Journal of Banking & Finance 36 (2012) 2438–2454
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Journal of Banking & Finance
journal homepage: www.elsevier.com/locate/jbf
Derivatives traders’ reaction to mispricing in the underlying equity
Darren K. Hayunga a, Richard D. Holowczak b, Peter P. Lung c,⇑, Takeshi Nishikawa d
a
Department of Finance and Real Estate, University of Texas Arlington, Arlington, TX 76019, USA
Department of Statistics/Computer Information Systems, Baruch College, New York, NY 10010, USA
c
Reiman School of Finance, University of Denver, Denver, CO 80208, USA
d
Business School, University of Colorado Denver, Denver, CO 80217, USA
b
a r t i c l e
i n f o
Article history:
Received 17 April 2011
Accepted 25 April 2012
Available online 18 May 2012
JEL classification:
G1
G13
G14
Keywords:
Asset pricing
Mispricing
Options
Information content
Price equilibrium
a b s t r a c t
This article examines trading behavior in the options market conditioned on mispricing in the underlying
stock. We investigate the price equilibrium between the observed equity asset and the options-implied
synthetic share as well as the relative divergence between the two prices. We find a consistently positive
relation between the level of stock mispricing and violations of the upper-boundary condition using
derivatives, along with an increase in price divergence. To control for the effect of shorting limitations
on mispricing, we further examine prices during the short-sale ban in 2008. The results hold and in many
instances are more significant during the ban period. Given the persistent disequilibria between the synthetic and observed stock prices, we argue the results are evidence of informed trading in the derivatives
market.
Published by Elsevier B.V.
1. Introduction
This article investigates options traders’ response to mispricing
in the underlying equity asset. By examining the options market’s
behavior, we extend the literature documenting price discovery in
the derivatives market. Consequently, this paper provides new evidence regarding the information linkage between the stock and options markets as well as pricing efficiency in the two markets.
Motivation for our analysis stems from the growing literature
examining the informational value of derivatives and informed
trading in the options market. A rationale for this literature is the
seminal Black and Scholes (1973) options pricing model. The oftemployed model assumes that derivatives are redundant assets.
Consequently, options trades, along with the behavior of the overall derivatives market, should be uninteresting.
In contrast, there is literature documenting reasons for traders
preferring the derivatives market over the stock market. For example, theories from Back (1992) and Biais and Hillion (1994) show
that informed traders may prefer to trade derivatives due to the
⇑ Corresponding author. Tel.: +1 817 272 0115; fax: +1 817 272 2252.
E-mail addresses: hayunga@uta.edu (D.K. Hayunga), richard.holowczak@
baruch.cuny.edu (R.D. Holowczak), lungpeip@du.edu (P.P. Lung), takeshi.nishikawa@
ucdenver.edu (T. Nishikawa).
0378-4266/$ - see front matter Published by Elsevier B.V.
http://dx.doi.org/10.1016/j.jbankfin.2012.04.018
increase in leverage. Black (1975) and Manaster and Rendleman
(1982) contend that informed traders may prefer the cost structure
in the options market.1
In addition to these features, there is an expanding literature
documenting informed trading by derivatives investors i.e., the options market provides price discovery to the underlying equity. The
longer literature strand examines the relation between options
trading volume—as well as volume imbalances or the ratio of volumes in the derivatives and stock markets—and the underlying asset’s price and trading volume.2 Overall, the evidence demonstrates
a systematic relation between options trading volume and future
stock price as well as pending earnings announcements, short-sales,
and corporate mergers.
While the trading volume literature offers insight, options
volume is not a direct measure of derivative or equity prices.
Alternatively, two more recent papers examine implied volatilities,
a metric closer to prices. Cremers and Weinbaum (2010) find
1
There may be other rationales for investors preferring the derivatives market.
Options traders may realize favorable implicit borrowing rates and lower margin
requirements relative to the equity market. Additionally, there is no up-tick rule
governing the short sales of options. Thus, it may be easier to take a short position by
trading options than by shorting the underlying stock.
2
See, for example, Anthony (1988), Amin and Lee (1997), Easley et al. (1998), Cao
et al. (2005), Pan and Poteshman (2006), and Roll et al. (2010).
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
informed trading in the options market using the implied volatility
spread to predict stock returns. Xing et al. (2010) uncover a trading
strategy using implied volatility skewness.
In contrast to trading volume or implied volatilities, this paper
focuses directly on prices by examining the relation between the
options-implied or synthetic price in the derivatives market relative
to the observed price in the stock market.3 Consideration for examining prices is the ability to investigate market efficiency and the law
of one price. A primary implication of the efficient market hypothesis
is that prices reflect fundamental values, which permits capital markets to allocate resources accurately. The most basic test of efficient
allocation is the law of one price, where two assets with the same
payoff structure will have the same price in frictionless markets. By
investigating the relative value of the stock and options assets, this
article provides results of a fundamental concept in finance.
We use two main variables to document the relative prices in
the two markets. One is violations of the American-options boundary conditions and the other is the degree of price divergence between the options-implied security and the observed share.
Accordingly, this article is the first to present direct evidence of
the effect mispricing in the underlying equity has on the equilibrium between synthetic and actual stock prices.
To the extent options traders identify mispricing in the underlying stock, we hypothesize an increase in boundary violations
and the magnitude of price divergence with an increase in mispricing. Further, we expect upper-boundary violations (when the observed stock price is greater than the options-implied price) to
occur more frequently than lower-boundary violations (when the
observed stock price is lower than the options-implied price) due
to short-sale constraints. In frictionless markets, theory shows that
when a firm’s equity is relatively overvalued such that the observed stock price is greater than the synthetic value in the options
market, an investor can realize an arbitrage profit by being short
the stock and long the synthetic. The arbitrage force helps to keep
the stock and options markets in sync. However, when the observed stock price is greater than the synthetic value but an arbitrager is limited in shorting the stock due to short-sale constraints,
the law of one price can break down.
Consequently, we add to our investigation the effect of shorting
limitations on the relation between mispricing and price disequilibria in the stock and options markets. To this end, we use the
shorting ban in the fall of 2008 as a clean test environment to conduct a model-free study. Since the short-sale ban is almost a total
shorting constraint on financial firms, we are able to examine the
mispricing relationship using a structural break between the preban and ban periods. Further, we can contrast mispricing behavior
with respect to the banned firms and a control group of industrial
firms that continue to trade under typical market conditions.
By combining mispricing with the shorting ban, a second contribution of this article addresses a gap in the literature that has been
alluded to but is not fully analyzed.4 Two existing papers investigate price disequilibria between the stock and options markets with
respect to mispricing and shorting constraints. Lamont and Thaler
(2003) examine firms that conduct equity carve-outs. They find
two firms that violate the law of one price with the synthetic short
selling for much less than the observed price of the stock. The second
paper is by Ofek et al. (2004). They look for violations of put-call
parity in the options market upon conditioning on the price-earnings
3
A recent example of comparing stock and synthetic share prices is Chen et al.
(2011).
4
Although our investigation examines price disequilibria between stock and
options markets, the test results lend support to the arguments of Miller (1977), who
examines stock prices in the presence of heterogeneous beliefs and short-sale
constraints. Lim (2011) discusses extension of Miller (1977) in dynamic markets.
Nilsson (2008) uses the restrictive shorting case in Sweden to example mispricing in
put-call parity.
2439
(P/E) ratio. However, Ofek et al. do not explicitly examine the interaction between the violations and equity mispricing. In contrast to
the small sample size in Lamont and Thaler (2003) and the noisy
proxy in Ofek et al. (2004), this article directly quantifies price disequilibria using two complementary mispricing measures along
with an almost complete shorting ban on a set of firms.
Prior to considering the banned financial firms, we first examine
a price-equilibrium baseline between the observed and synthetic
shares for the industrial firms using tick-by-tick derivatives transaction data. When we sort an ordered distribution using mispricing
terciles, we find few lower-boundary violations overall and across
mispricing terciles. Conversely, we observe a significant number of
upper-boundary violations, which increase in mispricing. Upperboundary violations occur in 0.92% of the observations for industrial
firms that exhibit low mispricing. For medium-mispricing industrial
firms, the upper-boundary violations increase to 1.05%. The violations burgeon to 6.00% for high-mispricing industrial firms.
Similarly, we find a positive relation between mispricing and
price divergence. For the low-mispricing industrial firms the
divergence measure is 1.06%, which increases to 2.03% for medium-mispricing industrial firms. The high-mispricing firms demonstrate a price divergence of 3.67%.
We next examine the financial firms that are banned from being
shorted during the fall of 2008. Prior to the ban, there is not a considerable difference in the test metrics between the industrial and
financial subsamples. For the upper-boundary condition, we find
the financial firms exhibit an average violation ratio of 2.67% preban, which is similar to the industrial firms pre-ban (2.62%) and
during the ban (2.57%).
Upon sorting financial firms into mispricing tranches, we
observe pricing disequilibria, which exhibits a positive relation with
mispricing. Prior to the ban, financial firms in the low-mispricing
group violate the law of one price in 0.25% of the subsample. The
violation ratio for medium-mispricing financial firms is 1.46%. For
high-mispricing financial firms, the violation ratio increases to
6.55%. This level is comparable to high-mispricing industrial firms,
both before and during the ban (6.00% and 6.09% respectively).
Upon implementation of the shorting ban, we observe an increase in upper-boundary violations and price divergence. Further,
the results maintain the positive relation with mispricing. During
the ban, low-mispricing financial firms violate 0.68% of the time.
We note that this magnitude is still less than the control group
of industrial firms before or during the shorting ban. We find medium-mispricing financial firms violate the upper-boundary condition 2.12% during the ban. In contrast, firms whose stock is
mispriced the greatest exhibit violations of the law of one price
in 10.31% of the observations. We also observe significant increases
in the magnitude of price divergence during the ban for financial
firms across all three mispricing tranches. When we compute daily
abnormal divergence, we find the largest divergence in the medium- and high-mispricing groups.
Overall, the evidence consistently demonstrates that derivatives
investors differentiate the levels of mispricing in the underlying
equity, and, consequently, adjust options prices such that the levels
violate the upper-boundary condition and increase the magnitude
of price divergence. We argue this is evidence of price discovery in
the options market, especially in times of greater market uncertainty during the sample period. We present the details supporting
our conclusion in the remainder of this article.
2. Methods and estimation
Our method of estimating boundary violations of the law of one
price and price divergence is different from previous studies. We
use American options boundaries instead of put-call parity.
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D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Consequently, our methods do not rely on the estimation of an option’s early exercise premium. We are also able to consider transaction costs in both stock and derivatives markets. Further, we
do not use midpoints but the actual bid and ask prices.
In addition to the boundary-violation and price-divergence test
measures, we are motivated to investigate options bid-ask spreads
as another pricing metric with respect to mispricing. Battalio and
Schultz (2011) examine spreads during the 2008 shorting ban;
however, the impact of stock mispricing on options spreads has
not been examined in the literature. In addition to finding the expected increase in spreads during the shorting ban, we observe a
positive relation between mispricing and spreads. This is especially
true for put options. Indeed, during the uncertain market conditions around the time of the ban, we find few abnormal increases
in daily spreads for the low- and medium-mispricing industrial
firms. Instead, the high-mispricing industrial firms drive the increase in mean spreads. These findings provide further detail
regarding options spreads during the shorting ban that has not
been discussed in the literature.
We detail the theory and econometrics for the test metrics in
the remainder of this section and the appendices. We begin by
describing the estimation of stock mispricing in the next subsection. Subsequently, Sections 2.2 and 2.3 discuss calculations for
the boundary violations and divergence, respectively.
2.1. Measures for stock mispricing
We employ two measures to proxy stock mispricing—the mispricing component from a vector autoregression model (VAR)
using the Vuolteenaho model (2002) and the dispersion of financial
analyst’ forecasts (DISP). These two measures complement each
other. The VAR approach provides a direct measure of mispricing
and has a solid theoretical model. Alternatively, any model may
suffer from measurement error or misspecification. Consequently,
we also use the standard deviation of financial analysts’ forecasts
on long-term earnings growth. This measure is intuitive and easy
to obtain. Conversely, it is an empirical proxy that indirectly measures mispricing and lacks a theoretical foundation. Thus, we use
the two measures as robustness checks for one another. Overall,
the results using either mispricing measure are quite similar and
the conclusions are the same.
is as follows. Investors observe different sets of information or
signals about the value of an asset. They form different beliefs
about the price, e.g., some investors can be more optimistic than
others. Because of short-sale constraints, pessimistic investors cannot act on their market views based on their negative information
and remain out of the market. As a consequence, stock prices are
driven up by the investors with the most optimistic opinion and
the negative opinion cannot be fully reflected in the stock prices.
As heterogeneous beliefs widen and the dispersion of financial analyst’ forecasts increases, the assets deviate from their fundamental
value. This, in effect, causes mispricing in stock prices.
Literature using DISP as an indirect proxy for heterogeneous beliefs and mispricing includes Diether et al. (2002), Boehme et al.
(2006), and Moeller et al. (2007).6 We follow Moeller et al. (2007)
to construct DISP. The measure is the standard deviation of analysts’
forecasts of the long-term earnings growth forecast, which I/B/E/S
defines as 3–5 years. We use the dispersion immediately preceding
the test sample period to classify the firms into three groups—low,
medium, and high.
2.2. Options boundary violations
We use American-options boundary conditions to examine the
violations of the law of one price. A study by Akram et al. (2009)
finds that the law of one price holds on average but there are instances of violations. The inequality for American options is
S D K 6 C P 6 S PVðKÞ;
where S is the stock price, D is the present value of dividends over
the life of the option, K is the strike price, C and P are the call and
put prices in the options market, and PV(K) is the present value of
the strike price. Inequality (1) provides a lower-boundary condition
for the stock price of C P + PV(K) 6 S and an upper-boundary condition of S 6 C P + D + K.
An advantage of using the inequality for American options is the
ability to incorporate market frictions into the arbitrage procedure.
We can decompose Inequality (1) into the two conditions as
ðPa þ Sa PVðKÞÞ þ ðT X þ T S þ T P Þ P C b T C ;
2.1.2. DISP measure
Building on Chen et al. (2002) and Scheinkman and Xiong
(2003), the intuition regarding heterogeneous beliefs and mispricing
5
Campbell and Shiller (1988), Vuolteenaho (2002), Campbell and Vuolteenaho
(2004), Coakley and Fuertes (2006), and Brunnermeier and Julliard (2008), among
others, follow this method to determine stock mispricing.
ð2Þ
and
ðC a Sb þ D þ KÞ þ T x þ T s þ T c P Pb T p ;
2.1.1. VAR model
We obtain the direct mispricing values using the dynamic valuation framework of Vuolteenaho (2002). The model shows the fundamental value for the log of market to book value ratio (MB)
should equal all the future returns on equity (ROE) minus all the future stock returns (r). Following previous studies, we define the
mispricing component as the difference between the observed
and the fundamental value.5 Appendix A details further the estimation method.
For each stock in our sample, we use the quarterly estimate of
stock mispricing immediately preceding the test sample period.
We use the mispricing measure to classify the firms into three
groups—low, medium, and high mispricing. Thus, by construction,
high-mispricing firms are relatively more overvalued than medium-mispricing stocks, and vice versa for low-mispricing versus
medium-mispricing firms.
ð1Þ
ð3Þ
where S, P, C, K, r, s, and D are as specified above. The superscripts a
and b denote ask and bid prices. TX, TS, TP, and TC are the transaction
costs for exercising options and trading stocks as well as specific
values for puts and calls. Rearranging Inequality (2) gives us the
lower-boundary condition for the underlying stock as
Sa P C b Pa þ PVðKÞ ðT X þ T S þ T P þ T C Þ:
ð4Þ
Similarly, rearranging Inequality (3), the upper-boundary condition is
Sb 6 C a Pb þ D þ K þ ðT X þ T S þ T P þ T C Þ:
ð5Þ
Inequality (4) shows that the lowest value of the underlying’s
ask price is the synthetic security on the right side. Likewise,
Inequality (5) shows that the maximum value of the underlying’s
bid price is the synthetic on the right side. If the bid price of the
underlying stock is greater than the synthetic in Inequality (5),
arbitragers will short the stock and take a long position in the synthetic. However, in the presence of short sale constraints, investors
6
Boehme et al. (2006) find a negative effect of opinion dispersion on monthly and
annual subsequent returns. Chen et al. (2002) find a negative effect of opinion
dispersion on quarterly subsequent returns. Desai et al. (2002) also document annual
negative subsequent returns.
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
cannot short the underlying stock as in frictionless market and,
thus, the upper-boundary condition as specified in Inequality (5)
should be more likely to be violated than the lower-boundary condition in Inequality (4).
2.3. Divergence
Inequalities (4) and (5) can also measure the deviation between
the options-implied price and actual stock price. From Inequality
(4), the difference between the two sides of the inequality,
[Cb Pa + PV(K) (TX + TS + TP + TC)] Sa, measures the distance
between the lower bound and the observed stock price, which
we denote by LD. The greater the distance, the higher the call premium is relative to the put premium and stock price. By the same
token, Sb minus the synthetic position and transaction costs on the
right side of Inequality (5) determines the distance between the
observed stock price and the upper bound. The greater the distance, the higher the put premium is relative to the call premium
and stock price. This distance we denote by UD. The divergence
measure is given by the difference between LD and UD.
Div ergence ¼ LD UD
¼ ðC a þ C b P a P b Þ ðSa þ Sb DÞ þ ðK þ PVðKÞÞ:
ð6Þ
Eq. (6) demonstrates that the more negative the Divergence, the
lower the options-implied price relative to the observed stock
price. As a relative measure of the synthetic versus actual stock
price, Divergence is not a direct result of boundary violations. Instead, we may observe divergence in the two prices when boundaries are not violated, and conversely, find no divergence even
though we observe boundary violations. To make Divergence comparable for different options, we scale the measure by the average
of derivatives prices and specify the divergence ratio as
DR ¼
Div ergence
a
ðC þ C b þ Pa þ P b Þ=4
:
ð7Þ
We provide more details regarding the test statistics in Appendix B.
3. Sample selection
We obtain intraday bid and ask quotes for both options and
stocks to calculate the upper and lower options boundary violations, divergences, and bid-ask spreads. The options data are from
Baruch Options Data Warehouse (BODW) based on the Options
Price Reporting Authority (OPRA). OPRA collects intraday options
quote and trade messages, adds a sequence number to each message, and subsequently tags each quote with a code indicating
whether the quote represents a national best bid and/or offer. At
any given point in the day, an options series’ National Best Bid
(NBB) is the highest bid price from all participating options exchanges. The National Best Offer (NBO) is the lowest posted offer.
We exclude non-firm quotes as well as those flagged as closing
quotes when calculating our National Best Bid and Offer (NBBO)
since these quotes are only indicative. BODW parses the standard
OPRA data formatted files into four main types of data files according to the OPRA message types: Quotes, Trades, Open Interest, and
End of Day. This paper employs the Quotes data, which include the
OPRA message sequence, options root, options series, expiration
date, strike price, NBBO quote, trading date, and trading time
(hour, minute and second).
For the underlying equity market, we obtain quote records from
the New York Stock Exchange’s Trade and Quote database. Following Bessembinder (2003) and others, we remove indicative quotes
and quotes associated with trading halts or designated order
imbalances. In addition, to ensure that our derivatives and stock
2441
quotes represent prices at which investors could actually trade,
we omit observations with the best bid quote equal to the best offer quote or the best bid quote higher than the best offer quote. For
each options series, we obtain the NBBO options quotes and stock
quotes at 4:00 pm (EST). In this manner, we eliminate any nonsynchronous trading problem.
To compute the interest rate we use the daily continuouslycompounded zero rates whose maturities match the expiration
dates on the options. Because the bid-ask quotes of interest rates
are not available for us, we consider the bid (ask) interest quote
is 5% less (more) than the interest rate we have when we estimate
the present value for exercise prices and dividends. We find the results are robust to varying the ratio from 5% to 20%. Each transaction cost (TX, TS, TP, or TC) is set at $0.01 per share.7
The sample period is from April 29, 2008 through October 22,
2008, which covers 100 trading days before the ban, the 14 trading
days during the ban period, and 10 trading days post-ban. We exclude all the stocks without options data during our sample period.
To be included in the sample, a stock must have data in both
COMPUSTAT and CRSP for 10 years immediately preceding the test
sample period. Thus, we have forty quarters of data to calculate the
mispricing component in the quarter immediately preceding the
test sample period for each firm.
As in Vuolteenaho (2002), we calculate mispricing for each
stock using quarterly COMPUSTAT data. The data are from the second calendar quarter of 1998 through the second calendar quarter
of 2008. In the VAR model, the market value is the stock price times
shares outstanding in the beginning of each quarter, the book value
of common equity is the sum of common equity, deferred taxes,
and taxes payable, (i.e., Book Value = CEQ + TXDC + TXP) according
to Fundamentals Quarterly in COMPUSTAT. If common equity,
CEQ, is not available, we use last period’s book value of common
equity plus earnings (NIQ), less dividends (DVTQ). If neither earnings nor book value are available, we assume that the market-tobook ratio has not changed and compute the book value proxy
from the last period’s market-to-book ratio and this period’s market value. We treat negative or zero book equity values as missing.
ROE is earnings over the last period’s book value, NIQ t/Book
Valuet1. If earnings are missing, we use the clean-surplus formula
to compute a proxy for earnings; that is, earnings equal the change
in book value of common equity plus dividends. For each case, we
do not allow the firm to lose more than its book value. We define
the range of the net income as a maximum of the reported net income (or surplus net income, if earnings are note reported) and
negative of the beginning of the period book equity as the
minimum.
4. Analysis
Table 1 reports descriptive statistics of the sample, which consists of 213 financial firms (Panel A) and 1996 industrial firms
(Panel B).8 We equally divide the stocks according to pre-ban mispricing into low, medium, and high tranches using both the VAR
and DISP measures. The financial-firm mispricing measure ranges
from negative 4.99% for the low-mispricing group to positive 6.19%
7
The transaction cost is based on the charges of a leading online trading firm,
TradeStation.com.
8
According to SEC RELEASE NO. 34-58592 on September 18, 2008, this list includes
banks, insurance companies, and securities firms identified by SICs 6000, 6011,
6020-22, 6025, 6030, 6035-36, 6111, 6140, 6144, 6200, 6210-11, 6231, 6282, 6305,
6310-11, 6320-21, 6324, 6330-31, 6350-51, 6360-61, 6712, and 6719. The short sales
ban list was modified over time after the initial announcement of 799 financial firms
on September 18, 2008. In an effort to reduce the influence of the modification on our
test results, we exclude all the stocks which are not on the original list or are removed
from the list. The excluded stocks are ACAP, ARCC, ATLS, BKCC, DHIL, GLRE, HTGC,
JMP, KCAP, NEWS, NITE, PCAP, SHLD, TAXI, and UNCL.
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D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Table 1
Descriptive statistics. This table presents summary statistics of the study sample split by firm type and two mispricing measures – VAR and Dispersion of Analysts’ Forecast
(DISP). Within each firm type, we divide the sample between low, medium, and high mispricing firms. We examine 90 trading days from April 29, 2008 through September 4,
2008 (pre-ban) in contrast to the short-sale ban period from September 19 to October 8, 2008.
Mispricing measure
Panel A. Financial firms
Number of firms
Mispricing measure
Pre-ban period
Number of put-call
pairs
Return
Return std. dev.
Ban period
Number of put-call
pairs
Return
Return std. dev.
Panel B. Industrial firms
Number of firms
Mispricing measure
Pre-ban period
Number of put-call
pairs
Return
Return std. dev.
Ban period
Number of put-call
pairs
Return
Return std. dev.
VAR
DISP
Full
sample
Low
mispricing
Medium
mispricing
High
mispricing
Full
sample
Low
mispricing
Medium
mispricing
High
mispricing
213
1.01%
71
4.99%
71
1.83%
71
6.19%
213
4.84%
71
2.17%
71
6.92%
71
7.42%
318,818
105,992
111,735
101,091
318,818
100,621
104,506
113,691
0.02%
4.97%
0.06%
3.09%
0.02%
4.46%
0.09%
7.36%
0.02%
4.97%
0.10%
2.77%
0.06%
4.57%
0.10%
7.57%
34,948
12,244
13,121
9583
34,948
11,941
12,905
10,102
1.44%
9.04%
1.03%
6.53%
1.33%
7.66%
1.96%
12.93%
1.44%
9.04%
1.03%
6.92%
1.35%
7.56%
1.94%
12.64%
1996
0.35%
665
4.43%
666
2.42%
665
3.06%
1996
3.71%
665
2.01%
666
3.61%
665
5.51%
3091,246
1014,778
1071,045
1005,423
3091,246
1002,326
1005,577
1083,343
0.07%
3.59%
0.13%
2.97%
0.05%
3.29%
0.03%
4.51%
0.07%
3.59%
0.13%
2.51%
0.06%
3.22%
0.02%
5.03%
342,294
120,083
115,641
106,570
342,294
125,182
115,410
101,702
1.05%
5.76%
0.92%
4.97%
1.06%
5.52%
1.17%
6.79%
1.05%
5.76%
0.90%
4.28%
0.98%
5.97%
1.27%
7.03%
for the high-mispricing class. The average mispricing measure is
lower for the full sample of industrial firms—0.35%, with a tighter
range from negative 4.43% for the low-pricing firms to positive
3.06 for the high-mispricing tranche.
Using the DISP measure, we observe a full-sample mean of 4.84%,
with a range from 2.17% for the low-mispricing group to 7.42% the
average of the high-mispricing tranche for financial firms. Note that
these magnitudes are not comparable to the VAR values as the DISP
measure is the standard deviation of analysts’ growth forecasts and
not deviations from a computed fundamental MB ratio.
Table 1 also reports the mean and standard deviation of returns
for the full firm-type subsamples as well as the mispricing groups.
Whether increasing or decreasing, returns exhibit the same monotonic relation across mispricing measures for either the VAR model
or DISP. This fact holds for both financial and industrial firms.
4.1. Options boundary violations
Table 2 reports violation ratios. Panel A details violations of the
lower boundary as specified in Inequality (4). The results demonstrate a relatively efficient market. Using either the VAR or DISP
mispricing measure, violations range from 0.01% to 0.02% of the total observations for both firm types. We observe no economically
significant differences between the pre-ban and ban periods for
either financial or industrial firms. The lack of difference holds
for the full sample of both firm types as well as across all the mispricing categories. When we compute difference across firm types
within either the pre-ban or ban periods, we find little significance.
In contrast, the upper-boundary violations as specified in
Inequality (5) demonstrate significant differences, both across
ban periods and between the two firm types. Focusing first on
the VAR results, the findings in Panel B of Table 2 demonstrate that,
on average, 2.67% of the 318,818 financial-firm observations violate the upper-boundary violation prior to the shorting ban period.
This proportion is quite comparable to the industrial firms for both
the pre-ban and ban time periods. The mean ratios for the industrial firms are 2.62% pre-ban and 2.57% during the ban. For comparison, the difference between firm types pre-ban is just marginally
significant with a t-statistic of 1.72.
Such consistent ratios offer a gauge of typical short-sale constraints and other market-friction costs. Alternatively, when the
SEC bans short selling in the financials, we find a significant increase in violation ratios. For the full sample of financial firms,
the violation ratio increases to 3.86%, an increase of approximately
45%. The difference between firm types during the ban period is
1.29%, which yields a t-statistic of 15.20.
Decomposing the sample into mispricing terciles, we find that
the full-sample averages are driven by the high mispricing subsample across both time periods and firm types. We note that
the full sample means are materially larger than the mediummispricing averages, which is consistent with a right-skewed
distribution due to the high-mispricing firms.
Focusing on the VAR results, we observe the upper-boundary
violation ratio is 0.25% for the low-mispricing financial firms and
1.46% for medium-mispricing financial firms prior to ban. However,
the high-mispricing financial firms violate the upper-boundary
condition in 6.55% of the observations. For industrial firms, we find
the low-mispricing firms violate in 0.92% of the observations while
the medium-mispricing firms violate 1.05%. The high-mispricing
industrial firms violate the boundary in 6.00% of the observations.
We note that low-mispricing financial firms actually violate
materially less than industrial firms pre-ban—the difference is a
statistically significant amount of 0.67%.
Upon the implementation of the ban, we find that financial-firm
violations increase across the mispricing tranches, but especially
for the high mispricing firms. The percentage violations are 0.68,
2.12, and 10.31 for the low-, medium-, and high-mispricing firms,
respectively. Clearly, the shorting ban produces increased
violations but much more for high-mispricing firms. Alternatively,
the industrial firms do not increase markedly after the
2443
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Table 2
Violation ratios of boundary conditions. The table reports lower-boundary violation ratios in Panel A and upper-boundary in Panel B. We examine ratios from April 29, 2008
through September 4, 2008 as well as during the short-sale ban from September 19 through October 8, 2008. We use two models of mispricing in the stock market – the
Vuolteenaho (2002) model and dispersion of analysts’ forecasts (DISP).
Mispricing measure
Full sample
VAR
Low mispricing
Panel A. Lower boundary violation ratios
Financial firms
Pre-ban period
0.02%
Ban period
0.01%
Difference
0.00%
t-Statistic
0.61
Industrial firms
Pre-ban period
0.01%
Ban period
0.01%
Difference
0.00%*
t-Statistic
1.72
Difference between firm types
Pre-ban period
Difference
0.00%*
t-Statistic
1.94
Ban period
Difference
0.00%
t-Statistic
0.57
*
***
High mispricing
Low mispricing
Medium mispricing
High mispricing
0.02%
0.01%
0.01%
0.75
0.02%
0.02%
0.00%
0.30
0.02%
0.00%
0.00%
0.01
0.02%
0.01%
0.01%
0.63
0.02%
0.02%
0.00%
0.35
0.02%
0.00%
0.00%
0.08
0.01%
0.01%
0.00%
0.85
0.02%
0.01%
0.00%
0.99
0.02%
0.00%
0.00%
1.00
0.01%
0.00%
0.00%
1.59
0.02%
0.01%
0.00%
1.02
0.02%
0.00%
0.00%
0.24
0.01%**
2.29
0.00%
0.80
0.00%
0.61
0.01%**
2.27
0.00%
0.79
0.00%
0.64
0.00%
0.09
0.00%
0.33***
0.01%
0.56**
0.00%
0.57*
0.00%
0.25
0.00%
0.24
6.55%
10.31%
3.76%***
14.90
0.12%
0.49%
0.37%***
11.69
Panel B. Upper boundary violation ratios
Financial firms
Pre-ban period
2.67%
0.25%
Ban period
3.86%
0.68%
***
Difference
1.19%
0.43%***
t-Statistic
13.82
9.48
Industrial firms
Pre-ban period
2.62%
0.92%
Ban period
2.57%
0.88%
Difference
0.05%*
0.04%
t-Statistic
1.74
1.50
Difference between firm types
Pre-ban period
Difference
0.05%*
0.67%***
t-Statistic
1.72
22.92
Ban period
Difference
1.29%***
0.20%**
t-Statistic
15.20
2.41
**
DISP
Medium mispricing
1.46%
2.12%
0.66%***
6.34
1.05%
1.08%
0.03%
1.05
6.00%
6.09%
0.09%
1.25
0.56%
0.99%
0.43%***
6.53
6.86%
11.14%
4.28%***
17.31
0.71%
0.68%
0.03%
1.28
1.07%
0.98%
0.09%***
3.03
5.83%
6.71%
0.89%***
12.07
0.51%***
15.92
1.03%***
14.87
0.01%
0.08
4.43%***
19.31
0.40%***
13.24
0.54%***
7.25
0.59%***
22.29
1.04%***
11.45
4.22%***
17.25
0.19%***
2.58
Statistical significance at the 10% levels.
Statistical significance at the 5% levels.
Statistical significance at the 1% levels.
implementation of the ban. Despite the turbulent market, low-mispricing industrial firms violate slightly less during the ban (0.88%)
than pre-ban (0.92%). Across all mispricing terciles, the differences
in industrial-firm violations during the shorting ban versus the preban period are not statistically significant.
The results demonstrate largely the same conclusions when we
use DISP as the proxy for mispricing. The effect the high-mispricing
firms have on percentage of violations is a bit more pronounced
using the DISP model. For instance, the relative number of violations
for low- and medium-pricing financial firms is lower than the VAR
findings, both before and during the ban. Conversely, the percentage
violations are higher than the VAR results for the high-mispricing
financial firms, 6.86% pre-ban and 11.14% during the ban.
Decomposing the upper-boundary results in Panel B of Table 2
into daily values, Figs. 1 and 2 provide expedient views of the
abnormal violation proportions from 100 days before the ban to
10 days after the ban using the mispricing subsamples. Since the
DISP values and graphs are quite similar, we report the results
using the VAR model. We note that in Fig. 1 the high-mispricing
financial firms increase markedly 10 days prior to the ban and during the ban period. In contrast to Fig. 1, the abnormal violation ratios of the industrial firms in Fig. 2 demonstrate no differences
across mispricing classes before, during, or after the shorting ban.
Table 3 details the values of the daily abnormal violation values
in Figs. 1 and 2. We include the 10 days before and after the ban for
comparison to the behavior during the ban. Panel A presents the
results using the VAR model. For the financial firms, Panel A demonstrates the positive relation between abnormal violations and
mispricing. For low-mispricing financial firms, we observe a significant increase in abnormal violation ratios the day of the ban. The
abnormal values continue until Day +5, and then do not exhibit
abnormal behavior during the remainder of the ban and postban. Medium-mispricing financial firms experience significant
abnormal violation ratios coinciding entirely with the ban period.
Likewise, high-mispricing financial firms also exhibit significant
abnormal violation ratios only during the ban period; however,
the magnitudes of the violations of the upper-boundary condition
are much larger than those for the medium-mispricing group. In
sum, Panel A of Table 3 strongly suggests that options traders
can determine the firms that are highly mispriced and trade
accordingly, thereby, violating the upper-boundary condition.
In contrast, we observe no systematic abnormal violation ratios
during the sample period for industrial firms. This holds across the
mispricing terciles.9 Despite the turbulent times, we find that there
are basically no abnormal ratios of the upper-boundary condition
9
Note that our asymmetric findings are robust to issues of parametric tests, market
microstructure, or nonsynchronous trading. As a robustness test, we compute the
results for the violations, divergence ratios, and bid-ask spreads using non-parametric
sign tests and find the conclusions to be the same.
2444
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Fig. 1. Abnormal violation ratios for financial firms. The figure reports abnormal violation ratios of the upper-boundary condition for a sample of financial firms. The
mispricing measures are from Vuolteenaho (2002). The period is from April 28, 2008 through October 22, 2008 with Day 0 the beginning of the shorting ban on September 19,
2008.
Fig. 2. Abnormal violation ratios for industrial firms. The figure reports abnormal violation ratios of the upper-boundary condition for a sample of industrial firms. The
mispricing measures are computed using the VAR model from Vuolteenaho (2002). The period is from April 28, 2008 through October 22, 2008 with Day 0 the beginning of
the shorting ban on September 19, 2008.
when the stock and options markets are allowed to transact
normally.
The conclusions are the same when we proxy for mispricing
using DISP. Panel B of Table 3 presents a positive relation between
violation ratios and DISP for financial firms. Low-mispricing financial firms exhibit some violations upon implementation of the ban
but reduce to normal within a few days after the beginning of the
ban. Medium-mispricing financial firms exhibit significant
violation ratios throughout the ban; however, the greatest ratio
magnitudes are found in the high mispricing tranche. In contrast,
the industrial firms exhibit generally no abnormal behavior across
mispricing terciles and across the three time periods.
4.2. Divergence ratios
In Table 4, we document the divergence ratios. As with the
upper-boundary violation ratios, we find that the high-mispricing
firms, both financial and industrial, drive the results. Whether be-
2445
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Table 3
Time series of upper boundary abnormal violation ratios. This table reports the abnormal violation ratios of the upper-boundary condition from 10 days before to 10 days after a
short-sale ban in the fall of 2008. Panel A details boundary violations conditioned on mispricing in the stock market using the Vuolteenaho (2002) model, while Panel B uses
dispersion of analysts’ forecasts as a mispricing proxy. Values are in percentages.
Date
*
***
Financial firms
Industrial firms
Low
Medium
High
Low
Medium
High
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
0.07
0.11
0.02
0.05
0.07
0.04
0.05
0.04
0.03
0.17
4.25***
0.59**
2.95***
1.47***
1.19**
0.78**
0.27
0.19
0.08
0.41
0.09
0.01
0.09
0.11
0.05
0.13
0.05
0.11
0.05
0.04
0.03
0.18
0.02
0.08
0.41
0.26
0.15
0.09
0.41
0.09
0.22
0.25
0.10
0.26
7.03***
1.96***
4.17***
1.72***
3.40***
1.04***
1.11***
1.12***
1.82***
1.37***
1.39***
1.06***
0.88***
0.67***
0.07
0.18
0.19
0.44
0.40
0.35
0.48
0.63
0.32
0.09
3.16
3.31
3.52
2.70
2.12
3.21
2.19
2.63
2.66
2.74
15.65***
6.97***
18.45***
8.13***
15.43***
4.79**
5.14***
4.09**
16.07***
7.92***
5.36***
3.98**
4.33**
3.85**
1.09
1.13
3.15
0.49
1.10
1.13
2.68
1.78
2.45
1.02
0.05
0.09
0.11
0.09
0.07
0.03
0.28
0.35
0.24
0.42
0.61
0.38
0.09
0.15
0.08
0.14
0.61
0.28
0.11
0.38
0.49
0.24
0.53
0.07
0.12
0.08
0.03
0.24
0.17
0.09
0.34
0.12
0.17
0.41
0.29
0.24
0.29
0.33
0.16
0.31
0.59
0.20
0.11
0.13
0.06
0.40
0.07
0.04
0.21
0.32
0.34
0.04
0.43
0.16
0.26
0.20
0.01
0.13
0.33
0.10
0.01
0.02
0.15
0.09
0.13
0.31
0.12
0.38
0.34
0.42
1.50
0.19
0.36
0.06
1.06
1.09
0.35
1.69
0.88
0.57
0.08
0.16
0.15
0.09
1.88
0.80
0.23
0.46
0.89
0.62
0.81
1.67
0.96
0.25
1.10
0.38
0.50
0.20
1.61
0.44
1.32
0.52
Panel B. Dispersion of analysts’ forecasts
September 05, 2008
10
September 08, 2008
9
September 09, 2008
8
September 10, 2008
7
September 11, 2008
6
September 12, 2008
5
September 15, 2008
4
September 16, 2008
3
September 17, 2008
2
September 18, 2008
1
September 19, 2008
0
September 22, 2008
1
September 23, 2008
2
September 24, 2008
3
September 25, 2008
4
September 26, 2008
5
September 29, 2008
6
September 30, 2008
7
October 01, 2008
8
October 02, 2008
9
October 03, 2008
10
October 06, 2008
11
October 07, 2008
12
October 08, 2008
13
October 09, 2008
14
October 10, 2008
15
October 13, 2008
16
October 14, 2008
17
October 15, 2008
18
October 16, 2008
19
October 17, 2008
20
October 20, 2008
21
October 21, 2008
22
October 22, 2008
23
0.06
0.10
0.02
0.04
0.07
0.05
0.05
0.04
0.03
0.18
4.78***
0.53***
3.38***
0.62**
0.20
0.77*
0.36
0.21
0.07
0.28
0.08
0.02
0.08
0.12
0.05
0.15
0.05
0.13
0.06
0.04
0.04
0.18
0.10
0.07
0.43
0.23
0.17
0.09
0.38
0.10
0.20
0.26
0.10
0.22
8.97***
3.15***
5.74***
2.21***
4.18***
1.04***
1.52***
1.11***
2.10***
1.63***
1.35***
1.06***
1.00***
0.63***
0.07
0.20
0.16
0.51
0.28
0.28
0.62
0.76
0.32
0.10
3.22
3.63
3.05
2.20
2.13
2.69
1.70
2.05
2.40
3.24
14.26***
6.36***
17.08***
7.53***
14.39***
4.22**
4.67***
3.74**
15.20***
7.04***
5.26***
3.86**
3.82**
3.67*
1.20
0.98
3.35
0.40
1.13
1.35
2.51
1.44
2.40
0.76
0.06
0.07
0.08
0.10
0.08
0.02
0.25
0.36
0.24
0.42
0.56
0.59
0.10
0.14
0.12
0.13
0.64
0.39
0.10
0.40
0.70
0.19
0.59
0.07
0.14
0.09
0.03
0.25
0.16
0.06
0.30
0.11
0.15
0.40
0.23
0.21
0.25
0.38
0.16
0.39
0.59
0.19
0.13
0.15
0.07
0.28
0.05
0.05
0.18
0.47*
0.31
0.05
0.63
0.16
0.28
0.20
0.01
0.12
0.29
0.10
0.01
0.02
0.11
0.12
0.13
0.39
0.11
0.34
0.34
0.40
1.94
0.21
0.35
0.07
1.27
1.06
0.30
1.44
0.98
0.68
0.10
0.17
0.20
0.07
1.64*
0.65
0.28
0.41
0.66
0.74
1.03
1.31
1.01
0.31
1.37
0.33
0.53
0.22
1.32
0.38
1.16
0.49
Panel A. VAR mispricing model
September 05, 2008
September 08, 2008
September 09, 2008
September 10, 2008
September 11, 2008
September 12, 2008
September 15, 2008
September 16, 2008
September 17, 2008
September 18, 2008
September 19, 2008
September 22, 2008
September 23, 2008
September 24, 2008
September 25, 2008
September 26, 2008
September 29, 2008
September 30, 2008
October 01, 2008
October 02, 2008
October 03, 2008
October 06, 2008
October 07, 2008
October 08, 2008
October 09, 2008
October 10, 2008
October 13, 2008
October 14, 2008
October 15, 2008
October 16, 2008
October 17, 2008
October 20, 2008
October 21, 2008
October 22, 2008
**
Day
Statistical significance at the 10% levels.
Statistical significance at the 5% levels.
Statistical significance at the 1% levels.
2446
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Table 4
Price divergence. This table exhibits the divergence ratio of the options-implied price compared to the actual stock price using two mispricing measures—the VAR model using
Vuolteenaho (2002) model and dispersion of analysts’ forecasts (DISP). The pre-ban period is from April 29, 2008 through September 4, 2008. The ban period is from September 19
to October 8, 2008.
Mispricing measure
Full sample
***
DISP
Medium mispricing
High mispricing
Low mispricing
Medium mispricing
High mispricing
Financial firms
Pre-ban period
Ban period
Difference
t-Statistic
4.02%
10.56%
6.54%***
57.88
1.32%
5.21%
3.90%***
26.66
3.42%
10.43%
7.00%***
53.40
6.96%
16.72%
9.76%***
43.91
0.74%
5.00%
4.26%***
33.08
3.01%
9.01%
6.00%***
78.79
7.84%
18.69%
10.85%***
43.30
Industrial firms
Pre-ban period
Ban period
Difference
t-Statistic
2.28%
3.21%
0.93%***
22.02
1.06%
1.15%
0.08%
1.11
2.03%
3.13%
1.10%***
17.83
3.67%
5.34%
1.67%***
15.08
0.54%
0.92%
0.38%***
4.07
1.73%
2.50%
0.78%***
12.79
4.46%
6.17%
1.71%***
41.38
0.25%***
6.58
1.39%***
41.00
3.29%***
46.98
0.20%***
7.42
1.29%***
35.36
3.38%***
61.29
4.07%***
25.53
7.30%***
51.81
11.39%***
47.79
4.08%***
25.95
6.51%***
72.12
12.52%***
50.52
Difference between firm types
Pre-ban period
Difference
1.74%***
t-Statistic
65.14
Ban period
Difference
7.35%***
t-Statistic
62.46
VAR
Low mispricing
Statistical significance at the10% levels.
Statistical significance at the 5% levels.
Statistical significance at the 1% levels.
Fig. 3. Abnormal divergence ratios for financial firms. The figure reports abnormal price divergence ratios for a sample of financial firms. The mispricing measures are
computed using Vuolteenaho (2002). The period is from April 28, 2008 through October 22, 2008 with Day 0 the beginning of the shorting ban on September 19, 2008.
fore or during the shorting ban, the full-sample mean divergence is
greater than the average of the medium-mispricing group. This is
true for both firm types.
For both the VAR model and DISP, we find a monotonically
increasing relation between mispricing and price divergence.10
This fact holds for both firm types as well as across ban periods.
10
Divergence coefficients becomes more negative with an increase in mispricing,
thus, the greater negative amount signifies an increase in divergence with an increase
in mispricing—a positive relationship.
As with the upper-boundary violation findings, actual stock prices
become relatively too expensive compared to the synthetic prices.
While the results demonstrate some increase in price divergence within the industrial-firm sample after the implementation
of the shorting ban, we find the greater increase in divergence in
the financial firms. The divergence for the low-mispricing financial
firms increases from negative 1.32% to negative 5.21%. Similarly,
the medium-mispricing sample increases in divergence from negative 3.42% to 10.43% while the high-mispricing tranche increases
from negative 6.96% to 16.72%. Additionally, the difference between the two periods is increasing in mispricing. The differences
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
2447
Fig. 4. Abnormal divergence ratios for industrial firms. The figure reports abnormal price divergence ratios for a sample of industrial firms. The mispricing measure is from
Vuolteenaho (2002). The period is from April 28, 2008 through October 22, 2008 with Day 0 the beginning of the shorting ban on September 19, 2008.
are negative 3.90, 7.00, and 9.76% for the low-, medium-, and highmispricing financial firms, respectively.
We next compute daily abnormal divergence ratios. Fig. 3 graph
the values for the two firm types. Fig. 3 shows how abnormal
divergence increases markedly on the ban date for financial firms.
After continued abnormal divergence during most of the ban, the
abnormal ratios trend back to pre-ban levels as the ban ceases.
Moreover, we observe greater abnormal price divergence with an
increase in mispricing.
Fig. 4 details the market for the industrial firms. While Table 4
indicates an increase in average levels for the industrial subsample,
the daily values in Fig. 4 suggest the price divergence is a function
of the high-mispricing firms initially after the implementation of
the shorting ban.
Table 5 reports the specific values shown in these figures. Panel
A presents the results using the VAR model. Clearly the financial
firms experience significant abnormal divergence during the ban
across all three mispricing types. Further, the abnormal behavior
extends for a longer period of time with greater mispricing. Lowmispricing financial firms exhibit price divergence through
approximately half of the ban period. Medium-mispricing financial
firms demonstrate divergence through Day +11. The high-mispricing
financial firms demonstrate the longest price divergence as well as
the greatest magnitudes.
We find another instance of the effect of mispricing in the
industrial firms using the VAR model in Panel A. The results
demonstrate no abnormal divergence in the low- and mediummispricing tranches. Conversely, the high-mispricing industrial
firms exhibit significant abnormal price divergence in the middle
of the shorting ban but not prior nor after.
Panel B of Table 5 details the findings using the DISP proxy. The
conclusions are the same as the results largely correspond to the
VAR model. For instance, for the financial firms, significant abnormal divergence ends on Day +7 for the low-mispricing firms, Day
+11 for the medium-mispricing group, and Day +13 for the highmispricing tranche. Also similar to the VAR findings, the results
demonstrate no abnormal divergence for the low- and mediummispricing industrial firms, but some abnormal behavior for the
high-mispricing industrial firms during the middle of the shorting
ban.
4.3. Bid-ask spreads
Table 6 reports the test results from interacting bid-ask spreads
with mispricing. Overall, spreads increase in mispricing, and the
positive correlation is evident in both call (Panel A) and put (Panel
B) options. We observe the monotonic relation before and during
the ban period.
The increase in spreads with greater mispricing holds for both
financial and industrial firms. We observe the greatest impact is
from the high-mispricing firms of either firm type. For instance,
the greatest increases between the pre-ban and ban period—
regardless of the firm type and for both the VAR model and
DISP—is found in the high mispricing tranche. Using the VAR model, the difference between periods is 8.03% for the high-mispricing
financial firms versus 5.19% and 6.25% for the low and mediummispricing financial firms.
Additionally, for call options using either the VAR model or
DISP, we find no significant difference between spreads for lowpricing firms across firm types within the two respective time
frames. The pre-ban spread is 12.82% for financial firms and
13.07% for industrial firms. The ban period spread is 18.01% for
financial firms and 17.90% for industrial firms. The same is true
for the pre-ban spread in put options when we compute mispricing
using the VAR method i.e., 13.37% for financial firms and 13.41% for
industrial firms.
We detail the daily abnormal options-spread values from
10 days prior to 10 days after the ban period in Table 7. Again,
since the results are quite similar and the conclusions the same,
we report the finding using the VAR model of mispricing. Table 7
is dominated by significant abnormal options-spread ratios. Panel
A presents the call options. The fact that we find significant abnormal values for both firm types beginning on September 15 coincides with Lehman Brothers filing for Chapter 11 bankruptcy on
that day. It is notable that abnormal ratios continue after the ban
is lifted. The results demonstrate elevated spread amounts across
2448
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Table 5
Times series of abnormal divergence. The table reports abnormal divergence ratios (in %). Larger negative values indicate greater divergence between synthetic and the actual
share prices. Panel A details divergence conditioned on mispricing in the stock market using the Vuolteenaho (2002) model, while Panel B uses dispersion of analysts’ forecasts as
a mispricing proxy.
Date
*
***
Financial firms
Industrial firms
Low
Medium
High
Low
Medium
High
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
0.67
2.19
2.41
1.17
2.18
2.56
2.53
3.00***
2.29
2.43**
6.15***
6.25***
7.04***
4.43***
7.02***
4.77***
2.73**
3.83***
2.13
1.88
1.08
1.71
1.03
1.41
1.35
2.16
1.10
1.92
1.33
1.93
2.54
0.67
2.11
0.66
1.45
0.75
2.21
1.26
3.39
1.43
2.54
3.39
1.08
0.97
18.43***
4.83***
6.85***
5.60***
7.01***
3.91***
5.41***
4.38***
5.81***
4.11***
4.52***
5.58***
2.62
2.74
0.63
0.51
0.77
0.96
0.17
1.94
0.57
0.36
1.65
2.32
1.51
1.72
0.61
1.01
1.06
1.51
0.46
4.61
1.21
1.63
25.23***
15.97***
14.63***
5.95***
12.82***
7.22**
7.24**
6.91**
16.28***
4.43**
10.53***
3.53**
6.68***
4.01**
1.46
1.52
1.71
1.32
0.74
0.32
0.23
1.15
0.59
0.47
0.51
0.36
1.37
0.45
1.30
0.46
1.47
1.63
1.63
1.25
0.51
1.96
1.19
1.95
2.08
0.45
1.04
0.70
0.96
0.63
0.14
1.60
1.33
1.35
1.26
1.36
1.90
1.89
1.99
1.78
0.65
1.38
0.45
0.34
0.34
0.35
1.52
1.67
1.05
0.02
1.21
1.63
1.43
1.60
1.15
1.62
1.05
1.38
0.99
2.06
2.02
1.50
1.10
0.85
0.15
1.40
1.33
0.87
0.99
0.85
1.89
1.02
1.35
1.33
1.65
1.28
1.17
1.29
0.02
1.03
0.79
0.93
1.54
0.29
1.17
1.95
1.94
1.70
2.25
2.35
1.70
2.01
2.21**
3.19**
2.93**
5.74***
4.51***
2.34*
0.85
2.18*
1.88
1.22
1.04
1.25
1.05
1.38
1.96
0.20
1.15
1.55
1.30
1.48
Panel B. Dispersion of analysts’ forecasts
September 05, 2008
10
September 08, 2008
9
September 09, 2008
8
September 10, 2008
7
September 11, 2008
6
September 12, 2008
5
September 15, 2008
4
September 16, 2008
3
September 17, 2008
2
September 18, 2008
1
September 19, 2008
0
September 22, 2008
1
September 23, 2008
2
September 24, 2008
3
September 25, 2008
4
September 26, 2008
5
September 29, 2008
6
September 30, 2008
7
October 01, 2008
8
October 02, 2008
9
October 03, 2008
10
October 06, 2008
11
October 07, 2008
12
October 08, 2008
13
October 09, 2008
14
October 10, 2008
15
October 13, 2008
16
October 14, 2008
17
October 15, 2008
18
October 16, 2008
19
October 17, 2008
20
October 20, 2008
21
October 21, 2008
22
October 22, 2008
23
0.81
2.72
2.31
1.33
1.96
3.26
3.09
3.01
2.03
2.24**
5.03***
7.46***
7.02***
5.36***
7.46***
3.92**
2.89**
2.70**
2.21
2.34
1.37
4.83
1.25
1.29
2.67
2.53
1.17
2.03
2.26
1.57
2.82
0.51
2.22
0.55
1.42
0.86
2.22
1.44
3.35
1.61
2.68
2.70
1.18
1.18
17.96***
4.28***
5.64***
5.92***
8.05***
5.27***
12.83***
4.92***
6.18***
4.87**
5.25***
5.43***
2.25
2.70
0.58
0.55
0.80
1.00
0.17
1.81
0.70
0.29
1.87
1.74
2.00
2.17
0.49
0.96
2.60
2.09
0.40
4.46
1.31
1.41
27.51***
13.49***
11.64***
6.55***
14.55***
8.91***
8.31**
6.41**
15.15***
3.71***
8.24***
3.74***
6.70**
3.36**
1.83
1.29
1.93
1.41
0.69
0.36
0.23
1.12
0.45
0.53
0.42
0.39
1.49
0.45
1.14
0.55
1.71
1.55
1.57
1.20
0.61
1.49
1.13
2.02
1.66
0.36
0.93
0.68
1.00
0.57
0.14
1.60
1.35
1.56
1.29
0.95
1.64
2.34
1.82
1.29
0.73
1.42
0.43
0.39
0.36
0.30
1.58
1.67
0.76
0.02
1.13
1.56
1.29
1.94
1.12
1.66
1.10
1.07
1.02
2.49
2.09
1.60
1.51
0.64
0.15
1.96
1.40
0.86
1.20
0.86
2.12
1.12
1.41
1.33
1.40
1.23
1.18
1.27
0.02
0.96
0.87
1.21
1.76
0.29
1.51
1.54
1.79
1.56
0.22
1.31
0.70
2.51
2.14**
7.75***
2.17***
5.00***
5.00***
2.75**
3.95***
5.68***
1.89
1.12
0.98
1.56
1.66
1.16
2.55
1.66
1.33
1.59
1.45
1.68
Panel A. VAR mispricing model
September 05, 2008
September 08, 2008
September 09, 2008
September 10, 2008
September 11, 2008
September 12, 2008
September 15, 2008
September 16, 2008
September 17, 2008
September 18, 2008
September 19, 2008
September 22, 2008
September 23, 2008
September 24, 2008
September 25, 2008
September 26, 2008
September 29, 2008
September 30, 2008
October 01, 2008
October 02, 2008
October 03, 2008
October 06, 2008
October 07, 2008
October 08, 2008
October 09, 2008
October 10, 2008
October 13, 2008
October 14, 2008
October 15, 2008
October 16, 2008
October 17, 2008
October 20, 2008
October 21, 2008
October 22, 2008
**
Day
Statistical significance at the 10% levels.
Statistical significance at the 5% levels.
Statistical significance at the 1% levels.
2449
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Table 6
Options bid-ask spreads. This table reports bid-ask spread ratios in options securities. The numerator is the bid-ask spread in dollars, which we scale by the midpoint. Thus, the
values are percentages. We measure mispricing using the VAR model from Vuolteenaho (2002) along with dispersion of analysts’ forecasts (DISP). The pre-ban period is from April
29, 2008 through September 4, 2008. The ban period is from September 19 to October 8, 2008.
Mispricing measure
Full sample
Panel A. Call options
Financial firms
Pre-ban period
15.25%
Ban period
21.74%
Difference
6.49%***
t-Statistic
9.75
Industrial firms
Pre-ban period
14.04%
Ban period
19.74%
Difference
5.70%***
t-Statistic
8.66
Difference between firm types
Pre-ban period
Difference
1.21%***
t-Statistic
6.14
Ban period
Difference
2.00%**
t-Statistic
2.19
Panel B. Put options
Financial firms
Pre-ban period
14.72%
ban period
20.07%
Difference
5.35%***
t-Statistic
7.06
Industrial firms
Pre-ban period
14.10%
Ban period
15.56%
Difference
1.46%***
t-Statistic
2.84
Difference between firm types
Pre-ban period
Difference
0.62%***
t-Statistic
3.29
Ban period
Difference
4.51%***
t-Statistic
5.04
*
**
***
VAR
DISP
Low mispricing
Medium mispricing
High mispricing
Low mispricing
Medium mispricing
High mispricing
12.82%
18.01%
5.19%***
6.78
15.38%
21.63%
6.25%***
7.93
17.55%
25.57%
8.03%***
12.17
13.33%
17.63%
4.30%***
5.61
14.76%
21.01%
6.25%***
7.93
17.66%
26.58%
8.92%***
13.53
13.07%
17.90%
4.83%***
8.22
13.97%
19.61%
5.65%
8.21
15.09%
21.71%
6.62%
9.23
13.32%
17.66%
4.34%***
7.39
14.10%
19.47%
5.37%***
7.81
14.70%
22.09%
7.39%***
10.30
0.66%***
3.22
2.97%***
13.24
1.42%***
6.96
2.46%***
10.98
0.01%
0.03
0.12%
0.12
2.02%**
1.97
3.86%***
4.07
0.03%
0.04
13.37%
17.61%
4.24%***
4.70
14.71%
19.93%
5.21%***
6.71
16.08%
22.67%
6.59%***
8.06
11.47%
16.12%
4.65%***
5.15
15.16%
20.10%
4.94%***
6.36
17.53%
23.99%
6.46%***
7.90
13.41%
14.44%
1.02%**
2.07
13.96%
15.30%
1.34%***
2.81
14.92%
16.94%
2.02%***
3.33
13.06%
13.93%
0.86%*
1.74
14.25%
15.51%
1.26%***
2.64
14.98%
17.24%
2.26%***
3.73
0.75%***
3.79
1.15%***
5.05
1.60%***
8.54
0.91%***
4.60
2.55%***
11.14
4.62%***
5.19
5.73%***
5.78
2.19%**
2.16
4.59%***
5.16
6.75%***
6.80
0.25%
1.18
0.04%
0.24
3.18%***
3.14
1.54%
1.50
4.49%***
4.74
Statistical significance at the 10% levels.
Statistical significance at the 5% levels.
Statistical significance at the 1% levels.
all mispricing classes while, in general, the high mispricing group
has higher abnormal spreads than the two lower mispricing
groups.
The put options in Panel B of Table 7 are similar to the calls with
regard to the financial firms. We observe an increase in abnormal
spread ratios on September 15 and into the ban period. This holds
across mispricing for the financial firms.
In contrast, the low and medium-mispricing industrial firms do
not exhibit systematically abnormal spreads. Given the market
uncertainty during the sample period, we find it noteworthy that
the low- and medium-mispricing industrial firms do not exhibit
systematically increased abnormal spreads before, during, or after
the short-sale ban. The tercile that does exhibit an increase in bidask spreads for put options is the high-mispricing firms. While all
of the results in this article demonstrate the effect of mispricing in
the equity market on prices in the options market, the industrial
firms clearly demonstrate the differentiation mispricing has on option values.
5. Robustness test
In order to investigate the sensitivity of our results to empirical
design choices, we examine a complimentary test. The analysis
uses a regression model to capture the effect mispricing has on
price equilibrium in the options markets after controlling for the
shorting ban, maturity, and moneyness. We use the general relationship of
Price Measures ¼ a þ bL Low Mispricing þ bH High Mispricing
þ bE Ev ent þ bM Maturity þ bD D þ ;
where Price Measures denotes the dependent variable as one of the
pricing metrics—options-boundary violations, divergences, and options bid-ask spreads. Low Mispricing is a dichotomous variable
equal to 1 if the mispricing measure falls in the lower tercile
relative to the other sample firms, and 0 otherwise. Similarly, High
Mispricing is a dummy variable equal to 1 if the mispricing measure
is in the high tercile and 0 otherwise. The control tercile is Medium
Mispricing. Event equals 1 during the ban period and 0 otherwise.
Maturity is the continuous value denoting time to maturity. Delta
is the delta of the call in each options pair when computing the
boundary violation ratios and divergences.
When the dependent variable is either lower or upper optionsboundary violations, we use a dichotomous variable coded to equal
1 for a violation, and 0 otherwise. Accordingly, we use a logistic
regression. When the regress and is either price divergence or
bid-ask spreads, we use least squares since the dependent variable
is continuous.
2450
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Table 7
Time series of abnormal options-spread ratios. This table presents the abnormal options-spread ratios (in %) for calls in Panel A and puts in Panel B from 10 days before to 10 days
after a shorting ban in the fall of 2008. Low, medium, and high indicate the relative level of mispricing, which we measure using the Vuolteenaho (2002) model.
Date
*
**
***
Panel A. Call options
September 05, 2008
September 08, 2008
September 09, 2008
September 10, 2008
September 11, 2008
September 12, 2008
September 15, 2008
September 16, 2008
September 17, 2008
September 18, 2008
September 19, 2008
September 22, 2008
September 23, 2008
September 24, 2008
September 25, 2008
September 26, 2008
September 29, 2008
September 30, 2008
October 01, 2008
October 02, 2008
October 03, 2008
October 06, 2008
October 07, 2008
October 08, 2008
October 09, 2008
October 10, 2008
October 13, 2008
October 14, 2008
October 15, 2008
October 16, 2008
October 17, 2008
October 20, 2008
October 21, 2008
October 22, 2008
Panel B. Put options
September 05, 2008
September 08, 2008
September 09, 2008
September 10, 2008
September 11, 2008
September 12, 2008
September 15, 2008
September 16, 2008
September 17, 2008
September 18, 2008
September 19, 2008
September 22, 2008
September 23, 2008
September 24, 2008
September 25, 2008
September 26, 2008
September 29, 2008
September 30, 2008
October 01, 2008
October 02, 2008
October 03, 2008
October 06, 2008
October 07, 2008
October 08, 2008
October 09, 2008
October 10, 2008
October 13, 2008
October 14, 2008
October 15, 2008
October 16, 2008
October 17, 2008
October 20, 2008
October 21, 2008
October 22, 2008
Day
Financial firms
Industrial firms
Low
Medium
High
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
0.61
1.37
0.21
0.31
1.09
0.80
2.70***
3.32***
6.48***
10.78***
10.18***
4.72***
5.98***
6.68***
4.12***
2.68***
10.01***
5.52***
4.77***
2.69**
6.95***
7.62***
7.57***
7.87***
10.32***
6.81***
7.49***
5.12***
7.07***
6.60***
6.32***
6.15***
4.93***
8.06***
1.04
1.40
0.71
0.08
1.09
0.25
2.43**
3.72***
6.13***
8.15***
11.64***
4.53***
6.55***
6.94***
4.19***
2.74**
10.52***
7.54***
5.26***
2.70***
8.06***
8.86***
8.04***
8.49***
10.56***
5.15***
11.28***
4.79***
7.91***
7.36***
7.41***
4.85***
2.50**
6.68***
2.06
2.12
0.61
0.50
1.66
0.10
3.36***
4.05**
5.69***
8.00***
11.78***
5.22***
8.45***
7.89***
4.11***
5.38***
11.22***
9.50***
5.29***
3.54**
8.97***
8.76***
10.28***
9.76***
10.06***
6.81***
11.04***
4.14***
5.74***
6.65***
3.11**
4.54***
2.80**
7.40***
0.57
0.79
1.12
1.48*
1.37*
1.32*
3.00***
3.62***
5.43***
8.24***
4.61***
3.78***
4.15***
3.42***
2.79***
3.23***
4.81***
5.70***
4.18***
6.15***
6.36***
8.14***
7.20***
7.93***
6.57***
5.54***
6.14***
6.73***
6.05***
6.40***
6.17***
5.88***
6.15***
8.41***
0.68
0.30
1.77
1.06
0.86
0.60
3.29***
3.13***
5.75***
7.88***
6.03***
4.42***
4.84***
4.03***
3.98***
4.37***
9.70***
6.96***
5.85***
8.02***
8.13***
9.93***
9.15***
9.66***
8.41***
7.11***
7.53***
8.04***
8.28***
7.62***
8.26***
6.93***
7.15***
6.07***
0.16
0.16
1.32
0.30
0.38
2.55**
2.83***
3.20***
6.67***
8.75***
7.95***
5.59***
5.25***
5.04***
4.68***
4.87***
9.51***
7.25***
6.43***
8.19***
8.88***
9.37***
9.79***
10.10***
8.67***
7.43***
8.37***
8.92***
7.02***
8.83***
9.16***
8.71***
8.09***
7.39***
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
1.81
0.23
1.84
0.25
0.68
0.98
1.65
1.02
4.43***
10.15***
12.11***
5.06***
6.21***
2.75**
5.35***
3.09***
9.91***
4.25***
3.40***
2.13*
4.09***
2.52***
4.10***
4.19***
4.09***
2.98***
10.69***
1.82*
0.92
0.72
1.54
1.37
1.43
1.24
1.66
1.09
2.54
0.84
1.79
0.32
3.40***
3.54***
6.16***
6.33***
13.75***
5.66***
7.36***
3.08**
4.89***
4.29***
9.70***
7.50***
4.77***
6.12***
7.37***
4.40***
3.95***
3.75**
3.92**
3.94***
2.93***
1.72
3.43*
2.73*
2.43*
3.02**
1.95
2.20**
2.20
0.21
2.07
1.19
1.76
1.06
5.31***
3.66***
6.75***
9.62***
15.26***
9.46***
12.29***
5.24***
8.24***
6.53***
9.83***
7.40***
7.57***
5.00***
8.40***
4.84***
6.69***
5.25***
7.74***
5.60***
9.99***
3.22**
7.99***
3.15**
3.72***
3.25**
3.95***
4.05***
1.58
1.62
0.65
1.35
1.20
1.02
0.33
0.06
1.99**
5.10***
3.31***
1.42
0.90
0.01
0.41
0.21
2.13
1.76
0.26
0.01
0.13
0.24
0.52
0.75
2.00
1.51
1.45
0.11
0.28
0.92
0.09
1.52
0.19
0.90
0.51
0.61
0.00
0.56
0.13
0.01
0.63
1.09
3.19***
5.92***
4.34***
1.94**
1.57*
0.47
0.94
0.91
3.56***
2.18
0.41
0.75
0.64
0.53
0.37
0.93
1.78
1.24
1.07
0.52
0.30
1.03
0.49
0.77
0.56
0.60
0.28
0.14
0.55
0.03
0.57
0.03
1.38*
1.98**
5.57***
8.50***
6.93***
3.94***
2.89***
2.10***
2.27***
1.93**
5.52***
3.81***
2.10***
2.48***
2.67***
3.13***
2.32***
2.46***
2.50**
2.69**
2.77**
1.36*
1.33*
0.60
1.51*
2.41***
1.07
2.26***
Statistical significance at the 10% levels.
Statistical significance at the 5% levels.
Statistical significance at the 1% levels.
Low
Medium
High
2451
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Table 8
Multivariate regressions. This table presents regression coefficients and t-statistics (in parentheses) for the sample of financial firms using three pricing measures
(options boundary violations, divergences, and options bid-ask spreads) as the dependent variable in the following specification. PriceMeasures ¼ a þ bL LowMispricingþ
bH HighMispricing þ bE Ev ent þ bM Maturity þ bD D þ e, Low Mispricing is a dichotomous variable equal to 1 if the mispricing measure falls in the lower tercile relative to the
other sample firms while High Mispricing is a dummy variable equal to 1 if the mispricing measure is in higher tercile. We measure mispricing using the VAR model (Panel A)
from Vuolteenaho (2002) along with dispersion of analysts’ forecasts (Panel B). Event is a dichotomous variable equal to 1 during the fall 2008 shorting ban. Maturity is the
continuous value of time to maturity in years. Delta is the delta for call options. The time period is 100 days before to 13 days after the short-sale ban—from April 29, 2008
through October 8, 2008.
Dependent variable
Panel A. VAR mispricing model
Upper-boundary violation
Lower-boundary violation
Divergence
Options-spread ratio
Calls
Puts
Panel B. Dispersion of analysts’ forecasts
Upper-boundary violation
Lower-boundary violation
Divergence
Options-spread ratio
Calls
Puts
***
Low mispricing
High mispricing
Event
Maturity
Delta
Adjusted R2
0.21***
(2.94)
0.01
(0.24)
0.05***
(4.14)
0.05***
(3.79)
0.04***
(5.08)
0.24***
(3.79)
0.01
(0.36)
0.05***
(5.27)
0.06***
(2.99)
0.03***
(3.27)
0.25***
(4.19)
0.08
(0.11)
0.06***
(4.29)
0.06***
(4.46)
0.03***
(3.36)
0.15***
(3.24)
0.07
(0.15)
0.04***
(3.46)
0.06***
(2.89)
0.06***
(3.81)
0.31***
(2.77)
0.09
(0.06)
0.05***
(4.56)
0.25***
(2.90)
0.19***
(2.68)
0.24
0.24***
(3.48)
0.01
(0.25)
0.06***
(4.90)
0.03***
(3.12)
0.06***
(4.99)
0.27***
(3.28)
0.01
(0.39)
0.07***
(6.18)
0.04***
(3.18)
0.05***
(3.28)
0.25***
(5.08)
0.05
(0.11)
0.04***
(4.77)
0.05***
(3.85)
0.04***
(3.40)
0.11***
(2.90)
0.04
(0.15)
0.06***
(3.28)
0.07***
(2.95)
0.04***
(3.42)
0.28***
(2.09)
0.07
(0.06)
0.07***
(4.71)
0.23***
(2.71)
0.21***
(2.77)
0.22
0.01
0.22
0.27
0.34
0.01
0.20
0.32
0.29
Statistical significance at the 1% level or less.
When upper-boundary violations is the dependent variable, we
expect bL to be negative and bH should be positive if upper options
boundary violations are positively related to stock mispricing as
demonstrated by our previous findings. When divergence is the
regressand, bL should be positive and bH should be negative if the
stocks with high mispricing have more price divergence than those
with low mispricing.
Table 8 reports the findings for the financial firms, which are
consistent with the prior results. Panel A details the coefficients
and t-statistics in parentheses using the VAR model. The results
demonstrate that the mispricing consistently demonstrates a significant impact on the pricing dependent variables, with the exception being, as before, the lower-boundary condition.
When upper-boundary violations is the dependent variable, the
coefficient of bL is negative 0.21 while the slope on bH is positive
0.24, both values significant at the 1% level. Given that the regression controls for the shorting ban, these coefficients demonstrate
the relation mispricing has in general on option prices. Highmispricing firms violate the upper-boundary condition more than
the medium-mispricing control group. Further, low-mispricing
firms violate the upper-boundary condition significantly less than
the medium-mispricing group. Given the asymmetric nature of
heterogeneous beliefs and shorting constraints, the negative value
on low-mispricing is not necessarily expected. Put differently, pessimistic investors cannot act on their market views such that Chen
et al. (2002), Desai et al. (2002) and Boehme et al. (2006) find the
negative effect of opinion dispersion on subsequent equity returns.
Similarly, when regressing on price divergence or bid-ask
spreads, the results demonstrate significant relationships between
low-mispricing firms and the dependent variable as well as highmispricing firms and the regressand. We find the low-mispricing
firms decrease price divergence and bid-ask spreads relative to
the medium-mispricing tranche. Conversely, high-mispricing firms
increase price divergence and bid-ask spreads for both calls and
puts. Again, these results hold after controlling for the shorting
ban.
Regarding the other variables, we find a positive and significant
coefficient on Maturity in the upper-boundary violations specification, which is consistent with the result in Ofek et al. (2004) demonstrating the maturity effect on put-call parity violations. The
results also demonstrate a negative and significant coefficient on
Delta for call options when the options-spread ratio is the dependent variable. This result suggests that the options boundary consisting of ITM calls and OTM puts is more likely to be violated,
which is consistent with the Xing et al. (2010) finding that informed traders prefer to trade OTM put options.
Examining the DISP results in Panel B, we observe coefficients
and standard errors on the independent variables similar to the
VAR model in Panel A. Consequently, the results demonstrate the
same direction or relation with the regressand as well as the same
statistical significance. Likewise, all the models display the similar
ability to explain the variation in the independent variable as in Panel A. Accordingly, the main conclusion for the all the regression
results is that both low- and high-mispricing firms affect price efficient in the stock and options markets.
6. Conclusion
We investigate the possibility of disequilibria between optionsimplied and observed equity prices when firms exhibit mispricing
in the underlying stock. Our main interest and contribution is to
investigate whether derivatives traders observe and correctly trade
in the direction of the mispricing. If they do, we will observe a positive relation between mispricing and our measures of price
disequilibria.
We first analyze violations of lower and upper options boundary conditions. We find few violations of the lower-boundary condition when arbitragers need to short the synthetic security and
hold a long position in the underlying equity to profit from
violations of the law of one price. Alternatively, when arbitragers
are short the firm’s equity and are long the synthetic, we find
significantly more violations of this upper-boundary condition.
2452
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Moreover, for those firms that are mispriced the most, approximately 6% of the sample violates the law of one price.
We also examine the level of divergence between the optionsimplied and actual share prices. The proportion of price divergence
demonstrates a consistently positive relation with mispricing.
Our pricing measures will be sensitive to short-sale constraints
since the underlying equity must be shorted to avoid violations of
the upper-boundary condition. Accordingly, we include in our
sample period the shorting ban in the fall of 2008 to study the effect of a nearly total ban on shorting. This helps us mitigate concern about the degree shorting constraints have on the results.
As expected, we find a significant increase in upper-boundary violations and the magnitude of price divergence.
Overall, the results demonstrate an ability by options investors
to recognize mispricing in the equity market and trade accordingly.
We suggest the test results support at least four implications. One,
we argue that our results document that derivatives traders demonstrate a degree of informed trading. The distinct and robust nature of the price disequilibria between the mispricing terciles
demonstrates a systematic trading behavior by options investors.
Another implication of the results is market efficiency. The options
market appears to comprehend a degree of price inefficiency in the
stock market and corrects for the mispricing to the extent permitted given shorting constraints.
A third implication is that our findings support the contention
by Miller (1977) that heterogeneous beliefs and shorting constraints work in tandem. In contrast to the upper-boundary condition, we do not find significant instances of the lower boundary
being violated as shorting constraints are not as material.
A final consideration is for public policy makers. While shortsale constraints may appear appropriate initially, and may even
ease volatility during uncertain periods, our findings demonstrate
that shorting restrictions have a negative impact on pricing
efficiency.
Acknowledgements
We thank an anonymous referee, John Adams, Don Chance, the
editor, Shane Johnson, and Vassil Mihov for helpful comments.
Appendix A. Mispricing estimation
This paper adopts the dynamic valuation framework of
Vuolteenaho (2002) to estimate stock mispricing. The model is
specified as:
mt bt ¼ c þ
1
X
1
X
s¼1
s¼1
qs1 Et ðROEtþs Þ qs1 Et ðrtþs Þ;
ðA1Þ
where mt is the log of the market value for a stock and bt is the log of
the book value. We calculate the book value as the sum of common
equity, deferred tax, and tax payable. c is a constant estimated as
p
is the average log
c = k/(1 q), where q ¼ 1=ð1 þ edp Þ; d
dividend-price ratio for the period, and k = log(q) (1 q) log(1/q 1).11 In Eq. (A1), rt is the excess stock return defined
as log return of the stock minus the risk-free rate, Et is the expectations operator, Et is the conditional expectations calculated using the
estimated VAR parameters, and
Net Incomet
ft ;
ROEt ¼ log 1 þ
Book Valuet1
11
As discussed in Vuolteenaho (2002), the value of q is an empirical question. We
maintain a constant value of 0.95 for all the stocks. By varying the value from 0.90 to
0.99, we find that our test results are not sensitive to q, which is consistent with the
conclusion in Vuolteenaho (2002).
where ft is the risk-free rate.
According to the theoretical model developed in Campbell and
Shiller (1988), the fundamental value of the log of price to dividend
P
s1 ½E ðDd Þ E ðr
ratio can be specified as 1
t
tþs
t tþs Þ, where Dd is
s¼1 q
the dividend growth rate and r is the stock return. The mispricing
component in Campbell and Vuolteenaho (2004) is estimated as
the difference between the actual price to dividend ratio and its
fundamental value. For the situation without any cash dividends,
Vuolteenaho (2002) modifies the original Campbell and Shiller
VAR model and derives an alternative model showing that the
fundamental value for the log of the market-to-book ratio can be
P
s1 ½E ðROE Þ E ðr
specified as 1
t
tþs
t tþs Þ, where ROE is the return
s¼1 q
on equity. In our sample, approximately 6% of the firms (130
out of 2209) pay quarterly cash dividends regularly; therefore,
Vuolteenaho’s (2002) model becomes our best choice.
Eq. (A1) represents the fundamental or intrinsic market-to-book
ratio. This intrinsic value, (mt bt), does not automatically equal
~t Þ. The ob~t ¼ b
the observed market-to-book ratio, denoted by ðm
served value depends on the investors’ expectations when they
price the stock. If all investors have identical objective expecta~t Þ.
~t ¼ b
tions, then, by construction, ðmt bt Þ ¼ ðm
Following the extant literature, we relax the objective expectations assumption and consider the possibility that some investors
use subjective expectations. We decompose the observed log market-to-book value ratio into:
~t ¼ mt bt þ ~et
~t b
m
1
1
X
X
¼cþ
qs1 Et ðROEtþs Þ qs1 Et ðrtþs Þ þ ~et :
s¼1
ðA2Þ
s¼1
The mispricing term, ~et , is the difference between the observed and
fundamental log market-to-book value ratio, which is consistent
with the mispricing calculation in the literature.
To obtain the stock mispricing in Eq. (A2) we estimate a VAR
model. We define xe as a 3 1 vector for the three variables at time
g t ; ROEt ; r t Þ0 , where MB
g t is the observed log of markett as xt ¼ ð MB
to-book value ratio. The VAR system with one lag is specified as
xt ¼ Bxt1 ; þnt ;
ðA3Þ
where B is a 3 3 matrix of VAR coefficients and nt is a 3 1 vector of
the VAR system. Given Eq. (A3), the multi-period forecast is determined as Et ðxtþs Þ ¼ Bs xt . Define further two vectors, e2 = (0, 1, 0)0
and e3 = (0, 0, 1)0 , then the discounted value of the expected future
P
s1 E ðROE Þ, is given by
excess return on equity, 1
t
tþs
s¼1 q
1
X
1
X
s¼1
s¼1
qs1 Et ðROEtþs Þ ¼
qs1 e20 Bx xt ¼ e20 BðI qBÞ1 xt :
Likewise, the discounted value of the expected future excess return,
P1 s1
Et ðr tþs Þ, is given by
s¼1 q
1
X
1
X
s¼1
s¼1
qs1 Et ðrtþs Þ ¼
qs1 e30 Bx xt ¼ e30 BðI qBÞ1 xt :
Before estimating the VAR, we examine the stationarity for MB,
ROE, and r. Jiang and Lee (2007) is one of the few studies that
examines the stationarity of the Vuolteenaho (2002) model and addresses the issue. We find ROE and r are stationary. However, MB is
non-stationary in 85% or more of our sample firms. Thus, we transform MB to a stationary process by first differencing.12 Subsequent
to obtaining stationarity, we estimate the VAR parameters. With the
estimated VAR parameters, we estimate the mispricing measure as
the difference between the realized and expected transformed MB.
12
We thank an anonymous referee for this invaluable suggestion.
2453
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
Appendix B. Test statistics
for stock i as DRi ¼
is given by
B.1. Boundary violations
DRt ¼
We compute options boundary violation ratios by scaling the
number of violations by total observations to obtain proportions
we can compare across samples. The ratio is
VR ¼
Number of Violations
:
Number of Total Observ ations
P11
t¼100 DRi;t =90.
N
1X
DRi;t
;
N i¼1 DRi
where N is the number of stocks for day t. The abnormal divergence
ratio for day t is then
ADRt ¼ DRt 1:
The test statistic is given by
The variance of the ADR is
ðVRban VRpreban Þ Nban
t stat ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
Nban VRpreban ð1 VRpreban Þ
r2ADR ¼
where VRban is the probability of violation during the ban period,
VRpre-ban is the probability of violation during the pre-ban period,
and Nban is the number of total observations during the ban window.13 The pre-ban period is from 100 to 11 trading days before
the ban, which is from April 29, 2008 through September 4, 2008.
The ban period is from September 19, 2008 through October 08,
2008.14
In addition to contrasting violation ratios by dividing average
percentages into pre-ban and ban periods, we detail the daily values from 10 days prior to the ban, through the ban period, and
10 days post-ban from September 5, 2008 through October 22,
2008. Using VR, we compute the average daily VR using the daily
options boundary violation ratio during the pre-ban period from
P
days 100 to 11 such that VR ¼ 11
t¼100 VRt =90. Any deviation
from the average violation ratio prior to the ban, we term the
abnormal violation ratio (AVR). The value on day t is given by
AVRt ¼ VRt VR:
11
1 X
ðADRt ADRÞ2 ;
89 t¼100
where ADR ¼ 1=90
divergence ratio is
The test statistic for the abnormal
B.3. Bid-ask spreads
To calculate spreads, we first compute the daily options spread
for stock i as
OSR ¼
ðOptionAsk OptionBid Þ
:
ðOptionAsk þ OptionBid Þ=2
The denominator is the midpoint of the bid and ask quotes, which
scales the dollar spread by the magnitude of the ask and bid price
quotes, and yields a percentage. Subsequently, we calculate the daily options-spread ratio as
N
X
OSRi;t =N;
i¼1
with the variance
11
1 X
ðAVRt AVRÞ2 ;
89 t¼100
where AVR ¼ 1=90
t¼100 ADRt .
r2ADR
OSRt ¼
r2AVR
r2AVR ¼
P11
ADRt
t stat ¼ qffiffiffiffiffiffiffiffiffiffi :
The test statistic for the abnormal violation ratio is
AVRt
t stat ¼ qffiffiffiffiffiffiffiffiffiffi ;
The aggregate daily DR for day t
P11
t¼100 AVRt .
B.2. Divergence ratios
The divergence ratio is given in Eq. (7) above. We compute the
t-statistic for the mean difference in the divergence ratio (DR) before and after the ban as
ðDRban DRpreban Þ
t stat ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
r2DRpreban =ðNpreban 1Þ þ r2DRban =ðNban 1Þ
where DRban and DRpre-ban are the averages of the divergence ratios.
r2DRban and r2DRpre-ban are the variances for the divergence ratios. N is
the number of observations.
As with boundary-condition violations, we examine the daily
values by comparing the average divergence ratio prior to the
ban to the values during the ban. We term these values the abnormal divergence ratios (ADR). Similar to Michaely et al. (1995), and
Liang (1999), we calculate the average of the daily divergence ratio
13
Alternatively, we can also use chi-square tests. We confirm the results as
qualitatively the same using this method.
14
The choice of the pre-ban period is arbitrary. We conduct a robust test using
alternative windows and find consistent results.
where N is the number of stocks on day t. The test statistic for the
options-spread ratio difference between the pre-ban and ban periods is
ðOSRban OSRpreban Þ
t-stat ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;
r2OSRpre-ban =89 þ r2OSRban =13
where OSR is the daily average of the spread ratios for the ban and
pre-ban periods and r2OSR is the variance of the spread ratios. The
number of days for the pre-ban period is 90 and 14 for the ban.
We compute the abnormal options bid-ask spread ratio (ASR) on
day t in a similar manner to the calculation of ADR above Given the
daily spread ratio for each firm, we calculate the average daily OSR
for stock i during the pre-ban period for days 100 to 11, such
that
OSRi ¼
11
X
OSRi;t
:
90
t¼100
The daily OSR for day t is specified as
OSRt ¼
N
1X
OSRi;t
;
N i¼1 OSRi
where N is the number of stocks for day t. The abnormal options
spread ratio for day t is then
ASRt ¼ OSRt 1:
The variance of the abnormal options bid-ask spread is
2454
r2ASR ¼
D.K. Hayunga et al. / Journal of Banking & Finance 36 (2012) 2438–2454
11
1 X
ðASRt ASRÞ2 ;
89 t¼100
P
where ASR ¼ 1=90 11
t¼100 ASRt . The test statistic for the abnormal
spread ratio is
ASRt
t-stat ¼ qffiffiffiffiffiffiffiffiffi :
r2ASR
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