Journal of Financial Economics 119 (2016) 578–598
Contents lists available at ScienceDirect
Journal of Financial Economics
journal homepage: www.elsevier.com/locate/finec
Nominal price illusionR
Justin Birru a,∗, Baolian Wang b,∗∗
a
b
Fisher College of Business, The Ohio State University, Columbus, OH 43215, USA
Gabelli School of Business, Fordham University, New York, NY 10023, USA
a r t i c l e
i n f o
Article history:
Received 9 October 2014
Revised 1 May 2015
Accepted 29 May 2015
Available online 22 January 2016
JEL classification:
G02
G11
G12
G13
G14
Keywords:
Nominal price
Skewness
Stock splits
Options
Behavioral finance
a b s t r a c t
We explore the psychology of stock price levels and provide evidence that investors suffer
from a nominal price illusion in which they overestimate the room to grow for low-priced
stocks relative to high-priced stocks. While it has become increasingly clear that nominal
price levels influence investor behavior, why prices matter to investors is a question that
as of yet has gone unanswered. We find widespread evidence that investors systematically
overestimate the skewness of low-priced stocks. In the broad cross-section of stocks, we
find that investors substantially overweight the importance of price when forming skewness expectations. Asset pricing implications of our findings are borne out in the options
market. A zero-cost option portfolio strategy that exploits investor overestimation of skewness for low-priced stocks generates significant abnormal returns. Finally, investor expectations of future skewness increase drastically on days that a stock undergoes a split to
a lower nominal price. Empirically, however, future physical skewness decreases following
splits.
© 2016 Elsevier B.V. All rights reserved.
1. Introduction
R
We are grateful for valuable comments received from Yakov Amihud,
Nicholas Barberis, Brian Boyer, Kalok Chan, Zhi Da, Sudipto Dasgupta,
David Hirshleifer, Nishad Kapadia, Bingxin Li, Laura Liu, Xuewen Liu, Abhiroop Mukherjee, Sophie Ni, Kasper Nielsen, Mark Seasholes, Rik Sen,
Bruno Solnik, John Wei, Joakim Westerhold, Chu Zhang, Guochang Zhang,
Feng Zhao, and seminar participants at AFA 2014, Yale Whitebox Advisors
Graduate Student Conference 2013, Asian Finance Association 2013, Financial Management Association Asia 2013, Financial Management Association Europe 2013, Financial Management Association 2013, Hong Kong
University of Science and Technology, Hong Kong Baptist University, The
Ohio State University, Sun Yat-sen University and Tsinghua University. We
also thank Patrick Dennis for providing the computational code. All errors
are our responsibility.
∗
Corresponding author. Tel.: +1 614 688 1299; fax: +1 614 292 2418.
∗∗
Corresponding author. Tel.: +1 212 636 6118; fax: +1 646 312 8295.
E-mail addresses: [email protected] (J. Birru), [email protected]
(B. Wang).
http://dx.doi.org/10.1016/j.jfineco.2016.01.027
S0304-405X(16)0 0 034-9/© 2016 Elsevier B.V. All rights reserved.
The level of a firm’s stock price is arbitrary as it can be
manipulated by the firm via altering the number of shares
outstanding. Nevertheless, nominal prices clearly influence
investor behavior. For example, individuals tend to hold
lower-priced stocks than institutions.1 Schultz (20 0 0) finds
additional evidence of investor price-level preferences,
showing an increase in the number of small shareholders
following a split to a lower price level. Fernando, Krishnamurthy, and Spindt (2004) find that IPO offer price plays
a strong role in determining investor composition. Finally,
Green and Hwang (2009) find particularly strong evidence
that investors categorize stocks based on price. They show
that similarly priced stocks move together; after a stock
split, splitting stocks experience increased comovement
1
See Gompers and Metrick (2001), Dyl and Elliot (2006), Kumar and
Lee (2006), and Kumar (2009).
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
with low-priced stocks and decreased comovement with
high-priced stocks.
Firms appear to be well aware of the important role
that nominal prices play in influencing investor perceptions, as they frequently engage in the active management of share price levels in an apparent effort to cater
to investor demand. For instance, despite the lack of a rational explanation, firms have proactively managed share
prices to stay in a relatively constant nominal range since
the Great Depression (Weld, Michaely, Thaler, and Benartzi,
2009). Baker, Greenwood, and Wurgler (2009) find that investors have time-varying preferences for stocks of different nominal price levels and that firms actively manage
their share price levels to maximize firm value by catering
to these time-varying investor preferences. Dyl and Elliot
(2006) also find evidence that firms manage share prices
to appeal to their investor base in an effort to increase
the value of the firms. The rationale for investors’ focus
on nominal prices is not well understood, as past work
has focused on the implications of these preferences while
only hypothesizing about the potential underlying drivers.
In short, while past research shows that nominal prices
clearly influence the behavior of investors, why prices matter to investors is an as of yet unanswered question.
The lack of empirical evidence has not dissuaded speculation as to why investors are influenced by nominal
prices. For example, Kumar (2009, p. 1890) states that “as
with lotteries, if investors are searching for ‘cheap bets’,
they are likely to find low-priced stocks attractive”. Green
and Hwang (2009, p. 38) hypothesize that “investors may
perceive low-priced stocks as being closer to zero and
farther from infinity, thus having more upside potential”.
While Baker, Greenwood, and Wurgler (2009, p. 2562)
state that “One question that the results raise, and that we
leave to future work, is why nominal share prices matter
to investors...Perhaps some investors suffer from a nominal
illusion in which they perceive that a stock is cheaper after
a split, has more ‘room to grow,’ or has ‘less to lose’”.
A great deal of anecdotal evidence also exists that investors believe low-priced stocks have more room to grow.
For example, a number of mutual fund families offer lowpriced stock funds that primarily invest in stocks trading
below a specified price per share (the cutoff for meeting
the low-priced definition varies by fund, but it is typically in the $15–35 range).2 This is often viewed as a marketing gimmick designed to appeal to investor psychology.3 The notion that low-priced stocks have more upside potential is often reinforced by the funds themselves.
2
Fidelity’s Low-Priced Stock Fund has operated since 1989, with an explicit strategy of investing in stocks priced below $35 per share. Royce’s
Low-Price Fund has been in operation since 1993, with the strategy of investing in stocks priced below $25. A now defunct low-priced fund was
launched by Robertson Stephens in 1995. Perritt launched its own version
in 2012, with a strategy of investing in stocks priced below $15.
3
Media sources that have referred to low-priced funds as gimmicks are
too plentiful to cite, but they include the New York Times, the Washington
Post, Mutual Fund Observer, Morningstar, Motley Fool, and even the manager of Fidelity’s Low-Priced Fund (Schiffres, 2002, p. 64): “I was an analyst at Fidelity when everyone was asked for ideas for new funds…They
accepted this one and added the low-price provision. It’s a bit of a
gimmick.”
579
The long-time manager of Fidelity’s Low-Priced Stock Fund
argues that “it’s easier for a $4 stock to go to $8 than for a
$40 stock to go to $80”. 4 And the Perritt Low Priced Stock
Fund claims that stocks priced under $15 have “plenty of
room to move”.5
In this paper, we provide evidence that investors exhibit
a psychological bias in the manner in which they relate
nominal prices to expectations of future return patterns.
We find evidence that investors suffer from the illusion
that low price stocks have more upside potential. In doing so, we identify one potential driver of investor demand
shifts that have been shown to lead to supply responses
from corporations.
In attempting to assess expectations of upside potential, the natural variable to focus on is skewness. We rely
empirically on the options market to extract investor skewness expectations. A key insight of our analysis is the use
of option-implied risk-neutral skewness (RNSkew), which
is a market-based ex-ante measure of investors’ expectations. By utilizing risk-neutral skewness extracted from option prices, we are able to circumvent the need for a long
time series of returns to estimate skewness. Instead, we
can assess how market expectations of an asset’s future
skewness change on a daily basis. In doing so, we follow
a number of recent papers in inferring investor expectations from option-implied skew.6 Importantly, in our analyses we either hold constant the firm and examine highfrequency (day-over-day) changes in RNSkew or hold time
constant and examine differences in option-implied skew
in the cross-section of firms.
Our empirical analysis consists of three tests, each of
which provides independent evidence that investors overestimate the skewness of low-priced stocks. First, examining the entire cross-section of stocks, we find that
when forming skewness expectations, investors substantially overweight the importance of price relative to its
true observed relation with physical skewness. Second, we
show mispricing in option portfolios that is consistent with
investor overestimation of skewness for low-priced stocks.
Third, we find that investor expectations of skewness drastically increase (decrease) on the date of a stock split (reverse split) to a lower (higher) price. We also present evidence that the findings are driven by investor expectational errors regarding the upside, but not the downside,
potential of low-priced stocks. Next, we discuss the findings in more detail.
Our initial analysis focuses on the cross-section of all
stocks. While a relatively strong univariate inverse relation exists between price and physical skewness, this relation is driven by the correlation of price with other firm
characteristics. After controlling for firm characteristics
such as size, no significant relation remains between price
and physical skewness. However, a quite strong inverse
relation remains between price and RNSkew, even after
4
See Schiffres (2002, p. 64).
http://www.prnewswire.com/news-releases/
perritt- capital- management- launches- perritt- low- priced- stock- fundplowx-248540111.html.
6
See Dennis and Mayhew (2002), Han (2008), Bali and Murray (2013),
and Conrad, Dittmar, and Ghysels (2013), among others.
5
580
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
controlling for firm characteristics. That is, in forming expectations of skewness, investors overweight the importance of price relative to its observed relation with physical
skewness (or rational model–predicted expected skewness
measures that we also employ in our analysis). Investors
appear to mistakenly extrapolate the observed univariate
relation between price and physical skewness when pricing options. The evidence supports the idea that investors
overestimate the lottery-like properties of low-priced
stocks.
Utilizing option open interest and volume data, we also
find evidence that investors display increased optimism toward low-priced stocks. The ratio of call to put open interest and volume is substantially higher for low-priced
stocks than it is for high-priced stocks. Past work shows
investor preferences for lottery-like assets (e.g., Barberis
and Huang, 2008; Kumar 2009; Boyer, Mitton, and Vorkink,
2010; Bali and Murray, 2013; Boyer and Vorkink, 2014). We
build upon this evidence by showing that investors also
have a preference for utilizing the leverage benefits options provide to take lottery-like bets on these lottery-like
stocks.
We complement the cross-sectional results by undertaking a second, separate test of the hypothesis that investors suffer from a nominal price illusion. We explore
the asset-pricing implications of investor biased beliefs regarding nominal prices. Overestimation of expected skewness for low-priced stocks relative to high-priced stocks
suggests the potential overpricing of a portfolio of outof-the-money (OTM) calls on low-priced stocks relative to
a similar portfolio of options for high-priced stocks. Following Bollen and Whaley (2004) and Goyal and Sarreto
(2009), we estimate the returns to option portfolios and
find that the overpricing of call options increases as underlying stock price decreases. The results are consistent
with relative investor overestimation of skewness for lowpriced stocks compared with high-priced stocks. To ascertain whether RNSkew changes are driven by investor beliefs
regarding upside potential, downside potential, or some
combination of both, we examine put and call returns separately. We find the underlying stock price is related to
only the returns of OTM calls, but not OTM puts, consistent
with the effect coming from biased investor expectations
regarding the upside potential instead of the downside potential of the stock. Other tests further support the explanation. As an added benefit, the option portfolio methodology does not rely upon extracting RNSkew from option
prices and, therefore, also serves as a robustness check of
the earlier results.
Finally, we explore investor expectations around stock
splits, as this is a setting with few, if any, confounding influences and, therefore, allows the ability to examine the
effects of large, exogenous price changes in a relatively
clean setting.7 We first find that investor expectations of
7
Many potential motives for stock splits have been suggested and explored, including signaling (Brennan and Copeland, 1988; McNichols and
Dravid, 1990; Ikenberry, Rankine, and Stice, 1996) and liquidity arguments
(Muscarella and Vetsuypens, 1996; Angel, 1997). However, splits do not
seem to be correlated with future corporate profitability (Lakonishok and
Lev, 1987; Asquith, Healy, and Palepu, 1989), and it is not clear that splits
skewness drastically increase on the day that a stock splits
to a lower price level. On the day of a split to a lower price,
we find that skewness expectations (RNSkew) increase by
over 40%. In sharp contrast to this increase reflecting a rational increase in expected skewness, we find a substantial physical skewness decrease following stock splits. This
is not surprising given that splits occur after a long runup in price and, therefore, periods of high past skewness,
making the observed expected skewness increase all the
more surprising. Importantly, we find no such increase in
RNSkew on the day of the stock split announcement. The
increase of RNSkew around the ex-date instead of the announcement date is consistent with investors reacting only
to the change in stock price and inconsistent with an information signaling story. We find similar evidence of investor expectational errors in a much smaller sample of
reverse splits. On the date of a reverse split, RNSkew decreases drastically. In contrast to investor expectations, future physical skewness actually increases. The evidence is
consistent with investors assigning greater upside potential
(or lower downside potential, or both) to stocks trading at
lower prices.
As a further effort to ascertain whether the effect is
driven by investor beliefs regarding upside or downside
potential, we examine changes in implied volatilities for
calls and puts around the split. Again, consistent with biased investor expectations reflecting biased beliefs regarding upside potential, not downside potential, we find that
the difference between out of the money and at the money
(ATM) implied volatility substantially increases for call options after a split, suggesting that out of the money calls
become more expensive. No accompanying change exists
in this difference for put options. The evidence suggests
that the skewness results do in fact reflect investors assigning greater upside potential to stocks trading at lower
prices, with no accompanying change in beliefs regarding
downside potential. Overall, the evidence is consistent with
investors suffering from a nominal price illusion in which
they overestimate the cheapness or room to grow of lowpriced stocks relative to high-priced stocks.
The paper proceeds as follows. Section 2 discusses the
methodology and introduces the data. Section 3 presents
evidence of investor nominal price biases in the crosssection of stocks. Section 4 examines option trading.
Section 5 assesses asset-pricing implications of investor
nominal price bias. Section 6 presents the stock split analysis, and Section 7 concludes.
2. Risk-neutral skewness and data
To examine whether nominal share price is systematically related to investors’ misperception of skewness,
increase liquidity (Conroy, Harris, and Benet, 1990; Schultz, 20 0 0; Easley,
O’Hara, and Saar, 2001). In contrast to information or microstructure motives, the prevailing view is that firms split their shares to return prices
to a normal trading range (Baker and Gallagher, 1980; Lakonishok and
Lev, 1987; Conroy and Harris, 1999; Dyl and Elliot, 2006; Weld, Michaely,
Thaler, and Benartzi, 2009) and thus provide a clean laboratory to examine the effect of nominal prices on investor expectations. In addition, by
examining the effect of the information free ex-date, our tests allow us to
exclude any potential signaling effect.
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
we need measures of investor expected skewness and
rational unbiased measures of expected skewness. We
use risk-neutral skewness implied from option prices to
capture investor expectations of skewness. We employ
two primary measures of unbiased expected skewness.
First, ex-post realized skewness (Skew) is calculated using daily return data over a one-year period. Second, the
expected skewness measure (E(Skew)) of Boyer, Mitton,
and Vorkink (2010) incorporates all relevant past and current information to formulate a best prediction of future skewness. We use the model-free methodology of
Bakshi, Kapadia, and Madan (2003) to measure risk-neutral
skewness (RNSkew).8
2.1. Risk-neutral skewness
Risk-neutral skewness is a prominent variable in our
analysis, utilized to capture changing investor expectations
of asymmetry in return distributions. Because the option
prices from which risk-neutral moments are extracted are
updated daily, they reflect an up-to-date measure of investor ex-ante expectations.
Bakshi, Kapadia, and Madan (2003) show that riskneutral skewness is
3
EtQ (R(t, τ ) − EtQ [R(t, τ )] )
RNSkewi,t (τ ) =
2
EtQ (R(t, τ ) − EtQ [R(t, τ )] )
3/2
erτ Wi,t (t, τ ) − 3μi,t (t , τ )erτ Vi,t (t , τ ) + 2μi,t (t, τ )
3
=
2 3/2
[erτ Vi,t (t, τ ) − μi,t (t, τ ) ]
(1)
where i, t, andτ represent stock, current time, and time
to maturity, respectively. r is the risk-free rate. EtQ (. )is
the expectation under the risk-neutral measure. R(t, τ )is
the return from time t to t + τ , and μi,t (t, τ ) = erτ − 1 −
er τ
er τ
er τ
2 Vi,t (t, τ ) − 6 Wi,t (t, τ ) − 24 Xi,t (t, τ ). Bakshi, Kapadia,
and Madan (2003) further show that Vi, t (t, τ ),Wi, t (t, τ ),
and Xi, t (t, τ ) can be extracted from OTM options and are
defined as
Vi,t (t, τ ) = EtQ {e−rτ R(t, τ ) }
∞
2(1 − ln[K/S(t )]
=
C (t, τ ; K )dK
K2
S(t )
S(t )
2(1 + ln[K/S(t )]
+
P (t, τ ; K )dK,
K2
0
2
Wi,t (t, τ ) = EtQ e−rτ R(t, τ )
∞
2
6 ln[K/S(t )] − 3(ln[K/S(t )] )
=
C (t, τ ; K )dK
2
K
S(t )
S(t )
2
6 ln[K/S(t )] + 3(ln[K/S(t )] )
−
P (t, τ ; K )dK,
2
K
0
3
(2)
(3)
8
The model-free risk neutral measure of Bakshi, Kapadia, and Madan
(2003) has been widely used in the literature (Dennis and Mayhew, 2002;
Han, 2008; Bali and Murray, 2013; and Conrad, Dittmar, and Ghysels,
2013, among others).
and
581
Xi,t (t, τ ) = EtQ e−rτ R(t, τ )
∞
2
3
12 ln [K/S(t )] − 4(ln[K/S(t )] )
=
C (t, τ ; K )dK
2
K
S(t )
S(t )
2
3
12 ln [K/S(t )] + 4(ln[K/S(t )] )
+
P (t, τ ; K )dK.
2
K
0
4
(4)
Ideally, Vi, t (t, τ ), Wi, t (t, τ ), and Xi, t (t, τ ) should be calculated based on a continuum of European options with
different strikes. However, in reality, only a limited number
of options are available for each stock-expiration combination, and individual equity options are not European. To
accommodate the discreteness of options strikes, we follow Dennis and Mayhew (2002) to estimate the integrals
in expressions (2) to (4) using discrete data.9
Price per se should not be mechanically related to
RNSkew because RNSkew is homogeneous of degree zero
with respect to the underlying price; that is, altering the
underlying price will increase or decrease the numerator
and denominator of Equation (1) by the same proportion.
However, options for stocks with different prices can have
different strike structures, potentially imposing a systematic bias to the calculation of RNSkew.
Dennis and Mayhew (2002) examine two potential
sources of bias in RNSkew estimation. The first arises
due to the use of discrete strike prices, and the second arises from the potential asymmetry in the domain
of integration. Dennis and Mayhew (2002) show that
the bias in RNSkew is negative and increasing in absolute magnitude when the relative option strike interval
strike increment
( option
) increases.10 In practice, standard stock
under lying stock pr ice
option strike prices are in increments of $2.50 for strikes at
or below $25, $5.00 for strikes above $25 but below $200,
and $10 for strikes at or above $200. However, Dennis and
Mayhew (2002) show that the bias in RNSkew induced by
the option strike interval is quite small. The bias is approximately −0.01, −0.05 and −0.07 when the relative option
strike increment
strike intervals ( option
) are 2%, 5%, and 10%, reunder lying stock pr ice
spectively.
Dennis and Mayhew (2002) also investigate the potential bias arising due to an asymmetric domain of integration. They show that RNSkew is biased upward when there
is a lesser number of OTM puts relative to OTM calls and is
biased downward when there is a greater number of OTM
puts relative to OTM calls. However, Dennis and Mayhew
(2002) show that the bias is essentially zero if there are at
least two OTM puts and two OTM calls. As a result, we require at least two OTM put options and at least two OTM
call options.11 Finally, we standardize RNSkew to 30 days
by linearly interpolating the skewness of the option with
9
We thank Patrick Dennis for providing us the code.
Dennis and Mayhew (2002) use simulations to evaluate the bias of
option discreteness. They choose the underlying stock price to be $50 and
evaluate the magnitude of bias induced by option strike increments from
$1 to $5.
11
In the cross-sectional analysis that follows, our sample has an average of four OTM puts and four OTM calls for each stock analyzed. The
depth of the option chain is also similar in our sample for puts and calls.
10
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J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
expiration closest to, but less than, 30 days and the option with expiration closest to, but greater than, 30 days.
If there is no option with maturity longer than 30 days
(shorter than 30 days), we choose the longest (shortest)
available maturity.12
2.2. Data
IvyDB’s OptionMetrics database provides data on option prices, volume, open interest, implied volatility (IV),
and Greeks for the period from January 1996 to December 2012. IVs and Greeks are calculated using the binomial tree model of Cox, Ross, and Rubinstein (1979). We include options on all securities classified as common stock.
To minimize the impact of data errors, we remove options
missing best bid or best offer prices, as well as those with
bid prices less than or equal to $0.05. We also remove options that violate arbitrage bounds, and options with zero
open interest. Unless stated explicitly, we also exclude options with special settlement arrangement, and options for
which the underlying stock price is under $10.13 The midquote of the best bid and best offer is taken as the option
price. Data on stocks are from the Center for Research in
Security Prices (CRSP). The stock split data used later in
the study are obtained from the CRSP distribution file. We
define stock splits as events with a CRSP distribution code
of 5523. We define regular splits as those with a split ratio
of at least 1.25 to 1 and reverse splits as those with a ratio
below 1. We obtain company accounting information from
Compustat.
For both the large cross-sectional sample (all optionable
stocks) and the stock split sample, we include observations
only for which we are able to calculate RNSkew and future
Skew. The full stock sample contains 203,974 firm-month
observations. To mitigate the effect of outliers, we winsorize all continuous variables at the 0.5% level.
3. Nominal price and skewness: all optionable stocks
In this section, we test our main hypothesis that investors suffer from a nominal price illusion. We do so by
examining the entire cross-section of optionable stocks.
3.1. Characteristics of stocks with different nominal prices
We first explore the relation between nominal price
and skewness in the cross-section of stocks, and then examine whether this is consistent with the relationship
The average maximum (minimum) call delta is 0.318 (0.098). The average
minimum (maximum) put delta is −0.311 (−0.09).
12
All results are robust to the use of 60-day or one hundred-day skewness.
13
OptionMetrics defines an option as having a standard settlement if
one hundred shares of the underlying security are to be delivered at exercise and the strike price and premium multipliers are $100 per tick. For
options with a non-standard settlement, the number of shares to be delivered could be different from one hundred, and additional securities or
cash, or both, could be required. We exclude stocks with prices under $10
to mitigate the effects of microstructure noise. This excludes 6.7% of the
sample. As we show in the Online Appendix, our results are qualitatively
similar if we instead include all stocks.
between price and RNSkew that investors price into options. As we are interested in isolating the effect of nominal price only on investor behavior, we first examine how
price is correlated with firm characteristics. Table 1 reports
the summary statistics of stocks that are sorted into price
quintiles based on stock prices at the end of month t − 1.
On average, our sample stocks have a large dispersion
in stock price. The average stock price for the lowest quintile is 14.505, and the average price for the highest quintile
is 74.106. Relative to high-priced stocks, low-priced stocks
are smaller, have higher betas, have lower book-to-market
(B/M) ratios, have worse past performance, and are more
likely to be listed on Nasdaq. Table 1 also shows that lowpriced stocks have higher past volatility, lower past skewness, and lower trading volume and liquidity.
The last rows of Table 1 examine the relation between
price and our main measures of skewness. Our first measure of unbiased skewness is future physical skewness
calculated from daily returns from month t + 1 to month
t + 12.14 We also utilize a measure of expected skewness
(E(Skew)). E(Skew) is a forward-looking measure of skewness first introduced by Boyer, Mitton, and Vorkink (2010)
that incorporates all relevant past information to best form
a future prediction of skewness. This methodology utilizes
the parameters from a cross-sectional regression of skewness on lagged firm characteristics to estimate expected
skewness. We use the following model specification to
construct our E(Skew) variable:
E (Skewt ,t + 11 ) = α + β1 log (Pricet− 1 ) + β2 Betat− 1
+
+
+
+
+
β3 log (Sizet− 1 ) + β4 log (B/Mt− 1 )
β5 Leveraget−1 + β6 Volatilityt− 12,t−1
β7 Skewnesst− 12,t−1 + β8 Rt− 12,t−1 + β9 Rt −60,t − 13
β10 log (Volumet− 1 ) + β11 log (Illiqt− 1 )
β12 NASDAQt−1 + Industry Fixed Effects + ε .
(5)
Both future realized skewness and expected skewness
from the model of Boyer, Mitton, and Vorkink (2010) are
decreasing in nominal stock price.15 However, the magnitude of the cross-sectional relation between each of
these measures and price is much smaller than that between price and investor expectations reflected in RNSkew.
To see this, note that the unconditional variance of realized skew is larger than that of the RNSkew measure
(Table 7). However, conditional on price, the difference
in RNSkew between the top and bottom price quintiles is
more than twice that of the difference for future realized
skew or E(Skew). The magnitude of the univariate relation
between price and RNSkew provides preliminary evidence
that investors perceive the relation between price and future skewness to be much stronger than the true ex-post
14
In the reported results, we calculate volatility and skewness from the
raw returns. Moments of raw returns are likely to be affected by outliers.
Therefore, we also verify that the results hold if we calculate volatility
and skewness based on logged returns. The results are robust to this specification and are available in the Online Appendix. The results are also robust to using physical skewness calculated using only a 30-day horizon.
These results can be found in the Online Appendix.
15
All of our results are robust to using size tercile dummies as in Boyer,
Mitton, and Vorkink (2010). Results are available in the Online Appendix.
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
583
Table 1
Summary statistics by price quintiles.
This table reports summary statistics for the main variables of interest, sorted on price at the end of month t−1. We
include all stocks for which RNSkew and Skew can be calculated and the underlying stock price is greater than $10. We
calculate the mean of the variables month by month and report the time series average of the cross-sectional means.
Beta is calculated following Fama and French (1992). For each month, we use the past 60 months’ monthly excess
return data and CRSP value-weighted excess market return to calculate Beta. Size is market capitalization (price times
shares outstanding). B/M is the ratio of book value of equity to market value. Leverage is the ratio of total liabilities to
total assets. Past volatility and Past skewness are calculated using past one year daily returns. Rt − 12 ,t − 1 is the cumulative
return from month t−12 to month t−1. Rt − 60 ,t − 13 is the cumulative return from month t−60 to t−13. Volume is the total
number of shares traded in the last month. ILLIQ is measured following Amihud (2002). Nasdaq is a dummy variable
that is equal to one if the firm is listed in Nasdaq and zero otherwise. RNSkew is calculated following Bakshi, Kapadia,
and Madan (2003). Skew is calculated using daily data from month t + 1 to month t + 12. E(Skew) is the expected future
physical skewness, calculated following Boyer, Mitton, and Vorkink (2010), specifically using Eq. (5) from the text. The
sample period is from January 1996 to December 2011.
Price portfolio
Variable
Price
log (Price)
Beta
log (Size)
log (B/M)
Leverage
Past volatility
Past skewness
Rt − 12 ,t − 1
Rt − 60 ,t − 13
log (Volume)
log (ILLIQ)
Nasdaq
RNSkew
Skew
E(Skew)
Lowest
2
3
4
Highest
14.505
2.659
1.684
13.652
−1.029
0.188
0.616
0.157
0.120
1.060
11.799
−19.869
0.599
−0.231
0.370
0.373
22.125
3.086
1.414
14.194
−1.129
0.198
0.523
0.176
0.212
1.267
11.809
−20.458
0.497
−0.469
0.265
0.300
31.214
3.429
1.277
14.733
−1.261
0.198
0.479
0.216
0.307
1.473
11.968
−21.078
0.418
−0.709
0.207
0.229
43.574
3.762
1.103
15.267
−1.310
0.209
0.420
0.231
0.363
1.483
12.047
−21.653
0.319
−0.796
0.173
0.173
74.106
4.231
1.012
15.968
−1.426
0.217
0.382
0.286
0.522
1.583
12.215
−22.434
0.236
−0.929
0.109
0.103
relation realized in the data, as well as much stronger
than is predicted by a rational model of expected skewness incorporating all relevant current and past information. While the univariate evidence is consistent with the
notion that investors overweight the informativeness of
price when predicting future skewness, we next employ a
multivariate analysis to control for firm characteristics potentially correlated with both skewness and price.
3.2. Nominal price and skewness: Fama-MacBeth regressions
We use Fama and MacBeth (1973) regressions to analyze the cross-sectional relation between our various
skewness measures and price, while controlling for a
number of variables that the literature finds to be important in explaining skewness. Motivated by Dennis and
Mayhew (2002), we include beta to control for systematic
risk. Beta is calculated using the past 60 months of excess
return data.
The leverage effect predicts that decreases in equity
value will result in volatility increases that are larger than
the decreases in volatility that occur after increases in
equity value. This asymmetry implies that the implied
volatility of out-of-money puts is higher than the implied
volatility of out-of-money calls. While existing empirical
findings do not support the leverage effect (e.g., Dennis
and Mayhew, 2002; Bakshi, Kapadia, and Madan, 2003), we
nevertheless include leverage in the model.
We also control for firm size, past skewness, volume,
Rt − 12 ,t − 1 , Rt − 60 ,t − 13 , and a dummy variable indicating
whether a firm is listed on Nasdaq. Chen, Hong, and Stein
(2001) motivate the inclusion of firm size, past volatility,
Rt − 12 ,t − 1 and volatility, as they find that each of these
variables is significantly correlated with future skewness.16
We also include past skewness in the model, as Boyer, Mitton, and Vorkink (2010) show that skewness is persistent.
Finally, following Boyer, Mitton, and Vorkink (2010) we include a dummy variable for Nasdaq firms.
Past work typically finds a positive relation between
RNSkew and valuation in the cross-section. For example, Bali and Murray (2013), Conrad, Dittmar, and Ghysels (2013), and Boyer and Vorkink (2014) find that
RNSkew is negatively related to future asset returns, and
Friesen, Zhang, and Zorn (2012) explicitly show that low
B/M is associated with higher RNSkew in the cross-section.
While this positive documented relation between valuation
and RNSkew goes in the opposite direction of our predicted
negative relation between price and RNSkew, we nevertheless control for multiple measures of valuation to isolate the effect of nominal price from valuation. We include
both book-to-market and assets-to-price per share.17 We
also include industry fixed effects to control for industry
16
Boyer, Mitton, and Vorkink (2010) find that allowing for a nonlinear
relation between firm size and skewness can fit the data better. They do
so by using two dummy variables indicating small firms and medium-size
firms based on the NYSE breakpoints. Our results are similar if we adopt
their nonlinear methodology (see Table A5 of the Online Appendix).
17
We have also used earnings/price ratio and sales/price ratio (both
measured in natural logarithm) as additional controls for valuation. The
results are similar with these other measures of valuation.
584
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
heterogeneity. Industry definitions are based on the FamaFrench 48 industry classification scheme.
Table 2 reports the results of monthly Fama-MacBeth
regressions of our different measures of skewness regressed on the beginning of period independent variables. Standard errors are corrected for heteroskedasticity and autocorrelation up to 12 lags (Newey and West,
1987). We separately examine RNSkew, Skew, RN skew premium (RNSkew − Skew), and RN expected skew premium
(RNSkew − E(Skew)).18 For each dependent variable, we analyze three different models: a univariate specification that
includes only log nominal price, a multivariate specification including all control variables, and a multivariate
specification that also includes industry fixed effects.
As expected, the first three columns of the table show
a strong inverse relation between price and RNSkew. The
fourth column shows that a univariate relation exists between price and Skew, albeit one that is weaker than the
relation between price and RNSkew. However, after controlling for other firm-level determinants of Skew, price
ceases to be significant at the 10% level in explaining Skew
and the magnitude of the effect diminishes by over 66%.
While a significant univariate relation exists between price
and physical skew, this relation is driven by the correlation
of price with other firm characteristics. Once other firm
characteristics are accounted for, price no longer has an
economically or statistically significant relation with physical skew. However, the relation between price and RNSkew
is quite strong, even in the presence of controls for the
other firm characteristics. The results suggest that, when
pricing options, investors mistakenly extrapolate the observed univariate relation between price and physical skew
and fail to realize that this relation is driven by other
factors.
In terms of economic magnitude, from Column 3,
a 100% increase in price is associated with a 0.151
(0.693∗0.218) decrease in RNSkew, which is equal to 16.9%
of one standard deviation of the RNSkew variable. In contrast, from column 6, a 100% increase in price is associated with a 0.034 (0.693∗0.049) decrease in Skew, which is
equal to only 2.5% of one standard deviation of the Skew
variable and is not statistically significant. The remaining
columns show that the difference in this relation is statistically and economically significant for RNSkew relative to
Skew and E(Skew). The multivariate results confirm the earlier univariate results. The negative coefficient implies that
the lower is price, the larger is RNSkew relative to Skew
or E(Skew), and this difference shrinks as price increases.
In short, investors seem to substantially overestimate the
skew of low-priced stocks. Expectations of future return
distribution asymmetry are biased, as investors allow price
to play an irrationally large role in the shaping of beliefs.
Aside from price, five other variables enter significantly
and in a consistent direction in explaining both the RN
skew premium and the RN expected skew premium. Stocks
that are small, low book-to-market, low leverage, low past
18
Our measure of skew premium is analogous to the manner in which
the literature measures variance risk premium. However, the relationship
between RNSkew and Skew may not be linear. We address this in a more
structural way later in this section.
skew stocks, or high long-run return have higher RNSkew
relative to expected skew or future realized skew. High
RNSkew relative to physical skew can reflect expectational
errors but also could indicate a lower risk premium. If high
book-to-market stocks are riskier than low book-to-market
stocks, the negative coefficient for book-to-market can be
a reflection of a book-to-market risk premium. Leverage is
negatively related to RNSkew while having a positive relation with Skew. This is potentially consistent with the
leverage effect discussed above. Large firm size predicts
low RNSkew, but it does not have predictive power for future Skew. High past Skew predicts high future Skew in the
cross-section, while having predictive power of the opposite sign for RNSkew. A risk premium-based interpretation
would need to argue that larger firms and more positively
skewed firms are riskier than smaller firms and less positively skewed firms. While we cannot completely exclude
these possibilities, it is not consistent with most of the existing studies.
4. Nominal price and option trading
The evidence presented thus far is consistent with investors overestimating the lottery-like qualities of lowpriced stocks. Mitton and Vorkink (2007) and Barberis and
Huang (2008) incorporate investor preferences for skewness into models explaining stock returns, and Kumar
(2009) finds empirical evidence that retail investors prefer
stocks with lottery features. Given the past evidence and
general view that investors exhibit a preference for skew,
combined with the evidence presented here that investors
perceive low-priced stocks to have high skew, we next examine investor trading behavior in the options market regarding low-priced stocks. We find evidence that investors
exhibit increased optimism, as well as gambling-like behavior, toward these lottery-like assets. We do so by examining the ratio of call to put volume and open interest.
The ratio of call to put option trading volume is commonly regarded as a sentiment measure, with more call
option trading volume indicating optimism (see for example, Lemmon and Ni, 2014). Given the documented investor preference for skew, if price does affect investor perceptions of upside potential, a negative correlation should
emerge between stock price and the call-put volume ratio.
Option volume reflects both option writing and position
closing. Open interest measures the total existing position
and thus can potentially better measure investor beliefs.
Therefore, we also examine the relation between nominal
price and the call to put open interest ratio.
We define our volume ratio (VolRatio) and open
interest
ratio
(OIRatio)
as
VolRatio = log (1 + call
option
volume) − log (1 + put
option
volume)
and
OIRatio = log (1 + call option open interest) − log (1 + put
option open interest). We add one to deal with instances
of zero trading volume or open interest.
The results in Table 3 show that price is strongly negatively related to the call-to-put volume ratio and open
interest ratio. That is, investors are more optimistic about
low-priced stocks and take more lottery-like bets on lowpriced stocks. The evidence is consistent with investors
perceiving low-priced stocks to be lottery-like and with
Table 2
The relationship between nominal price and skewness: Fama-MacBeth regressions.
For each month, the following cross-sectional regression is run.
DVi,t = α + β1 Log(Pricei,t−1 ) + Xi,t−1 + εi,t−1 ,
where DV is one of the following: RNSkew, Skew, RNSkew−Skew, or RNSkew−E(Skew). RNSkew is calculated following Bakshi, Kapadia, and Madan (2003). Skew is calculated using daily data from month t + 1 to
month t + 12. E(Skew) is a forward looking skewness measure constructed following the method of Boyer, Mitton, and Vorkink (2010), using EQ. (5) from the text.
Beta is calculated following Fama and French (1992). For each month, we use the past 60 months’ monthly excess return data and CRSP value weighted excess market return to calculate Beta. Size is market
capitalization (price times shares outstanding). B/M is the ratio of book value of equity to market value. Leverage is the ratio of total liabilities to total assets. Past volatility and Past skewness are calculated using
past one year daily returns. Rt − 12 ,t − 1 is the cumulative return from month t − 12 to month t − 1. Rt − 60 ,t − 13 is the cumulative return from month t−60 to t−13. Volume is the total number of shares traded in
the last month. ILLIQ is measured following Amihud (2002). Industry is defined based on the 48 Fama-French industry classification. We include all stocks for which RNSkew and Skew can be calculated and the
underlying stock price is higher than $10. The sample period is from January 1996 to December 2011. Reported t-statistics are in parentheses and are Newey-West adjusted with 12 lags. ∗ , ∗∗ , and ∗∗∗ denote
significance at the 10%, 5%, and 1% levels, respectively. Reported adj.-R2 is the average adj.-R2 across all the months.
Variable
(1)
(2)
∗∗∗
(3)
∗∗∗
(4)
∗∗∗
Yes
No
No
Yes
No
No
Yes
No
No
Yes
0.217
192
0.007
192
0.032
192
0.087
192
0.014
192
0.052
192
0.105
192
0.037
192
0.140
192
0.249
192
Rt − 60 ,t − 13
log (Volume)
log (ILLIQ)
Nasdaq
0.741∗∗∗
(9.84)
0.124
(1.32)
0.033
(0.33)
−0.132
(−3.29)
0.011
(0.90)
−0.012
(−0.66)
−0.066∗∗∗
(−4.22)
−0.084∗∗∗
(−4.04)
−0.295∗∗∗
(−8.38)
0.302∗∗
(2.56)
−0.070∗∗∗
(−9.92)
0.051∗∗
(2.05)
0.020∗∗∗
(3.17)
0.069∗∗
(2.63)
0.027
(1.18)
0.055∗∗∗
(2.96)
0.145
(0.76)
−0.137∗∗∗
(−3.58)
0.057∗∗∗
(5.79)
0.002
(0.09)
−0.045∗∗∗
(−3.36)
−0.118∗∗∗
(−6.44)
−0.357∗∗∗
(−8.87)
0.292∗∗∗
(2.68)
−0.046∗∗∗
(−6.29)
0.051∗∗
(2.25)
0.020∗∗∗
(3.35)
0.044∗
(1.82)
0.023
(1.19)
0.084∗∗∗
(4.54)
−0.044
(−0.21)
0.163
192
Rt − 12 ,t − 1
−0.262
(−9.84)
(12)
∗∗∗
No
Past skewness
−0.170
(−4.20)
0.018
(1.23)
0.013
(0.70)
−0.089∗∗∗
(−4.47)
−0.073∗∗∗
(−3.65)
−0.383∗∗∗
(−8.31)
0.060
(0.30)
−0.038∗∗∗
(−3.60)
0.039
(1.46)
0.021∗∗∗
(3.56)
0.036
(1.44)
−0.002
(−0.09)
0.026
(1.23)
0.560
(1.62)
(11)
∗∗∗
0.085
192
Past volatility
−0.154
(−3.84)
−0.007
(−0.57)
0.007
(0.39)
−0.106∗∗∗
(−4.98)
−0.059∗∗∗
(−3.03)
−0.384∗∗∗
(−10.55)
0.058
(0.27)
−0.043∗∗∗
(−4.03)
0.041
(1.39)
0.018∗∗∗
(3.17)
0.063∗∗∗
(2.41)
0.007
(0.35)
0.008
(0.39)
0.650∗
(1.85)
(10)
∗∗∗
No
Leverage
−0.286
(−10.83)
(9)
∗∗∗
Adj.-R2
N
log (BM)
−0.049
(−1.48)
0.039∗∗∗
(3.86)
−0.021
(−1.17)
−0.009
(−0.62)
0.0 0 0
(0.01)
0.096∗
(1.82)
0.238∗
(1.68)
0.019∗
(1.94)
−0.086∗∗∗
(−4.00)
−0.017∗∗∗
(−4.14)
−0.010
(−0.48)
0.024
(1.18)
0.001
(0.05)
1.006∗∗∗
(4.03)
(8)
∗∗∗
Industry FE
log (Size)
−0.059
(−1.89)
0.030∗∗∗
(2.71)
−0.012
(−0.62)
0.002
(0.12)
0.007
(0.60)
0.167∗∗∗
(4.49)
0.249∗
(1.77)
0.023∗∗
(2.51)
−0.092∗∗∗
(−4.22)
−0.015∗∗∗
(−4.42)
−0.025
(−1.13)
0.018
(0.90)
0.006
(0.25)
0.971∗∗∗
(4.07)
(7)
0.865∗∗∗
(8.51)
Beta
−0.150
(−7.42)
(6)
∗
Constant
log (AT/Price)
−0.218
(−5.28)
0.057∗∗∗
(5.82)
−0.009
(−1.42)
−0.098∗∗∗
(−6.84)
−0.073∗∗∗
(−6.15)
−0.287∗∗∗
(−11.95)
0.298∗∗∗
(3.50)
−0.020∗∗∗
(−4.27)
−0.047∗∗∗
(−4.39)
0.005
(1.20)
0.026
(1.17)
0.022
(1.15)
0.028∗∗∗
(3.79)
1.566∗∗∗
(7.92)
(5)
∗∗∗
RNSkew−E(Skew)
−0.213
(−4.88)
0.022∗
(1.74)
−0.005
(−0.62)
−0.105∗∗∗
(−6.96)
−0.052∗∗∗
(−4.05)
−0.217∗∗∗
(−8.47)
0.307∗∗∗
(3.27)
−0.020∗∗∗
(−4.03)
−0.051∗∗∗
(−3.99)
0.003
(0.80)
0.038
(1.51)
0.025
(1.15)
0.014∗
(1.71)
1.621∗∗∗
(8.15)
log (Price)
−0.437
(−15.72)
RNSkew−Skew
Skew
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
RNSkew
585
586
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
Table 3
Nominal price and option trading.
For each month, we run the cross-sectional regression
DVi,t = α + β1 Log(Pricei,t−1 ) + Xi,t−1 + εi,t−1 ,
where DV is either call relative to put option trading volume ratio (VolRatio) or the call relative to put option open interest
ratio (OIRatio). VolRatio = log (1 + volume of call options) − log (1 + volume of put options). OIRatio = log (1 + options interests of
call options) − log (1 + options interests of put options). The independent variables are log price, beta, log of firm market capitalization, log of B/M, leverage, past volatility, past skewness, Rt − 12 ,t − 1 , Rt − 60 ,t − 13 , log volume, log ILLIQ, a Nasdaq dummy
indicating listing exchange, and industry dummies. Beta is calculated following Fama and French (1992). For each month, we
use the past 60 months’ monthly excess return data and CRSP value-weighted excess market return to calculate Beta. Size is
market capitalization (price times shares outstanding). B/M is the ratio of book value of equity to market value. Leverage is the
ratio of total liabilities to total assets. Past volatility and Past skewness are calculated using past one year daily returns. Rt − 12 ,t − 1
is the cumulative return from month t − 12 to month t − 1. Rt − 60 ,t − 13 is the cumulative return from month t − 60 to t − 13. Volume is the total number of shares traded from the last month. ILLIQ is measured following Amihud (2002). Industry is defined
based on the 48 Fama-French industry classification. We include all stocks for which RNSkew and Skew can be calculated and
the underlying stock price is greater than $10. The sample period is from January 1996 to December 2011. Reported t-statistics
are in parentheses and are Newey-West adjusted with 12 lags. ∗ , ∗∗ , and ∗∗∗ denote significance at the 10%, 5%, and 1% levels,
respectively. Reported adj.-R2 is the average adj.-R2 across all the months
VolRatio
Variable
OIRatio
(1)
(2)
∗∗∗
∗∗∗
(4)
∗∗∗
(5)
∗∗∗
(6)
0.919∗∗∗
(10.13)
−0.228
(−14.51)
−0.037∗∗∗
(−4.27)
−0.011
(−1.21)
0.103∗∗∗
(11.54)
0.027∗∗∗
(3.03)
−0.052
(−1.37)
−0.230∗∗∗
(−3.17)
0.010
(1.61)
0.218∗∗∗
(7.88)
0.006∗∗
(2.08)
−0.067∗∗∗
(−4.61)
0.060∗∗∗
(4.19)
0.016∗
(1.82)
1.929∗∗∗
(10.60)
−0.247∗∗∗
(−15.53)
−0.052∗∗∗
(−5.76)
−0.016∗
(−1.89)
0.088∗∗∗
(9.75)
0.023∗∗∗
(2.76)
−0.023
(−0.73)
−0.280∗∗∗
(−3.57)
0.013∗∗
(2.18)
0.205∗∗∗
(8.11)
0.003
(1.34)
−0.062∗∗∗
(−4.25)
0.053∗∗∗
(4.18)
0.011
(1.31)
2.041∗∗∗
(10.62)
Constant
1.582∗∗∗
(12.09)
Industry FE
No
No
Yes
No
No
Yes
0.030
192
0.071
192
0.135
192
0.009
192
0.061
192
0.122
192
log (AT/Price)
Beta
log (Size)
log (B/M)
Leverage
Past volatility
Past skewness
Rt − 12 ,t − 1
Rt − 60 ,t − 13
log (Volume)
log (ILLIQ)
Nasdaq
Adj.-R
N
2
investors possessing more optimistic perceptions of the
upside potential of low-priced stocks relative to highpriced stocks. We next examine the asset-pricing implications of investor biases regarding nominal prices.
5. Asset pricing implications
Expectational errors regarding the upside potential of
low- relative to high-priced stocks will also have asset
pricing implications. One implication of investor overestimation of skewness of low-priced relative to high-priced
stocks is that call options will be more overvalued for lowpriced stocks than for high-priced stocks. A portfolio taking a long position in calls of high-priced stocks and a
−0.399
(−16.67)
−0.005
(−0.56)
−0.018∗
(−1.86)
0.083∗∗∗
(7.26)
−0.006
(−0.70)
−0.082∗∗∗
(−2.90)
−0.129∗
(−1.74)
0.017∗∗∗
(3.15)
0.168∗∗∗
(8.35)
0.009∗∗∗
(3.06)
−0.077∗∗∗
(−4.84)
0.034∗∗
(2.32)
0.019
(1.62)
2.453∗∗∗
(9.77)
−0.113
(−7.80)
∗∗∗
−0.384
(−19.06)
−0.006
(−0.51)
−0.011
(−1.07)
0.103∗∗∗
(8.77)
0.007
(0.73)
−0.045
(−1.33)
−0.062
(−0.95)
0.015∗∗
(2.52)
0.181∗∗∗
(7.60)
0.013∗∗∗
(3.07)
−0.085∗∗∗
(−4.92)
0.041∗∗
(2.52)
0.020
(1.49)
2.338∗∗∗
(10.19)
log (Price)
−0.264
(−13.31)
(3)
short position in calls of low-priced stocks should therefore make positive returns. This will be particularly true
for OTM options. To test this hypothesis, we construct both
delta-hedged and non-hedged call portfolios and examine
whether call portfolio returns are systematically related
to underlying stock price. Our methodology and analysis
largely follows that of Goyal and Saretto (2009). This analysis also doubles as a robustness check of the earlier results
relying on RNSkew.
To be clear, the purpose of this analysis is to illustrate the existence of mispricing in the options market
in a manner that is consistent with investor overestimation of skewness for low-priced stocks. As a means to this
end, we utilize a portfolio strategy as a tool to identify
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
whether mispricing exists. In undertaking this analysis, we
measure option prices as the midpoint of the bid and
ask quotes. Whether the mispricing can be profitably exploited in practice once option trading costs are considered
is a separate question, which is beyond the scope of this
paper.
Portfolios are formed on the expiration Friday (or the
previous trading day if Friday is a public holiday) of the
month, and the option portfolio strategies are initiated on
the first trading day (typically a Monday) after the expiration Friday of the month. On each portfolio formation
day, we sort all stocks with available options into quintiles based on the stock price on the portfolio formation
day and choose only options expiring within one month.
For each option series, we construct portfolios that are
held until option expiration. All the portfolios are equal
weighted. As in Goyal and Saretto (2009), we use the absolute position value as the reference beginning price to calculate portfolio return. For delta-hedged portfolios we use
the following formula to calculate returns.19
T
(cT − c0 ) − 0 ST + Dt er (T −t ) − S0
Rcall =
t=1
| 0 S 0 − c 0 |
(6)
Relative to ATM and in-the-money (ITM) options, OTM
options better reflect investor skewness beliefs. Thus, we
focus on the OTM options. As a comparison, we also report results for ATM options. We define option moneyness
following Bollen and Whaley (2004). ATM options are defined as call options with delta greater than 0.375 and not
greater than 0.625. OTM options are call options with delta
above 0.02 and not greater than 0.375. Options with absolute delta below 0.02 are excluded due to the distortions
caused by price discreteness.
Table 4 reports the results of the portfolio return analysis by price quintile. Panel A reports monthly average
returns for delta-hedged portfolios, and Panel B displays
results for non-delta-hedged portfolios. In addition to reporting the raw return of the delta-hedged option portfolios, we follow Goyal and Saretto (2009) and report riskadjusted returns using model (7).
Rcall =
α + β Ft + εt
(7)
To obtain risk-adjusted returns, we regress the portfolio returns on a linear pricing model consisting of the
three Fama-French factors, the momentum factor (Fama
and French, 1993; Carhart, 1997), and an additional aggregate factor reflecting the average call return of S&P 500
index options. 20 The average call return of S&P 500 index options can capture the compensation to jump risk
(Pan, 2002). We use the same method [Eq. (6)] to calculate
the average call return of S&P 500 index options by moneyness. The call index option returns are matched to the
call individual stock option returns with the same moneyness. The intercept from the regression can be interpreted
19
We consider stock splits and use the adjustment factor given by OptionMetrics for the adjustment.
20
We compound the daily factor return to get monthly factor returns to
match the timing of the option strategy.
587
as mispricing relative to the factor model. We refer to the
adjusted return as the five-factor adjusted return.
For both delta-hedged and non-delta-hedged portfolios, the portfolio return increases when moving from the
lowest to highest price quintile. In Panel A, the call returns for the delta-hedged sample increase from −2.901%
in the lowest price quintile to −0.908% in the highest
price quintile. The five-factor adjusted call return increases
from −2.02% in the lowest price quintile to 0.368% in the
highest price quintile. The high minus low portfolio raw
return and the five-factor adjusted return is 1.993% and
2.387%, respectively. Both are statistically significant at 1%
level. The results indicate that OTM calls are substantially
more overpriced for low-priced stocks than for high-priced
stocks. We also find some supportive evidence from ATM
options. For ATM options, the raw and five-factor adjusted
return of the call portfolio increases as a function of the
underlying stock price. The magnitude of the highest minus lowest price quintile difference in call portfolio returns
is larger for OTM options than for the ATM options, consistent with the view that skewness misperception most
greatly affects the price of the OTM options. The patterns
are similar and of larger magnitude for the non-deltahedged analysis in Panel B.
Lastly, we attempt to isolate the effect of price on returns to examine whether other factors could be driving
the observed relation between price and option returns.
To examine this question, we rely on Fama-MacBeth regressions to control for other variables the literature finds
to affect option pricing. Columns 1–3 of Table 5 confirm
the positive relation between OTM call option returns and
price shown in Table 4.
In Table 5, we also test two further predictions of our
main hypothesis. First, if the relations shown thus far are
driven by investor overestimation of upside potential and
not downside potential, then we should expect to see an
effect only for calls, but not for puts. In Columns 4–6, we
examine whether a relation exists between put returns and
price. The put portfolios are formed with OTM options,
defined as those with delta greater than 0.375 and not
greater than 0.02. Columns 4–6 of Panel A and Panel B
in Table 5 confirm that no significant relation exists between put portfolio returns and the price of the underlying, consistent with the effects shown being driven by
investor overestimation of upside potential, not downside
potential.
A further implication of investor overestimation of
skewness of low-priced relative to high-priced stocks is
that, relative to put options, call options are more overvalued for low-priced stocks than for high-priced stocks.
We formally test this in Columns 7–9 and confirm that
this difference is increasing in price and that this relation
is statistically significant. Importantly, the results confirm
that the abnormal returns we find are completely driven
by the mispricing of calls, and not puts. After controlling
for other variables that are known to be related to option
returns, we find that OTM call returns are increasing in the
nominal price of the underlying stock and OTM put returns
are unrelated to nominal price. This is consistent with investors overestimating the upside potential of low-priced
stocks.
588
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
Table 4
Option trading strategy.
Portfolios are formed on the expiration Friday (or last trading day if Friday is a public holiday) of the month, and the
option portfolio strategies are initiated on the first trading day (typically a Monday) after the expiration Friday of the
month. OTM options are call options with delta greater than 0.02 and not greater than zero. As in Bollen and Whaley
(2004), we exclude options with absolute delta below 0.02 due to the distortions caused by price discreteness. On
each portfolio formation day, we choose the options for any given moneyness that will expire within one month. On
each portfolio formation day, all stocks with available options are sorted into quintiles based on the stock price on the
portfolio formation day. For each option series, we construct delta-hedged and non-delta hedged portfolios and hold
them until option expiration. As in Goyal and Sarreto (2009), we use the absolute value of the position as the reference
price to calculate the delta-hedged return. The formula is
(cT − c0 erT ) − 0 (ST +
Rcall =
T
t=1
Dt er (T −t ) − S0 erT )
,
| 0 S 0 − c 0 |
We also report the average abnormal return adjusting for Fama-French three factors, the Carhart (1997) momentum
factor, and a factor reflecting the market-level call return. The portfolio returns are equal weighted. The sample extends
from January 22, 1996 to December 21, 2012. There are 203 months in total. ∗ , ∗∗ , and ∗∗∗ denote significance at the
10%, 5%, and 1% levels, respectively.
Panel A: delta-hedged portfolio returns
Price portfolio
Option
Lowest
2
3
4
Highest
Highest − lowest
−2.901∗∗∗
(−6.67)
−2.020∗∗∗
(−6.97)
−2.734∗∗∗
(−6.41)
−1.974∗∗∗
(−7.15)
−2.208∗∗∗
(−5.14)
−0.506∗∗
(−2.60)
−1.296∗∗∗
(−3.17)
−0.270
(−0.98)
−0.908∗
(−1.94)
0.368
(1.15)
1.993∗∗∗
(5.53)
2.387∗∗∗
(7.25)
−0.373
(−1.34)
0.104
(0.58)
−0.033
(−0.13)
0.372∗∗∗
(3.66)
−0.007
(−0.03)
0.356∗∗
(2.20)
0.101
(0.40)
0.537∗∗∗
(3.18)
0.043
(0.16)
0.484∗∗
(2.37)
0.416∗∗
(2.25)
0.380∗∗
(2.07)
OTM
Call
Call adjusted
ATM
Call
Call adjusted
Panel B: non-delta-hedged portfolio returns
OTM
Call
Call adjusted
−21.498∗∗∗
(−4.70)
−20.423∗∗∗
(−6.40)
−27.849∗∗∗
(−6.45)
−27.453∗∗∗
(−8.77)
−25.099∗∗∗
(−5.32)
−24.518∗∗∗
(−6.93)
−18.928∗∗∗
(−3.24)
−17.049∗∗∗
(−3.87)
−10.190
(−1.08)
0.914
(0.17)
11.308∗∗
(2.55)
21.338∗∗∗
(4.25)
2.484
(0.59)
−0.361
(−0.19)
5.855
(1.41)
2.257
(1.25)
6.192
(1.46)
2.382
(1.26)
5.502
(1.21)
1.722
(0.82)
8.889
(1.57)
5.095∗∗
(2.01)
6.405∗
(1.92)
5.457∗∗
(2.35)
ATM
Call
Call adjusted
While the results are consistent with investor overestimation of skewness for low-priced stocks leading to overpricing of low-priced OTM call options, the tests cannot
definitively attribute overpricing of low-priced OTM calls
to investor overestimation of skewness. To further support
a causal story, we next examine a double-sort of returns
on both price and RNSkew. If investor overestimation of
skewness for low-priced stocks is driving our results, we
would expect to see the effect show up most strongly in
the high RNSkew quantiles. Options of low-priced stocks
with high RNSkew are potentially overvalued, and options
of high-priced stocks with high RNSkew are less likely to be
overvalued according to our story. However, within the low
RNSkew quantile, we are unlikely to see substantial overestimation of skew and, therefore, would not expect to see as
large of an effect. We utilize independent sorts to ensure
that the sample of low- and high-priced stocks and spread
in high and low price is similar across RNSkew terciles and
is comparable with the analysis in Table 4. Table 6 examines this hypothesis. Consistent with the above arguments,
we find that the difference in OTM call returns between
top and bottom price quintiles is large and statistically significant for the highest RNSkew tercile and smaller and insignificant for the lowest RNSkew tercile. While not proving
causality, the results are consistent with a causal story.
A further question is whether the abnormal returns that
this strategy generates result from irrational beliefs on the
part of investors or whether options on low-priced stocks
are more difficult for dealers to hedge and, therefore, trade
at a premium. A dealer hedging story is not consistent
with our finding of an effect only for call options on
low-priced underlying stocks, but not put options, as it
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
589
Table 5
Option trading strategy: Fama-MacBeth regressions.
For each month, we run the cross-sectional regression
DVi,t = α + β1 Log(Pricei,t−1 ) + Xi,t−1 + εi,t−1 ,
where DV is either the delta-hedged OTM call return, OTM put return, or the difference between OTM call and OTM put return (OTM call − OTM put). OTM
options are call options with delta greater than 0.02 and not greater than 0.375, and put options with delta greater than −0.375 and not greater than
−0.02. As in Bollen and Whaley (2004), we exclude options with absolute delta below 0.02 due to the distortions caused by price discreteness. We choose
the put and call options for any given moneyness that will expire within one month. Delta-hedged option return is calculated from the first trading day
(typically a Monday) after the expiration Friday of the month until option expiration. As in Goyal and Sarreto (2009), we use the absolute value of the
position as the reference price to calculate the delta-hedged return. The formulas are
(cT − c0 erT ) − 0 (ST +
Rcall =
T
t=1
Dt er (T −t ) − S0 erT )
( pT − p0 erT ) − 0 (ST +
, R put =
| 0 S 0 − c 0 |
T
t=1
Dt er (T −t ) − S0 erT )
.
| 0 S 0 − p 0 |
If there is more than one option series for one stock-moneyness group, we calculate the equal-weighted delta-hedged option return. The independent
variables are log price, beta, log of firm market capitalization, log of B/M, leverage, past volatility, past skewness, Rt − 12 ,t − 1 , Rt − 60 ,t − 13 , implied volatility,
log(volume), log(ILLIQ), a Nasdaq dummy indicating listing exchange, and industry dummies. Beta is calculated following Fama and French (1992); for
each month, we use the past 60 months’ monthly excess return data and CRSP value-weighted excess market return to calculate Beta. Size is market
capitalization (price times shares outstanding). B/M is the ratio of book value of equity to market value. Leverage is the ratio of total liabilities to total
assets. Past volatility and Past skewness are calculated using past one year daily returns. Rt − 12 ,t − 1 is the cumulative return from month t − 12 to month
t − 1. Rt − 60 ,t − 13 is the cumulative return from month t − 60 to t − 13. Volume is the total number of shares traded from the last month. ILLIQ is measured
following Amihud (2002). Industry is defined based on the 48 Fama-French industry classification. We include all stocks for which we have both OTM call
and OTM put options and the underlying stock price is greater than $10. The sample period is from January 22, 1996 to December 21, 2012. There are 203
months in total. Reported t-statistics are in parentheses and are Newey-West adjusted with 12 lags. ∗ , ∗∗ , and ∗∗∗ denote significance at the 10%, 5%, and 1%
levels, respectively. Reported adj.-R2 is the average adj.-R2 across all the months.
Panel A: delta-hedged
OTM call
Variable
log (Price)
(1)
∗∗∗
0.010
(6.01)
RNSkew
RNVar
RNKurt
log (AT/Price)
Beta
log (Size)
log (BM)
Leverage
Past volatility
Past skewness
Rt − 12 ,t − 1
Rt − 60 ,t − 13
log (Volume)
log (ILLIQ)
Nasdaq
Industry FE
Adj.-R2
N
(2)
∗∗∗
0.011
(4.37)
0.001
(0.57)
0.101∗∗
(2.64)
−0.0 0 0
(−0.32)
0.0 0 0
(0.19)
0.001
(0.44)
−0.007∗∗∗
(−4.01)
0.001
(0.60)
−0.002
(−0.51)
−0.017∗∗∗
(−3.07)
0.001
(0.71)
0.002
(1.13)
−0.0 0 0
(−0.47)
0.002
(0.94)
−0.004∗
(−1.83)
−0.003∗
(−1.90)
OTM call − OTM put
OTM put
(3)
∗∗∗
0.011
(4.83)
0.001
(0.45)
0.101∗∗∗
(2.74)
−0.0 0 0
(−0.30)
−0.0 0 0
(−0.15)
−0.001
(−0.42)
−0.007∗∗∗
(−4.45)
0.001
(1.08)
−0.001
(−0.14)
−0.017∗∗∗
(−3.11)
0.0 0 0
(0.50)
0.001
(0.50)
−0.0 0 0
(−0.78)
0.003
(1.37)
−0.004∗
(−1.76)
−0.003∗
(−1.89)
(4)
(5)
∗∗∗
0.005
(4.02)
0.005
(1.48)
0.004∗∗
(2.14)
0.042
(0.96)
0.001∗∗
(2.45)
0.0 0 0
(0.54)
0.001
(0.67)
−0.007∗∗∗
(−4.38)
0.0 0 0
(0.11)
0.002
(0.35)
−0.013∗∗
(−2.11)
−0.001∗∗
(−2.56)
−0.001
(−0.70)
0.0 0 0
(0.63)
−0.0 0 0
(−0.04)
−0.006∗∗∗
(−3.22)
−0.002
(−1.25)
(6)
(7)
∗
0.005
(1.74)
0.004∗∗
(2.29)
0.053
(1.23)
0.001∗∗
(2.63)
0.0 0 0
(0.18)
0.0 0 0
(0.25)
−0.007∗∗∗
(−4.33)
−0.0 0 0
(−0.24)
0.005
(1.29)
−0.012∗∗
(−2.03)
−0.001∗∗∗
(−2.98)
−0.002
(−1.11)
0.0 0 0
(0.56)
−0.0 0 0
(−0.16)
−0.006∗∗∗
(−3.62)
−0.002
(−1.27)
(8)
∗∗∗
0.004
(3.27)
(9)
∗∗
0.006
(2.05)
−0.003∗
(−1.76)
0.059
(1.51)
−0.001∗∗
(−2.57)
−0.0 0 0
(−0.15)
−0.001
(−0.27)
0.0 0 0
(0.15)
0.001
(0.31)
−0.004
(−0.61)
−0.005
(−0.70)
0.002∗∗
(2.34)
0.004∗
(1.78)
−0.0 0 0
(−0.79)
0.002
(0.85)
0.002
(0.94)
−0.0 0 0
(−0.21)
0.006∗∗
(2.02)
−0.004∗∗
(−2.00)
0.048
(1.16)
−0.001∗∗∗
(−2.79)
−0.0 0 0
(−0.32)
−0.001
(−0.54)
−0.001
(−0.31)
0.001
(0.89)
−0.006
(−0.98)
−0.005
(−0.75)
0.002∗∗
(2.26)
0.003
(1.29)
−0.0 0 0
(−1.02)
0.003
(1.22)
0.002
(0.86)
−0.001
(−0.57)
No
No
Yes
No
No
Yes
No
No
Yes
0.006
203
0.062
203
0.133
203
0.004
203
0.057
203
0.134
203
0.003
203
0.045
203
0.115
203
(continued on next page)
590
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
Table 5 (Continued.)
Panel B: non-delta-hedged
log (Price)
0.089∗∗
(2.14)
RNSkew
RNVar
RNKurt
log (AT/Price)
Beta
log (Size)
log (B/M)
Leverage
Past volatility
Past skewness
Rt − 12 ,t − 1
Rt − 60 ,t − 13
log (Volume)
log (ILLIQ)
Nasdaq
Industry FE
Adj.-R2
N
No
0.004
203
0.186∗∗∗
(5.17)
−0.227∗∗∗
(−7.97)
−1.311∗∗
(−2.37)
−0.009∗∗∗
(−2.94)
0.001
(0.04)
0.037∗
(1.77)
−0.150∗∗∗
(−6.79)
0.015
(0.69)
−0.143∗∗
(−2.58)
−0.251∗∗
(−2.36)
−0.013∗
(−1.78)
0.034
(0.62)
−0.006
(−1.37)
−0.024
(−0.54)
−0.128∗∗∗
(−3.16)
−0.040
(−1.21)
No
0.048
203
0.178∗∗∗
(5.21)
−0.222∗∗∗
(−8.79)
−1.174∗∗
(−2.27)
−0.009∗∗∗
(−2.70)
0.034
(1.46)
0.021
(1.25)
−0.153∗∗∗
(−7.67)
−0.006
(−0.32)
−0.204∗∗∗
(−3.04)
−0.170∗∗
(−1.97)
−0.016∗∗
(−2.52)
0.015
(0.29)
−0.008
(−1.57)
−0.054∗
(−1.73)
−0.156∗∗∗
(−5.35)
−0.032
(−1.14)
Yes
0.134
203
−0.011
(−0.59)
No
0.004
203
cannot explain why only call options on low-priced stocks,
but not put options on low-priced stocks, would be difficult to hedge.21 The evidence presented here is consistent with the effects we see being driven by biased beliefs
regarding the upside potential, but not downside potential, of the stock. In Section 6, we provide additional evidence that investor biased beliefs are driven solely by biased beliefs regarding upside potential and not downside
potential.
−0.073
(−1.10)
0.222∗∗∗
(10.54)
2.198∗∗∗
(4.71)
0.007∗∗∗
(3.18)
−0.005
(−0.27)
0.020
(0.90)
−0.064∗∗
(−2.28)
0.012
(0.63)
0.069
(1.08)
0.055
(0.58)
−0.002
(−0.25)
−0.020
(−0.89)
0.010∗∗
(2.17)
0.045
(1.50)
−0.020
(−0.73)
−0.043∗∗
(−2.06)
No
0.050
203
−0.079
(−1.52)
0.215∗∗∗
(11.24)
2.209∗∗∗
(4.72)
0.008∗∗∗
(3.76)
−0.022
(−1.06)
0.015
(0.97)
−0.064∗∗
(−2.49)
0.020
(1.42)
0.143∗∗
(2.11)
0.068
(0.73)
−0.002
(−0.24)
−0.028
(−1.22)
0.010∗∗∗
(2.75)
0.043
(1.59)
−0.021
(−0.79)
−0.041∗∗
(−2.15)
Yes
0.131
203
0.099∗
(1.90)
No
0.005
203
0.259∗∗∗
(3.09)
−0.448∗∗∗
(−12.25)
−3.509∗∗∗
(−5.84)
−0.016∗∗∗
(−4.51)
0.006
(0.17)
0.017
(0.50)
−0.087∗∗
(−2.04)
0.003
(0.11)
−0.213∗∗
(−2.34)
−0.306∗
(−1.79)
−0.011
(−0.96)
0.054
(0.77)
−0.016∗∗
(−2.08)
−0.068
(−1.17)
−0.108∗∗
(−2.02)
0.004
(0.09)
No
0.062
203
0.257∗∗∗
(3.66)
−0.437∗∗∗
(−14.21)
−3.383∗∗∗
(−5.69)
−0.017∗∗∗
(−4.91)
0.056
(1.49)
0.005
(0.22)
−0.089∗∗
(−2.37)
−0.026
(−1.02)
−0.347∗∗∗
(−3.32)
−0.237
(−1.64)
−0.014
(−1.42)
0.043
(0.64)
−0.018∗∗∗
(−2.68)
−0.097∗∗
(−2.11)
−0.136∗∗∗
(−3.16)
0.010
(0.27)
Yes
0.156
203
is information free, as any potential information revelation occurs at the announcement date. Stock splits therefore provide a clean environment to examine the effect of
nominal price changes on investor expectations.22
Fig. 1 provides a preview of the main split results.
RNSkew is plotted against days relative to ex-date. From
Fig. 1, stocks undergoing splits experience a large jump
in RNSkew on the ex-date, and those undergoing reverse
splits experience a large decrease in RNSkew that again occurs precisely on the ex-date.23
6. Nominal price and skewness: stock splits
In this section, we examine the effect of nominal price
on investor skewness expectations in a setting where nominal price changes are seemingly exogenous to changes in
expectations of future return distributions. The prevailing
view is that stock splits are motivated by an effort to return prices to a normal trading range (Baker and Gallagher,
1980; Lakonishok and Lev, 1987; Conroy and Harris, 1999;
Dyl and Elliot, 2006; Weld, Michaely, Thaler, and Benartzi,
2009). Regardless of the rationale for splitting, the ex-date
21
One hedging story that does predict an asymmetry in hedging costs is
put forth by Ofek, Richardson, and Whitelaw (2004). Shorting costs, particularly the fact that short sales constraints are more binding for lowpriced stocks, make hedging OTM puts more difficult relative to OTM
calls, particularly for low-priced stocks. This effect, however, works in the
opposite direction of our results.
6.1. Skewness around regular splits
Table 7 displays summary statistics for the full sample
of stocks alongside summary statistics for the split sample and the reverse split sample.24 The regular split sample has 1,940 observations, and the reverse split sample
22
Recent papers by Green and Hwang (2009) and Baker, Greenwood,
and Wurgler (2009) also use stock splits as an instrument to test behavioral theories, arguing that the lack of a relationship between splits and
firm fundamentals allows for a particularly clean experimental setting.
23
The confidence intervals are narrow, especially for regular splits.
In the Online Appendix, we show a version of Fig. 1 with confidence
intervals.
24
See Table A1 for detailed definitions of all variables.
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
591
Table 6
Double sorts: option trading strategy.
Portfolios are formed on the expiration Friday (or last trading day if Friday is a public holiday) of the month, and the option
portfolio strategies are initiated on the first trading day (typically a Monday) after the expiration Friday of the month. OTM
options are call options with delta greater than 0.02 and not greater than zero. As in Bollen and Whaley (2004), we exclude
options with absolute delta below 0.02 due to the distortions caused by price discreteness. On each portfolio formation day, we
choose the options for any given moneyness that will expire within one month. On each portfolio formation day, all stocks with
available options are sorted into quintiles based on the stock price on the portfolio formation day. For each option series, we
construct delta-hedged and non-delta-hedged portfolios and hold them until option expiration. As in Goyal and Sarreto (2009),
we use the absolute value of the position as the reference price to calculate the delta-hedged return. The formula is
(cT − c0 erT ) − 0 (ST +
Rcall =
T
t=1
Dt er (T −t ) − S0 erT )
,
| 0 S 0 − c 0 |
We also report the average abnormal return adjusting for Fama-French three factors, the Carhart (1997) momentum factor and
a factor reflecting the market-level call return. The portfolio returns are equal weighted. The sample extends from January
22, 1996 to December 21, 2012. There are 203 months in total. Reported t-statistics are in parentheses. ∗ , ∗∗ , and ∗∗∗ denote
significance at the 10%, 5%, and 1% levels, respectively.
Panel A: delta hedged portfolio returns
Price portfolio
RNSkew tercile
Low
Medium
High
Lowest
∗∗
−1.327
(−2.35)
−2.173∗∗∗
(−3.88)
−2.007∗∗∗
(−4.23)
2
3
∗∗∗
−1.680
(−3.17)
−1.865∗∗∗
(−4.25)
−1.753∗∗∗
(−3.85)
4
∗∗∗
−1.671
(−3.68)
−1.948∗∗∗
(−4.45)
−1.670∗∗∗
(−3.68)
∗∗∗
−1.243
(−3.19)
−1.288∗∗∗
(−3.25)
−0.874∗∗
(−2.06)
Highest
Highest − lowest
−0.674
(−1.62)
−0.859∗∗
(−2.22)
−0.391
(−0.97)
0.653
(1.44)
1.314∗∗∗
(3.23)
1.616∗∗∗
(4.72)
Panel B: non-delta hedged portfolio returns
Low
Medium
High
9.869
(1.09)
−11.862∗
(−1.67)
−40.872∗∗∗
(−6.52)
2.494
(0.32)
−13.023∗∗
(−2.19)
−41.153∗∗∗
(−8.81)
−0.123
(−0.02)
−17.902∗∗∗
(−3.05)
−48.656∗∗∗
(−11.95)
5.465
(0.71)
−11.197∗
(−1.78)
−38.165∗∗∗
(−9.77)
16.827∗∗
(2.51)
1.517
(0.19)
−27.888∗∗∗
(−7.08)
6.958
(1.49)
13.379∗∗
(1.97)
12.984∗∗∗
(3.23)
Fig. 1. RNSkew around stock splits. The solid line with square markers represents reverse splits, and the solid line without square markers represents
regular splits. The graph plots RNSkew against trading days relative to the split. Day 0 is the ex-date. RNSkew is calculated following Bakshi, Kapadia, and
Madan (2003). The sample period is from January 1996 to December 2012.
has 130 observations.25 The average pre-split price of regular splits is 79.982, which is more than double the price
25
Unsurprisingly, a significant number of stocks in the reverse split
sample have prices under $10. We include options on these stocks in the
reverse split analysis.
of an average optionable stock. The average split ratio is
1.985. The average post-split stock price is 39.534. The average pre-split price of a stock undergoing a reverse split is
10.984, which is much lower than the average stock price.
The average split ratio for reverse splits is 0.258, resulting
in an average post-split price of 38.534.
592
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
Table 7
Summary statistics.
This table reports summary statistics for three different samples. All optionable stocks are all stocks for which RNSkew and Skew can be calculated and the
underlying stock price is greater than $10. Regular splits and reverse splits are the split stocks that are also covered by OptionMetrics. Regular splits and
reverse splits are defined as splits with split ratio greater than or equal to 1.25-to-1 and lower than 1-to-1, respectively. Beta is calculated following Fama
and French (1992). For each month, we use the past 60 months’ monthly excess return data and CRSP value-weighted excess market return to calculate
Beta. Size is market capitalization (price times shares outstanding). B/M is the ratio of book value of equity to market value. Leverage is the ratio of total
liabilities to total assets. Past volatility and Past skewness are calculated using past one year daily returns. We annualize volatility by multiplying daily
standard deviation by the square root of 252. Momentum is the cumulative return from month t − 12 to month t − 1. Rt − 60 ,t − 13 is the cumulative return
from month t − 60 to t − 13. Volume is the total number of shares traded from the last month. ILLIQ is measured following Amihud (2002). Nasdaq is a
dummy variable that is equal to one if the firm is listed in Nasdaq and zero otherwise. RNSkew is calculated following Bakshi, Kapadia, and Madan (2003).
Skew is calculated using daily data from month t + 1 to month t + 12. E(Skew) is expected skewness. We calculate E(Skew) following the method of Boyer,
Mitton, and Vorkink (2010), specifically using Eq. (5) from the text. The sample period is from January 1996 to December 2011.
All optionable stocks (N = 203,974)
Variables
Price
log (Price)
Beta
log (Size)
log (B/M)
Leverage
Past volatility
Past skewness
Rt − 12 ,t − 1
Rt − 60 ,t − 13
log (Volume)
log (ILLIQ)
Nasdaq
RNSkew
Skew
E(Skew)
Price after split
Split ratio
Regular splits (N = 1,940)
Reverse splits (N = 130)
Mean
Std. dev.
Mean
Std. dev.
Mean
Std. dev.
37.285
3.436
1.300
14.778
−1.216
0.203
0.483
0.218
0.300
1.357
12.005
−21.146
0.414
−0.634
0.231
0.249
NA
NA
33.454
0.576
0.818
1.486
0.870
0.184
0.235
1.093
0.889
2.818
1.280
1.753
0.493
0.893
1.346
0.350
NA
NA
79.982
4.238
1.311
15.126
−1.774
0.185
0.499
0.429
1.324
2.234
11.679
−21.538
0.440
−0.732
0.163
0.100
39.534
1.985
85.484
0.498
0.998
1.379
0.940
0.193
0.271
0.921
2.261
3.957
1.190
1.527
0.497
0.848
1.066
0.348
18.691
1.201
10.984
1.519
1.848
12.866
−0.152
0.261
0.836
0.604
−0.247
−0.093
12.826
−19.281
0.392
0.553
0.487
0.337
38.534
0.258
14.735
1.515
1.233
2.188
1.346
0.240
0.520
1.569
0.769
1.347
2.069
2.731
0.490
0.942
1.176
0.275
46.313
0.189
Not surprisingly, relative to the large sample, regular
splits are larger, have higher market valuation ratios (lower
B/M), and have higher past performance (both Rt − 12 ,t − 1
and Rt − 60 ,t − 13 ), and reverse splits are smaller, have lower
market valuation ratios, and have lower past performance.
Furthermore, regular splits have similar past volatility as
compared with the larger sample, and the past volatility
of reverse splits is much higher than the average optionable stock. The average RNSkew for the full sample, regular
split sample, and reverse split sample is −0.634, −0.732,
and 0.553, respectively. Future physical skewness (Skew)
of these three groups of stocks is 0.231, 0.163, and 0.487,
respectively.
Panel A of Table 8 more rigorously examines whether
risk neutral skewness expectations are affected by splitinduced changes in nominal price. It explores changes in
risk-neutral skewness around the ex-date. The results indicate that investor expectations are greatly affected by the
nominal change in price. RNSkew increases from −0.732 to
−0.464 on the day of the stock split. The effect is economically large and statistically significant, and it persists in
the weeks and months after the split. That investors respond immediately to a split-induced change in price is
not unexpected, as substantial investor ex-date responses
have been shown in the equity literature. Schultz (20 0 0)
finds that small shareholders are very active on the day of
a split. He shows a large and immediate increase in small
shareholders at the ex-date, as net small trade buy volume
increases from slightly above zero in the day prior to the
split to about two million shares on the ex-date. Further-
more, ex-date price responses are seen in the equity market as there is a large abnormal return accruing to stocks
on the ex-date, a finding first shown by Grinblatt, Masulis,
and Titman (1984).
Past work finds that splits do not seem to be motivated
by factors correlated with firm fundamentals. To further
verify that our results reflect a response to the change in
price, not a change in fundamentals, we examine whether
there is a skewness response at the announcement date.
If splits signal changes in fundamentals and the change
in RNSkew is driven by this change in investors’ information set, then an effect should be evident at the announcement date instead of the ex-date. Panel A shows that this
is not the case. We examine the change of RNSkew around
the announcement date. To avoid the effect of the ex-date,
we examine only the period before the ex-date in the announcement date analysis. We see no statistically or economically significant effect on the date of the announcement. There is some increase in RNSkew beginning on
the day after announcement, but the magnitude is much
smaller than the change at the actual date of the split. Instead, RNSkew reaches its max on the day of the ex-date.26
26
In unreported results, we examine the change of RNSkew around the
announcement date separately for options with maturity before the exdate and options with maturity after the ex-date. We find that RNSkew
does not change around the announcement date if the options used to
calculate RNSkew expire before the ex-date. This finding also suggests that
the RNSkew change around stock splits is not driven by release of new
information.
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
593
Table 8
RNSkew around regular stock splits.
Regular stock splits are events with a CRSP distribution code of 5523 and a split ratio greater than or equal to 1.25-for-1.
RNSkew is calculated following Bakshi, Kapadia, and Madan (2003) and is detailed in the paper. Panel A reports changes in
RNSkew, and Panel B reports the change in physical skewness. For RNSkew, we report the change in RNSkew around the ex-date
and the announcement date. For the analysis of RNSkew change around announcement dates, to avoid the effect of ex-date,
we examine only the period before the ex-date. We use a paired t-test to gauge statistical significance. Different windows are
chosen. For example, for (−3, 0) the before column displays RNSkew on day −3 relative to the event date, and the after column
displays RNSkew on day 0 relative to the event date (i.e., the date of the event). For each given window, we require RNSkew to
be available on both days. In Panel B, for each split stock, we require the matched stock and the split stock to be in the same
size quintile, B/M quintile, momentum quintile, and skewness quintile. We also require that the skewness difference between
the matched stock and the split stock is lower than 0.25. If more than one stock satisfies the above criteria, we choose the
one with the smallest price difference with the split stock. All the information used for matching is available at the time of
split. B/M is the ratio of book value of equity to market value. Rt − 12 ,t − 1 is the cumulative return from month t − 12 to month
t − 1. Size is equal to the log market capitalization at the end of month t − 1. Pre-matching skewness is calculated using daily
returns from month t − 11 to month t − 1. Post-matching skewness is calculated using daily returns from month t + 1 to month
t + 12. The sample period is January 1996 to December 2012. ∗ , ∗∗ , and ∗∗∗ denote significance at the 10%, 5%, and 1% levels,
respectively.
Panel A: RNSkew change
Window
N
(−10, 0)
(−5, 0)
(−3, 0)
(−1, 0)
(−1, 1)
(−1, 3)
(−1, 5)
(−1, 10)
Before
After
Change
∗∗∗
t
1,847
1,898
1,917
1,940
1,936
1,926
1,916
1,901
−0.734
−0.705
−0.706
−0.732
−0.733
−0.733
−0.730
−0.730
−0.474
−0.469
−0.468
−0.464
−0.461
−0.481
−0.502
−0.452
0.260
0.236∗∗∗
0.238∗∗∗
0.268∗∗∗
0.272∗∗∗
0.252∗∗∗
0.229∗∗∗
0.277∗∗∗
11.06
10.68
11.20
13.02
13.10
11.38
10.00
12.02
1,674
1,707
1,722
1,749
1,727
1,663
1,615
1,403
−0.863
−0.876
−0.881
−0.883
−0.880
−0.878
−0.882
−0.879
−0.872
−0.869
−0.865
−0.860
−0.837
−0.800
−0.785
−0.722
−0.009
0.007
0.016
0.022
0.043∗∗
0.078∗∗∗
0.096∗∗∗
0.158∗∗∗
−0.36
0.29
0.78
1.23
2.28
3.68
4.14
6.14
Announcement
(−10, 0)
(−5, 0)
(−3, 0)
(−1, 0)
(−1, 1)
(−1, 3)
(v1, 5)
(−1, 10)
Panel B: physical skewness change: matched sample analysis (N = 1,520)
Before
After
Variable
Treated
Matched
Difference
t
Treated
Matched
Difference
t
log (B/M)
log (Price)
Rt − 12 ,t − 1
log (Size)
Skew
−1.771
4.145
1.242
15.096
0.384
−1.762
4.049
1.114
15.261
0.386
−0.008
0.096∗∗∗
0.128∗∗∗
−0.165∗∗∗
−0.273
−0.43
14.47
2.81
−4.67
−0.85
0.151
0.190
−0.039
−1.23
The evidence suggests that nominal changes in price levels
around stock splits affect investor beliefs about the future
distribution of returns.
A potential story not excluded by the lack of an effect on the announcement date is that a change in liquidity occurs on the ex-date. The increase in RNSkew on the
ex-date could reflect a decrease in risk premium driven
by an increase in liquidity. The previous literature typically finds little evidence that splits increase liquidity. For
instance, Schultz (20 0 0), Easley, O’Hara, and Saar (2001),
Maloney and Mulherin (1992), Gray, Smith, and Whaley
(2003), and Conroy, Harris, and Benet (1990) all find that
spreads as a percentage of price increase following splits,
suggesting lower liquidity. In the Online Appendix, we report results showing that this is also true for our sample of
splits. We find that, consistent with the past research, rel-
ative spreads and relative effective spreads increase in the
post-split period, suggesting that liquidity changes cannot
explain the observed RNSkew change we see. Further alleviating the concern of a relation between RNSkew and liquidity changes, we find that in the cross-section of splits
no statistically significant relation exists between RNSkew
changes and liquidity changes. The results are reported in
the Online Appendix.
What prevents smart money from anticipating this increase in RNSkew and trading ahead of the ex-date?27 It
is well known that substantial limits to arbitrage exist in
27
An analogous effect occurs contemporaneously in the equity market.
A large abnormal return accrues to stocks on the ex-date (Grinblatt, Masulis, and Titman, 1984), suggesting that, even in the more liquid stock
market, the market lacks the ability to fully absorb the demand from investors who prefer lower-priced stocks.
594
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
options markets (for example, options are relatively illiquid
as compared with equities; also one would want to have a
delta-hedged position to hedge the risk of price changes in
the underlying, but this requires nearly constant rebalancing).28 The extent of limits to arbitrage in the options market is perhaps most clearly illustrated by the recent finding that end-user demand for options plays a strong role in
influencing options prices (Bollen and Whaley, 2004; Garleanu, Pedersen, and Poteshman, 2009).29 Despite the limits to arbitrage in the options market, in the case of the
stock split evidence, Panel A of Table 8 does show that the
RNSkew begins to increase slightly in the period between
the announcement date and ex-date, consistent with some
effort to arbitrage this phenomenon.
The lack of an effect on the announcement day suggests
that changes in investor expectations are not driven by expectations of changes in fundamentals. However, to completely rule out the possibility that the observed change in
RNSkew reflects a rational expectation of change in future
expected skewness, we assess whether physical skewness
does change in the period following splits. A priori, expectations of post-split increases in skewness seem especially hard to rationalize given that splits occur after a runup in stock price and therefore a period of above-average
skewness.
The last row of Panel B in Table 8 displays the change
in physical skew. In contrast to the expected increase in
skewness exhibited in the options market, physical skewness decreases in the period after the split. The decrease
is substantial, with daily skewness in the year following a
split decreasing by over 50% (from 0.384 to 0.151, t = 8.93)
relative to the year leading up to the split. That physical
skewness decreases following splits makes the evidence of
investor expectations of increased upside potential for recently split stocks all the more compelling.
To further ensure that the change in physical skewness
that we observe post-split does not reflect a change in
fundamentals, we compare splitting firms with a matched
sample of non-splitting firms. The matched firms are similar in that they have experienced a similar price run-up
and are of similar size, book-to-market, and past skewness.
A detailed discussion of the matching procedure is in the
table description.30 These are firms that can reasonably be
expected to have equal expectations ex-ante of undergoing a split. After matching, our sample of firms reduces to
1,520 due to missing Compustat or CRSP data.
Alleviating the concern that splitting stocks undergo a
change in fundamentals, we find no difference in future
skewness between the split sample and the matched sample. The lack of a difference in skewness provides reassurance that the post-split decrease in skewness is not an unpredictable artifact of the split. The findings provide evidence strongly supporting the theory that a nominal price
28
Buraschi and Jackwerth (2001), Coval and Shumway (2001), and Jones
(2006) show that limits to arbitrage exist between options and stocks and
that options are not simply redundant assets.
29
Cao and Han (2013) also clearly illustrate limits to arbitrage in the
options market.
30
The results are similar if we vary the matching method.
bias leads investors to attribute irrationally high skewness
to low-priced stocks relative to high-priced stocks.
6.2. Skewness around reverse splits
As an additional test of our hypothesis, we examine the
RNSkew response around reverse splits. The results are reported in Table 9. Despite a much smaller sample size, the
results are clear. Consistent with nominal prices inducing
a bias in investor expectations, we find that RNSkew drastically decreases on the day that prices increase due to a
reverse split taking effect (Panel A of Table 9).31 Panel B of
Table 9 confirms that the future physical skewness changes
do not reflect the risk-neutral changes in expectations. In
fact, we find that, in contrast to the observed decrease
in risk-neutral skewness, physical skewness increases substantially in the year following a reverse split (from 0.552
to 0.674, t = 2.09).
6.3. Robustness
In this robustness subsection, we address two concerns.
The first regards the estimation of physical skew. Physical skew is sensitive to tail events and typically requires
a long time series of return data for estimation. Even with
a sufficiently long time series, it is still subject to a peso
problem; that is, extremely good or bad events do not occur as frequently as rationally expected, leading to an error
in the measured physical skew. The problem can be more
severe for the reverse split sample than the regular split
sample, as the reverse split sample is much smaller. We
attempt to address this fear by examining a much larger
sample of splits. We extend the analysis of physical skew
to all splits occurring after 1963. This sample has 11,005
regular splits (a five-fold increase in sample size relative to
the earlier analysis) and 1,607 reverse splits (a 12-fold increase in sample size relative to the earlier analysis). This
much larger split sample confirms the earlier results: Physical skew drastically decreases following stock splits. The
results are available in the Online Appendix.
A second concern is that somehow the split process
mechanically induces a change in RNSkew as a result of the
necessary adjustment of the strike price by the split factor.
While it is not at all clear how this would induce a change
in RNSkew, we are fortunate to encounter a natural experiment during our sample period that we exploit to rule
out this possibility. Historically, option contracts have been
adjusted accordingly by the split factor. For example, if a
stock undergoes a two-for-one split, an option with a strike
price of 50 calling for the delivery of one-hundred shares
would be divided into two options with a strike price of
25, each calling for the delivery of one-hundred shares.
Effective September 4, 2007, the Options Clearing Corporation (OCC) adopted a new rule to govern the postsplit administration of options contracts. At the time, strike
prices were denominated in one-eighth increments, which
led to small inaccuracies in adjustments for certain stock
31
We are not able to compare the RNSkew change around the announcement date, as the announcement dates in CRSP are missing for
most reverse splits.
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
595
Table 9
RNSkew around reverse stock splits.
Reverse stock splits are events with a CRSP distribution code of 5523 and a split ratio less than 1-for-1. RNSkew is
calculated following Bakshi, Kapadia, and Madan (2003) and is detailed in the paper. Panel A reports changes in RNSkew
and Panel B reports the change in physical skewness. We report the change in RNSkew around the ex-date. A paired
t-test is used to gauge statistical significance. Different windows are chosen. For example, for (−3, 0) the before column
displays RNSkew on day −3 relative to the event date, and the after column displays RNSkew on day 0 relative to the
event date (i.e., the date of the event). For each given window, we require RNSkew to be available on both days. In Panel
B, for each split stock, we require the matched stock and the split stock to be in the same size quintile, BM quintile,
momentum quintile and skewness quintile. We also require that the skewness difference between the matched stock
and the split stock is lower than 0.25. If more than one stock satisfies the above criteria, we choose the one with the
smallest price difference with the split stock. All the information used for matching is available at the time of split. B/M
is the ratio of book value of equity to market value. Rt − 12 ,t − 1 is the cumulative return from month t − 12 to month
t − 1. Size is equal to the log market capitalization at the end of month t−1. Pre-matching skewness is calculated using
daily returns from month t − 11 to month t − 1. Post-matching skewness is calculated using daily returns from month
t + 1 to month t + 12. The sample period is January 1996 to December 2012. ∗ , ∗∗ , and ∗∗∗ denote significance at the 10%,
5%, and 1% levels, respectively.
Panel A: change of RNSkew
Window
N
(−10, 0)
(−5, 0)
(−3, 0)
(−1, 0)
(−1, 1)
(−1, 3)
(−1, 5)
(−1, 10)
Before
123
126
127
130
128
128
128
126
After
Change
−0.426
−0.434
−0.455
−0.470
−0.408
−0.407
−0.423
−0.321
0.255
0.320
0.332
0.553
0.566
0.566
0.566
0.570
t
∗∗∗
−0.681
−0.754∗∗∗
−0.787∗∗∗
−1.023∗∗∗
−0.974∗∗∗
−0.974∗∗∗
−0.989∗∗∗
−0.891∗∗∗
−5.11
−6.00
−6.18
−7.81
−7.94
−7.81
−8.08
−7.75
Panel B: Change of physical skewness (N = 61)
Before
log (B/M)
log (Price)
Rt − 12 ,t − 1
log (Size)
Skew
After
Treated
Matched
Difference
t
Treated
Matched
Difference
t
−0.509
1.507
−0.184
13.566
0.552
−0.401
2.223
−0.153
13.321
0.538
−0.108∗
−0.716∗∗∗
−0.030
0.245∗∗
0.013
−1.76
−8.06
−1.61
2.07
0.76
0.674
0.731
−0.057
−0.24
splits. For example, an option contract with a strike price
of $40 for which the underlying stock experienced a threefor-two split should have a new strike of $26.67. However, because strikes are denominated only in one-eighth
increments, the new strike would instead be $26.625. To
avoid this inaccuracy, the new OCC rule effective leaves
option contracts untouched and instead recalculates the
stock price to the hypothetical price it would trade had
the split not occurred. The new rule affected all splits except two-for-one or four-for-one splits, and was extended
until February 12, 2010, at which time decimalization of
strike prices was set to take place (for more details of the
rule change, see Memos 23484 and 26853 of OCC). We exploit this window of time in which strike prices were unadjusted by splits to confirm that our results are not somehow mechanically driven by adjustments to the strike price
itself.
The empirical findings we show are robust in both the
pre and post-rule change periods. Table 10 shows the effect
of splits on RNSkew for the subsample of splits occurring
between September 4, 2007 and February 12, 2010. The results are consistent with those from our earlier analysis.32
32
Because the sample is not large, we also examine median changes to
ensure that the results aren’t driven by a few outliers. The median change
The sample is small for multiple reasons. First, two-for-one
and four-for-one splits were exempt from the rule change.
Second, relatively few splits occurred during this time period, which is not surprising given the generally poor market performance. Finally, because two-for-one and four-forone splits are excluded, the sample is tilted toward very
large splits. The average split in the rule change sample
is slightly greater than six-for-one. Because investors react more strongly to larger price changes, the magnitude of
the effect seen here is much larger than in the entire sample of splits. The results for the regular splits are overall
consistent with the results over the entire sample period,
suggesting that the relation between RNSkew and price is
not driven by options market microstructure concerns.33
In summary, we find sharp changes in RNSkew at the
date that a stock splits to a lower price. In stark contrast
ranges from 1.822 to 2.343, larger than the mean change. All the median
changes are significant at the 1% level (based on Wilcoxon ranked-sign
test).
33
To further exclude the possibility of a mechanical relation, we examine a sample of ETF splits. Typically, ETFs try to mimic the return of the
whole market or a very broad group of stocks. It is very unlikely that investors would think that lower-priced ETFs have more upside potential,
especially considering that when an ETF splits, there is no split of the underlying index. We find no change in RNSkew for this sample. Results are
available in the Online Appendix.
596
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
Table 10
New rule period: regular splits.
This table reports the change of RNSkew around regular stock splits for
the new rule period. Before September 4, 2007, option strike prices were
adjusted to the nearest one-eighth to undo the effect of stock splits. The
new rule effective September 4, 2007 leaves option contracts untouched,
and instead recalculates the stock price to the hypothetical price it would
trade had the split not occurred. The new rule affected all splits except
two-for-one or four-for-one splits and extended until February 12, 2010.
Regular stock splits are events with a CRSP distribution code of 5523 and
a split ratio greater than or equal to 1.25-for-1. The new rule period is
from September 4, 2007 to February 12, 2010. We exclude two-for-one
and four-for-one splits. RNSkew is calculated following Bakshi, Kapadia,
and Madan (2003). We report the change in RNSkew around the ex-date
and the announcement date. A paired t-test is used to gauge statistical
significance. Different windows are chosen. For example, for (−3, 0) the
before column displays RNSkew on day −3 relative to the event date, and
the after column displays RNSkew on day 0 relative to the event date (i.e.,
the date of the event). For each given window, we require RNSkew to be
available on both days. ∗ , ∗∗ , and ∗∗∗ denote significance at the 10%, 5%,
and 1% levels, respectively
Window
(−10, 0)
(−5, 0)
(−3, 0)
(−1, 0)
(−1, 1)
(−1, 3)
(−1, 5)
(−1, 10)
N
22
22
22
22
21
22
22
22
Before
−0.613
−0.413
−0.381
−0.606
−0.606
−0.606
−0.606
−0.606
After
1.186
1.186
1.186
1.186
1.226
1.273
1.286
1.307
Mean change
∗∗∗
1.799
1.599∗∗∗
1.567∗∗∗
1.792∗∗∗
1.833∗∗∗
1.879∗∗∗
1.893∗∗∗
1.913∗∗∗
t
6.45
5.73
5.69
6.72
7.32
7.12
7.62
7.63
to investor expectations of post-split skewness increases,
we find a drastic decrease in physical skewness following
a split. The evidence is further strengthened by supportive results among the smaller sample of stocks undergoing reverse splits. The evidence is consistent with nominal
price levels biasing investor expectations of future return
distributions.
6.4. Increased upside or decreased downside?
The observed option-implied skew increase following
splits could reflect an increase in the perceived upside potential for the stock or a decrease in the perceived downside potential for the stock, or a combination of both. To
answer this more precise question we separately examine
puts and calls. That is, we examine changes to OTM put
and OTM call implied volatilities.
Investor perception of increased upside potential manifests itself in option-implied skew through the increased
valuation of OTM calls. Perception of decreased downside potential shows up in option-implied skew via decreased valuations of OTM puts. Intuitively, implied volatility changes allow us to identify valuation changes to assess
whether it is changing beliefs regarding upside potential or
downside potential (or some combination of the two) that
drives the observed RNSkew changes.
To examine how the relative valuations of puts and calls
change, we compare OTM implied volatilities with ATM
implied volatilities. We focus on the difference in OTM
and ATM implied volatilities, and we examine whether
this quantity changes after the split. Table 11 displays the
Table 11
Implied volatility change around stock splits.
This table reports the change of implied volatility (measured in percent)
around stock splits, both regular splits (Panel A) and reverse splits (Panel
B). Regular stock splits are events with a CRSP distribution code of 5523
and a split ratio greater than or equal to 1.25-for-1. Reverse stock splits
are events with a CRSP distribution code of 5523 and a split ratio less
than 1-for-1. Implied volatility is obtained from OptionMetrics. ATM options are defined as call options with delta higher than 0.375 and not
higher than 0.625 and put options with delta higher than −0.625 and
not higher than −0.375. OTM options are call options with delta greater
than 0.02 and not greater than 0.375 and put options with delta greater
than −0.375 and not greater than −0.02. As in Bollen and Whaley (2004),
we exclude options with absolute delta below 0.02 due to the distortions caused by price discreteness. We report changes in the difference
between the implied volatility of OTM call options and ATM call options
and the difference between the implied volatility of OTM put options and
ATM put options, around the ex-date. A paired t-test is used to gauge statistical significance. Different windows are chosen. For example, for (−3,
0) the before column displays IV differences on day −3 relative to the
event date, and the after column displays IV differences on day 0 relative
to the event date (i.e., the date of the event). For each given window, we
require the IV difference to be available on both days. The sample period
is January 1996 to December 2012. ∗ , ∗∗ , and ∗∗∗ denote significance at the
10%, 5%, and 1% levels, respectively.
Panel A: regular splits
Window
N
Before
After
Change
t
−1.493
−1.675
−1.677
−1.699
−1.699
−1.709
−1.703
−1.698
−0.597
−0.591
−0.647
−0.657
−0.780
−0.302
−0.216
0.189
0.896∗∗∗
1.084∗∗∗
1.030∗∗∗
1.042∗∗∗
0.919∗∗∗
1.407∗∗∗
1.487∗∗∗
1.888∗∗∗
2.73
3.27
3.29
3.75
3.04
4.64
4.70
5.94
2.335
2.401
2.551
2.508
2.508
2.508
2.508
2.512
2.122
2.121
2.121
2.121
2.206
2.260
2.398
2.622
−0.213
−0.281
−0.430∗∗
−0.387∗∗
−0.303
−0.248
−0.110
0.110
−1.01
−1.38
−2.28
−2.01
−1.65
−1.21
−0.52
0.55
OTM call − ATM call
(−10, 0)
(−5, 0)
(−3, 0)
(−1, 0)
(−1, 1)
(−1, 3)
(−1, 5)
(−1, 10)
2,171
2,181
2,193
2,196
2,196
2,195
2,193
2,191
OTM put − ATM put
(−10, 0)
(−5, 0)
(−3, 0)
(−1, 0)
(−1, 1)
(−1, 3)
(−1, 5)
(−1, 10)
2,171
2,181
2,193
2,196
2,196
2,195
2,193
2,191
Panel B: reverse splits
OTM call − ATM call
(−10, 0)
(−5, 0)
(−3, 0)
(−1, 0)
(−1, 1)
(−1, 3)
(−1, 5)
(−1, 10)
197
199
199
200
200
197
196
193
5.335
6.864
5.029
9.346
9.346
8.846
8.931
8.641
−1.171
−1.289
−1.289
−1.288
−4.480
0.059
−1.051
−0.149
−6.505∗∗∗
−8.152∗∗∗
−6.318∗∗∗
−10.634∗∗∗
−13.825∗∗∗
−8.787∗∗∗
−9.982∗∗∗
−10.128∗∗∗
−3.11
−3.67
−2.90
−5.04
−6.88
−4.12
−5.11
−5.03
197
199
199
200
200
197
196
193
1.053
−2.570
2.073
2.036
2.036
1.590
0.984
0.945
−0.585
−0.663
−0.663
−0.679
−0.438
−0.481
−0.665
−1.745
−1.638
1.907
−2.735
−2.716
−2.474
−2.071
−1.649
−2.690
−0.73
0.87
−1.25
−1.33
−1.17
−0.98
−0.85
−1.31
OTM put− ATM put
(−10, 0)
(−5, 0)
(−−3, 0)
(−1, 0)
(−1, 1)
(−1, 3)
(−1, 5)
(−1, 10)
J. Birru, B. Wang / Journal of Financial Economics 119 (2016) 578–598
results. Panel A examines the case of regular splits, and
Panel B examines reverse splits.34
The results are clear. Panel A shows that the difference
in OTM and ATM implied volatilities substantially increases
following the split. That is, OTM calls become more expensive following the split, reflecting investors’ increased perception of upside potential. However, the same is not true
for puts. Panel A also shows that while on average OTM
puts do seem to get slightly cheaper following a split, this
change is small and generally not statistically significant.
The smaller sample of reverse split results (Panel B of Table
11) are also consistent with the regular split evidence. After price increases due to a reverse split, OTM calls become relatively cheaper, reflecting investor beliefs of decreased upside potential. However, there is no significant
change in the case of puts. The evidence suggests that the
skew change does in fact reflect changes in investor perceptions regarding upside potential, with no accompanying
change in investor perceptions regarding downside potential. This evidence complements the earlier option return
results that look differentially at OTM calls and OTM puts,
and provides additional evidence that the general effects
we see are driven by investor-biased beliefs regarding the
upside potential, but not downside potential, of low-priced
stocks.
7. Conclusion
We provide the first evidence that investors link nominal share price to return skewness and systematically
overestimate the skewness of low-priced stocks relative to
high-priced stocks. The evidence presented is consistent
with investors suffering from the illusion that low-priced
stocks have more upside potential.
Analysis of option volume data provides further evidence that investors exhibit greater optimism toward lowpriced stocks than high-priced stocks. Finally, investor
overweighting of nominal prices in forming return distribution expectations has asset-pricing implications. We find
that OTM call options of low-priced stocks are more overvalued than those of high-priced stocks.
Firms have long engaged in the costly management of
share price through stock splits, despite nominal prices
lacking real economic content. Recent work provides evidence that investors view stocks of similar price as sharing similar attributes. Green and Hwang (2009) find that
investors categorize stocks by nominal price, and Baker,
Greenwood, and Wurgler (2009) find that firms exploit
the fact that nominal prices matter to investors by increasing the supply of low-priced securities when investors
are willing to pay a premium for them. While nominal
prices clearly matter to investors, it has thus far been less
clear why prices matter. We provide the first empirical insight into why prices matter to investors. We find evidence
34
We have slightly more observations for this analysis than for the
RNSkew estimation as we do not require a minimum number of OTM options as we do in the RNSkew analysis. Furthermore, it is not necessary for
all OTM options to have the same maturity as is the case in the RNSkew
analysis.
597
that is consistent with the notion that investors view lowpriced stocks as cheap assets with more room to grow
Appendix A
Table A1
Table A1
Definition of variables.
Variable
Description
Price
log (Price)
Beta
Stock price
Natural log of stock price
Calculated using monthly return data over the past
five years, following Fama and French (1992)
Size is the product of stock price and number of
shares outstanding, calculated at the end of
month t−1
Book-to-market is the ratio of book value of
common equity to market value of common
equity and is matched to CRSP data following
Fama and French (1992). Book equity of common
equity is calculated following Fama and French
(2008). It is equal to total assets (Compustat data
item 6), minus liabilities (181), plus balance
sheet deferred taxes and investment tax credit
(35) if available, minus preferred stock
liquidating value (10) if available, or redemption
value (56) if available, or carrying value (130).
Negative book-to-market observations are
deleted
Ratio of total liabilities divided by total assets and
matched to CRSP data following Fama and
French (1992)
Volatility is calculated using past daily returns over
the past year. Daily standard deviation is
annualized by multiplying by the square root of
252
Skewness is calculated using daily returns from
month t − 12 to month t − 1
Cumulative return from month t − 12 to month t − 1
Cumulative return from month t − 60 to month
t − 13
Natural log of total shares traded
Natural log of the illiquidity measure of Amihud
(2002) calculated using daily data in month t − 1
A dummy variable equal to one if the firm is listed
in NASDAQ, and zero otherwise
Risk-neutral skewness implied from option prices,
calculated following Bakshi, Kapadia, and Madan
(2003)
Future skewness, calculated using future one year
daily returns from month t + 1 to month t + 12
log (Size)
log(B/M)
Leverage
Past volatility
Past skewness
Rt − 12, t − 1
Rt − 60, t − 13
log (Volume)
log (ILLIQ)
Nasdaq
RNSkew
Skew
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