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.

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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.

15

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

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0.8

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41.7

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81.5

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80.8

56.3

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99-0

.08

12.9

2-0

.01

10.8

40.2

110.3

20.3

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7.5

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420

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12.4

90.4

211.9

10.5

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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)

inn

ovat

ion

ofw

eekly

call

(pu

t)im

pli

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51