thly
retu
rns
for
mom
entu
man
dca
llop
tion
imp
lied
vola
tili
tygro
wth
dou
ble
-sort
edp
ort
foli
os.
Pan
elA
rep
ort
sth
ere
sult
sof
dep
end
ent
two-
way
sort
ing
(firs
tso
rtst
ock
sb
ase
don
thei
rp
ast
cum
ula
tive
retu
rns,
an
dth
enso
rtb
ase
don
imp
lied
vola
tili
tygro
wth
),an
dP
anel
Bp
rese
nts
the
mar
gin
alco
ntr
ibu
tion
of
sort
ing
on
the
imp
lied
vola
tili
tygro
wth
.F
or
the
win
ner
port
foli
oP
10,VS
conta
ins
stock
sw
ith
the
larg
est
wee
kly
imp
lied
vola
tili
tygr
owth
.F
or
the
lose
rp
ort
foli
oP
1,VS
conta
ins
stock
sw
ith
the
small
est
wee
kly
imp
lied
vola
tili
tygro
wth
.W
efi
xJ
(=6)
for
pas
tcu
mu
lati
vere
turn
calc
ula
tion
,sk
ipS
(=1)
month
,an
dh
old
port
foli
os
for
K(=
1,
3,
6)
month
s.M
om
entu
mra
nkin
gla
sts
for
Km
onth
s,an
dop
tion
ran
kin
gis
reca
lcu
late
dat
the
beg
inn
ing
of
each
hold
ing
month
base
don
imp
lied
vola
tili
tygro
wth
of
30-d
ayto
mat
uri
tyat
-th
e-m
oney
call
opti
ons.
We
excl
ud
est
ock
sw
ith
mark
etca
pit
aliza
tion
less
than
the
10%
NY
SE
cuto
ffor
ash
are
pri
cele
ssth
an
$5
inth
efo
rmat
ion
mon
thto
ensu
reli
qu
idit
y.W
eals
ow
inso
rize
the
data
each
month
by
excl
ud
ing
stock
sth
at
hav
eim
pli
edvo
lati
lity
gro
wth
inth
eto
pan
db
otto
m1%
.W
ere
por
tu
nad
just
edex
cess
retu
rns
an
dri
sk-a
dju
sted
alp
has
rela
tive
toth
eC
AP
M,
the
Fam
a-F
ren
chth
ree-
fact
or
mod
el,an
dth
eF
ama-
Fre
nch
thre
e-fa
ctor
plu
ssh
ort
-ter
mre
vers
al(S
TR
)fa
ctor
mod
el.
New
ey-W
est
fou
r-la
gad
just
edt-
stati
stic
sare
inp
are
nth
eses
.
Pan
elA
:M
onth
lyre
turn
sfo
rm
omen
tum
and
impli
edvo
lati
lity
grow
thd
oub
le-s
orti
ng
por
tfol
ios
K=1
K=3
K=6
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
P1
P10
P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1
VF
0.47
0.85
0.39
0.64
0.51
0.64
0.00
0.20
0.09
0.20
-0.04
0.13
0.11
0.18
(0.57)
(1.35)
(0.54)
(1.04)
(0.83)
(1.08)
(0.01)
(0.35)
(0.15)
(0.35)
(-0.06)
(0.25)
(0.21)
(0.35)
VM
0.33
1.27
0.94
1.14
1.12
1.26
1.02
1.18
1.20
1.33
0.89
1.01
1.06
1.15
(0.49)
(1.98)
(1.26)
(1.65)
(1.59)
(1.86)
(1.48)
(1.88)
(1.87)
(2.15)
(1.51)
(1.86)
(1.93)
(2.13)
VS
-0.17
1.38
1.55
1.73
1.68
1.78
1.32
1.47
1.45
1.52
1.04
1.16
1.19
1.25
(-0.21)
(2.20)
(2.16)
(2.66)
(2.57)
(2.69)
(2.01)
(2.54)
(2.54)
(2.61)
(1.78)
(2.26)
(2.35)
(2.38)
Pan
elB
:M
argi
nal
contr
ibu
tion
ofso
rtin
gon
the
impli
edvo
lati
lity
grow
th
K=1
K=3
K=6
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
VF
VS
VS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
F
P1
0.47
-0.17
-0.63
-0.57
-0.60
-0.60
-0.66
-0.62
-0.65
-0.64
-0.52
-0.48
-0.49
-0.49
(0.57)
(-0.21)
(-2.04)
(-1.97)
(-2.02)
(-2.04)
(-2.28)
(-2.45)
(-2.59)
(-2.64)
(-2.00)
(-2.02)
(-2.13)
(-2.17)
P10
0.85
1.38
0.53
0.51
0.57
0.55
0.66
0.65
0.70
0.69
0.55
0.55
0.59
0.59
(1.35)
(2.20)
(2.16)
(2.15)
(2.35)
(2.32)
(2.97)
(2.90)
(3.12)
(3.07)
(2.84)
(2.68)
(2.87)
(2.86)
P10-P
10.39
1.55
1.16
1.09
1.17
1.15
1.32
1.27
1.35
1.33
1.07
1.03
1.08
1.07
(0.54)
(2.16)
(2.80)
(2.68)
(2.84)
(2.87)
(3.43)
(3.54)
(3.85)
(3.91)
(3.05)
(2.93)
(3.12)
(3.18)
35
Tab
le5:
Mon
thly
Ret
urn
sfo
rP
ortf
olio
sB
ased
onM
omen
tum
and
Put
Opti
onIm
plied
Vol
atilit
yG
row
th:
Wee
kly
,D
epen
den
tSor
ting
Th
ista
ble
pre
sents
mon
thly
retu
rns
for
mom
entu
man
dp
ut
op
tion
imp
lied
vola
tili
tygro
wth
dou
ble
-sort
edp
ort
foli
os.
Pan
elA
rep
ort
sth
ere
sult
sof
dep
end
ent
two-
way
sort
ing
(firs
tso
rtst
ock
sb
ase
don
thei
rp
ast
cum
ula
tive
retu
rns,
an
dth
enso
rtb
ase
don
imp
lied
vola
tili
tygro
wth
),an
dP
anel
Bp
rese
nts
the
mar
gin
alco
ntr
ibu
tion
of
sort
ing
on
the
imp
lied
vola
tili
tygro
wth
.F
or
the
win
ner
port
foli
oP
10,VS
conta
ins
stock
sw
ith
the
smal
lest
wee
kly
imp
lied
vola
tili
tygr
owth
.F
or
the
lose
rp
ort
foli
oP
1,VS
conta
ins
stock
sw
ith
the
larg
est
wee
kly
imp
lied
vola
tili
tygro
wth
.W
efi
xJ
(=6)
for
pas
tcu
mu
lati
ve
retu
rnca
lcu
lati
on
,sk
ipS
(=1)
month
,an
dh
old
port
foli
os
for
K(=
1,
3,
6)
month
s.M
om
entu
mra
nkin
gla
sts
for
Km
onth
s,an
dop
tion
ran
kin
gis
reca
lcu
late
dat
the
beg
inn
ing
of
each
hold
ing
month
base
don
imp
lied
vola
tili
tygro
wth
of
30-d
ay-t
o-m
atu
rity
at-t
he-
mon
eyp
ut
opti
ons.
We
excl
ud
est
ock
sw
ith
mark
etca
pit
ali
zati
on
less
than
the
10%
NY
SE
cuto
ffor
ash
are
pri
cele
ssth
an
$5
inth
efo
rmat
ion
mon
thto
ensu
reli
qu
idit
y.W
eals
ow
inso
rize
the
data
each
month
by
excl
ud
ing
stock
sth
at
hav
eim
plied
vola
tili
tygro
wth
inth
eto
pan
db
otto
m1%
.W
ere
por
tu
nad
just
edex
cess
retu
rns
an
dri
sk-a
dju
sted
alp
has
rela
tive
toth
eC
AP
M,
the
Fam
a-F
ren
chth
ree-
fact
or
mod
el,an
dth
eF
ama-
Fre
nch
thre
e-fa
ctor
plu
ssh
ort
-ter
mre
vers
al(S
TR
)fa
ctor
mod
el.
New
ey-W
est
fou
r-la
gad
just
edt-
stati
stic
sare
inp
are
nth
eses
.
Pan
elA
:M
onth
lyre
turn
sfo
rm
omen
tum
and
impli
edvo
lati
lity
grow
thd
oub
le-s
orti
ng
por
tfol
ios
K=1
K=3
K=6
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
P1
P10
P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1
VF
0.22
1.19
0.96
1.16
1.06
1.16
1.05
1.21
1.13
1.21
0.82
0.94
0.94
0.99
(0.28)
(1.89)
(1.37)
(1.78)
(1.65)
(1.83)
(1.65)
(2.07)
(1.96)
(2.09)
(1.48)
(1.88)
(1.85)
(1.89)
VM
0.44
1.18
0.75
0.96
0.95
1.08
0.57
0.73
0.74
0.85
0.55
0.68
0.72
0.80
(0.53)
(1.84)
(1.01)
(1.50)
(1.44)
(1.71)
(0.84)
(1.21)
(1.21)
(1.44)
(0.91)
(1.26)
(1.34)
(1.49)
VS
0.12
1.28
1.16
1.37
1.28
1.41
0.88
1.04
0.99
1.11
0.57
0.72
0.72
0.81
(0.15)
(2.03)
(1.60)
(2.10)
(1.95)
(2.20)
(1.30)
(1.76)
(1.60)
(1.83)
(0.95)
(1.40)
(1.35)
(1.55)
Pan
elB
:M
argi
nal
contr
ibu
tion
ofso
rtin
gon
the
impli
edvo
lati
lity
grow
th
K=1
K=3
K=6
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
VF
VS
VS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
F
P1
0.22
0.12
-0.11
-0.11
-0.17
-0.18
0.11
0.11
0.09
0.06
0.06
0.05
0.05
0.01
(0.28)
(0.15)
(-0.42)
(-0.49)
(-0.77)
(-0.78)
(0.43)
(0.44)
(0.36)
(0.25)
(0.26)
(0.21)
(0.20)
(0.06)
P10
1.19
1.28
0.09
0.09
0.05
0.08
-0.06
-0.06
-0.05
-0.04
-0.18
-0.17
-0.17
-0.16
(1.89)
(2.03)
(0.36)
(0.41)
(0.22)
(0.32)
(-0.25)
(-0.26)
(-0.25)
(-0.20)
(-0.86)
(-0.83)
(-0.82)
(-0.8)
P10-P
10.96
1.16
0.20
0.21
0.22
0.25
-0.17
-0.17
-0.14
-0.11
-0.25
-0.22
-0.22
-0.18
(1.37)
(1.60)
(0.52)
(0.59)
(0.64)
(0.72)
(-0.45)
(-0.44)
(-0.38)
(-0.28)
(-0.69)
(-0.60)
(-0.58)
(-0.48)
36
Table 6: Fama-MacBeth Cross-Sectional Regressions with Options Implied Volatility Growth
This table presents the results of the Fama-MacBeth regressions. Independent variables include the pastsix-month cumulative return, option implied volatility growth, and their interaction, after controlling for
an array of firm characteristics. The interaction term PastCumRet × ∆ ˆIVC
(PastCumRet × ∆ ˆIVP
) isconstructed as the product of PastCumRet and ∆IV C (−∆IV P ) for stocks with cumulative returns abovethe median, and the product of PastCumRet and −∆IV C (∆IV P ) for stocks with cumulative returns belowthe median. Control variables include stock size, stock price, book-to-market ratio; stock trading volume,number of analyst coverage, the maximum daily return, market beta, Amihud illiquidity measure, realizedvolatility, idiosyncratic volatility, options’ open interest growth, and options’ trading volume change. Weexclude stocks with market capitalization less than the 10% NYSE cutoff or a share price less than $5 at theend of formation month to ensure liquidity. We also winsorize the data by excluding stocks that have impliedvolatility growth in the top and bottom 1%. Regressions are performed on the full sample as well as on thewinner/loser groups. The average slope coefficients and their Newey-West four-lag adjusted t-statistics arereported in parentheses.
(1) (2) (3) (4) (5) (6) (7)
PastCumRet -0.001 -0.004 -0.004 -0.006 -0.005(-0.28) (-0.86) (-0.8) (-0.96) (-0.91)
∆IV C 0.018 0.016 0.009(4.45) (3.88) (0.89)
PastCumRet×∆ ˆIVC
0.011 0.017(3.89) (2.35)
∆IV P -0.004 -0.007 -0.026(-1.43) (-2.02) (-2.64)
PastCumRet×∆ ˆIVP
-0.012 -0.022(-4.79) (-3.16)
Intercept 0.008 -0.010 -0.007 0.011 0.015 -0.005 0.027(1.69) (-1.50) (-1.03) (2.15) (2.91) (-0.37) (2.47)
Winner and loser x x
Adj. R2 0.10 0.09 0.10 0.09 0.10 0.12 0.12
37
Tab
le7:
Char
acte
rist
ics
ofP
ortf
olio
sSor
ted
by
Mom
entu
man
dIm
plied
Vol
atilit
yG
row
th:
Wee
kly
,D
epen
den
tSor
ting
Th
ista
ble
pre
sents
the
char
acte
rist
ics
for
mom
entu
man
dim
pli
edvo
lati
lity
gro
wth
dou
ble
-sort
edp
ort
foli
os.
Ch
ara
cter
isti
cs,
mea
sure
das
the
med
ian
valu
eof
stock
sw
ith
inea
chp
ortf
olio
,in
clu
de:
stock
size
(in
mil
lion
of
US
D),
stock
pri
ce(i
nU
SD
),st
ock
trad
ing
volu
me,
the
aver
age
nu
mb
erof
anal
yst
cove
rage
,fo
rmat
ion
per
iod
cum
ula
tive
retu
rn,
reali
zevo
lati
lity
,id
iosy
ncr
ati
cvo
lati
lity
,th
em
axim
um
dail
yre
turn
,th
eop
enin
tere
stgr
owth
,an
dth
ech
ange
inop
tion
trad
ing
volu
me.
Panel
Are
port
sth
ere
sult
sw
hen
call
op
tion
sare
use
d,
an
dP
an
elB
rep
ort
sth
ere
sult
sw
hen
pu
top
tion
sar
eu
sed
.Sto
cks
wit
hin
each
mom
entu
m-s
ort
edgro
up
are
sort
edin
toth
ree
equ
al
gro
up
sb
ase
don
thei
rim
pli
edvo
lati
lity
grow
th(s
mal
l,m
edia
n,
larg
e).
We
fix
J(=
6)
for
past
cum
ula
tive
retu
rnca
lcu
lati
on
,sk
ipS
(=1)
month
,an
dh
old
port
foli
os
for
K(=
1)
month
.W
eu
seca
ll(p
ut)
opti
ons
wit
h30
-day
tom
atu
rity
an
da
del
taof
0.5
(-0.5
).W
eex
clu
de
stock
sw
ith
mark
etca
pit
ali
zati
on
less
than
the
10%
NY
SE
cuto
ffor
ash
are
pri
cele
ssth
an$5
atth
een
dof
form
ati
on
month
toen
sure
liqu
idit
y.W
eals
ow
inso
rize
the
data
by
excl
ud
ing
stock
sth
at
hav
eim
pli
edvol
atil
ity
grow
thin
the
top
an
db
ott
om
1%
.P
ort
foli
os
sele
cted
as
the
win
ner
-min
us-
lose
rm
om
entu
mp
ort
foli
oare
ind
icate
din
bol
d.
Pan
elA
:P
ortf
olio
sfo
rmed
by
dep
end
entl
yso
rtin
gon
mom
entu
man
dca
llop
tion
imp
lied
vol
atil
ity
gro
wth
Size
Price
Stock
volume
Analyst
coverage
Cumulativeretu
rnSmall
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Loser-1
741
822
740
15
16
14
0.012
0.013
0.012
7.9
8.3
7.9
-0.36
-0.36
-0.36
21222
1471
1326
20
23
21
0.009
0.009
0.009
8.0
8.8
8.4
-0.21
-0.21
-0.21
31721
1942
1798
25
27
24
0.007
0.007
0.007
8.7
9.2
8.9
-0.12
-0.12
-0.12
42058
2385
2189
28
30
27
0.007
0.007
0.007
9.0
9.7
9.3
-0.06
-0.06
-0.06
52239
2653
2515
29
33
30
0.006
0.006
0.006
9.2
10.1
9.6
0.00
0.00
0.00
62461
2760
2671
30
34
31
0.006
0.006
0.006
9.2
10.0
9.7
0.06
0.06
0.06
72454
2846
2595
32
34
32
0.006
0.007
0.007
9.1
9.9
9.5
0.12
0.12
0.12
82357
2759
2507
33
36
32
0.007
0.007
0.007
8.9
9.8
9.5
0.21
0.21
0.20
92069
2374
2207
32
35
32
0.008
0.009
0.008
8.5
9.1
8.8
0.33
0.33
0.33
Winner
-10
1530
1698
1569
30
33
30
0.012
0.012
0.012
7.2
7.9
7.4
0.63
0.63
0.62
Rea
lizedvolatility
Idiosyncraticvolatility
Maxdailyretu
rnOIgrowth
Optionvolumech
ange
Small
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Loser-1
0.63
0.64
0.63
0.54
0.54
0.54
0.078
0.075
0.075
0.06
0.05
0.05
0.26
-5.41
-0.79
20.49
0.48
0.49
0.41
0.40
0.41
0.060
0.059
0.058
0.05
0.05
0.05
-0.67
-3.12
-1.45
30.42
0.41
0.42
0.34
0.33
0.35
0.052
0.050
0.050
0.05
0.05
0.05
-0.70
-3.54
-2.02
40.38
0.38
0.39
0.31
0.31
0.32
0.046
0.045
0.046
0.05
0.04
0.05
-0.55
-3.01
-1.44
50.37
0.36
0.37
0.30
0.29
0.30
0.044
0.043
0.044
0.05
0.04
0.05
-0.49
-2.10
-0.74
60.35
0.35
0.36
0.29
0.29
0.29
0.043
0.042
0.043
0.05
0.04
0.05
-0.64
-2.53
-1.24
70.36
0.36
0.37
0.30
0.29
0.30
0.044
0.042
0.044
0.05
0.04
0.05
-0.38
-3.34
-0.46
80.38
0.38
0.38
0.32
0.31
0.32
0.046
0.044
0.045
0.05
0.04
0.05
-0.83
-2.05
-1.05
90.43
0.42
0.42
0.36
0.35
0.35
0.051
0.050
0.049
0.05
0.05
0.05
1.69
-2.71
1.21
Winner
-10
0.54
0.54
0.54
0.46
0.45
0.46
0.064
0.062
0.063
0.06
0.05
0.06
-0.32
-4.66
2.36
38
Tab
le7
(con
t.)
Pan
elB
:P
ortf
olio
sfo
rmed
by
dep
end
entl
yso
rtin
gon
mom
entu
man
dp
ut
opti
onim
pli
edvol
atilit
ygr
owth
Size
Price
Stock
volume
Analyst
coverage
Cumulativeretu
rnSmall
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Loser-1
757
790
748
15
16
15
0.012
0.013
0.012
7.8
8.1
8.0
-0.36
-0.36
-0.36
21292
1419
1361
21
22
20
0.009
0.009
0.009
8.1
8.7
8.5
-0.21
-0.21
-0.21
31787
1962
1920
25
27
25
0.007
0.008
0.007
8.6
9.3
8.9
-0.12
-0.12
-0.12
42130
2342
2242
28
30
27
0.007
0.007
0.007
9.0
9.5
9.4
-0.06
-0.06
-0.06
52201
2655
2610
29
32
30
0.006
0.006
0.006
9.1
10.0
9.7
0.00
0.00
0.00
62534
2688
2800
31
33
32
0.006
0.006
0.006
9.4
9.8
10.0
0.06
0.06
0.06
72398
2792
2844
32
34
33
0.006
0.007
0.006
9.1
9.9
9.8
0.12
0.13
0.12
82310
2760
2658
33
35
33
0.007
0.007
0.007
9.0
9.8
9.5
0.21
0.21
0.21
91997
2335
2338
31
35
33
0.008
0.009
0.008
8.5
9.2
8.8
0.33
0.33
0.33
Winner
-10
1569
1633
1630
31
33
31
0.012
0.013
0.013
7.4
7.6
7.6
0.63
0.62
0.62
Rea
lizedvolatility
Idiosyncraticvolatility
Maxdailyretu
rnOIgrowth
Optionvolumech
ange
Small
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Small
Med
ian
Large
Loser-1
0.63
0.63
0.63
0.54
0.53
0.54
0.077
0.075
0.075
0.03
0.03
0.03
0.30
-1.11
-0.18
20.49
0.48
0.49
0.40
0.40
0.41
0.061
0.058
0.058
0.03
0.03
0.03
-0.40
-1.13
-0.20
30.42
0.42
0.42
0.34
0.34
0.35
0.052
0.051
0.051
0.03
0.03
0.03
-1.02
-0.70
-0.42
40.38
0.38
0.39
0.31
0.31
0.32
0.046
0.045
0.046
0.03
0.03
0.04
-0.27
-0.86
0.34
50.37
0.36
0.37
0.30
0.29
0.30
0.045
0.043
0.044
0.03
0.03
0.03
-0.54
-1.09
-0.22
60.36
0.35
0.36
0.29
0.29
0.29
0.043
0.042
0.042
0.03
0.03
0.03
-0.45
-0.84
-0.09
70.36
0.36
0.36
0.30
0.29
0.30
0.044
0.043
0.043
0.03
0.03
0.03
-0.33
-1.03
-0.02
80.38
0.38
0.38
0.32
0.31
0.32
0.046
0.044
0.045
0.04
0.03
0.04
-0.49
-1.19
0.63
90.42
0.43
0.42
0.35
0.35
0.35
0.051
0.049
0.050
0.04
0.04
0.04
-0.28
-1.14
-0.02
Winner
-10
0.54
0.54
0.54
0.46
0.45
0.46
0.064
0.062
0.063
0.05
0.05
0.05
-0.54
-1.48
1.67
39
Tab
le8:
Mon
thly
Ret
urn
sfo
rP
ortf
olio
sB
ased
onM
omen
tum
and
Opti
onIm
plied
Vol
atilit
yG
row
th:
Sto
cks
wit
hou
tE
arnin
gsA
nnou
nce
men
ts
Th
ista
ble
pre
sents
mon
thly
retu
rns
for
mom
entu
man
dop
tion
imp
lied
vola
tili
tygro
wth
dou
ble
-sort
edp
ort
foli
os.
We
excl
ud
est
ock
sth
at
hav
eea
rnin
gan
nou
nce
men
tsin
the
hol
din
gm
onth
.P
anel
A(c
all
)an
dP
an
elB
(pu
t)re
port
the
resu
lts
of
dep
end
ent
two-w
ayso
rtin
g(s
ort
stock
sb
ased
onth
eir
pri
cem
omen
tum
,an
dth
enso
rtth
emb
ase
don
imp
lied
vola
tili
tygro
wth
).F
or
the
win
ner
port
foli
oP
10,VS
conta
ins
stock
sw
ith
the
larg
est
(sm
alle
st)
wee
kly
call
(pu
t)im
pli
edvo
lati
lity
gro
wth
.F
or
the
lose
rp
ort
foli
oP
1,VS
conta
ins
stock
sw
ith
the
small
est
(larg
est)
wee
kly
call
(pu
t)im
pli
edvo
lati
lity
grow
th.
We
fix
J(=
6)
for
past
cum
ula
tive
retu
rnca
lcu
lati
on
,sk
ipS
(=1)
month
,an
dh
old
equ
al-
wei
ghte
dp
ortf
olio
sfo
rK
(=1)
mon
th.
Op
tion
sw
ith
30-d
ayto
matu
rity
wit
ha
del
taof
0.5
(-0.5
)are
use
d.
We
excl
ud
est
ock
sw
ith
mark
etca
pit
ali
zati
on
less
than
the
10%
NY
SE
cuto
ffor
ash
are
pri
cele
ssth
an
$5
inth
efo
rmati
on
month
toen
sure
liqu
idit
y.W
eals
ow
inso
rize
the
data
each
month
by
excl
ud
ing
stock
sth
ath
ave
imp
lied
vola
tili
tygro
wth
inth
eto
pan
db
ott
om
1%
.W
ere
port
un
ad
just
edex
cess
retu
rns
an
dri
sk-a
dju
sted
alp
has
rela
tive
toth
eC
AP
M,
the
Fam
a-F
ren
chth
ree-
fact
or
mod
el,
an
dth
eF
am
a-F
ren
chth
ree-
fact
or
plu
ssh
ort
-ter
mre
ver
sal
(ST
R)
fact
or
mod
el.
New
ey-W
est
fou
r-la
gad
just
edt-
stat
isti
csar
ein
pare
nth
eses
.
Pan
elA
:D
epen
den
tly
sort
ing
onca
llop
tion
sim
pli
edvo
lati
lity
gro
wth
Un
adju
sted
CA
PM
alp
ha
FF
3F
alp
ha
FF
3F
+S
TR
alp
ha
VF
VS
VS−VF
VF
VS
VS−VF
VF
VS
VS−VF
VF
VS
VS−VF
P1
0.14
-0.4
1-0
.55
-0.6
4-1
.14
-0.5
0-0
.69
-1.2
3-0
.54
-0.8
0-1
.32
-0.5
2(0
.17)
(-0.
51)
(-1.
5)(-
1.4
2)
(-2.6
1)
(-1.5
0)
(-1.5
7)
(-3.0
4)
(-1.5
8)
(-1.9
2)
(-3.2
7)
(-1.6
1)
P10
0.77
1.11
0.34
0.2
50.5
90.3
30.1
10.4
50.3
40.1
70.4
70.3
1(1
.19)
(1.7
8)(1
.08)
(0.5
4)
(1.4
9)
(1.0
2)
(0.3
1)
(1.3
6)
(1.0
2)
(0.4
8)
(1.3
9)
(0.9
5)
P10
-P1
0.64
1.53
0.89
0.8
91.7
30.8
40.8
01.6
80.8
80.9
61.7
90.8
3(0
.82)
(2.0
1)(1
.73)
(1.3
6)
(2.7
2)
(1.6
2)
(1.2
2)
(2.5
9)
(1.6
8)
(1.5
7)
(2.7
3)
(1.6
7)
Pan
elB
:D
epen
den
tly
sort
ing
onp
ut
op
tion
sim
pli
edvola
tili
tygro
wth
Un
adju
sted
CA
PM
alp
ha
FF
3F
alp
ha
FF
3F
+S
TR
alp
ha
VF
VS
VS−VF
VF
VS
VS−VF
VF
VS
VS−VF
VF
VS
VS−VF
P1
0.11
-0.1
2-0
.22
-0.6
2-0
.86
-0.2
4-0
.67
-0.9
9-0
.32
-0.7
4-1
.07
-0.3
3(0
.13)
(-0.
15)
(-0.
70)
(-1.3
1)
(-2.0
8)
(-0.8
4)
(-1.5
7)
(-2.5
3)
(-1.1
7)
(-1.7
6)
(-2.7
6)
(-1.1
9)
P10
0.89
1.15
0.26
0.3
70.6
10.2
40.2
30.4
80.2
40.2
50.5
20.2
7(1
.41)
(1.7
8)(0
.85)
(0.8
6)
(1.3
4)
(0.8
0)
(0.6
4)
(1.2
8)
(0.7
6)
(0.6
7)
(1.4
1)
(0.8
7)
P10
-P1
0.78
1.26
0.49
0.9
91.4
70.4
80.9
01.4
60.5
60.9
91.6
00.6
0(1
.07)
(1.6
7)(1
.06)
(1.5
4)
(2.2
4)
(1.1
2)
(1.4
2)
(2.2
2)
(1.3
4)
(1.5
5)
(2.4
4)
(1.4
5)
40
Tab
le9:
Mon
thly
Ret
urn
sfo
rP
ortf
olio
sB
ased
onM
omen
tum
and
Cal
lO
pti
onIm
plied
Vol
atilit
yG
row
th:
the
Impac
tof
Tra
nsa
ctio
nC
osts
Th
ista
ble
pre
sents
mon
thly
retu
rns
for
mom
entu
man
dca
llop
tion
imp
lied
vola
tili
tygro
wth
dou
ble
-sort
edp
ort
foli
os
aft
erco
nsi
der
ing
tran
sact
ion
cost
s.T
ran
sact
ion
cost
sar
eex
pre
ssed
inte
rms
of
are
stri
ctio
non
the
fract
ion
of
stock
sth
at
can
be
reb
ala
nce
dev
ery
month
:fo
rst
ock
sth
atre
qu
ire
reb
alan
cin
g,on
lyon
esw
ith
mark
etca
pit
ali
zati
on
inth
eto
px
per
centi
leca
nb
eso
ld/p
urc
hase
d.
We
con
sid
erth
ree
diff
eren
tsi
zere
stri
ctio
nle
vels
:80
%,
50%
,an
d20
%.
Th
ista
ble
pre
sents
the
retu
rns
of
the
win
ner
-min
us-
lose
rp
ort
foli
o.
We
fix
J(=
6)
for
past
cum
ula
tive
retu
rnca
lcu
lati
on,
skip
S(=
1)m
onth
,an
dh
old
equ
al-
wei
ghte
d/va
lue-
wei
ghte
dp
ort
foli
os
for
K(=
1)
month
.O
pti
ons
wit
h30-d
ay-t
o-m
atu
rity
wit
ha
del
taof
0.5
(-0.
5)ar
eu
sed
.W
eex
clu
de
stock
sw
ith
mark
etca
pit
ali
zati
on
less
than
the
10%
NY
SE
cuto
ffor
ash
are
pri
cele
ssth
an
$5
inth
efo
rmat
ion
mon
thto
ensu
reli
qu
idit
y.W
eals
ow
inso
rize
the
data
each
month
by
excl
ud
ing
stock
sth
at
hav
eim
pli
edvo
lati
lity
gro
wth
inth
eto
pan
db
otto
m1%
.W
ere
por
tu
nad
just
edex
cess
retu
rns
an
dri
sk-a
dju
sted
alp
has
rela
tive
toth
eC
AP
M,
the
Fam
a-F
ren
chth
ree-
fact
or
mod
el,an
dth
eF
ama-
Fre
nch
thre
e-fa
ctor
plu
ssh
ort
-ter
mre
vers
al(S
TR
)fa
ctor
mod
el.
New
ey-W
est
fou
r-la
gad
just
edt-
stati
stic
sare
inp
are
nth
eses
.
Equ
al-
wei
ghte
dV
alu
e-w
eighte
dU
nad
j.C
AP
MF
F3F
FF
4F
Un
ad
j.C
AP
MF
F3F
FF
4F
P10
-P1
P10-P
1P
10-P
1P
10-P
1P
10-P
1P
10-P
1P
10-P
1P
10-P
1
80%
1.32
1.4
91.4
71.5
71.4
91.7
41.6
11.
72
(1.9
9)(2
.56)
(2.5
4)
(2.6
4)
(1.7
3)
(2.1
5)
(1.9
9)
(2.0
9)
50%
0.98
1.1
31.1
11.2
11.4
21.6
71.5
51.
65
(1.7
7)(2
.39)
(2.3
5)
(2.5
8)
(1.7
2)
(2.1
6)
(2.0
0)
(2.1
1)
20%
0.66
0.7
40.8
20.8
61.0
61.2
61.2
41.
31
(1.6
5)(1
.89)
(2.0
8)
(2.1
5)
(1.4
9)
(1.8
3)
(1.8
2)
(1.9
0)
41
Tab
le10
:M
onth
lyR
eturn
sfo
rP
ortf
olio
sB
ased
onM
omen
tum
and
Cal
lO
pti
onIm
plied
Vol
atilit
yG
row
th:
Wee
kly
,D
epen
-den
tso
rtin
g,O
pti
onT
ime-
to-m
aturi
tyM
atch
esw
ith
Hol
din
gH
oriz
on
Th
ista
ble
pre
sents
mon
thly
retu
rns
for
mom
entu
man
dca
llop
tion
imp
lied
vola
tili
tygro
wth
dou
ble
-sort
edp
ort
foli
os.
Pan
elA
rep
ort
sth
ere
sult
sof
dep
end
ent
two-
way
sort
ing
(sor
tst
ock
sb
ase
don
thei
rp
rice
mom
entu
m,
an
dth
enso
rtth
emb
ase
don
imp
lied
vola
tili
tygro
wth
),an
dP
an
elB
pre
sents
the
mar
gin
alco
ntr
ibu
tion
ofso
rtin
gon
the
imp
lied
vola
tili
tygro
wth
.F
or
the
win
ner
port
foli
oP
10,VS
conta
ins
stock
sw
ith
the
larg
est
wee
kly
imp
lied
vola
tili
tygr
owth
.F
orth
elo
ser
port
foli
oP
1,VS
conta
ins
stock
sw
ith
the
small
est
wee
kly
imp
lied
vola
tili
tygro
wth
.W
efi
xJ
(=6)
for
pas
tcu
mu
lati
vere
turn
calc
ula
tion
,sk
ipS
(=1)
month
,an
dh
old
port
foli
os
for
K(=
2,
3,
6)
month
s.M
om
entu
mra
nkin
gfo
rea
chst
ock
last
sfo
rK
mon
ths,
and
opti
onra
nkin
gis
calc
ula
ted
at
the
end
of
the
skip
pin
gm
onth
acc
ord
ing
tow
eekly
imp
lied
vola
tili
tygro
wth
for
call
op
tion
sw
ith
ad
elta
of0.
5an
da
tim
eto
mat
uri
tyof
Km
onth
s.W
eex
clu
de
stock
sw
ith
mark
etca
pit
ali
zati
on
less
than
the
10%
NY
SE
cuto
ffor
ash
are
pri
cele
ssth
an$5
inth
efo
rmat
ion
mon
thto
ensu
reli
qu
idit
y.W
eals
ow
inso
rize
the
data
by
excl
ud
ing
stock
sth
at
hav
eim
pli
edvo
lati
lity
gro
wth
inth
eto
pan
db
otto
m1%
.W
ere
por
tu
nad
just
edex
cess
retu
rns
an
dri
sk-a
dju
sted
alp
has
rela
tive
toth
eC
AP
M,
the
Fam
a-F
ren
chth
ree-
fact
or
mod
el,an
dth
eF
ama-
Fre
nch
thre
e-fa
ctor
plu
ssh
ort
-ter
mre
vers
al(S
TR
)fa
ctor
mod
el.
New
ey-W
est
fou
r-la
gad
just
edt-
stati
stic
sare
inp
are
nth
eses
.
Pan
elA
:M
onth
lyre
turn
sfo
rm
omen
tum
and
impli
edvo
lati
lity
grow
thd
oub
le-s
orti
ng
por
tfol
ios
K=2
K=3
K=6
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
P1
P10
P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1
VF
0.53
0.91
0.38
0.60
0.50
0.61
0.35
0.53
0.45
0.55
0.32
0.48
0.48
0.56
(0.65)
(1.51)
(0.56)
(0.97)
(0.80)
(1.00)
(0.52)
(0.90)
(0.74)
(0.92)
(0.54)
(0.9)
(0.88)
(1.02)
VM
0.33
1.18
0.85
1.03
0.99
1.11
0.77
0.94
0.90
1.02
0.53
0.66
0.67
0.76
(0.42)
(1.85)
(1.22)
(1.61)
(1.49)
(1.74)
(1.17)
(1.61)
(1.48)
(1.74)
(0.91)
(1.30)
(1.27)
(1.44)
VS
-0.14
1.21
1.35
1.51
1.45
1.54
1.03
1.17
1.16
1.26
0.77
0.89
0.93
0.98
(-0.17)
(1.94)
(2.04)
(2.60)
(2.49)
(2.60)
(1.62)
(2.17)
(2.09)
(2.25)
(1.38)
(1.80)
(1.84)
(1.91)
Pan
elB
:M
argi
nal
contr
ibu
tion
ofso
rtin
gon
the
impli
edvo
lati
lity
grow
th
K=2
K=3
K=6
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
VF
VS
VS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
F
P1
0.53
-0.14
-0.66
-0.64
-0.67
-0.66
-0.58
-0.56
-0.60
-0.60
-0.28
-0.25
-0.60
-0.26
(0.65)
(-0.17)
(-3.24)
(-3.18)
(-3.29)
(-3.3)
(-3.69)
(-3.83)
(-4.13)
(-4.16)
(-2.12)
(-2.30)
(-4.13)
(-2.54)
P10
0.91
1.21
0.30
0.28
0.28
0.27
0.10
0.08
0.11
0.10
0.17
0.16
0.11
0.16
(1.51)
(1.94)
(1.60)
(1.41)
(1.49)
(1.40)
(0.64)
(0.55)
(0.80)
(0.76)
(1.55)
(1.44)
(0.80)
(1.52)
P10-P
10.38
1.35
0.97
0.92
0.95
0.93
0.68
0.64
0.71
0.70
0.45
0.41
0.71
0.42
(0.56)
(2.04)
(3.24)
(2.96)
(3.17)
(3.15)
(2.98)
(2.90)
(3.37)
(3.34)
(2.40)
(2.49)
(3.37)
(2.69)
42
Tab
le11
:M
onth
lyR
eturn
sfo
rP
ortf
olio
sB
ased
onM
omen
tum
and
Put
Opti
onIm
plied
Vol
atilit
yG
row
th:
Wee
kly
,D
epen
-den
tso
rtin
g,O
pti
onT
ime-
to-m
aturi
tyM
atch
esw
ith
Hol
din
gH
oriz
on
Th
ista
ble
pre
sents
mon
thly
retu
rns
for
mom
entu
man
dp
ut
op
tion
imp
lied
vola
tili
tygro
wth
dou
ble
-sort
edp
ort
foli
os.
Pan
elA
rep
ort
sth
ere
sult
sof
dep
end
ent
two-
way
sort
ing
(sor
tst
ock
sb
ase
don
thei
rp
rice
mom
entu
m,
an
dth
enso
rtth
emb
ase
don
imp
lied
vola
tili
tygro
wth
),an
dP
an
elB
pre
sents
the
mar
gin
alco
ntr
ibu
tion
ofso
rtin
gon
the
imp
lied
vola
tili
tygro
wth
.F
or
the
win
ner
port
foli
oP
10,VS
conta
ins
stock
sw
ith
the
small
est
wee
kly
imp
lied
vola
tili
tygr
owth
.F
orth
elo
ser
port
foli
oP
1,VS
conta
ins
stock
sw
ith
the
larg
est
wee
kly
imp
lied
vola
tili
tygro
wth
.W
efi
xJ
(=6)
for
pas
tcu
mu
lati
vere
turn
calc
ula
tion
,sk
ipS
(=1)
month
,an
dh
old
port
foli
os
for
K(=
2,
3,
6)
month
s.M
om
entu
mra
nkin
gfo
rea
chst
ock
last
sfo
rK
mon
ths,
and
opti
onra
nkin
gis
calc
ula
ted
at
the
end
of
the
skip
pin
gm
onth
acc
ord
ing
tow
eekly
imp
lied
vola
tili
tygro
wth
for
pu
top
tion
sw
ith
ad
elta
of-0
.5an
da
tim
eto
mat
uri
tyof
Km
onth
s.W
eex
clu
de
stock
sw
ith
mark
etca
pit
ali
zati
on
less
than
the
10%
NY
SE
cuto
ffor
ash
are
pri
cele
ssth
an$5
inth
efo
rmat
ion
mon
thto
ensu
reli
qu
idit
y.W
eals
ow
inso
rize
the
data
by
excl
ud
ing
stock
sth
at
hav
eim
pli
edvo
lati
lity
gro
wth
inth
eto
pan
db
otto
m1%
.W
ere
por
tu
nad
just
edex
cess
retu
rns
an
dri
sk-a
dju
sted
alp
has
rela
tive
toth
eC
AP
M,
the
Fam
a-F
ren
chth
ree-
fact
or
mod
el,an
dth
eF
ama-
Fre
nch
thre
e-fa
ctor
plu
ssh
ort
-ter
mre
vers
al(S
TR
)fa
ctor
mod
el.
New
ey-W
est
fou
r-la
gad
just
edt-
stati
stic
sare
inp
are
nth
eses
.
Pan
elA
:M
onth
lyre
turn
sfo
rm
omen
tum
and
impli
edvo
lati
lity
grow
thd
oub
le-s
orti
ng
por
tfol
ios
K=2
K=3
K=6
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
P1
P10
P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1
VF
0.15
0.93
0.77
0.94
0.91
1.01
0.63
0.77
0.77
0.88
0.47
0.60
0.63
0.70
(0.20)
(1.48)
(1.18)
(1.62)
(1.58)
(1.74)
(0.99)
(1.45)
(1.40)
(1.61)
(0.84)
(1.24)
(1.27)
(1.38)
VM
0.35
1.36
1.01
1.19
1.11
1.23
0.89
1.08
1.02
1.12
0.63
0.76
0.76
0.85
(0.46)
(2.18)
(1.49)
(1.95)
(1.76)
(1.98)
(1.38)
(1.79)
(1.62)
(1.83)
(1.06)
(1.47)
(1.41)
(1.60)
VS
0.41
1.13
0.72
0.91
0.82
0.94
0.63
0.81
0.74
0.84
0.43
0.57
0.56
0.63
(0.50)
(1.81)
(1.01)
(1.46)
(1.28)
(1.48)
(0.92)
(1.35)
(1.20)
(1.36)
(0.72)
(1.06)
(1.01)
(1.13)
Pan
elB
:M
argi
nal
contr
ibu
tion
ofso
rtin
gon
the
impli
edvo
lati
lity
grow
th
K=2
K=3
K=6
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
VF
VS
VS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
FVS-V
F
P1
0.15
0.41
0.26
0.22
0.27
0.25
0.14
0.11
0.16
0.17
0.08
0.06
0.09
0.09
(0.20)
(0.50)
(1.36)
(1.32)
(1.63)
(1.58)
(0.81)
(0.78)
(1.23)
(1.25)
(0.60)
(0.54)
(0.86)
(0.81)
P10
0.93
1.13
0.20
0.20
0.18
0.18
0.15
0.15
0.13
0.12
0.03
0.03
0.03
0.02
(1.48)
(1.81)
(1.07)
(1.06)
(0.98)
(1.03)
(0.95)
(0.94)
(0.82)
(0.78)
(0.26)
(0.19)
(0.15)
(0.15)
P10-P
10.77
0.72
-0.06
-0.03
-0.09
-0.07
0.01
0.04
-0.03
-0.04
-0.04
-0.03
-0.07
-0.06
(1.18)
(1.01)
(-0.21)
(-0.10)
(-0.36)
(-0.29)
(0.02)
(0.17)
(-0.16)
(-0.21)
(-0.22)
(-0.13)
(-0.30)
(-0.28)
43
Appendices
A Additional tables
44
Tab
leA
.1:
Bet
aL
oadin
gsan
dA
dju
sted
R-s
quar
esof
Win
ner
-Min
us-
Los
erP
ortf
olio
sB
ased
onM
omen
tum
and
Implied
Vol
atilit
yG
row
th:
Wee
kly
,D
epen
den
tSor
ting
Th
ista
ble
pre
sents
bet
alo
adin
gsan
dad
just
edR
2fo
rm
om
entu
man
dim
pli
edvo
lati
lity
gro
wth
dou
ble
-sort
edp
ort
foli
os
un
der
the
Fam
a-F
ren
chth
ree-
fact
orp
lus
the
shor
t-te
rmre
vers
al(S
TR
)fa
ctor
mod
el.
Pan
elA
rep
ort
sth
ere
sult
sth
at
use
call
op
tion
s,an
dP
an
elB
rep
ort
sth
ere
sult
sth
atu
sep
ut
opti
ons.
We
fix
J(=
6)fo
rpas
tcu
mu
lati
vere
turn
calc
ula
tion
,sk
ipS
(=1)
month
,an
dh
old
port
foli
os
for
K(=
1)
month
s.O
pti
on
ran
kin
gis
calc
ula
ted
bas
edon
imp
lied
vola
tili
tygro
wth
of
30-d
ayto
matu
rity
call
(pu
t)op
tion
sw
ith
ad
elta
of
0.5
(-0.5
).W
eex
clu
de
stock
sw
ith
mar
ket
cap
ital
izat
ion
less
than
the
10%
NY
SE
cuto
ffor
ash
are
pri
cele
ssth
an
$5
inth
efo
rmati
on
month
toen
sure
liqu
idit
y.W
eals
ow
inso
rize
the
dat
aby
excl
udin
gst
ock
sth
ath
ave
imp
lied
vola
tili
tygro
wth
inth
eto
pan
db
ott
om
1%
.N
ewey
-Wes
tfo
ur-
lag
ad
just
edt-
stati
stic
sare
inp
are
nth
eses
.
Panel
A:Dep
enden
tlydouble-sortingbymomen
tum
andca
lloptionim
plied
volatility
growth
βm
kt
βsm
bβhm
lβstr
Adj.
R2
VF
VS
VS-V
FVF
VS
VS-V
FVF
VS
VS-V
FVF
VS
VS-V
FVF
VS
VS-V
F
P1
1.59
1.46
-0.13
0.47
0.52
0.05
-0.19
-0.13
0.06
0.45
0.42
-0.03
0.69
0.67
0.01
(10.05)
(10.35)
(-2.1)
(2.03)
(2.03)
(0.41)
(-0.84)
(-0.49)
(0.65)
(2.12)
(2.00)
(-0.35)
P10
1.04
1.06
0.02
1.15
1.04
-0.11
-0.18
-0.27
-0.09
-0.21
-0.14
0.07
0.76
0.75
0.00
(12.37)
(12.00)
(0.39)
(8.07)
(6.70)
(-2.18)
(-1.06)
(-1.43)
(-1.11)
(-1.59)
(-1.20)
(1.12)
P10-P
1-0.55
-0.40
0.15
0.68
0.52
-0.16
0.01
-0.14
-0.15
-0.67
-0.56
0.11
0.21
0.13
0.02
(-2.42)
(-1.84)
(1.94)
(2.04)
(1.32)
(-1.19)
(0.02)
(-0.32)
(-1.08)
(-2.02)
(-1.80)
(0.81)
Panel
B:Dep
enden
tlydouble-sortingbymomen
tum
andputoptionim
plied
volatility
growth
βm
kt
βsm
bβhm
lβstr
Adj.
R2
VF
VS
VS-V
FVF
VS
VS-V
FVF
VS
VS-V
FVF
VS
VS-V
FVF
VS
VS-V
F
P1
1.50
1.53
0.03
0.47
0.47
0.00
-0.27
-0.11
0.16
0.44
0.48
0.04
0.68
0.69
0.01
(10.1)
(10.04)
(0.61)
(1.96)
(1.87)
(0.04)
(-0.94)
(-0.49)
(1.46)
(2.08)
(2.39)
(0.37)
P10
1.07
1.10
0.03
1.01
1.07
0.06
-0.25
-0.15
0.10
-0.11
-0.24
-0.13
0.74
0.75
0.01
(11.69)
(12.75)
(0.62)
(8.78)
(7.50)
(0.89)
(-1.35)
(-0.81)
(1.50)
(-0.98)
(-1.81)
(-1.59)
P10-P
1-0.42
-0.42
0.00
0.54
0.59
0.06
0.02
-0.04
-0.06
-0.55
-0.72
-0.16
0.13
0.18
0.00
(-1.89)
(-1.86)
(0.04)
(1.66)
(1.57)
(0.44)
(0.05)
(-0.10)
(-0.47)
(-1.84)
(-2.24)
(-1.16)
45
Table A.2: Monthly Returns for Portfolios Sorted by Implied Volatility Growth
This table presents monthly excess returns, risk-adjusted alphas, risk loadings, and adjusted R2 forportfolios sorted by implied volatility growth. V3 contains stocks with the largest (smallest) weekly call(put) implied volatility growth. Rankings are assigned at the beginning of each month and stocks are heldfor one month. Option ranking is calculated according to weekly implied volatility growth of 30-day tomaturity call (put) options with a delta of 0.5 (-0.5). We exclude stocks with market capitalization lessthan the 10% NYSE cutoff or a share price less than $5 in the formation month to ensure liquidity. We alsowinsorize the data by excluding stocks that have implied volatility growth in the top and bottom 1%. Weuse the Fama-French three-factor plus the short-term reversal (STR) model. Newey-West four-lag adjustedt-statistics are in parentheses.
Panel A: Portfolio sorted by call option implied volatility growth
Ex. return Alpha βmkt βsmb βhml βstr Adj. R2
V1 0.28 -0.38 1.08 0.46 0.14 0.11 0.94(0.62) (-3.06) (32.65) (5.74) (2.42) (2.32)
V2 0.68 0.00 1.14 0.47 0.13 0.08 0.95(1.45) (0.00) (45.45) (7.62) (2.69) (1.89)
V3 0.85 0.18 1.13 0.43 0.10 0.14 0.93(1.80) (1.34) (34.07) (5.31) (1.79) (2.19)
V3-V1 0.57 0.56 0.05 -0.03 -0.04 0.03 0.01(4.25) (3.87) (1.65) (-0.55) (-0.66) (0.35)
Panel B: Portfolio sorted by put option implied volatility growth
Ex. return Alpha βmkt βsmb βhml βstr Adj. R2
V1 0.59 -0.07 1.10 0.46 0.15 0.08 0.95(1.31) (-0.58) (34.88) (6.62) (2.59) (2.98)
V2 0.70 0.03 1.12 0.46 0.11 0.10 0.96(1.51) (0.30) (43.32) (8.12) (2.32) (2.25)
V3 0.56 -0.13 1.13 0.45 0.11 0.14 0.94(1.18) (-1.03) (37.03) (5.74) (2.35) (2.53)
V3-V1 -0.04 -0.06 0.03 -0.01 -0.04 0.06 0.03(-0.34) (-0.57) (0.97) (-0.21) (-0.99) (1.09)
46
Table A.3: Monthly Returns for Portfolios Based on Momentum and Out-of-the-moneyOption Implied Volatility Growth: Weekly, Dependent Sorting
This table presents monthly returns for momentum and option implied volatility growth double-sortedportfolios. Panel A reports the results that use call options, and Panel B reports the results that use putoptions. For the winner portfolio P10, VS contains stocks with the largest (smallest) weekly call (put)implied volatility growth. For the loser portfolio P1, VS contains stocks with the smallest (largest) weeklycall (put) implied volatility growth. We fix J (= 6) for past cumulative return calculation, skip S (= 1)month, and hold portfolios for K (= 1, 3, 6) months. Momentum ranking for each stock lasts for K months,option ranking is recalculated each month using 30-day to maturity call/put options. Two sets of averageimplied volatility growth are considered: the average across options with a delta of 0.35, 0.40, and 0.45 andthe average across options with a delta of 0.2, 0.25, and 0.30 (negative for put options). We exclude stockswith market capitalization less than the 10% NYSE cutoff or a share price less than $5 in the formationmonth to ensure liquidity. We also winsorize the data by excluding stocks that have implied volatilitygrowth in the top and bottom 1%. We report risk-adjusted alphas for the hedged winner-minus-loserportfolios (P10-P1) relative to the Fama-French three-factor plus the short-term reversal (STR) factormodel. Newey-West four-lag adjusted t-statistics are in parentheses.
Panel A: Out-of-the-money call option implied volatility growth
Delta = 0.35, 0.40, 0.45 Delta = 0.20, 0.25, 0.30K=1 K=3 K=6 K=1 K=3 K=6
VF 0.56 0.19 0.08 0.55 0.12 0.06(0.95) (0.33) (0.14) (0.89) (0.20) (0.12)
VM 1.21 1.32 1.06 1.19 1.44 1.10(1.73) (2.11) (2.02) (1.78) (2.37) (1.94)
VS 1.96 1.66 1.53 1.68 1.40 1.33(2.87) (2.65) (2.72) (2.55) (2.27) (2.46)
VS-VF 1.40 1.47 1.45 1.13 1.29 1.27(3.22) (3.65) (3.68) (2.96) (3.34) (3.25)
Panel A: Out-of-the-money call option implied volatility growth
Delta = 0.35, 0.40, 0.45 Delta = 0.20, 0.25, 0.30K=1 K=3 K=6 K=1 K=3 K=6
VF 1.09 1.18 0.90 1.03 1.05 0.79(1.76) (2.04) (1.79) (1.55) (1.79) (1.51)
VM 0.87 0.71 0.66 1.01 1.00 0.80(1.16) (1.11) (1.14) (1.66) (1.77) (1.54)
VS 1.55 1.17 0.92 1.28 0.91 0.77(2.52) (2.00) (1.76) (1.92) (1.43) (1.39)
VS-VF 0.46 -0.01 0.02 0.25 -0.14 -0.02(1.29) (-0.02) (0.04) (0.64) (-0.40) (-0.05)
47
Table A.4: Momentum Profit and Marginal Contribution of Double-sorting on Momentumand Implied Volatility Growth: Value-weighted Portfolios
This table presents monthly value-weighted portfolio returns for momentum and option implied volatilitygrowth double-sorted portfolios. Panel A report risk-adjusted alphas for momentum strategy, and PanelB reports marginal contribution of implied volatility growth sorting for momentum profit. We fix J(= 6) for past cumulative return calculation, skip S (= 1) month, and hold portfolios for K (= 1)month. We use options with 30-day to maturity and a delta of 0.5 (-0.5) for call (put). We excludestocks with market capitalization less than the 10% NYSE cutoff or a share price less than $5 in theformation month to ensure liquidity. We also winsorize the data by excluding stocks that have impliedvolatility growth in the top and bottom 1%. Alphas are calculated based on the Fama-French(1993) three-factor plus the short-term reversal factor model. Newey-West four-lag adjusted t-statistics are in parentheses.
Panel A: Risk-adjusted alphas of momentum winner-minus-loser portfolios
Call (P10-P1) Put (P10-P1)K=1 K=3 K=6 K=1 K=3 K=6
VF 0.29 -0.09 0.02 0.91 1.10 1.220.33 -0.12 0.03 1.41 1.86 2.27
VM 1.21 1.11 0.96 1.64 1.15 1.151.59 1.57 1.68 2.22 1.67 1.85
VS 1.73 1.96 1.94 0.89 0.67 0.352.10 2.83 3.13 1.02 0.79 0.51
Panel B: Marginal contribution of sorting on the implied volatility growth
Call (VS-VF ) Put(VS-VF )K=1 K=3 K=6 K=1 K=3 K=6
P1 -0.88 -1.03 -1.14 -0.13 0.13 0.45-1.43 -1.79 -2.15 -0.25 0.23 0.89
P10 0.56 1.02 0.78 -0.14 -0.31 -0.431.45 2.71 2.50 -0.37 -0.92 -1.53
P10-P1 1.43 2.05 1.92 -0.02 -0.44 -0.881.86 2.79 2.79 -0.03 -0.64 -1.53
48
Tab
leA
.5:
Mon
thly
Ret
urn
sfo
rP
ortf
olio
sSor
ted
by
Mom
entu
man
dIm
plied
Vol
atilit
yG
row
th:
Mon
thly
,D
epen
den
tly
Sor
ting
Th
ista
ble
pre
sents
mon
thly
exce
ssre
turn
sfo
rm
om
entu
man
dim
pli
edvola
tility
gro
wth
dep
end
entl
yd
oub
le-s
orte
dp
ort
foli
os.
Imp
lied
vola
tili
tygr
owth
isca
lcu
late
dfo
rth
esk
ipp
ing
mon
th.
Pan
elA
rep
ort
sth
ere
sult
sth
at
use
call
op
tion
s,an
dP
an
elB
rep
ort
sth
ere
sult
sth
at
use
pu
top
tion
s.F
orth
ew
inn
erp
ortf
olio
P10
,VS
conta
ins
stock
sw
ith
the
larg
est
(sm
all
est)
month
lyca
ll(p
ut)
impli
edvo
lati
lity
gro
wth
.F
or
the
lose
rp
ortf
olio
P1,VS
conta
ins
stock
sw
ith
the
small
est
(larg
est)
month
lyca
ll(p
ut)
imp
lied
vola
tili
tygro
wth
.W
efi
xJ
(=6)
month
sfo
rp
ast
cum
ula
tive
retu
rnca
lcu
lati
on,
skip
S(=
1)
month
,an
dhold
port
foli
os
for
K(=
1,
3,
6)
month
s.M
om
entu
mra
nkin
gfo
rea
chst
ock
last
sfo
rK
mon
ths,
and
opti
onra
nkin
gis
reca
lcu
late
dea
chm
onth
base
don
imp
lied
vola
tili
tygro
wth
of
30-d
ayto
matu
rity
at-
the-
month
op
tion
s.W
eex
clu
de
stock
sw
ith
mar
ket
cap
ital
izat
ion
less
than
the
10%
NY
SE
cuto
ffor
ash
are
pri
cele
ssth
an
$5
inth
efo
rmati
on
month
toen
sure
liqu
idit
y.W
eal
sow
inso
rize
the
dat
aby
excl
ud
ing
stock
sth
at
hav
eim
pli
edvo
lati
lity
gro
wth
inth
eto
pan
db
otto
m1%
.W
ere
port
un
ad
just
edex
cess
retu
rns
and
risk
-ad
just
edal
ph
asre
lati
veto
the
CA
PM
,th
eF
am
a-F
ren
chth
ree-
fact
or
mod
el,
an
dth
eF
am
a-F
ren
chth
ree-
fact
or
plu
sth
esh
ort-
term
rever
sal
(ST
R)
fact
orm
od
el.
New
ey-W
est
fou
r-la
gad
just
edt-
stati
stic
sare
inp
are
nth
eses
.
Pan
elA
:D
epen
den
tly
sort
edby
mom
entu
man
dca
llop
tion
imp
lied
vol
atil
ity
grow
th
K=1
K=3
K=6
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
P1
P10
P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1
VF
0.49
1.05
0.56
0.78
0.73
0.89
0.25
0.45
0.41
0.57
0.07
0.24
0.28
0.40
(0.58)
(1.61)
(0.73)
(1.11)
(1.05)
(1.41)
(0.35)
(0.7)
(0.64)
(0.97)
(0.12)
(0.45)
(0.51)
(0.78)
VM
0.42
1.19
0.78
0.98
0.92
1.03
0.64
0.80
0.77
0.88
0.67
0.80
0.80
0.88
(0.53)
(1.94)
(1.12)
(1.73)
(1.56)
(1.84)
(0.95)
(1.44)
(1.32)
(1.57)
(1.08)
(1.56)
(1.51)
(1.71)
VS
-0.26
1.15
1.41
1.58
1.50
1.57
1.29
1.42
1.38
1.41
1.11
1.20
1.22
1.22
(-0.35)
(1.86)
(2.15)
(2.51)
(2.33)
(2.39)
(2.14)
(2.48)
(2.38)
(2.37)
(2.08)
(2.35)
(2.4)
(2.34)
Pan
elB
:D
epen
den
tly
sort
edby
mom
entu
man
dp
ut
opti
onim
pli
edvol
atil
ity
grow
th
K=1
K=3
K=6
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
Unadj.
CAPM
FF3F
FF3F+STR
P1
P10
P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1P10-P
1
VF
-0.09
0.84
0.93
1.12
1.07
1.14
0.84
0.98
0.95
0.99
0.60
0.70
0.73
0.75
(-0.12)
(1.31)
(1.38)
(1.76)
(1.72)
(1.82)
(1.31)
(1.66)
(1.64)
(1.68)
(1.06)
(1.39)
(1.45)
(1.46)
VM
0.50
1.40
0.90
1.11
1.03
1.13
0.65
0.81
0.76
0.86
0.54
0.66
0.66
0.72
(0.64)
(2.27)
(1.26)
(1.76)
(1.57)
(1.77)
(1.02)
(1.44)
(1.28)
(1.51)
(0.94)
(1.34)
(1.28)
(1.45)
VS
0.10
1.16
1.06
1.25
1.23
1.40
0.89
1.08
1.05
1.20
0.74
0.91
0.93
1.04
(0.12)
(1.83)
(1.39)
(1.86)
(1.8)
(2.19)
(1.29)
(1.77)
(1.71)
(2.07)
(1.23)
(1.69)
(1.71)
(1.97)
49
Table A.6: Decomposition of Implied Volatility Growth
This table presents the distribution of estimation results of implied volatility growth decomposition. Wedecompose the daily log implied volatility growth into three parts: the lagged call and put implied volatilitygrowths, and an innovation term, by running the following set of VAR:
log(∆IV Ct ) = βC
1 log(∆IV Ct−1) + βC
2 log(∆IV Pt−1) + εCt
log(∆IV Pt ) = βP
1 log(∆IV Ct−1) + βP
2 log(∆IV Pt−1) + εPt ,
where ∆IV Ct (∆IV P
t ) is the daily implied volatility growth calculated using 30 day-to-maturity call (put)options with a delta of 0.5(-0.5). We run this set of two regressions for each security using its full sample.A security is required to have at least thirty observations to ensure estimation accuracy. Distribution ofcoefficients, t-statistics, and adjusted R2 averaged across all stocks are presented in the table. t-statisticsare in absolute values.
Call regressionsMean Min P25 P50 P75 Max
βC1 -0.3624 -0.7966 -0.4190 -0.3632 -0.3087 0.2698
(14.08) (0.02) (8.85) (13.90) (18.70) (44.17)βC2 0.0815 -0.7270 0.0259 0.0806 0.1371 0.8845
(3.50) (0.00) (1.21) (2.77) (5.15) (16.26)Adj. R2 14.2% -3.6% 9.7% 14.0% 18.3% 58.1%
Put regressionsMean Min P25 P50 P75 Max
βP1 0.1126 -0.4649 0.0587 0.1146 0.1659 1.7225
(4.94) (0.00) (1.94) (4.11) (7.42) (19.03)βP2 -0.3657 -0.8946 -0.4255 -0.3656 -0.3084 0.0751
(14.22) (0.01) (9.08) (13.86) (18.68) (55.10)Adj R2 15.2% -4.4% 10.5% 15.2% 19.5% 67.9%
50
Tab
leA
.7:
Mon
thly
Ris
k-a
dju
sted
Ret
urn
sfo
rp
ortf
olio
sso
rted
onM
omen
tum
and
Innov
atio
nof
Wee
kly
Implied
Vol
atilit
yG
row
th
Th
ista
ble
pre
sents
the
resu
lts
for
dou
ble
-sort
edp
ort
foli
os
on
mom
entu
man
din
nov
ati
on
of
wee
kly
call
(pu
t)op
tion
imp
lied
vola
tili
tygro
wth
,ca
lcu
late
dfr
oma
dai
lyon
e-la
gve
ctor
auto
regre
ssio
n(V
AR
).P
an
els
Aand
Bre
port
the
Fam
a-F
ren
chth
ree
fact
ors
plu
ssh
ort
-ter
mre
vers
al
(ST
R)
faco
trad
just
edal
ph
asfo
rp
ortf
olio
su
sin
gca
llan
dp
ut
info
rmati
on
,re
spec
tivel
y.F
or
the
win
ner
port
foli
oP
10,VS
conta
ins
stock
sw
ith
the
larg
est
(sm
alle
st)
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9)
51
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