D
EVIATIONS FROM
P
UT
–C
ALL
P
ARITY AND
V
OLATILITY
P
REDICTION
: E
VIDENCE FROM
THE
T
AIWAN
I
NDEX
O
PTION
M
ARKET
CHIN‐HO CHEN*, HUIMIN CHUNG and SHU‐FANG YUAN
This study examines whether deviations from put–call parity are informative about future volatility in the underlying index. Using the difference in implied volatility between call and put options to measure these deviations, wefind that deviations from put–call parity predict future volatility. The predictability becomes stronger as option liquidity increases and the liquidity of the underlying index decreases. The results for volatility prediction remain significant even after controlling for implied volatility, information shocks, other information variables on return and volatility used widely in the literature, and short sales constraints. In addition, our results also show that deviations from put–call parity contain information about the future trading volume of options and the underlying index. © 2014 Wiley Periodicals, Inc. Jrl Fut Mark 34:1122–1145, 2014
1. INTRODUCTION
Put–call parity (PCP) is a simple no‐arbitrage relation in which the payoff of an asset can be
synthetically replicated using a call option, a put option, and a bond. Most empirical studies of
violations of PCP find that apparent violations do not always represent tradable arbitrage
opportunities because of factors such as transaction costs, dividend payments, the early
exercise value of American options, nonsynchronous trades, and margin requirements.1Ofek
and Richardson (2003), Ofek, Richardson, and Whitelaw (2004), and Shleifer and Vishny
(1997)find that violations of PCP may also occur when limits on arbitrage exist, such as those
based on short sales constraints and the ability of professional arbitrageurs. The trading activity of informed investors is another potential reason for violations (Cremers &
Weinbaum, 2010; Easley, O’Hara, & Srinivas, 1998; Finucane, 1991).
Recent research reports that deviations from PCP can predict asset returns as a consequence of informed trading in the option market. Atilgan (2010), Cremers and
Weinbaum (2010), and Finucane (1991)find that PCP deviations predict stock and index
returns. Easley et al. (1998) demonstrate that if some informed investors favor trading in
Chin‐Ho Chen and Huimin Chung are collocated at the Institute of Finance, National Chiao Tung University, Hsinchu, Taiwan. Shu‐Fang Yuan is at the Department of Business Administration, Nanhua University, Chiayi County, Taiwan. The authors thank Robert Webb (the editor), Wen‐Liang Hsieh, Wei‐Peng Chen, and especially an anonymous referee for helpful comments and suggestions.
JEL Classification: G10; G12; G13
*Correspondence author, Institute of Finance, National Chiao Tung University, No. 1001 Ta‐Hsueh Road, Hsinchu 30050, Taiwan. Tel:þ886‐3‐5712121 ext. 57075, Fax: þ886‐3‐5733260, e‐mail: chinho.rrriver.chen6@gmail.com Received June 2013; Accepted January 2014
1
See, for example, Battalio and Schultz (2006), Klemkosky and Resnick (1980), Liu and Longstaff (2000), and Nisbet (1992).
The Journal of Futures Markets, Vol. 34, No. 12, 1122–1145 (2014) © 2014 Wiley Periodicals, Inc.
Published online 12 February 2014 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fut.21655
options instead of underlying assets, option prices may contain information about future
asset prices. Because of private information, option prices are not fully efficient and deviate
from PCP in the direction of the informed investors’ private information.2 However, in
practice, volatility trading is popular and options are uniquely suited to investors with private volatility information (Back, 1993). Despite the evidence for the presence of informed volatility trading in the option market (Chang, Hsieh, & Wang, 2010; Ni, Pan, & Poteshman, 2008), much less is known about whether deviations from PCP contain information about future volatility.
This study investigates the extent to which deviations from PCP predict future volatility realized by the underlying index. We adopt deviations from PCP, proposed by Cremers and Weinbaum (2010), to measure the quantity of information and separate the information into
positive and negative volatility signals. The deviations are quantified as the difference in
implied volatility between call and put options with the same maturity (i.e., volatility spread). Increased (decreased) volatility in the subsequent 2 days following large volatility spreads with positive (negative) information provides evidence on the predictability of volatility by volatility spreads. Findings in this study also suggest that deviations from PCP may be the outcome of volatility information trading.
In addition, a linkage between volatility spreads and trading volume in the option and underlying asset is also examined. If positive (negative) volatility information embedded in volatility spreads is realized in the spot market, the hedging and speculative uses of options arising from increased (decreased) price volatility will lead to a higher (lower) option trading volume because of large (small) potential gains and losses. Similarly, high uncertainty in the
underlying asset also affects its trading volume.3 The volatility‐induced change in trading
volume implies that volatility spreads and volume are related. Thus, this study further explores
the information content of volatility spreads for future volume.4
This study analyzes the Taiwan index option (TXO) written on the Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX), which is one of the most liquid index
options in the world.5 Taiwan is the focus of the analysis for two reasons. First, volatility
information trading is present in the TXO market. For example, Chang et al. (2010) show that foreign institutional investors and a few individual investors trade on private volatility information in the TXO market.
Second, Taiwan’s market has no derivatives to trade volatility. Investors must exploit
their volatility information directly in the option market. In the TXO market, volatility trades constructed on an option/futures combination are favored by informed volatility investors (Chang et al., 2010). Thus, as positive (negative) volatility information is realized by buying (selling) a call or put option and hedging by the index futures, the option trades by informed volatility investors may widen the relative position of call and put prices, subsequently leading to a widened volatility spread. This study in the Taiwan market sheds light on many other
markets in a similar stage of development.6
2Although any tradable violation of PCP is expected to be quickly arbitraged away, in the real‐world market imperfections and transaction costs may widen the range of the relative position of call and put prices so as not to violate arbitrage restrictions.
3Many studies show that a contemporaneous two‐way relation exists between volatility and trading volume in the option and underlying asset (Andersen, 1996; Lamoureux & Lastrapes, 1990, 1994; Sarwar, 2003; Sears, 2000; Tauchen & Pitts, 1983).
4
The authors thank the referee for this insightful suggestion. 5
On a global scale, the TXO is ranked thefifth most frequently traded index option in 2010. The detailed statistics are available at the World Federation of Exchanges website, http://www.world‐exchanges.org/.
6
For instance, India and Korea have very active index options, the NIFTY 50 index option, and KOPI 200 index option, respectively, but the derivatives for trade the volatility index have yet to be launched.
The results show that volatility spreads with positive and negative information have predictive power with respect to future volatility. In particular, the degree of predictability is stronger when option liquidity is high and liquidity in the underlying index is low, a result consistent with the implication of the Easley et al. (1998) model. In addition, higher rewards for selling volatility following large volatility spreads with negative information also support the notion that deviations from PCP contain information about future volatility.
The predictability of volatility is robust after controlling for implied volatility, information shocks, short sales constraints, and other information variables on return
and volatility. Thefinding that the option market leads the spot market by days, not minutes,
provides evidence in support of information discovery in options. In addition, deviations from
PCP also contain information about future trading volume. Specifically, following large
volatility spreads with positive (negative) information, we find an increase (decrease) in
option trading volume, option buy volume, and option sell volume. Similar results are found for trading dollar volume, buy dollar volume, and sell dollar volume of the underlying index.
This study is related to the recent literature showing that option price and volume contain predictive information about asset returns. An, Ang, Bali, and Cakici (2013) and Bali
and Hovakimian (2009) use information from the cross‐section of options, including the
change in call and put implied volatilities, and the realized/implied volatility spread and the
call–put implied volatility spread.7Bali and Murray (2013), Conrad, Dittmar, and Ghysels
(2013), Rehman and Vilkov (2012), and Xing, Zhang, and Zhao (2010) adopt information on
cross‐sectional risk‐neutral skewness.8 Bollerslev, Tauchen, and Zhou (2009) use the
information of implied–realized variance spread and find a positive relation between variance
spread and stock market returns. In addition, Pan and Poteshman (2006), Johnson and So (2012), and Roll, Schwartz, and Subrahmanyam (2010) show that option trading volume and the option/stock trading volume ratio predict stock returns.
Other studies focus on predicting option return or price volatility. Goyal and Saretto
(2009) find that delta‐hedged options with a large positive difference between realized
volatility and implied volatility have low average returns. Ni et al. (2008) and Chang et al. (2010) examine the predictive ability of volatility by option trading volume.
The remainder of this study is organized as follows. Section 2 contains a formal development of the method. Section 3 provides a brief description of the empirical data. Section 4 reports the empirical results derived from the models. Finally, Section 5 provides the key results of the study and concluding remarks.
2. METHOD
2.1. Volatility Information Variables
This study’s theoretical framework extends the sequential trade model of Easley et al. (1998),
allowing investors with private volatility information to trade in options first. In practice,
straddles, strangles, and an option/futures combination are the three most commonly used
7An et al. (2013) show that stocks with a large increase in call (put) implied volatility have high (low) future returns. Bali and Hovakimian (2009)find a negative relation between realized–implied volatility spread and future stock returns. They alsofind high future average returns in stocks with large call–put implied volatility spread. 8Bali and Murray (2013)find a negative relation between risk‐neutral skewness and the skewness asset returns. Rehman and Vilkov (2012) and Xing et al. (2010)find a positive relation between risk‐neutral skewness and future stock returns while Conrad et al. (2013) report the opposite relation.
volatility trading strategies (Chaput & Ederington, 2005). This suggests that informed investors who are privy to a positive (negative) volatility signal usually conduct their volatility trades by long (short) straddles, strangles, or an option/futures combination, that is, a long (short) position for a call or put option and a short (long) position for futures. Importantly, as a
result of private information, option prices are not fully efficient and deviate from PCP in the
direction of the informed investors’ private information. Each of the call and put options can
carry information about subsequent volatility.
In the model, buying a call (put) in the option/futures strategy is a trade that carries positive volatility information and that raises a call (put) price relative to a put (call) price. Similarly, selling a call (put) that conveys negative volatility information decreases a call (put) price relative to a put (call) price. These trades widen the range of the relative position of call and put prices. A strangle trade based on positive (negative) volatility information, which requires that investors simultaneously buy (sell) both call and put options with different
exercise prices to construct an approximately delta‐neutral option portfolio, also has a similar
effect whereas a straddle trade does not lead to PCP deviations.
This study adopts deviations from PCP, as proposed by Cremers and Weinbaum (2010), to measure the quantity of information. Following Amin, Coval, and Seyhun (2004) and
Figlewski and Webb (1993), the deviations are quantified as the call–put implied volatility
spread. The rationale is that the PCP for European options equivalently states that the Black–
Scholes (1973) implied volatilities of pairs of call and put options are equal, even if option
prices do not conform to the Black–Scholes formula. The difference between call and put
implied volatilities thus can be interpreted as deviations from model values. However, in practice, deviations caused by informed trading likely occur for options with different exercise prices. The deviations are calculated as the implied volatility spread between call and put options with the same maturity.
Under the measure of deviations, a widened volatility spread may be generated by volatility trading based on an option/futures strategy. Nevertheless, volatility trades with a long (short) straddle and strangle tend to drive both call and put options to a relatively higher (or lower) level of implied volatility rather than to produce a larger volatility spread. Since volatility trades through straddles and strangles only account for a small fraction of option trading volume (Chang et al., 2010; Lakonishok, Lee, Pearson, & Poteshman, 2007), the dominant role of volatility trading with an option/futures strategy suggests that the informed volatility
trading often results in a widened volatility spread. This widened volatility spread reflects
private volatility information.
The information implicit in volatility spreads is separated into positive and negative volatility signals depending on the option implied volatility and the level of PCP on the prior trading day. Buying a call that carries positive volatility information raises implied volatility of
calls (Vc) relative to the implied volatility of puts (Vp). Subsequently, the implied volatility of
calls rather than the implied volatility of puts deviates more from the prior level of PCP, which is measured by the average implied volatility of call and put options, denoted by OV. The quantity of positive private information implicit in calls on day t (i.e., the volatility spread with
positive volatility information) is quantified as
VScb¼ V c;t Vp;t; if Vc;t> Vp;tand V c;t OVt1 > V p;t OVt1;
0; if otherwise; ð1Þ
where Vc,tand Vp,tdenote the implied volatilities of calls and puts on day t, respectively. OVt1
denotes the level of PCP on day t 1, which is the average of Vcand Vpon day t 1.
Similarly, buying a put that has positive volatility information raises Vprelative to Vc,
puts on day t is quantified as
VSpb¼ V c;t Vp;t; if Vc;t< Vp;tand V c;t OVt1 < V p;t OVt1;
0 if otherwise: ð2Þ
The negative volatility signals are also obtained by abstracting them separately from call
and put options. Volatility trades through selling a call lessen Vcrelative to Vp, and drive Vcto
deviate more from OVt1. The quantity of negative private information on day t implicit in calls
is quantified as
VScs ¼ V c;t Vp;t; if Vc;t< Vp;tand V c;t OVt1 > V p;t OVt1;
0; if otherwise: ð3Þ
Likewise, the quantity of negative private information on day t implicit in puts is quantified as
VSps ¼ V c;t Vp;t; if Vc;t> Vp;tand V c;t OVt1 < V p;t OVt1;
0; if otherwise: ð4Þ
Straddle and strangle trades are more likely to raise or lessen both the implied volatilities of calls and puts and drive them to deviate more from prior level of PCP. The percentage
change of level of PCP (DOV) is used to gauge the information released by straddle and
strangle trades. However, if a strangle trade generates a large deviation from PCP, perhaps arising from a portfolio position in both a call and a put options with a large difference in strike
prices, the positive volatility information is incorporated in VScbor VS
p
bwhereas the negative
information is reflected in VScs or VSps.
2.2. Option Implied Volatility
Without the use of any option‐pricing model, this study adopts the approach proposed by Jiang and Tian (2005, 2007) to compute the implied volatilities of call and put options directly from option prices. This method corrects the inherent methodological problem in the Black– Scholes (1973) model for deriving the option‐implied volatility, which assumes that the underlying asset’s return follows a lognormal distribution that is almost too fat‐tailed to be lognormal.
Following the approach of Jiang and Tian (2005, 2007), the model‐free implied variance under the assumption of deterministic interest rates is written as
2 ZKmax Kmin Cðt; KÞ=Bð0; tÞ max½0; S0=Bð0; tÞ K K2 dK XM i¼1½f ðt; K iÞ þ f ðt; Ki1ÞDK; ð5Þ
where S0and C(t, K) are the asset price and option price, respectively. K is the exercise price. t
denotes the expiration date of option. Bðt; tÞ is the time t price of a zero‐coupon bound that
pays $1 at time t. fðt; KiÞ ¼ ½Cðt; KiÞ=Bð0; tÞ maxð0; S0=Bð0; tÞ KiÞ=K2i,DK ¼ (Kmax
Kmin)/M, and Ki¼ Kminþ iDK for 0 i M. The truncation interval [Kmin, Kmax] denotes the
range of available exercise prices, in which Kminand Kmax are referred to as left and right
truncation points, respectively.
To avoid the bid–ask bounce problem, the midpoint of the quote rather than the transaction price is used to compute the implied volatility (Bakshi, Cao, & Chen, 1997, 2000).
In each day, the implied volatilities at the two nearest maturities are linearly interpolated to
obtain the implied volatility at afixed 22 trading‐day horizon, denoted by IV, which is used as a
proxy for the implied volatility of the whole option market. For the implied volatility at each
contract month, the average implied volatility of calls and puts isfirst calculated for every
5‐minute interval and then averaged across intervals in a day. The 5‐minute implied volatility
of calls (and puts) is backed out from call (put) prices by using the right‐hand side of
Equation (5).9
Figure 1 shows the plot of implied volatility and volatility spread in the near‐month
options. The daily implied volatility of puts (Vp) during the time to maturity is found to be
frequently above the implied volatility of calls (Vc). Consistent with thefindings of Ofek et al.
(2004), deviations from PCP are more likely to occur in the direction of puts, which are relatively more expensive than the corresponding calls. The higher implied volatility of puts
implies that volatility traders realize their positive private information using a delta‐neutral
portfolio of a long position in puts and a hedged position in futures. By contrast, a trade that sells calls and buys futures is favored by volatility sellers. These two volatility trading strategies account for 87.71% of the sample. In addition, a higher level of volatility spread during the
FIGURE 1
Time‐series plots of the implied volatilities of near‐month options and volatility spreads. This figure depicts the time‐series relation between daily implied volatilities of calls and puts (Vcand Vp) in the
near‐month and volatility spread (VS). The time period is from January 1, 2005 through December 31, 2009. Vcand Vpare separately backed out from call and put prices by using Equation (5). VS is measured
as the size of difference in implied volatilities between call and put options with the same maturity. VSmis
the median of VS during the sample period.
9The steps are specified as follows. At first, to obtain the not‐traded option prices between available exercise prices, a cubic splines method is used to interpolate the Black–Scholes implied volatilities per 20 index points (DK ¼ 20). Next, for options with exercise prices beyond the available range, we follow the suggestion of Jiang and Tian (2007) to adjust the slope of the extrapolated segment to match the corresponding slope of the interior segment at the minimum or maximum available exercise price. Finally, the extracted implied volatilities are translated into option prices by using the Black–Scholes model, and the implied volatility is further computed from these option prices.
period offinancial crisis in 2008 indicates that larger deviations from PCP are induced by
more informationflows impounded in the option market. Further, we have similar findings for
options in the second month (correspondingfigures are not tabulated but are available upon
request).
Figure 1 also show that the size of volatility spread changes over time and reverts to its
long‐term median (VSm). This positive median (or mean) of volatility spreads likely arises from
the widened range between call and put prices due to market imperfection and transaction
costs. Facing a mean‐reversion volatility spread, the absolute deviations from median rather
than the size of volatility spread can better reflect the quantity of volatility information. Thus,
the absolute deviations are used from the median, denoted by VSM, as a proxy of volatility
information variables. As specified in Section 2.1, VSMcb and VSMpb indicate the volatility
spreads with positive volatility information implicit in call and put options, respectively, and
VSMcs and VSMps indicate volatility spreads with negative volatility information implicit in call
and put options, respectively.
2.3. Model Specifications
2.3.1. Volatility prediction by volatility spreads
If informed investors do trade on private volatility information in the option marketfirst, their
private information would be subsequently reflected in the spot market. We can expect that an
increase in subsequent volatility will follow volatility spreads with a positive signal (VSMb),
and a decrease in subsequent volatility will follow volatility spreads with a negative signal
(VSMs). The empirical model is specified as
RVt¼ a þ b1VSMb;tjþ b2VSMs;tjþ b3DOVtjþ b4IVt1þ b5ISt1þ et; ð6Þ
where RVtdenotes the realized volatility of underlying index on day t, which is calculated as
the difference of the intraday highest and lowest prices divided by the closing price.10VSMbis
the volatility spread with positive volatility information embedded in both call and put options,
VSMcband VSM
p
b, respectively. VSMs denotes the volatility spread with negative volatility
information involving VSMcs and VSMps.
Three control variables are considered in Equation (6). IVt1 denotes the implied
volatility of the whole option market on day t 1, which the literature, in general, finds to be
best predictor for future volatility. This variable controls for publicly available information
already contained in the option prices. DOVtjcontrols the effect of straddle and strangle
trades on day t j for future volatility, where OV is the average implied volatility of call and put
options. ISt1indicates the information shocks on day t 1, which may lead to persistence in
volatility.11 This variable is incorporated in the return caused by the information flows to
capture the dynamic impact of news on volatility. The information shocks are calculated as absolute return innovations by applying the AR(1) model to the daily return.
The regression in Equation (6) is estimated separately for different values of j, from 1 to k, to capture the potential predictive power of volatility spreads for future realized volatility. In
addition, the asymptotic t‐statistics of estimated parameters are calculated by using Newey
and West’s (1987) autocorrelation correction.
10
Based on the same realized volatility measure, Alizadeh, Brandt, and Diebold (2002)find that the results are robust to several alternative definitions of realized volatility.
11Evidence of volatility changes due to information shocks is provided by Andersen (1996), who shows that return volatility dynamics is governed by informationflows, and Chen and Ghysels (2010), who find that both very good news and bad news increase volatility, with the latter having a more serve impact.
Unlike investors with positive information about asset prices who only choose to buy calls, an investor with a positive volatility signal can buy either calls or puts due to the positive values of vega in both. The calls and puts can thus carry positive information about
subsequent volatility. To further test whether the directional volatility information is reflected
in both calls and puts, a predictive regression of volatility spreads with the positive and negative information implicit separately in calls and puts on future volatility is run as
RVt¼ a þ b1VSMcb;tjþ b2VSM p b;tjþ b3VSMcs;tjþ b4VSM p s;tjþ b5DOVtjþ b6IVt1 þ b7ISt1þ et; ð7Þ where VSMc b and VSM p b (VSM c
s and VSMps) denote the volatility spreads with positive
(negative) volatility information implicit separately in calls and puts. 2.3.2. Interaction between predictability in volatility and liquidity
In the Easley et al. (1998) model, the documented predictability is pronounced for the specific
condition under which informed trades in options should occur more frequently. The
condition is satisfied as options have higher liquidity relative to the underlying assets. This
implication is tested to determine whether the predictability in volatility for relatively liquid options with relatively illiquid underlying index can be improved. A liquid dummy variable (D) to control the level of liquidity in both options and the underlying index is included into
Equation (8) to analyze the influence of liquidity on the degree of predictability in volatility as
RVt¼ a þ b1VSMb;tjþ b2DtjVSMb;tjþ b3VSMs;tjþ b4DtjVSMs;tj
þ b5DOVtjþ b6IVt1þ b7ISt1þ et: ð8Þ
For daily option liquidity (QSPR) of each contract month, the average of percentage
quoted spread of calls and puts in every 5‐minute interval is first calculated and then averaged
across intervals in a day, in which the percentage quoted spread in calls (and puts) is equally weighted across all individual call (put) options. As for the percentage quoted spread of individual call (put) option, it is calculated as the difference between ask and bid prices
dividing the mid‐quote. In addition, the liquidity of underlying index is gauged by the
logarithmic daily trading volume of the underlying index (ILiq). Under these measures of liquidity, option liquidity increases as the QRSP decreases whereas liquidity in the underlying index decreases as the ILiq decreases.
2.3.3. Predictability by controlling for other information variables
To examine further whether the predictability is robust after controlling for short sales constraints and other information variables about return and volatility, several control
variables, each of which may have some influence on the volatility prediction, are added to the
following regression model:
RVt¼ a þ b1VSMcb;tjþ b2VSM p b;tjþ b3VSMcs;tjþ b4VSM p s;tjþ b5DOVtjþ b6IVt1 þb7ISt1þ X4 i¼1
giRVtiþ c1OIVtjþ c2DVtjþ c3IDVtjþ c4FVtjþ c5SBLtjþ et; ð9Þ
where RVtifor i¼ 1, 2, …, 4 is the i‐day lagged volatility used to control for the volatility
clustering and possible patterns (Lakonishok et al., 2007). IDVtjrefers to the logarithmic
trading volume of the underlying asset on day t j. FVtjis the logarithmic open interest of
underlying index futures on day t j.
OIV is the put–call ratio of options used to proxy for informed directional
informa-tion trades. Empirical studies show that investors trade on private direcinforma-tional informainforma-tion in the option market (e.g., Ahn, Kang, & Ryu, 2008; Chakravarty, Gulen, & Mayhew,
2004; Chang, Hsieh, & Lai, 2009; Easley et al., 1998; Pan & Poteshman, 2006). The put–
call ratio of options widely used as a proxy for private information is adopted to control for the potential impact of directional information trades on future volatility. OIV is calculated
as the open‐interest put contracts divided by summing the open‐interest put and call
contracts.
DV is the option demand for volatility used to proxy for informed volatility trading. In addition to trading on directional information, informed investors may also trade on volatility information in the option market, as reported in Ni et al. (2008) and Chang et al. (2010). The
possible influence of informed volatility trading on future volatility is controlled by using order
imbalances weighted by the Black–Scholes vega across all options to proxy the option demand
for volatility (DV).12
Following Chordia, Roll, and Subrahmanyam (2008), the order imbalance is calculated
as the difference between the number of buyer‐initiated and seller‐initiated trades divided by
the total number of trades within every 5‐minute interval, in which the Lee and Ready’s
(1991) algorithm is used to classify each option trade as buyer initiated or seller initiated. For
daily order imbalance, the order imbalance isfirst calculated for every 5‐minute interval by
weighting order imbalances across all options according to the Black–Scholes vega and then
averaged across intervals in a day.
SBL is a security borrowing and lending fee in the stock‐lending market used to proxy for
short sales constraints. Several studies argue that deviations from PCP may arise in the presence of short sales constraints on the underlying stocks (Lamont & Thaler, 2003; Ofek &
Richardson, 2003; Ofek et al., 2004). Bali and Hovakimian (2009)find that the deviations
caused by short sales constraints may reflect the expected future price change of the
underlying stock. The possible influence of the difficulty of selling short the underlying index
on volatility is controlled by using SBL to proxy short sales constraints. That is, a high SBL fee
rate in stock‐lending market makes shorting the stocks costly for the borrower of the stocks
and can thus be interpreted as a signal of short sale constraints.13The SBL fee rate of the
Taiwan Top 50 Tracker Fund, which is an exchange‐traded fund, is used to proxy for the short
sales constraints on the whole stock market.14
12The order imbalance is widely used in the literature as a proxy for the nonmarket maker net demand. Chordia, Roll, and Subrahmanyam (2002) and Chordia and Subrahmanyam (2004) use order imbalance to measure both direction and degree of buying or selling pressure. Bollen and Whaley (2004) gauge the net demand using order imbalance between the number of buyer‐initiated and seller‐initiated trades.
13In June 2003, the TWSE launched a centralized SBL system to meet the needs of qualified institutional investors while TWSE serves as an intermediary. This SBL system provides three types of transactions:fixed‐rate, competitive bid, and negotiated transaction. This study adopts a competitive bid as a proxy of short sales constraints because the SBL fee rate is determined by the bids and offers quoted by the borrower and the lender.
14
The Taiwan Top 50 Tracker Fund (0050), which is managed by Polaris International Securities and Investment Trust Company Ltd, uses full replication to track the performance of the TWSE Taiwan Top 50 Index. The fund’s constituents are selected from the top 50 listed stocks on the TWSE by market weight, after meeting the criteria to be included in the index. The fund allows investors to invest in all 50 constituents with a single investment and is adjusted quarterly to closely reflect the index and market.
2.3.4. Volume prediction by volatility spreads
The unexpected variation in volatility may generate a change in trading volume of options and the underlying index (Andersen, 1996; Lamoureux & Lastrapes, 1990, 1994; Sarwar, 2003; Sears, 2000; Tauchen & Pitts, 1983). The contemporaneous and lagged relation between price volatility and trading volume implies that volatility spreads with private information about future volatility and trading volume are related. This study further explores the information content of volatility spreads with positive and negative information for future option trading volume and trading dollar volume in the underlying index.
The daily option trading volume in the near and second month (OPVOL) is examined as a consequence of options with the best liquidity and largest trading volume. The OPVOL is
further separated into signed buy volume (OPVOLbuy) and signed sell volume (OPVOLsell) by
using the Lee and Ready’s (1991) algorithm. For daily trading dollar volume in the underlying
index, we use trading dollar volume (IDVOL) of dealers, securities investment trust companies, and foreign investors as a proxy and further separate IDVOL into buy dollar
volume (IDVOLbuy) and sell dollar volume (IDVOLsell).15This method is followed because the
underlying index of TXO, a stock market index for companies traded on the TWSE, is not traded directly whereas these three types of investors account for almost all trading volume in the Taiwanese spot market.
The volatility spreads with positive and negative information are used separately to test
for the predictability in OPVOL, OPVOLbuy, and OPVOLsell (and IDVOL, IDVOLbuy, and
IDVOLsell). Each of them is individually included in the following regression model:
VOLt ¼ a þ Tt1þ b1VSMctjþ b2VSM p tjþ b3FVtjþ b4IDVtjþ b5IStj þX4 i¼1 giVOLtiþ et; ð10Þ
where VSMctjand VSMptj denote either volatility spreads with positive information (VSMcb
and VSMpb) or negative information (VSMcs and VSMps) on day t j of calls and puts,
respectively. VOL is either of OPVOL, OPVOLbuy, or OPVOLsell(and IDVOL, IDVOLbuy, or
IDVOLsell). T is a linear time trend used to control for the time effect.
Figure 2 presents the time‐series trading volume and trading dollar volume. The graph
shows a slightly negative time trend in option trading volume (OPVOL), signed buy volume
(OPVOLbuy), and signed sell volume (OPVOLsell) in the left part offigure but a slightly positive
time trend in trading dollar volume (IDVOL), buy dollar volume (IDVOLbuy), and sell dollar
volume (IDVOLsell) in the right part offigure.
3. DATA DESCRIPTION
The intraday data on the TXO, which is traded on the Taiwan Futures Exchange (TAIFEX),16
are obtained from the Taiwan Economic Journal (TEJ) database. The data contain quote and
tradefiles of options. We extract transaction prices and volumes for every trade from the trade
15Dealers are defined by dealers’ proprietary account and securities investment trust companies mean domestic mutual funds managed by securities investment trust companies. Foreign investors are defined by regulations governing investment in securities by overseas Chinese and foreign nationals and regulations governing securities investment and futures trading in Taiwan by Mainland area investors.
16The TAIFEX introduced the European style TXO on December 24, 2001, which is written on the TAIEX. The contract matures on the third Wednesday of the delivery month. The contract months involvefive contracts with different maturities in the nearby month, the next 2 calendar months, and the following 2 quarterly months.
file, and the bid and ask prices from the quote file for January 1, 2005 through December 31, 2009. The daily open interest of index option and futures is collected from TAIFEX database covering the period from January 1, 2005 through December 31, 2009. Daily closing prices, highest prices, lowest prices, and volume of Taiwan stock index for January 1, 2005 through
December 31, 2009 and the 3‐month time deposit of postal saving system for the risk‐free
interest rate are obtained from TEJ database. In addition, the security borrowing and lending fee of the Taiwan Top 50 Tracker Fund, and the trading dollar volume, buy dollar volume, and sell dollar volume of dealers, securities investment trust companies, and foreign investors are collected from the TWSE database.
Several datafilters are applied to select the final sample. First, options with quote price
less than 0.1, the minimum tick size, are excluded from the sample. These prices do not reflect
true option value. Next, options with less than 5 trading days remaining to maturity are eliminated from the sample due to potential liquidity concerns. Finally, options violating the
PCP boundary conditions are deleted from the sample. These options are significantly
undervalued and have negative Black–Scholes implied volatilities.
A nonsynchronic question occurs because the conventional trading time periods for Taiwanese stock and option markets differ slightly, running weekdays from 9:00 a.m. to 1:30
p.m. and from 8:45 a.m. to 1:45 p.m., respectively. Battalio and Schultz (2006)find that such
nonsynchronicity can lead observations to violate PCP where none exists. To resolve the possible concern for nonsynchronicity, the daily volatility spreads and realized volatility are calculated by using the data of options and underlying index with the same trading time period on the day, formed on the trading time of TAIEX stock market from 9:00 a.m. to 1:30 p.m. Table I presents the summary statistics of common variables used for the empirical analysis. On average, the implied volatility (IV) is slightly above the realized volatility (RV) by
FIGURE 2
Time‐series plots of trading volume and trading dollar volume. This figure depicts the time‐series option trading volume, signed option buy volume, and signed option sell volume in the left offigure and trading dollar volume, buy dollar volume, and sell dollar volume of foreign and other investors in the right part of figure. The time period is from January 1, 2005 through December 31, 2009. The dotted line indicates a linear time trend for trading volume and trading dollar volume.
109 basis points, with average annualized volatilities of 0.2312 and 0.2203, respectively. This positive spread between implied volatility and actual volatility (i.e., volatility risk premium) compensates liquidity providers and motivates them to accept the unhedgeable risks of options, as described by Bollen and Whaley (2004) and Gârleanu, Pedersen, and Poteshman (2009). The volatility spreads with positive (negative) information implicit separately in calls
and puts, VSMcb and VSMpb (VSMcs and VSMps), average 0.0052 and 0.0181 (0.0156 and
0.0035), respectively, and are positively skewed and exhibit leptokurtosis. Overall, the put–
call ratio (OIV) is below 0.5 (0.4600), showing that the open‐interest position on calls is
relatively higher than that of puts.
The correlation coefficients between realized volatility and implied volatility and between
realized volatility and information shock are 0.6236 and 0.4941, respectively. The positive relation between information shock and realized volatility provides evidence that return
volatility is driven by information inflows (Andersen, 1996; Chen & Ghysels, 2010). In
addition, a negative correlation coefficient of 0.2469 between IV and volatility demand (DV)
indicates that the high level of implied volatility can generate more demand for selling volatility. 4. EMPIRICAL RESULTS
4.1. Predictability in Volatility
Table II shows the results of volatility prediction by volatility spreads with positive and negative
information. The volatility spreads are calculated separately from the near‐month and
second‐month options with the corresponding results reported in the left part and in the right
TABLE I
Summary Statistics of Common Variables
Mean Std Median Skewness Kurtosis Min Max
RV 0.2203 0.1401 0.1834 2.0535 9.2118 0.0476 1.1394 V SMc b 0.0052 0.0232 0.0000 6.9835 73.4701 0.0000 0.3290 V SMpb 0.0181 0.0393 0.0000 4.5710 32.6352 0.0000 0.4116 V SMc s 0.0156 0.0308 0.0000 3.7149 23.8299 0.0000 0.3139 V SMps 0.0035 0.0153 0.0000 4.4929 23.2038 0.0000 0.1108 DOV 0.0023 0.0648 0.0010 0.0899 6.8997 0.3416 0.3161 IV 0.2312 0.0864 0.2112 1.2508 4.8280 0.1241 0.6363 IS 0.0099 0.0103 0.0069 2.1273 8.6657 0.0000 0.0675 OIV 0.4600 0.0715 0.4668 0.4317 2.7751 0.2619 0.6342 DV 0.0295 0.0311 0.0278 1.1871 13.1943 0.3376 0.0717 IDV 15.1756 0.3146 15.1742 0.2683 2.8982 14.3222 16.1610 FV 10.7190 0.2278 10.7037 0.3489 2.8192 10.0905 11.3621 QSPR1 0.0319 0.0191 0.0281 0.7615 3.0750 0.0035 0.0971 QSPR2 0.0415 0.0259 0.0373 0.9279 3.7221 0.0054 0.1366 SBL 0.0379 0.0022 0.0380 0.5901 2.0368 0.0320 0.0400
Note. This table presents the summary statistics of the common variables, including realized volatility (RV), four volatility spreads with positive and negative volatility information respectively implicit in calls and puts (V SMc
b,V SMpb,V SMcs, andV SMps), percentage change of option‐implied volatility (DOV), implied volatility (IV), information shock (IS), option put–call ratio (OIV), option demand for volatility (DV), logarithmic trading volume of the underlying index (IDV), logarithmic open interest of the underlying index futures (FV), option liquidity in the near‐month (QSPR1), option liquidity in the second‐month (QSPR2), and short sales constraints
part of Table II, respectively. The left part of the table shows that the coefficients of volatility
spreads with positive (negative) information, VSMb(VSMs), are all positive (negative) in each
regression and are significant at 2 days ahead with t values of 2.35 (2.04).17This positive
(negative) relation from lagged volatility spreads with positive (negative) information to volatility indicates that deviations from PCP have some degree of predictive power for future volatility. This result also provides evidence for informed volatility trading in the TXO market,
consistent with the finding of Chang et al. (2010) who examine the predictive ability of
volatility by option trading volume.
The coefficients of lagged IV and information flows (IS) are all significant and positive,
with the former having a more pronounced influence on realized volatility (all coefficients are
at the 1% significance level). Consistent with the majority of prior empirical evidence, implied
volatility is a better predictor of future volatility due to private information about the volatility
brought into option prices through the trading process. A positive impact of informationflows
on volatility shows that return volatility is governed by information inflows, as described in
Andersen (1996) and Chen and Ghysels (2010). In particular, these results provide no
evidence of significant and positive DOV coefficients, used as a proxy for the effect of straddle
and strangle trades on future volatility. The result that volatility spreads rather than DOV
provide strong predictive power on volatility supports thefindings of Chang et al. (2010) that
volatility traders often adopt an option/future strategy within Taiwan’s option market.
TABLE II
Volatility Prediction by Volatility Spreads
j
VSM calculated from near‐month options
VSM calculated from second‐month options
1 2 1 2
Coeff. t Coeff. t Coeff. t Coeff. t
a 0.0003 0.37 0.0002 0.22 0.0003 0.35 0.0003 0.42 V SMb;tj 0.0146 1.53 0.0252 2.35 0.0122 1.03 0.0113 0.87 V SMs;tj 0.0058 0.49 0.0192 2.04 0.0017 0.14 0.0119 0.76 DOVtj 0.0041 1.12 0.0104 2.05 0.0019 0.40 0.0001 0.01 IVt1 0.0567 13.17 0.0554 14.31 0.0564 13.07 0.0557 13.47 ISt1 0.0815 1.95 0.1009 2.32 0.0845 2.01 0.0891 2.07 Adj.R2 0.3911 — 0.4044 — 0.3884 — 0.3885 — RVt¼ a þ b1V SMb;tjþ b2V SMs;tjþ b3DOVtjþ b4IVt1þ b5ISt1þ et
Note. This table represents the estimation results of the prediction regression, shown in Equation (6), of lagged volatility spreads with positive and negative information on volatility. The regression is separately estimated for different values ofj, from 1 to 2, to capture the potential predictive power of volatility spreads for future realized volatility. The volatility spreads are calculated respectively from near‐month and second‐month options with the corresponding results reported in the left part and in the right part of table. The volatility spreads are used to measure the quantity of volatility information and are further separated into positive and negative signals.VSMb;tjandV SMs;tjdenote volatility spreads with positive and negative volatility information on dayt j, respectively.RVtis the realized volatility, which is calculated as the difference of the intraday highest and lowest prices divided by the closing price on dayt. IVt1is implied volatility for the whole option market on dayt 1. DOVtjis used to control the effect of the
straddle and strangle trades on dayt j for future volatility, in which OV is the average implied volatility of call and put options. ISt1
denotes the absolute return innovations controlled for the impact of informationflows on day t 1. Newey–West standard errors are used to calculate thet‐statistics of the estimated parameters.,, andindicate thatt values are significant at the 0.01, 0.05, and 0.1 level, respectively.
17
To save space, Tables II, III, and VI only report the predictive results within two days because the results for the key variable, volatility spreads, are either weak or insignificant for j ¼ 3 and 4.
The right side of Table II shows no evidence to support the predictability of volatility
spreads to predict future volatility. One likely reason is that options in the second‐month (with
average liquidity of 0.0415 shown in Table I) are less liquid relative to the near‐month options
(with average liquidity of 0.0319), whereas informed traders are more likely to trade in options
with more liquidity.18Thus, the subsequent analysis focuses only on the near‐month options.
To further test whether the directional (positive or negative) volatility information is
reflected in both call and put options, four volatility spreads, which carry positive and negative
volatility information separately implicit in calls and puts, are adopted to run the predictive regression model in Equation (7). Table III reports the results.
Table III shows that both lagged volatility spreads with positive information separately abstracted from calls and puts are positively related to volatility in the subsequent 2 days and
are especially significant for the 2‐day‐ahead volatility. This finding is supported by a
significantly positive coefficient of VSMcb at 2 days ahead (with t¼ 3.74) and significantly
positive coefficients of VSMpb within 2 days (with t¼ 1.67 and 1.65). Similarly, volatility
spreads with negative information, VSMcs and VSMps, embedded in calls and puts have
significant predictive power on the 2‐day‐ahead volatility (with t ¼ 1.83 and 1.97).
Accordingly, for each of four volatility spreads, predictability lasts at least 2 days into the
future, which provides evidence that call and put options may contain same‐directional
information about future volatility.
TABLE III
Volatility Prediction by Controlling for Other Information Variables
j 1 2 Coeff. t Coeff. t a 0.0001 0.19 0.0003 0.36 V SMc b;tj 0.0039 0.50 0.0303 3.74 V SMpb;tj 0.0189 1.67 0.0229 1.65 V SMc s;tj 0.0058 0.42 0.0200 1.83 V SMp s;tj 0.0007 0.06 0.0179 1.97 DOVtj 0.0042 1.16 0.0103 2.01 IVt1 0.0559 11.65 0.0560 13.06 ISt1 0.0821 1.94 0.0995 2.26 Adj.R2 0.3924 — 0.4047 — RVt¼ a þ b1V SMcb;tjþ b2V SMpb;tjþ b3V SMcs;tjþ b4VSMps;tjþ b5DOVtjþ b6IVt1þ b7ISt1þ et
Note. This table represents the estimation results of the prediction regression, shown in Equation (7), of lagged volatility spreads with private information implicit in calls and puts on realized volatility. The regression is separately estimated for different values ofj, from 1 to 2, to capture the potential predictive power of volatility spreads for future volatility. The volatility spreads are used to measure the quantity of volatility information and are further separated into positive and negative signals.V SMc
bandV SMpb (V SMc
sandV SMps) denote volatility spreads with positive (negative) information separately implicit in calls and puts on dayt j, respectively.RVtis the realized volatility on dayt. IVt1is implied volatility for the whole option market on dayt 1. DOVtjis used to
control the effect of the straddle and strangle trades on dayt j for future volatility. ISt1denotes the absolute return innovations used to control for the impact of informationflows on day t 1. Newey–West standard errors are used to calculate the t‐statistics of the estimated parameters.,, andindicate thatt values are significant at the 0.01, 0.05, and 0.1 level, respectively.
18The average daily trading volumes of near‐month and second‐month options are 319,749 and 27,387 contracts for January 1, 2005 through December 31, 2009, respectively.
4.2. Predictability in Volatility and Liquidity
This study considers three cases of different levels of liquidity in both options and the underlying index, which is determined by a liquidity dummy variable D, including D
(QSPR< 50%, ILiq < 50%), D (QSPR < 30%, ILiq < 30%), and D (QSPR < 15%, ILiq
< 15%). In Case 1, D (QSPR < 50%, ILiq < 50%) indicates that options are liquid and its
underlying index is illiquid. In Case 2, D (QSPR< 30%, ILiq < 30%) represents higher
liquidity in the options and lower liquidity in the underlying index. In Case 3, D (QSPR< 15%,
ILiq< 15%) indicates the best liquidity in options and the worst liquidity in the underlying
index. Dtequals 1 if both the QRSP and ILiq are below the k% of the sample on day t, and zero
otherwise, in which k¼ 50, 30, and 15 for Case 1, Case 2, and Case 3, respectively. Panels A–
C of Table IV reports the results for Cases 1–3.
TABLE IV
Interaction Between Predictability in Volatility and Liquidity
j
1 2 3
Coeff. t Coeff. t Coeff. t
Panel A: Case 1 is D (QRSP< 50%, ILiq < 50%)
b1 0.0040 0.41 0.0225 2.77 0.0033 0.34
b2 0.0282 1.67 0.0063 0.38 0.0064 0.45
b3 0.0091 0.93 0.0209 1.55 0.0135 0.98
b4 0.0381 3.08 0.0039 0.24 0.0142 0.58
Adj.R2 0.4027 — 0.4047 — 0.3917 —
Panel B: Case 2 is D (QRSP< 30%, ILiq < 30%)
b1 0.0065 0.69 0.0272 2.61 0.0056 0.51
b2 0.0491 2.60 0.0112 0.81 0.0280 2.49
b3 0.0074 0.78 0.0196 1.58 0.0158 1.17
b4 0.0478 3.40 0.0025 0.14 0.0093 0.28
Adj.R2 0.4085 — 0.4049 — 0.3940 —
Panel C: Case 3 is D (QRSP< 15%, ILiq < 15%)
b1 0.0080 0.81 0.0239 2.30 0.0047 0.47 b2 0.0712 7.64 0.0142 1.72 0.0478 5.39 b3 0.0037 0.30 0.0194 1.79 0.0209 1.82 b4 0.0284 2.34 0.0002 0.01 0.0209 1.72 Adj.R2 0.4035 — 0.4048 — 0.3963 — RVt¼ a þ b1V SMb;tjþ b2DV SMb;tjþ b3V SMs;tjþ b4DV SMs;tjþ b5DOVtjþ b6IVt1þ b7ISt1þ et Note. This table represents the estimation results of the prediction regression, shown in Equation (8), in different levels of liquidity in both options and the underlying index. The regression is separately estimated for different values ofj, from 1 to 3, to capture the potential predictive power of volatility spreads for future realized volatility (RV). This study considers three cases of different liquidity levels, which is determined by a liquidity dummyD, involving D (QSPR < 50%, ILiq < 50%), D (QSPR < 30%, ILiq < 30%), and D (QSPR < 15%, ILiq < 15%); the results are separately reported in Panels A–C, respectively. In Case 1, a liquidity dummy variable D (QSPR < 50%, ILiq < 50%) indicates that options are liquid and its underlying index is illiquid. In Case 2, a dummy variable D (QSPR < 30%, ILiq < 30%) represents higher liquidity in options and lower liquidity in the underlying index. In Case 3, D (QSPR < 15%, ILiq < 15%) indicates the best liquidity in options and the worst liquidity in the underlying index. Dtis equal to 1 if both the QRSP and ILiq are below thek% of the sample on day t, and zero otherwise, in which k is 50, 30, and 15, respectively corresponding to the three cases.QSPR denotes option liquidity in the near month and ILiq is the liquidity of underlying index. IVt1
is implied volatility on dayt 1. DOVtjandISt1control the effect of the straddle and strangle trades and the impact of information
flows on day t 1 for future volatility, respectively. For brevity, only the b1, b2, b3, and b4are reported. Newey–West standard errors
are used to calculate thet‐statistics of the estimated parameters.,, andindicate thatt values are significant at the 0.01, 0.05, and 0.1 level, respectively.
As reported in Panel A of Table IV, the coefficients for b2and b4are significant at 1 day
ahead with t values of 1.67 and3.08, respectively, but are only significant for b1at 2 days
ahead with a t value of 2.77. The greater number of significant coefficients on b2 and b4
indicates that the predictive ability of volatility improves on relatively liquid options with a relatively illiquid underlying index. The results in Panel B are similar to those of Panel A except
that volatility spread with a positive signal (b2) has additional predictive ability at 3 days ahead
(evidenced by the t‐value of 2.49).
Compared with Panels A and B, the increased number of significant coefficients on b2
and b4(from two to three) shows that predictability increases as option liquidity increases and
liquidity in the underlying index decreases. Specifically, when liquidity in the options is at its
best and liquidity of its underlying index is at its worst, as shown in Panel C, predictability
becomes stronger. Thisfinding is supported by the five significant coefficients for b2and b4,
that is, significant b2coefficients at 1, 2, and 3 days ahead with t ¼ 7.64, 1.72, and 5.39,
respectively, and significant b4coefficients at 1 and 3 days ahead with t ¼ 2.34 and 1.72,
respectively. These results support the implication of Easley et al.’s (1998) model that
predictability is stronger in options with high liquidity and an underlying asset with low liquidity.
4.3. Volatility Trades Formed on Levels of Volatility Spreads
If deviations from PCP are driven by informed volatility trading, investors can profit from
observed large volatility spreads due to the preceding private volatility information embedded in volatility spreads. To test this conjecture, we construct a trading simulation of selling
volatility with a delta‐neutral option/futures strategy formed on levels of volatility spreads. The
rewards for selling volatility are separated into two classes based on the daily volatility spreads
with a negative signal (VSMs) at the spot market closing time (1:30 p.m.): volatility spreads
below 75% of the sample and volatility spreads over 75% of the sample.
To conduct the trading simulation, each call and put option in the near month is assumed to be sold at the midpoint price of quotes at the option closing time (1:45 p.m.).
Further, price risk is dynamically hedged every 30 minutes to maintain the delta‐neutrality of
the option positions in the subsequent four trading days by buying (selling) |delta| units of the
underlying futures in the case of a call (put).19The daily delta‐hedged gains for writing an
option (call/put) are given as
pt;tþ1 Ct Ctþ1þ Xn1 k¼0 DtkðFtkþ1 FtkÞ þ Xn1 k¼0 rkðCt DtkFtkÞ 1 n; ð11Þ
where the risk‐free interest rate, rk, and option delta,Dtk, are updated on a 30‐minute basis. n,
the hedged discretely times during 1 day, is set to 9. The rebalancing frequency, 1/n, is set to
30 minutes. Ctand Ftare the mid‐quotes of option and futures at the time t, respectively.
In the trading simulation, all the transaction costs are ignored because our focus is on the predictability of volatility. If volatility spreads are informative about future volatility, the
largest profit for selling volatility should occur at the 2‐day ahead window rather than at other
days, as suggested by the results in Tables II and III, which show that the private volatility
information is often incorporated into market prices at the 2‐day ahead interval.
19The delta is calculated by constructing on the Black–Scholes model, in which the implied volatility on day t is used to forecast the realized volatility on day tþ 1. In addition, every TXO can be hedged only by a quarter of TXF contract because the multipliers for the futures and options contracts are NT$200 and NT$50 per index point, respectively.
Table V provides the average of daily delta‐hedged gains in the subsequent four trading days from applying the volatility spread trading strategy to index options for January 1, 2005
through December 31, 2009. Panels A and B report the profits of the two classes, volatility
spreads over 75% of the sample (with 236 observations) and below 75% of the sample (with
708 observations), respectively. For each class, the delta‐hedged gains are grouped into three
categories by option moneyness: in‐the‐money (m < 0.975), at‐the‐money (0.975 m
1.025), and out‐the‐money (m > 1.025). The option moneyness is defined as
m¼ K=S ert, where K is strike price and S is underlying index price. The measure unit for
hedged gains is an index point, with each index point valued at NT$50.
Panel A of Table V shows that almost all the average delta‐hedged gains for calls and puts
are positive in the three moneyness groups, with the puts having large delta‐hedged gains. The
larger delta‐hedged gains for puts likely arise from the fact that the average implied volatility of
puts (0.2603) is higher than the average implied volatility of calls (0.2082) at the time of
selling index options. Thus, volatility sellers can capture more profits from puts because the
targeted profits are roughly the size of spread between implied volatility and realized volatility.
The sum of average delta‐hedged gains of calls and puts in the subsequent 4 days, shown
in the last column of Table V, are all positive with values of 74.18, 138.18, 23.83, and 76.44,
respectively. Specifically, the largest profit, 138.18, occurs at 2‐day ahead, the day predicted
by volatility spreads with private information as the day that informed volatility information is often incorporated in the spot market (see Tables II and III).
By contrast, the average delta‐hedged gains for calls and puts, as shown in Panel B of
Table V, are smaller and more negative. Only the sum of the average delta‐hedged gain at
TABLE V
Performance of Selling Volatility Formed on Levels of Volatility Spreads
Day Out‐the‐money (m> 1.025) At‐the‐money (0.975 m 1.025) In‐the‐money (m< 0.975) Total C P Cþ P C P Cþ P C P Cþ P
Panel A: Volatility spreads over 75% of the sample (V SMb 75%)
1 16.41 16.89 0.48 16.19 21.90 38.08 13.47 23.10 36.57 74.18
2 16.23 45.24 61.47 18.46 23.95 42.41 7.58 26.73 34.30 138.18
3 13.92 7.53 21.45 8.38 40.75 49.14 82.84 36.09 46.75 23.83
4 16.36 47.08 63.45 17.42 33.73 51.14 59.07 20.91 38.16 76.44
Panel B: Volatility spreads below 75% of the sample (V SMb< 75%)
1 12.99 5.61 7.39 10.79 27.44 38.24 55.79 25.55 30.24 15.39
2 13.46 40.51 27.05 10.23 27.85 38.08 77.44 21.58 55.86 44.82
3 15.07 72.21 57.14 18.68 18.20 36.88 16.30 11.84 4.46 24.72
4 12.64 99.22 86.58 16.92 23.24 40.16 32.09 17.50 14.60 61.01
Note. This table represents the performance of selling volatility formed on the levels of volatility spreads with negative information (VSMs). Panel A and Panel B, respectively, report the profits of two classes, volatility spreads over the 75% of the sample (with 236
observations) and below the 75% of the sample (with 708 observations). For each class, the delta‐hedged gains (p) are grouped into three categories by option moneyness (m): in‐the‐money options (m < 0.975), at‐the‐money options (0.975 m 1.025), and out‐the‐money options (m > 1.025). The option moneyness is defined as m ¼ K =S ert, whereK is strike price, r is risk‐free interest rate, andS is underlying index price. For each of three categories, the average daily delta‐hedged gains are calculated in the subsequent 4 days. Each of call and put options in the near‐month is assumed to be sold at the midpoint prices of quotes at the close time of options. Further, price risk is dynamically hedged every 30 minutes to keep delta‐neutral of option positions. The delta‐hedged gains of every option are calculated each day. In addition, the unit of measurement for delta‐hedged profits is an index point, which is valued at NT$50 per index point.
1‐day ahead, 15.39, is positive. A comparison of the results in Panels A and B shows higher
profits for selling volatility following large volatility spreads and again confirms that deviations
from PCP contain information about future volatility.
4.4. Incremental Predictive Ability of Volatility Spreads
Table VI provides the results of the prediction regression for realized volatility by adding
several control variables in Equation (9), including lagged RVs, option put–call ratio (OIV),
TABLE VI
Volatility Prediction by Controlling for Short Sales Constraints and Other Information Variables
j 1 2 Coeff. t Coeff. t a 0.0062 0.32 0.0096 0.50 V SMc b;tj 0.0066 0.82 0.0305 3.87 V SMpb;tj 0.0200 1.66 0.0180 1.18 V SMc s;tj 0.0110 0.89 0.0255 2.43 V SMp s;tj 0.0022 0.23 0.0215 2.22 DOVtj 0.0047 1.23 0.0108 2.10 IVt1 0.0462 7.12 0.0469 6.85 ISt1 0.0489 0.98 0.0742 1.38 RVt1 0.0717 1.34 0.0541 0.99 RVt2 0.0655 1.55 0.0404 0.93 RVt3 0.0579 1.21 0.0490 1.04 RVt4 0.0134 0.31 0.0181 0.44 OIVtj 0.0025 0.52 0.0042 0.85 DVtj 0.0187 3.25 0.0153 1.97 IDVtj 0.0003 0.18 0.0003 0.19 FVtj 0.0010 0.88 0.0016 1.35 SBLtj 0.0149 0.13 0.0745 0.74 Adj.R2 0.4084 — 0.4179 — RVt¼ a þ b1V SMcb;tjþ b2V SMpb;tjþ b3V SMcs;tjþ b4V SMps;tjþ b5DOVtjþ b6IVt1þ b7ISt1 þX4 i¼1
giRVtiþ c1OIVtjþ c2DVtjþ c3IDVtjþ c4F Vtjþ c5SBLtjþ et
Note. This table represents the estimation results of prediction regression, shown in Equation (9), for realized volatility by controlling on short sales constraints and other information variables. The regression is separately estimated for different values of j, from 1 to 2, to capture the potential predictive power of volatility spreads for future volatility. The volatility spreads are used to measure the quantity of volatility information and are further separated into positive and negative signals.V SMc
b;tjandVSMpb;tj (V SMc
s;tjandV SMps;tj) denote volatility spreads with positive (negative) signals separately implicit in calls and puts on dayt j. A security borrowing and lending fee in the stock‐lending market is used to proxy the short sales constraints (SBL). OIVt1andDVt1
are the put–call ratio of options and option demand for volatility separately used to proxy for informed directional trades and informed volatility trades. IDV andFV are logarithmic trading volume of the underlying index and logarithmic open interest of the underlying index futures, respectively.RVtis the realized volatility on dayt. IVt1is implied volatility for the whole option market on dayt 1. DOVtjis used to control the effect of the straddle and strangle trades on dayt j for future volatility. ISt1denotes the
absolute return innovations used to control for the impact of informationflows on day t 1. Newey–West standard errors are used to calculate thet‐statistics of the estimated parameters.,, andindicate thatt values are significant at the 0.01, 0.05, and 0.1 level, respectively.
volatility demand (DV), logarithmic trading volume of the underlying index (IDV), logarithmic open interest of underlying index futures (FV), and short sales constraints (SBL). The results show that volatility spreads with positive and negative information implicit in calls and puts
continue to predict 1‐ and 2‐day‐ahead volatility. This result also indicates that deviations
from PCP contain incremental information beyond implied volatility and other information variables about return and volatility used widely in the literature.
The option demand for volatility (DV) calculated from option trading volume has
significant predictive power for realized volatility in the subsequent 2 days. This result is
consistent with Ni et al. (2008) and Chang et al. (2010) whofind that option trading volume
contains future volatility information. In addition, the predictability continues to last at least
2 days after controlling for short sales constraints, suggesting that the difficulty of selling the
underlying stocks short has less influence on the predictability of volatility by deviations from
PCP. The finding that the option market leads the spot market by days, not minutes, also
provides evidence in support of information discovery in options.
4.5. Volume Prediction by Volatility Spreads
Table VII presents the results of the predictive regression for OPVOL, OPVOLbuy, and
OPVOLsell. The results of volume prediction by volatility spreads with positive and negative
volatility information are reported in Panels A and B, respectively. Panel A shows that the
coefficients of volatility spreads with positive information implicit in calls and puts (VSMcband
VSMpb, respectively) are all positive in each regression for option trading volume (OPVOL),
signed buy volume (OPVOLbuy), and signed sell volume (OPVOLsell). Specifically, the volatility
information abstracted from puts has a significantly positive effect on OPVOL, OPVOLbuy,
and OPVOLsellat 2‐day ahead interval (respectively, with t values of 3.46, 3.29, and 3.59)
whereas the significant and positive effect from the volatility information implicit in calls
occurs at 1‐day ahead (with t values of 1.99, 2.03, and 1.91, respectively).
This positive relation from lagged volatility spreads to volume indicates that volatility spreads have forecasting ability for option trading volume and option buy and sell volume. The
possible cause is that when an unexpected increase in market volatility is realized—exactly as
the volatility forecast by volatility spreads with positive information shown in Tables II, III,
and VI—the hedging and speculative uses of option not only enlarge option buy and sell
volume but also increase option trading volume because of large potential gains and losses
associated with greater volatility. This result also implies a positive volume–volatility relation,
consistent with Sarwar (2003) and Sears (2000) who cite a positive relation between volatility and option trading volume.
Panel B of Table VII shows a relatively weak negative relation from lagged volatility
spreads to volume: Most of the coefficients for volatility spreads with negative information are
negative, whereas only the volatility information abstracted from calls has a significant effect
on OPVOL, OPVOLbuy, and OPVOLsellat 2 days ahead. A possible explanation for this result is
that a decrease in market volatility predicted by volatility spreads with negative information reduces option buy and sell volume and option trading volume because of the limited use of
options for hedging and speculation. In addition, the coefficients of T are all significantly
negative in Panels A and B, indicating a negative time trend for OPVOL, OPVOLbuy, and
OPVOLsell. Results of a positive (negative) volatility‐induced change in option trading volume
following large volatility spreads with positive (negative) information provide evidence that deviations from PCP are informative about future trading volume in options.
Table VIII reports similar but relatively weak prediction results for IDVOL, IDVOLbuy,
and IDVOLsell. Panel A shows that volatility spreads with positive information implicit in calls
TABLE VII Predictability in Option Trading Volume by Volatility Spreads j OPVOL OPVOL buy OPVOL sell 1 212 12 Panel A: Volatility spread with positive information T 0.0701 ( 2.76 ) 0.0578 ( 2.53 ) 0.0368 ( 2.86 ) 0.0307 ( 2.65 ) 0.0336 ( 2.67 ) 0.0272 ( 2.40 ) VS M c b;t j 1.2781 (1.99 ) 0.8053 (1.14) 0.6452 (2.03 ) 0.3755 (1.05) 0.6336 (1.91 ) 0.4307 (1.22) VS M p b;t j 0.2833 (0.60) 1.5675 (3.46 ) 0.1576 (0.64) 0.7631 (3.29 ) 0.1286 (0.55) 0.8048 (3.59 ) Adj. R 2 0.3445 0.3453 0.3583 0.3587 0.3264 0.3274 Panel B: Volatility spread with negative information T 0.0764 ( 3.17 ) 0.0537 ( 2.37 ) 0.0398 ( 3.25 ) 0.0284 ( 2.46 ) 0.0368 ( 3.09 ) 0.0254 ( 2.28 ) VS M c s;t j 0.5642 (1.11) 1.8250 ( 2.82 ) 0.3044 (1.14) 0.8533 ( 2.61 ) 0.2545 (1.02) 0.9696 ( 3.00 VS M p s;t j 1.2784 ( 0.82) 2.4502 ( 1.46) 0.6014 ( 0.76) 1.1399 ( 1.40) 0.6783 ( 0.87) 1.3167 ( 1.51) Adj. R 2 0.3442 0.3462 0.3579 0.3591 0.3261 0.3289 VO Lt ¼ a0 þ a1 Tt 1 þ b1 VS M c tj þ b2 VS M p tj þ b3 FV t j þ b4 ID Vt j þ b5 ISt j þ P 4 i¼1 gi VO Lt i þ et Note . This table represents the estimation results of prediction regression, shown in Equation (10), for option trading volume by controlling on a linear time trend (T ), logarithmic open interest of the index futures (FV ), logarithmic trading volume of the underlying index (IDV ), the impact of information flows (IS ), and four lagged option trading volume. The regression is estimated for different values of j, from 1 to 2 , to capture the potential predictive power of volatility spreads for future volume. Volatility spreads with positive information and nega tive information are used separately to test for the predictability in OPVOL ,OPVOL buy , and OPVOL sell . The results of volume prediction by volatility spreads with positive and negative information are reported in Panel A and Panel B, respectively. VS M c tj and VS M p tj denote either volatility spreads with positive information (VS M c band VS M p b) o r negative information (VS M c sand VS M p s) o n day t jof calls and puts, respectively. VOL is either of OPVOL ,OPVOL buy ,o rOPVOL sell .OPVOL denotes the option trading volume in the near and second month. OPVOL buy and OPVOL sell are the signed buy and sell volume of options, respectively. T is linear time trend used to control for the time effect. For brevity, only a1 ,b 1 ,and b2 are reported. Newey – West standard errors are used to calculate the t‐ statistics of the estimated parameters. , ,and indicate that t values (in parentheses) are signi ficant at the 0.01, 0.05, and 0.1 level, respectively.
