Asymmetric Information, Perceived Risk and Trading Patterns: The Options Market Guy Kaplanski * Haim Levy** March 2012 * Bar-Ilan University, Israel, Tel: 972 50 2262962, Fax: 972 153 50 2262962, email: [email protected]. ** The Hebrew University of Jerusalem, 91905, and the Academic Center of Law and Business, Israel. Tel: 972 2 58831011 Fax: 972 2 5881341 email: [email protected] (Corresponding author).
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Asymmetric Information, Perceived Risk and Trading Patterns: The Options Market
**The Hebrew University of Jerusalem, 91905, and the Academic Center of Law and Business, Israel. Tel: 972 2 58831011 Fax: 972 2 5881341 email: [email protected] (Corresponding author).
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Asymmetric Information, Perceived Risk and Trading
Patterns: The Options Market
Abstract
Asymmetric information models are tested using options implied volatility and volume of trade in eight international markets. We explore the relations between the trading break time duration, the quality of public information, the discretion of options liquidity traders to postpone their trades, and the interday and intraday implied volatility and volume of trade in options. Although asymmetric information is generally related to the underline asset, we find that it strongly affects the investment strategies adopted by the various options traders which, in turn, affect implied volatility and options’ volume of trade. The current analysis sheds new light on those strategies and their interrelations with the stock market. The introduction of futures on implied volatility in 2004 is also explored. JEL Classification Numbers: D82, G12, G14
Stock market studies provide compelling empirical evidence for systematic
interday and intraday patterns in stock price volatility and volume of trade. Several
theoretical adverse selection models with asymmetric information have been
employed to explain these phenomena; each entails different predictions
corresponding to the intertemporal stock price and volume of trade behavior,
depending on the underline assumptions.
In this study, we focus on the options market by studying the intertemporal
trading patterns in eight international options markets. While the existing theoretical
models and the relevant empirical studies mainly focus on the effect of information
asymmetry corresponding to the underline asset on the asset itself, we study this effect
on the options written on this asset. The effect of asymmetric information
corresponding to the underline asset on options is not trivial, as asymmetric
information is expected to simultaneously increase the risk and decrease the price of
the underline asset. These effects have a contradicting influence on the price of call
options, but enhance effects in the same direction in regard to the price of put options.
Employing data on options written on various assets, we test several
hypotheses that shed light on the alternate asymmetric information models suggested
in the literature for the stock market, and on the implied investment strategies adopted
by the various options traders. By incorporating implied volatility into the analysis,
we add another dimension to the existing models: that of investor perceived risk (for,
say, the next 30 calendar days) which, to the best of our knowledge, has not been
previously explored in this context. We analyze the perceived risk relations of
uninformed options liquidity traders, the flow of public information that resolves the
information asymmetry, and the investment strategies employed by the various parties
which, in turn, systematically affect the interday and intraday options volume of trade.
Does the options market reveal interday and intraday trade patterns similar to
those observed in the stock market? How do uninformed options traders protect
themselves against traders who possess private information? Are the trading strategies
adopted by various traders affected by the quality of public information? Do the
futures on the U.S. volatility index (the VIX), introduced in 2004, mitigate the risk
induced by information asymmetry? Are the empirical results unique to the U.S.
market? The aim of this study is to answer these and other related questions. To
achieve this goal, we use Foster and Viswanathan’s (1990) theoretical model as a
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springboard for postulating the hypotheses regarding the options market. They
suggest a rich theoretical model with several predictions as regards stock market
behavior. As their alternate set of assumptions implies different predictions, we
empirically examine their (and other suggested models’) various sets of assumptions
and infer which set of assumptions (namely, the theoretical model) best conforms to
the options market.
As a trading break is a major cause of asymmetric information, we explore the
relations between implied volatility and volume of trade in options, and the trading-
break time duration during which private information is accumulated. Thus, we go
beyond the weekend and also test holiday and overnight trading breaks, as well as the
reversal during trading hours—a reversal which occurs due to the revealing process of
private information through trade. These relations shed light on the quality of public
information, the way uninformed options traders protect themselves against private
information, and whether they have the discretion to postpone their trade activities—
an action which depends on the quality of the public information. This analysis also
shows when, and how quickly, private information is revealed. As implied volatilities
corresponding to subsequent days include overlapping days, we suggest tests which
measure the daily differences in implied volatility, net of the overlapping days’ effect.
Finally, the introduction of futures on implied volatility in 2004 enables us to
separately explore the role of private information before and after 2004. This analysis
indicates that the various options traders use the futures market to either protect
themselves or exploit private information, thereby mitigating the asymmetric
information perceived risk which, in turn, improves market efficiency.
In a non-rigorous manner, Figure 1 illustrates the highlight of this study with
the U.S. VIX, which corresponds to the S&P 500 Index’s implied volatility (a similar
figure is obtained with other markets’ indices). The figure presents the average VIX at
market opening and market closing times, the average trading volume in the CBOE
corresponding to index options, and the actual price volatility calculated from realized
returns on the S&P 500 Index, as a function of the day of the week.
<< Insert Figure 1 >>
As can be seen from the figure, the average VIX and volume reveal systematic
patterns across the weekdays. The average VIX in Figure 1a is highest on Monday; it
decreases during the week, where the opening VIX is higher than the closing VIX,
especially at the beginning of the week. In contrast, the average trading volume,
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presented in Figure 1b, is at its lowest on Monday; it increases until Thursday and
then decreases on Friday. Finally, the average realized price volatility (as measured by
the GARCH model), presented in Figure 1c, is almost the same, with only minor non-
monotonic changes across the days.
Figure 1 reveals inverse patterns in implied volatility and trading volume of
options across the weekdays. These patterns are not induced by actual price volatility,
as no particular pattern is observed in this variable. The more rigorous statistical
analysis reveals that these patterns are not related to the day of the week, but rather to
the weekend trading-break. This trading-break effect is neither due to changes in
economic fundamentals nor to mechanical and statistical biases related to the
volatility index calculation method. It is rather nicely explained by the existing
theoretical models dealing with private information accumulated during trading
breaks, and the investment strategies employed by the various options traders in the
presence of asymmetric information. The private information accumulated during the
trading break is another risk component that uninformed traders face; hence, this
factor is also taken into account when establishing their investment strategies.
Moreover, we find that the trading-break effect is a global phenomenon which is not
unique to the U.S. market. Finally, we show that the options market results are
consistent with the results reported by French and Roll (1986), corresponding to
actual stock price volatility during trading and non-trading days.
The structure of this paper is as follows: Section 2 presents the existing
theoretical models and the empirical evidence regarding intemporal stock price
volatility and trading volume, and posits the hypotheses that are relevant to the
options market. Section 3 presents the data and methodology. Section 4 reports the
empirical results. Section 5 reports the results corresponding to alternative models and
robustness checks, while Section 6 concludes. Some technical, albeit important, tests
are relegated to the Appendix.
2. Existing theory, the empirical evidence and hypotheses of this study
The discovery of intemporal systematic patterns in stocks’ realized price
volatility goes back to Fama (1965), Granger and Morgenstern (1970), Christie (1981)
and French and Roll (1986), all of whom find that stock price volatility is
significantly higher during trading days than during non-trading days. French and Roll
(1986) provide compelling evidence showing that this phenomenon is due to private
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information that affects prices when informed traders trade (see also Barclay,
Litzenberger and Warner, 1990; and Stoll and Whaley, 1990).
Stock price volatility also reveals an intraday U-shape. Wood, McInish and
Ord (1985) and Harris (1986) find that volatility is higher at market opening and
closing times than during the middle of the day. Amihud and Mendelson (1987),
Lockwood and Linn (1990), Foster and Viswanathan (1993), and Stoll and Whaley
(1990) show that this U-shape is not symmetric, as volatility is larger at market
opening times than at market closing times.
Stock trading volume also reveals systematic interday and intraday patterns;
however, there are conflicting views and empirical evidence regarding the correlation
between stock price volatility and volume. Jain and Joh (1988) find that stock trading
volume is lower on Mondays and Fridays than on other days, which implies a
negative correlation between volume and volatility. On the other hand, during trading
hours, Stoll and Whaley (1990) find that higher volatility is accompanied by high
volume, indicating a positive correlation. Foster and Viswanathan (1993) find that for
the more actively traded stocks, volume and volatility are positively correlated as
regards intraday activity, but negatively correlated as regards interday activity.
Several theoretical models are employed to explain these stock market
empirical results. Kyle (1985) shows that in a market with three types of traders—
informed traders, noise traders, and competitive market makers—private information
is gradually incorporated into prices. Glosten and Milgrom (1985) show that adverse
selection can account for the existence of the bid-ask spread and that transaction
prices are informative in the presence of adverse selection; thus, spreads tend to
decline with trade. Admati and Pfleiderer (1988) expand the model to include
discretionary liquidity traders, who can time their trade activities. These traders lead
to trading concentrations during the day, which can explain the volatility intraday
asymmetric U-shape.
Foster and Viswanathan (1990) suggest that in the presence of private
information uniformed discretionally liquidity traders have an incentive to postpone
their trade activities to other days, while waiting for public information. This model
explains both the intraday and interday patterns in stock price volatility and volume,
and the correlations in these patterns. The main predictions of Foster and
Viswanathan’s model as regards the stock market, which also have implications that
relate to the options market, are as follows:
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1. As private information is received during all times, but revealed only during
trading hours, stock price volatility is expected to be higher after trading breaks, in
particular when the market is open and private information is at its highest level.
2. Uninformed discretionary liquidity traders will avoid trading on days
following non-trading days, in order to steer clear of the adverse high costs implied by
private information. Thus, stock trading volume is expected to be lower on Mondays
and after holidays when private information is high.
3. The incentive to postpone trading depends on the process by which private
information is revealed. It is predicted that with the regular release of high quality
public information, there will be two days before Friday with concentrated trading. In
contrast, poor public information is expected to lead to only one day (Friday) of
concentrated trading each week. Of course, the market aggregate results also depend
on the proportion of discretionary liquidity traders in the market.
Based on these predictions about the stock market, below we posit and test
several hypotheses regarding implied volatility and volume of trade in options.
Generally, an increase in uncertainty of the uniformed liquidity trades regarding the
value of the underline asset is expected to decrease its price, due to the increase in the
required risk premium. Therefore, there are two effects on the option price: The
increase in uncertainty (due to the asymmetric information risk) increases the price of
all options, and the decrease in the underline asset price decreases the price of call
options and increases the price of put options.1 Thus, while the total effect on the
option price depends on the option type and the relative magnitude of the two effects,
in both cases the increased uncertainty regarding the underline asset is expected to
1Jones and Shemesh (2010) show that the rate of return on options is relatively low over the weekend,
a phenomenon that is not related to the change in the price of the underline asset. They also show that the “total implied volatility” decreases over the weekend—in contradiction to what is reported by French and Roll (1986), the predicted results given by the private information and asymmetric information models, and the results reported here. There are several possible reasons for the different results in the two studies. First, Jones and Shemesh focus on options of individual stocks, while we focus on options of stock indices, e.g., the S&P 500 Index. Support for this possible reason for the differences is that when they report some results on indices options, they obtain inconclusive results. Other possible sources for the differences are the different periods covered, the different implied volatility measures employed (calendar versus total implied volatility, Model-free versus Black-Scholes), the different methodologies employed to measure the implied volatility on different days of the week, and finally, their use of options closing values, which overshadows the higher implied volatility at market open.
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increase the option implied volatility.2 To explore this prediction as regards implied
volatility, we test the following hypothesis with options data:
H1. The trading-break implied volatility (TBIV) hypothesis: Implied volatility
after trading breaks is not significantly different from that during trading hours. The
alternative hypothesis asserts that implied volatility is relatively higher after trading
breaks, due to higher risk as perceived by uninformed traders, a risk which decreases
when public signals are received.
The TBIV hypothesis has several spinoffs. First, it is separately tested for
weekend, holiday and overnight trading-breaks. According to the TBIV hypothesis,
implied volatility is expected to be higher, albeit not with the same magnitude, after
all types of trading breaks. Second, the longer the trading break the greater the
expected amount of private information; hence, the larger the risk perceived by the
uninformed traders. Therefore, we also test whether the higher implied volatility is
correlated with the trading-break time duration. To test whether private information is
gradually revealed during trading hours, we test whether implied volatility decreases
during trading hours. Finally, to test whether discretionary liquidity options traders
postpone their trades to other days—as suggested by Foster and Viswanathan (1990)
in regard to traders in underline assets—we test for possible patterns in implied
volatility across all weekdays.
We now turn to the hypothesis regarding the interday pattern in trade volume.
If, indeed, discretionary liquidity traders in the options market postpone their trades to
other days, then according to Foster and Viswanathan’s model, they will decrease
their trading on Mondays and after holidays. This leads to the following hypothesis:
H2. The trading-break volume (TBV) hypothesis: The options trading volume
after non-trading days is not significantly different from that on other days. The
alternative hypothesis asserts that the volume is lower after non-trading days due to
uninformed discretionary liquidity options traders who postpone their trades.
While the volume on Mondays is expected to be relatively low, the exact
trading pattern on the other days of the week depends on the quality of public
information. In the case of regular release of accurate and high quality public
information, discretionary liquidity traders will pool their trades into two days before
2Since the well-known Monday effect in returns has significantly attenuated over the last decades
(Schwert, 2003), the decline in the underline asset price during the period covered in this study probably does not reflect the increase in uncertainty.
7
Friday, whereas in the case of poor public information they will pool their trades on
Friday. To explore this issue and the role of the quality of public information, we test
the following hypothesis:
H3. The quality of public information (QPI) hypothesis: The options trading
volume pattern over the weekdays does not depend on whether the underline asset is
an individual stock or an index. The alternative hypothesis asserts that the trading
pattern is different for options written on individual stocks and indices, as accurate
public information corresponding to indices is released more regularly, on average,
than that on individual stocks.
If the higher implied volatility after trading breaks is due to higher risk
induced by private information, the introduction of futures on the VIX in 2004 has
possibly served to mitigate this phenomenon. This is because new instruments are
generally expected to improve market efficiency. In particular, these futures enable
traders to hedge against private information risk and also provide another relatively
low-cost channel for informed traders to exploit their private information which, in
turn, expedites the flow of private information to the market. To test whether the
futures on implied volatility have indeed mitigated the effect of private information,
we test the following hypothesis:
H4.The market efficiency (ME) hypothesis: The ability to trade implied
volatility in the futures market did not significantly change the interday pattern in
implied volatility. The alternative hypothesis asserts that the ability to trade implied
volatility mitigated the interday patterns in implied volatility.
Although the empirical results in this study reject the null hypotheses
presented above, there is always a possibility that the observed significant phenomena
are caused by economic factors or technical biases, which are correlated with the
predictions of the theoretical models. Therefore, we conduct several robustness tests.
These tests reject the hypotheses asserting that the patterns in the options market are
due to the following factors: economic fundamentals which are incorporated in actual
price volatility (where price volatility is measured by various methods) and in the
underline asset price returns; statistical and methodological biases including the
distinction between trading days and calendar days corresponding to the calculation of
implied volatility, and various numbers of trading days due to holidays; implied
volatility calculation methods (in particular, the volatility index time interpolation and
methodology); the type of options underline assets and, most importantly, the options
8
expiration day. Finally, we test whether the results are affected by specific
characteristics of the local markets and cross-border inefficiencies like the different
trading hours, currency effects and biases related to trading methods and the market’s
various settlement procedures.
3. Data and methodology
To measure implied volatility, we employ the well-known volatility indices
(VIs). The VI measures the volatility expectation as implied by the option prices. The
daily data of the following eight primaries’ VIs and their underline stock indexes are
employed: The U.S. VIX (S&P 500); Dutch VAEX (AEX); French VCAC (CAC 40);
Eurozone’s VSTOXX (EURO STOXX 50); and the German VDAX-NEW (DAX
30).3 Panel A in Table 1 presents the main characteristics of the eight VIs.
<< Insert Table 1 >>
All of the VIs were calculated backwards into the past, providing us with at
least 10 years of daily data, with 21 and 19 years of data in the case of the VIX and
the VDAX-NEW, respectively. The VIX, VAEX, VCAC, VFTSE and VXJ employ
the “New-VIX” methodology, and the VSMI, VSTOXX and VDAX-NEW are also
based on this methodology with some modifications. This methodology—first
adopted by the CBOE in 2003, when the VIX was recalculated backward into the
past—is based on Britten-Jones and Neuberger’s (2000) model-free methodology,
which estimates volatility expectations by averaging the weighted prices of put and
call options over a wide range of strike prices.
To explore whether the results reported in this study are affected by the VIs’
calculation method, the underline stock index or the options’ time to expiration, we
also study the following alternative VIs, presented in Panel B: The VXD (Dow Jones
Industrial), VXN (NASDAQ 100), and RVX (Russell 2000) are used to verify that the
results are general, rather than confined to a specific underline stock index. The
VDAX (DAX 30) and CSFI-VXJ (Nikkei 225) are used to verify that the results are
3The data on the U.S. VIs, the S&P 500 Index, and the options trading volume are provided by the
CBOE. The data on the VSMI, VSTOXX, and VDAX-NEW, as well as their alternative indexes, are provided by the SIX Swiss Exchange, STOXX Limited Company and the Deutsche Börse exchange, respectively. The data on the Japanese indexes is provided by The Center for the Study of Finance and Insurance (CSFI), Osaka University. Finally, the data on the VAEX, VCAC and the VFTSE, as well as their alternative indexes, are provided by the NYSE Euronext Group.
9
not technically induced by the New-VIX methodology.4 The VSMI6M (SMI),
VSTOXX6M (EURO STOXX 50) and VDAX-NEW6M (DAX 30) measure the
floating six-month expectation volatility from one options series whose expiration day
is the closest to six months, without time interpolation. These VIs are used to verify
that results are neither induced by the options’ expiration day nor by the VI’s time
interpolation procedure. As for the U.S. market, there is no six-month floating VI. For
comparison purposes, we also report the VXV (S&P 500), which measures the fixed
three-month expectation volatility.
As implied volatility is affected by economic fundamentals, it is also
important to measure the VI relative to the actual price volatility to verify that the
observed results are not induced by economic fundamentals, which are accounted for
in price volatility. Therefore, we employ the daily time series of the VI as well as the
daily price volatility.
To conduct an analysis of these time series, one first needs to choose the
appropriate econometric model. The choice of the model is important because the
VI’s time series incorporate several well-known econometric issues that may bias the
results. First, like the volatility time series, which may have a unit root (Pagan and
Schwert, 1990), the VI may also have a unit root. Second, volatility is serially
correlated and the VI is inherently serially correlated.5 Finally, like actual volatility,
the VI may also reveal “memory” in response to shocks, and a correspondingly high
degree of heteroskedasticity (for the existence of these phenomena in volatility, see
e.g. Poterba and Summers, 1986, French, Schwert, and Stambaugh, 1987 and
Schwert, 1990). To handle these issues, our first task is to choose the appropriate time
series statistical model, which takes into account all these problematic issues.
Comparing the various alternate models, presented in more detail in Appendix
A, we find that the Exponential Generalized Autoregressive Conditional
Heteroskedastic(1,1) model with Student’s t-distribution (EGARCH-t) and 12
autoregressive lag variables best handles the statistical issues mentioned above.
Therefore, this model is employed in the main analysis.6
4The VDAX employs the Deutsche Börse’s old methodology, which is based on the Black-Scholes
option pricing model, near-the-money options and corresponds to 45 calendar days, while the CSFI-VXJ employs the Center for the Study of Finance and Insurance novel model-free methodology.
5On each day, the VI measures the expected volatility for the next 30 calendar days; hence, the index values corresponding to day t and day t-1 include 29 common days.
6For the advantage of the EGARCH model as regards volatility time series see, for example, Nelson (1991), Pagan and Schwert (1990) and Hentschel (1995). Alternatively, we also employed a GARCH
10
To estimate actual price volatility, like many other studies we use a GARCH
model (for a review, see Poon and Granger, 2003). We use the GARCH(1,2) model
which, unlike the GARCH(1,1), eliminates significant autocorrelations corresponding
to all lags. Furthermore, comparing other models we find that in seven markets the
GARCH(1,2) is the best fitting model as measured by Schwarz’s (1978) BIC and
Akaike’s (1974) AIC criteria.7 Finally, in the robustness tests, price volatility is also
directly estimated from realized returns, where the analysis incorporates both the ex-
post and ex-ante price volatility corresponding to the VI period.
4. Empirical results
In this section, we report on several significant trading patterns in the options
market. The possibility that the results are artifacts induced by technical biases is
explored in Section 5.
4.1 The trading-break implied volatility (TBIV) hypothesis
Based on the results reported in Appendix A, to analyze the VIs we employ
the following EGARCH-t(1,1) model, while assuming that the residuals follow the
Student-t distribution. Specifically, we employ the following model:
ti
itii
itii
ititi
itit RRVTBREAKDAYV
22
0
2,5
22
0,4
12
1,32
5
1,,1 ,
ttt σzε ,
)log()()log( 21111
2 ttttt σβγzzEzαωσ , (1)
where tV is the volatility index (or a function of it) on day t; )5...1(, iDAY it are
dummies corresponding to the weekdays; tTBREAK is a dummy corresponding to
days other than Mondays after non-trading days; tR is the percentage rate of return on
the options underline stock index on day t, and t , tz and t are the innovation,
standardized innovation and the conditional standard deviation, respectively.
Market volatility and returns are correlated in a complex manner (e.g.,
Glosten, Jagannathan, and Runkle, 1993; French, Schwert, and Stambaugh, 1987;
model in which the innovations follow either the Student-t or the normal distribution as well as the Autoregressive Integrated Moving Average (ARIMA) (3,0,3) model which, according to the BIC and AIC information criteria, is the best fit ARIMA model. As the results with these models are very similar to those reported in this study, for brevity’s sake they are not reported, but are available upon request.
7In the U.S., the GARCH(2,2) model reveals slightly better results. As the differences are small, for the sake of consistency, to calculate the U.S market volatility we also employ the GARCH(1,2) model.
11
Campbell and Hentschel 1992; Brandt and Kang, 2004; and Avramov, Chordia and
Goyal, 2006). Specifically, as the patterns in the VI coincide with the well-known
weekend effect in returns, the results corresponding to the VIs may be induced by the
effect in returns. To account for this possibility and to control for any other bias
induced by returns, the regressions also include the returns variable ( tR ) and its lags
over a full month (22 trading days) as explanatory variables. As the dependent
variable is volatility, in the main tests we also include the squared returns ( 2tR ) and its
22 lags as explanatory variables.8
Table 2 reports Eq. (1) results, with the U.S. VIX.
<< Insert Table 2 >>
Test 1 examines the VIX opening values, where the Monday coefficient corresponds
to days subsequent to the weekend trading break, and the TBREAK coefficient
corresponds to non-Monday days subsequent to the trading break. Test 1 reveals that
both the Monday and TBREAK coefficients are several times larger than the other
days’ coefficients. The Friday coefficient, on the other hand, is substantially smaller
than the other coefficients. Finally, the Log-likelihood statistic for equal days
indicates that the differences across the days are highly significant ( 0001.0p ).
The results with the VIX closing values in Test 2 are very similar. The
Monday and TBREAK coefficients are, once again, several times larger than the other
days’ coefficients, while the Friday coefficient is smaller than the other coefficients,
and the differences across the days are highly significant.
Can the high VIX on Monday be attributed to specific characteristics of the
Monday or two-day weekend trading break? To answer this question, Tests 3–6
include dummies that correspond to days subsequent to one-, two- and more than two-
day trading breaks. Tests 3 and 4, which do not include the weekdays’ dummies,
examine the effect of the duration of the trading break on the regression coefficient.
For both opening and closing VIX, the three trading-break coefficients are
significantly positive. Moreover, the coefficients increase with the trading break time
duration, and the hypothesisthat the trading break coefficients are equal is rejected as
regards the VIX closing values ( 0001.0p ). As the two-day trading break
8In unreported tests, we verified that excluding the returns and squared returns variables do not
change the main results. In separate tests, we also include yearly dummy variables which control for outlier years with particularly high and low VIs. As these variables are found to be insignificant, these tests are not reported.
12
observations mainly consist of weekends, their number is much larger than the
number of one- and more than two-day observations, which explains the relatively
high t-value corresponding to the two-day trading break.
Tests 5 and 6 also include the weekdays’ dummies. Thus, the two-day trading
break variable includes all of the weekend effects, as measured by the Monday
variable, plus other two-day trading breaks that do not end on Mondays. Hence, these
two variables are highly correlated, which decreases the t-value of these two variables
due to multicollinearity. Indeed, we find that the t-value corresponding to these two
variables substantially decreases in comparison to the t-values reported in the
previous tests. This phenomenon is most profound in Test 6, where the Monday
coefficient turns out to be insignificant. This result indicates that the trading breaks
affect the increase in the VIX, rather than various Monday-specific factors.
Finally, in Tests 5 and 6 all the coefficients corresponding to trading breaks
are larger than the coefficients corresponding to weekdays; this supports the TBIV
hypothesis. After non-trading days, uninformed options traders face additional risk,
due to private information accumulated during the trading break reflected in the
higher VIX. As the longer the trading break the more private information is expected
to be accumulated, this risk—and correspondingly, the VIX—increase with the
trading break time duration.
Two results reported in Tests 1–6 require some further explanation. First, the
VIX is significantly lower on Fridays than on other days. Second, although the results
are similar with both the opening and closing VIX, they differ in magnitude.
The lower VIX on Fridays can be explained by means of the TBIV
hypothesis, as well as by a mechanical bias related to calendar days, and which
conforms to the findings of French and Roll (1986). According to Foster and
Viswanathan’s model, when high quality public information is regularly released,
discretionary liquidity traders pool their trade into two days before Friday. Dealing
with options written on the S&P 500 Index, the regularly released public information
is probably of high quality (relative to information on individual stocks). Therefore, if
a large portion of traders pool their trade, say, on Wednesday and Thursday, then all
private information is revealed in Thursday’s closing prices. Hence, on Friday, all
traders are informed, uncertainty due to private information vanishes, and the VIX is
relatively low. Although the lower VIX on Friday conforms, under reasonable
assumptions, to the TBIV hypothesis it may also be induced by a mechanical bias.
13
The VIX reflects the implied volatility corresponding to the next 30 calendar days. As
a result, the VIX on Friday relates to a smaller number of trading days.9 As according
to French and Roll (1986) price volatility over non-trading days is lower than that on
trading days, the smaller number of trading days corresponding to the VIX on Friday
may account for the lower VIX on Fridays. Of course, it is also possible that both the
release of public information and the mechanical bias, which operate in the same
direction, may account for the lower VIX on Friday.
Let us now address the differences between the opening and closing VIX. The
two hypotheses below test whether private information is also accumulated overnight;
hence, the VIX increases, and whether during trading hours private information is, at
least partially, revealed, leading to a decline in the VIX. As the time periods
corresponding to overnight and trading hours are relatively short, the effects, if they
exist, are expected to be less profound in comparison to those corresponding to
weekends and holidays.
To test for the existence of an overnight trading break effect, the dependent
variable in Test 7 is the overnight change in the VIX, which is calculated as the
opening VIX less the previous day’s closing VIX. As previously, the Monday and
TBREAK coefficients are positive and highly significant. However, the other
coefficients are relatively small, and the Friday coefficient is significantly negative.
Thus, the VIX increases after weekends and holidays, decreases on Friday mornings,
and does not significantly change over the other nights. The increase in the VIX after
weekends and holidays conforms to the TBIV hypothesis. The decrease in the VIX on
Friday mornings is also in line with this hypothesis. As with high quality public
information—which is more relevant for the VIX and the underline S&P 500 Index—
virtually all private information is revealed by the end of Thursday. Hence, on Friday
mornings all traders are informed, no risk premium is required for private
information, and the VIX decreases.
Finally, the other weekdays’ insignificant coefficients suggest that there is no
significant overnight trading-day effect. This is probably because not much
information is received over the relatively short overnight trading break, which is also
9A 30-calendar-day window, starting on Friday, includes the subsequent four weeks plus two non-
trading days: Saturday and Sunday. In contrast, a 30-calendar-day window starting on the other days includes the next four weeks plus either one non-trading (Thursday) or two trading days (Monday-Wednesday).
14
in line with the general result of French and Roll (1986): that during trading breaks
information is received at a slower pace than during trading hours.
As private information is revealed during trading hours, according to the TBIV
hypothesis the risk induced by private information is expected to diminish during
trading hours; hence, the closing VIX is expected to be lower than the opening VIX.
Indeed, Figure 1 shows that the average closing VIX is lower than the average
opening VIX, in particular on Mondays and Tuesdays, where a simple t-test rejects
the hypothesis of equal means ( 0001.0p ). To further test this prediction, the
dependent variable in Test 8 is the change in the VIX during trading hours, which is
calculated as the closing VIX less the opening VIX on the same day. In line with the
TBIV hypothesis, apart from Thursdays the days’ coefficients are negative and on
Mondays and Tuesdays they are relatively large, where the latter is also significant.
Thus, it seems that the VIX decreases during trading hours, in particular on Mondays
and Tuesdays when a greater amount of private information accumulated during the
weekend is revealed. Yet, the significance of this result depends on whether the
squared returns control variables are included or not in the regression.10
To complete the description of the tests in Table 2, note that in line with the
results reported in Appendix A regarding the VIs’ time series, in all the tests the
EGARCH coefficients (α, β and γ) are highly significant. As expected, the return and,
to some extent also the squared return variables, are significantly negatively and
positively correlated, respectively, at various lags (to avoid a complex table these
coefficients are not reported in the table). However, the effects in the VIX are highly
significant after controlling for returns.
4.2 .Overlapping period in the VIX calculation
The significant intraday and interday patterns reported so far are found in the
VIX values, which include overlapping days. For example, the opening VIX on
Monday and the subsequent Tuesday, which corresponds to 30 calendar days, i.e. to
the period that ends on the fifth Tuesday and Wednesday, respectively, include 29
overlapping days. These overlapping days are not expected to systematically bias the
results because they are common to both VIX values and have a similar effect or,
more precisely, a random effect rather than a systematic one, which is expected to be
10In unreported tests, we found that without the squared return variables the Monday and Tuesday coefficients are highly significantly negative. Thus, the relatively small t-values in Test 8 are probably due to the correlation between the daily difference in the VIX and squared returns variable and its lags.
15
canceled out on average. Therefore, if the opening VIX on Mondays is higher than
that on Tuesdays, it implies that the perceived volatility corresponding to Mondays is
higher than the perceived volatility corresponding to Wednesdays.
To supplement the overlapping analysis, we also measure the pairwise
differences in the VIX, after deducing the overlapping days’ volatility. For example,
the opening VIX on Mondays and Tuesdays correspond to the periods ending on the
fifth Tuesday and Wednesday, respectively. Hence, the opening VIX on Monday less
Tuesday measures the difference in daily perceived volatility corresponding to
Monday and Wednesday, where there is a time period of 30 calendar days between
these two days.11 Similarly, the opening VIX on Monday less Wednesday corresponds
to the perceived volatility on Monday-Tuesday less that on Wednesday-Thursday,
which comes 30 calendar days later. Finally, the opening VIX on Monday less
Thursday corresponds to Monday-Wednesday less Wednesday-Friday, which comes
30 calendar days later. Ignoring the common Wednesday, it actually measures the
difference in the perceived volatility corresponding to Monday-Tuesday and
Thursday-Friday. By the same logic, the opening VIX on Monday less that on Friday
measures the difference in the perceived volatility corresponding to (after ignoring the
common days) Monday-Tuesday less Friday-Saturday.
Finally, as we are interested in the interday effect across the weekdays to
control for the long-term trend across the weeks, we normalize the VIX values
according to the weekly mean. Thus, all observations for each week are divided by the
relevant weekly mean, which reduces the possibility that the results are biased by the
long-term trend and outlier periods during which the VIX was very high or very low.
This also reduces the possibility that the 30-day time period between the daily VIX
values biases the results.12
11The perceived volatility corresponding to the eliminated days is not necessarily the same on each
day. Therefore, although there is no reason to believe that there are more than random changes across the many years covered in this study, we look at both the coefficients corresponding to Monday less other days (e.g., Wednesday) and the coefficients corresponding to other days less Monday, where in the first case Monday precedes the other days by 30 days and in the latter case the other days precede Monday by 30 days. Thus, if the results are biased, due to the overlapping days in favor of the TBIV hypothesis in one case, they are expected to be biased against it in the other case. This is because in both cases almost the same days are eliminated. For example, when comparing Monday less Wednesday in 30 days and Wednesday of the same week less Monday in 30 days there are 28 common days among the 30 eliminated days. Hence, obtaining the same results in both cases reduces the possibility that the results are spurious.
12The results with the raw VIX (i.e. without normalization) are generally similar with only slightly smaller t-values probably due to trend biases.
16
Table 3 reports the results of Eq. (1), where the dependent variable is the
difference in one of the VIX pairs.13
<< Insert Table 3 >>
The first column in the table reports the coefficient corresponding to the VIX on
Mondays or the VIX on Mondays and Tuesdays less the VIX on other days. In all the
tests, the coefficient is significantly positive, indicating that the perceived volatility on
Mondays is significantly higher than that on Wednesdays (Test 1), and the perceived
combined volatility on Mondays and Tuesdays is significantly higher than that on the
combined Wednesdays and Thursdays (Test 2), Thursdays and Fridays (Test 3) and
Fridays and Saturdays (Test 4). The result in Test 1—that the daily perceived volatility
on Mondays is higher than that on Wednesdays—is particularly important, as
comparing the perceived volatility corresponding to Mondays and Wednesdays
completely bypasses the mechanical bias, due to a varied number of trading days.
This is because Mondays and Wednesdays have the same number of trading days in a
forward-looking 30-calendar-day window; hence, they are not exposed to the lower
volatility on non-trading days reported by French and Roll (1986).
Consistent with the results reported above, all the coefficients which
correspond to non-Monday days less Monday are negative and most of them are also
significant. For example, the fifth coefficient in Test 2 indicates that the combined
perceived volatility on Friday and Saturday is significantly lower than that on Sunday
and Monday (a t-value of 01.9 ). Thus, in line with the TBIV hypothesis, the daily
perceived volatility on Monday, and possibly also on Tuesday, is higher than that on
other days and these results are intact whether Monday precedes the other day or vice
versa which, as previously explained, reduces the possibility that the results are biased
by the eliminated overlapping days.
Higher perceived volatility at the beginning of the week is a general
phenomenon, which is not confined to Mondays. For example, the second and third
coefficients in Test 1 show that the perceived volatility on Tuesdays and Wednesdays
are significantly higher than that on Thursdays and Fridays, respectively. Consistent
results are obtained in all other tests. Thus, in line with the TBIV hypothesis and the
previous results with overlapping periods, the daily perceived volatility is at its
13Although the EGARCH coefficient, γ, is not significant, for the sake of consistency Table 3 reports
the results corresponding to the EGARCH model (the GARCH model results are very similar). The
17
highest level on Mondays, when private information is high; it then decreases over
the week as private information is revealed through trade.
Two other results emerge from Table 3, which are consistent across the
various tests. First, the daily perceived volatility on Saturdays is lower than that on
other days (e.g. the fourth coefficient in Test 1), which conforms to French and Roll’s
result that volatility on Saturdays is lower than that on the weekdays. In contrast, the
daily perceived volatility on Sundays is higher than that on other days (e.g. the fifth
coefficient in Test 1). Although this last result seems to contradict the fact that
volatility on non-trading days is smaller than on trading days, it probably reflects the
highest level of accumulated private information before the Monday trading and the
high volatility as recorded on Monday morning.
To summarize, the results reported so far reveal that in line with the TBIV
hypothesis, the VIX is significantly higher on days after non-trading days than on
other days, in particular when the market opens and private information is at its
highest level. These results are robust to serial correlation, overlapping days in the
VIX, mechanical bias due to a varied number of trading days, and stock market
returns. Finally, the VIX is significantly lower on Fridays than on other weekdays.
This result can be explained by both a mechanical bias due to the number of trading
days (which conforms to the findings of French and Roll, 1986), and by Foster and
Viswanathan’s (1990) model.
4.3. The trading break volume (TBV) hypothesis
In this section, we test the TBV hypothesis for various types of options. As the
distribution of the volume data is unknown, we employ Hansen’s (1982) Generalized
Method of Moments (GMM) analysis to estimate the following system of equations:
jti
Njitjitj
iitji
Njt VOLUMETBREAKDAYVOLUME ,
12
1,,,3,2
5
1,,,1,
, (3)
where )4...1(, jVOLUME Njt is the normalized daily volume of traded options in the
CBOE corresponding to index call options ( 1j ), index put options ( 2j ),
individual stock call options ( 3j ), and individual stock put options ( 4j ) on day
t; tTBREAK is a dummy corresponding to days other than Monday after non-trading
days; and )5...1(, iDAY it are dummies corresponding to the weekdays. To be able to
model in Table 3 does not include autoregressive variables to avoid multicollinearity, due to the correlation between the daily volatility and 2
tR and its lags.
18
compare the coefficients across the equations, the daily volume corresponding to each
type of options is normalized by the relevant all-day mean. The data on volume is
provided by the CBOE and covers the period from 2003 to 2010.
Table 4 reports the results of the regression corresponding to Eq. (3).
<< Insert Table 4 >>
Let us first discuss the results corresponding to the index options. The Monday and
TBREAK coefficients in Tests 1 and 2 are significantly negative, whereas the other
days’ coefficients are positive and most of them are highly significant. Indeed the
Wald statistics in both tests, reported in the last column of the table, reject the
hypothesis of equal weekdays coefficients (p<0.0001). Moreover, the TBREAK
coefficients are larger in absolute terms than the Monday coefficients and the
hypothesis of equal Monday and TBREAK coefficients is rejected at p=0.0105 (see
the last row in the table). Finally, the Friday coefficients are smaller than those
corresponding to other non-Monday weekdays.
The lower volume after weekend and holiday trading breaks conforms to
Foster and Viswanathan’s model and is consistent with the TBV and TBIV
hypotheses. As after trading breaks private information is at its highest level,
discretionary options liquidity traders postpone their trade to other days; hence, a
relatively low volume is recorded. Moreover, the higher TBREAK coefficients (in
absolute terms) in comparison to the Monday coefficients suggests that, as with
implied volatility, so, too with volume the intensity of the effect increases with the
trading break’s time duration. This is because 68% of TBREAK observations (which
do not include regular weekends) correspond to more than two-day trading breaks.14
Further in line with this model, as accurate public information corresponding
to indices is probably regularly released, discretionary liquidity options traders are
expected to pool their trades into two days, prior to Friday. Hence, the volume trade
on Friday is also expected to be lower than that on the other weekdays, which is
precisely what we obtain.
The results in Tests 3 and 4, which correspond to individual stock options, are
similar to those with indices, but less profound. The most important difference which
emerges from the comparison of index and individual stock options is that with
individual stock options the Monday coefficients are low, but not as low as with the
14In the U.S., most holidays fall on Mondays. Therefore, most of the TBREAK observations correspond to Tuesdays after three-day trading breaks.
19
index options. This result is significant as the hypothesis that the days’ coefficients
across the four types of options are equal is rejected for all weekdays. The relatively
weaker results corresponding to options on individual stocks conform to the quality of
information hypothesis. As the quality of regularly released public information is
expected to be higher in regard to indices compared to individual stocks, it is likely
that discretionary liquidity traders in index options postpone their trades more often
than those trading individual stock options. Hence, a larger decline is expected on
Mondays in the volume trade corresponding to index options.
Finally, obtaining similar patterns in the volume of trade corresponding to
both put and call options reduces the possibility that the results are technical in nature,
induced by the expected decline in the price of the underline asset, due to asymmetric
information risk. This is because the decline in the underline asset price is not
expected to have a symmetrical effect on put and call options. However, the increase
in uncertainty due to asymmetric information, which is our main explanation for the
results, is expected to have a symmetrical effect on both types of options.
4.4. The market efficiency (ME) hypothesis: The futures trade on the VIX
In April 2004, CBOE introduced futures on the VIX.15 As previously
explained, according to the EM hypothesis, it is expected that the futures on the VIX
mitigate market inefficiencies due to asymmetric information. To test the EM
hypothesis, Test 1 in Table 5 repeats the Eq. (1) analysis, while including additional
weekdays’ dummies corresponding to the period during which the futures on the VIX
were traded. This procedure covers the longest possible time period for which data is
available; hence, it contains a relatively large number of observations. For brevity’s
sake, in Table 5 and in the remainder of the study the VI data only corresponds to
closing values.
<< Insert Table 5 >>
The days’ coefficients corresponding to the whole period reveal a pattern that
is very similar to the one obtained so far. However, the Monday and Friday
coefficients corresponding to the period during which the futures were traded are
significantly negative and positive, respectively. Thus, the total effect on Mondays
15In February 2006, the CBOE introduced options on the VIX. As this period is already incorporated
in the period corresponding to the futures, and as the options on the VIX are expected to further mitigate the effect in implied volatility, we focus on the futures market and the period starting from 2004.
20
and Fridays during the period the futures were traded, which is equal to the sum of the
two Monday and two Friday coefficients, respectively, has significantly attenuated.16
Tests 2 and 3, which separately test the two sub-periods (with a lower number
of observations in comparison to Test 1), reveal similar results. As the number of
observations in each sub-period is different, let us focus on the regression coefficients
rather than on the t-values. While the coefficients corresponding to the more recent
period are generally smaller, due to a lower VIX on average, the Monday coefficient
decreased by 0.3652, whereas the Friday coefficient increased by 0.0075 (the other
days’ coefficients decreased by 0.1336, 0.0949 and 0.0623, respectively). Thus,
consistent with the results of Test 1, the decrease on Monday is the highest, while on
Friday the coefficient increased.
The results reported in Table 5 suggest that the interday pattern in implied
volatility has significantly attenuated since the introduction of futures on the VIX, but
it still remains highly significant. These results demonstrate how derivative
instruments improve market efficiency, presumably by reducing the asymmetric
information risk. Obviously, causality is not proven and further research is required to
determine the exact relations between the trade in options and futures on implied
volatility, which is beyond the scope of this study.
4.5. Alternative economic explanations: The international evidence
Figure 2 presents the average VIs and price volatilities corresponding to the
eight markets covered in this study. All the VIs, presented in Figure 2a, are highest on
Mondays and lowest on Fridays, whereas the actual price volatilities, presented in
Figure 2b, are very similar across the days with the exception that they are only
slightly higher on Tuesdays. Thus, a similar pattern in implied volatility is observed in
all markets, while no such phenomenon is observed in regard to price volatility.
<< Insert Figure 2 >>
To test whether the interday pattern is significant and similar across markets,
we employ a GMM analysis to estimate the following system of equations:
jti
jitjijtji
itjijt VTBREAKDAYV ,
12
1,,,3,,2
5
1,,,1,
, (4)
16For example, while the Monday coefficient corresponding to the whole period is equal to 0.6838,
the combined Monday coefficient corresponding to the futures period is equal to 4074.02764.06838.0 .
21
where jtV , is either the volatility index (Panel A) or the GARCH(1,2) price volatility
(Panel B) in market j ( 8...1j ) on day t; jtTBREAK , are dummies corresponding to
days other than Monday after non-trading days in market j; and )5...1(, iDAY it are
dummies corresponding to the weekdays. Table 6 reports the results of this analysis.
<< Insert Table 6 >>
Like with the U.S. VIX, the Monday coefficients corresponding to all the VIs
in Panel A are several times larger, and the Friday coefficients are smaller, than the
others days’ coefficients. The TBREAK coefficients are also larger than the others
days’ coefficients, apart from the one corresponding to Japan. Finally, in all the
markets the hypothesis asserting that the coefficients are equal is significantly rejected
(see the last column in the table).
In sharp contrast to the results in Panel A, the Monday coefficients in Panel B,
which correspond to the GARCH price volatility, are the same size order as the other
days’ coefficients, smaller than the Tuesday coefficients, and five of them are even
smaller than the Friday coefficients. The TBREAK coefficients are all negative and
most of them are highly significant.
The results in Table 6 show that the higher implied volatility after trading
breaks and the lower implied volatility on Fridays is a global phenomenon that exists
in all eight markets. The coexistence of this phenomenon in all the markets suggests
that this phenomenon cannot be explained by specific characteristics of the local
markets. This result also eliminates the possibility that the interday pattern is induced,
for example, by the different trading hours across the markets, any currency effects
and biases related to trading methods and market settlement procedures, all of which
have been proposed in the past as potential explanations for the weekend effect in
returns. Finally, this phenomenon does not exist in regard to stock price volatility,
which reduces the possibility that it is induced by economic fundamentals
incorporated in stock prices.
5. Rejecting technical and methodological explanations
In this section, we show that the results reported above are not merely artifacts
induced by some technical biases. To avoid unnecessary repetitions, when the tests
are straightforward we analyze the U.S. VIX, which is the most mature index with the
longest data history.
22
5.1. Rejecting the options’ expiration day as a potential explanation
As can be seen from Table 1, all the VIs’ underline options expire on Fridays.
Thus, the VI interday patterns may be induced by technical biases in the VI
calculations when shifting from one option series that has expired to another series, or
due to unique trading patterns around the expiration day. Note, however, that the
options’ expiration day cannot explain the interday patterns in volume, as the volume
data incorporates the transactions corresponding to all options series.
Wang, Li and Erickson (1997) explore the possibility that the options’
expiration day induces the weekend effect in returns. Following their methodology,
we break the month into five weeks17 and test whether the interday pattern in the VIs
differs in regard to the remaining time to options’ expiration. Thus, we employ a
GMM analysis to estimate the following system of equations:
(5) ,
))((
,
12
1,,,5
4
1,,,4
4
1,,,3,,2
5
1,,,1,
jti
jitjii
itji
iittjijtj
iitjijt
VWEEK
WEEKMONDAYTBREAKDAYV
where jtV , is the volatility index in market j ( 8...1j ) on day t; )5...1(, iDAY it are
dummies corresponding to the weekdays; jtTBREAK , are dummies corresponding to
days other than Monday after non-trading days; )4...1(),)(( , iWEEKMONDAY itt are
dummies corresponding to Mondays within the particular weeks of the month
excluding the fifth Monday, if it exists; and )4...1(, iWEEK it are dummies
corresponding to week of the month excluding the fifth week.
The hypothesis tested by Eq. (5) asserts that the patterns in the VIs are related
to the options’ expiration day; therefore, they differ across the weeks of the month,
depending upon the remaining time period until expiration. Presumably, the closer the
underline options to expiration, the greater/lesser the interday pattern. Focusing on the
higher VIs on Mondays, to test this hypothesis we add four Monday dummies,
))(( ,itt WEEKMONDAY , which allow the Monday coefficient to vary depending on the
remaining time period until expiration. As the options’ expiration day may induce a
17The first week of the month is defined as the week that contains the first trading day of the month.
If the first trading day of the month is a Monday, then it will be the Monday in the first week of the month; otherwise, there is no Monday observation for the first week of the month. As Wang, Li and Erickson (1997) note, this definition ensures that the Monday of the fourth week of the month always follows the options’ expiration day (where in Japan, the Monday of the third week of the month always follows the options’ expiration day).
23
systematic pattern across the weeks, we control for this possibility by adding four
dummies, itWEEK , , which capture any weekly pattern over the month that is not
unique to the days within the week.
Table 7 reports the results of the GMM coefficients estimated from Eq. (5). In
all the markets, the interday pattern is robust to the remaining time until expiration.
The Monday coefficients are significantly larger than the other days’ coefficients, the
TBREAK coefficients are also larger than the other days’ coefficients (apart from
Japan) and the Friday coefficients are smaller than the other days’ coefficients.
<< Insert Table 7 >>
In addition, the four Mondays’ week coefficients, ))(( ,itt WEEKMONDAY , are
generally insignificant and in four markets the hypothesis of equal Monday
coefficients within the month is not rejected at a 1% significance level. Thus, higher
VIs on Mondays is common to all weeks and no significant pattern as regards
Mondays across the weeks is found, which indicates that this phenomenon is not
induced by the options expiration day on a particular week.
Interestingly, the third and the fourth week coefficients are negative and most
of them are significant, where the differences across the week coefficients are highly
significant (see the last column in the table). Thus, it seems that the shift to a new
options series and the expiration of the options systematically affect the VIs.
However, this week-of-the-month bias does not change the main results regarding the
interday pattern in the VIs.
The results in Table 7 show that the interday pattern in implied volatility is
robust to the options’ expiration day. However, the VIs are timely interpolated to
reflect the volatility over a time period of 30 calendar days,18 a procedure which may
induce hidden biases that affect the interday results. To univocally determine that the
results are robust to the options’ expiration day, as well as to the VIs’ time
interpolation, Tests 1, 2 and 3 in Table 8 report the results of Eq. (1), where the
dependent variable is one of the Vis, which measures the floating six-month implied
18The VIX time interpolation formula, for example, is as follows:
)/()/()()/()(100 303653022230
211 121122
NNNNNNTNNNNTVIX TTTTTT ,
where T1 and T2 are the remaining time periods to expiration (in annual terms calculated in resolution of minutes) corresponding to the two options series, whose expiration time period is closer to 30 days;
21 and 2
2 are the volatilities of the two options series as derived from their prices; and TN is the
24
volatility: the VSMI6M, VSTOXX6M and VDAX-NEW6M; “floating” means that
the VI corresponds to a floating period without time interpolation.
<< Insert Table 8 >>
Like with the regular, fixed 30-day VIs, the Monday and TBREAK
coefficients corresponding to the floating six-month VIs are larger than the other
coefficients and the Friday coefficients are smaller, where the differences are highly
significant (see the last column). Thus, a similar interday pattern also exists in regard
to six-month VIs, which are not timely interpolated; in addition, they are substantially
less sensitive to the remaining time until expiration and to a one-week time shift from
Monday to Friday, which is only a fraction of the remaining time until expiration.
As expected, the differences across the days’ coefficients in this case are
smaller than those corresponding to the 30-day VIs because the risk implied by
private information accumulated over the two-day trading break is expected to be
smaller, on average, for six-month expected volatility than for 30-day expected
volatility. To show that the smaller coefficients are not related to the time
interpolation, but rather to the longer expected volatility period, in Test 4 the
dependent variable is the VXV, which measures the fixed, rather than the floating,
three-month expectation volatility in the U.S. market. Like with the other VIs, the
interday pattern is significant and the coefficients are also smaller than those
corresponding to the 30-days VIs. Thus, although this index is timely interpolated,
like the other long-term VIs, it reveals a smaller in magnitude, yet significant,
interday pattern.
5.2. Rejecting holidays as a potential explanation
As the VIX reflects the volatility over 30 calendar days and, as according to
French and Roll (1986), price volatility is lower during non-trading days, any
systematic pattern in the number of trading days within a rolling 30-calendar-day
window may induce a systematic pattern in the VIX. Holidays create systematic
patterns in the number of trading days; this is particularly important because many
U.S. holidays fall on Monday. Therefore, Test 5 includes two holiday variables that
explore whether the reduced number of trading days due to holidays affects the VI.
number of minutes in time period T. Note that when one series expires, this formula is used to extrapolate the VIX from two series whose expiration time period is longer than 30 days.
25
The first variable is a dummy for the last trading day before holidays, which
tests whether there is a holiday effect similar to the holiday effect in returns, in which
returns are relatively high on the last trading day before the holiday (Lakonishok and
Smidt, 1988; and Kim and Park, 1994). The second variable is a dummy for all of the
days within the month before the holiday, i.e. the days followed by an unusually small
number of trading days. As can be seen, the interday pattern in the VIX is robust to
holidays, as the results are similar to those obtained in the previous tests. Thus, a
varying number of trading days due to holidays does not account for the interday
pattern. This result, together with the fact that the interday pattern is also observed in
other markets characterized by different holidays, rule out holidays as a possible
explanation for the interday pattern in implied volatility.
Interestingly, the coefficient corresponding to the pre-holiday month is
negative and highly significant. This result suggests that the market recognizes the
lower volatility during non-trading days in comparison to that during trading days
(French and Roll, 1986). Hence, a smaller number of trading days within a forward-
looking 30-calendar-day induces a lower expected volatility.
5.3. Rejecting actual risk and economic fundamentals as a possible explanation
So far, we have found that the interday pattern in implied volatility does not
exist in the GARCH price volatility and is also robust to market returns. Thus, if one
assumes that price volatility and market returns account for economic fundamentals
relevant to implied volatility, then economic fundamentals do not account for the
observed interday pattern in the VI. However, while the VI reflects the volatility
expectation for the next 30 calendar days, which may include a varied number of
trading days, price volatility is calculated from trading-day realized returns. This
distinction between calendar and trading days may reduce the ability of price
volatility to timely reflect a daily resolution of the economic fundamentals relevant to
the VI. Moreover, so far we have employed price volatility from realized returns
known on day t (the GARCH method). One may reasonably argue that because the VI
measures volatility expectation, the future actual price volatility is more relevant in
accounting for the economic fundamentals that affect the VI. To test this argument, as
well as the possible bias due to the distinction between calendar and trading days, we
also measure actual volatility directly from realized returns while relying on both past
returns (the ex-post method) and future returns (the ex-ante method).
26
To calculate the forward-looking 30-calendar-day future volatility directly
from realized returns, we use the following well-known formula (e.g., Bakshi and
Kapadia, 2003):
N
ttt RR
NVOL
1
2)(252100 , (6)
where Rt is the return on the S&P 500 Index on day t; R is the mean return; and N is
the precise number of trading days in a forward-looking 30-calendar-day window.
Similarly, to calculate the past volatility, the parameter t in Eq. (6) runs from Nt
to 0t , where N is the precise number of trading days in a backward-looking 30-
calendar-day window. The division by N should, in principle, correct the bias due to
the changing number of trading days within the 30-calendar-day window.19
Tests 6 and 7 in Panel C of Table 8 report the results of Eq. (1) where either
the GARCH volatility, or the past and future price volatility calculated by Eq. (6), are
also included as control variables, respectively. In line with the results in Table 2, the
interday pattern in the VI is robust to past and future price volatilities; hence, this
pattern is not explained by economic fundamentals which are realized in past or future
actual price volatilities.
5.4 Other robustness tests
The New-VIX methodology may suffer from hidden biases of which we are
not aware. A first indication that such possible biases do not account for the interday
patterns is the fact that the patterns also exists in the VSMI, VSTOXX and VDAX-
NEW, whose methodologies differ from those of the other indexes (see Table 1).
To further refine the analysis, the dependent variable in Tests 8 and 9 is one of
the VIs whose methodology is different from the NEW-VIX methodology: the CSFI’s
novel methodology CSFI-VXJ (Japan), and the Deutsche Börse’s old methodology
VDAX (Germany).20 As can be seen, the two VIs reveal significant interday patterns.
Thus, the New-VIX methodology does not induce the interday pattern, as these
19In unreported tests, we also calculated the forward-looking 21-trading-day past and future
volatilities, by forcing 21N in Eq. (6). As the results in this case are very similar to those obtained with the forward-30-calendar-day volatilities, these tests are not reported. Note that to obtain the annualized volatility, the daily volatility is multiplied by 252, which is consistent with the definition over trading days. Multiplying the daily standard deviation by 365 calendar days (which is common in the industry) has only a constant affect, which does not change the interday pattern results.
20The CBOE’s old methodology VXO (U.S.) is not included in the analysis as it is calculated for 21 trading days; hence, each observation corresponds to a varied number of calendar days. Therefore, a comparison across the weekdays is biased. This is one of the reasons why the CBOE replaced it with the new VIX.
27
indices are based on different methodologies. The VDAX results also reinforce the
results in Table 7, which show that the interday pattern is not induced either by the
VIs’ time interpolation or by the options’ expiration day, as the VDAX employs a
different time interpolation procedure corresponding to market volatility over 45
calendar days.
While we find interday patterns in eight VIs that correspond to different stock
indices, these indices share one thing in common: they are all major indices that
include stocks from large firms and diversified industries. To univocally determine
that the patterns are not unique to major stock indices, in Tests 10, 11 and 12 the
dependent variable is one of the VIs whose option underline stock index either
belongs to particular industries (Dow Jones Average Industrial and NASDAQ 100) or
includes stocks from relatively small firms (Russell 2000). As can be seen, the
interday patterns are highly significant in all tests, suggesting that they are not related
to the stock index characteristics.
6. Concluding Remarks
Several theoretical models suggest that various distinct groups of investors are
involved in trading risky assets. The main distinction between the various traders
corresponds to the available level of information.
Informed traders have the advantage over uninformed liquidity traders; hence,
uninformed liquidity traders’ perceived risk increases with the accumulation of
private information by the informed traders. These asymmetric information models
predict several results regarding intertemporal behavior of stock price volatility and
volume, depending on the investment strategies adopted by the various parties.
Generally speaking, these theoretical predictions have been empirically confirmed by
employing data corresponding to the stock market.
In this paper, we analyze the intertemporal behavior of the implied volatility
and volume of trade in options, covering eight international markets. The implied
volatility reflects the perceived risk by traders, including the increase in risk, due to
the knowledge that there are informed investors who can take advantage of their
private information. We find that there are systematic intraday and interday patterns
in volume of trade in options and in the options’ implied volatility. The main result is
that implied volatility significantly increases, while the volume of trade significantly
decreases after weekend and holiday trading breaks. Furthermore, the longer the
28
trading break the sharper these phenomena, presumably because more private
information is accumulated, which creates a trade with asymmetrical information.
Uninformed liquidity traders who have the discretion to postpone their trade
activities are the main reason for the observed decrease in volume of trade in options
after a trading break. Some private information is revealed during trading hours,
which induces a decrease in the closing implied volatility, relative to the opening
implied volatility as measured on the first two days after the trading break.
We find that the investment strategies adopted by the various parties are
associated with the quality of the revealed public information during trading hours.
The higher the quality of information revealed during the trade the larger the
motivation of uninformed traders to postpone their trade while waiting for this
information. The introduction of trade in futures on the VIX in 2004 has mitigated the
observed phenomena, probably because liquidity traders use this instrument to hedge
their risk, while informed traders use it to expedite the exploitation process of their
private information.
Finally, the results are robust to economic fundamentals, which may account
for the observed phenomena, the fact that the implied volatility in both of the two
days under comparison include many overlapping days, the various methods of
calculating the implied volatility, the options’ expiration day, the various underline
assets of the options, the various international option markets covered in this study,
and other possible mechanical biases.
29
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31
Figure 1a
Figure 1b
Figure 1c
Figure 1. Average VIX, volume and actual price volatility as a function of the day of the week The figures present the average VIX at market opening and market closing (Figure 1a), volume of traded index options (Figure 1b), and index price volatility (Figure 1c) as a function of the day of the week. The VIX is the CBOE implied volatility index corresponding to options written on the S&P 500 Index. Volume is the number of index options traded in the CBOE. Index price volatility is a GARCH(1,2) model standard deviation corresponding to the S&P 500 Index. The data on the VIX and volatility covers the period of 1992–2010, while the data on volume covers the period of October 2003 – 2010. For comparison purposes, the left-hand y-axis in Figures 1a and 1c is scaled to be the same (60 basis points). The y-axis on the right-hand side translates the values on the left-hand side into a percentage deviation from the all-day mean.
32
Figure 2a
Figure 2b
Figure 2. Average volatility index and price volatility in eight markets The figure presents the average volatility index (Figure 1a) and the average index price volatility (Figure 2b) in eight markets as a function of the day of the week. The volatility indices are the VIX (U.S.), VAEX (Netherlands), VCAC (France), VFTSE (U.K.), VXJ (Japan), VSMI (Switzerland), VSTOXX (Eurozone), and the VDAX-New (Germany). Index price volatility is calculated as a GARCH (1,2) model standard deviation on the relevant stock index returns. The first year’s data ranges from 1990 to 2000, depending on the index (see Table 1), and the last year reported is 2010. For illustration purposes, the y-axis values are centered by subtracting the relevant index all-day mean from all values.
33
Table 1: The various volatility indices The table reports the descriptive statistic of the volatility indices employed in this study. Panel A corresponds to the eight markets’ main volatility indices, while Panel B corresponds to the alternative indices which employ different methodologies.
A. International volatility indices
Index Name: VIX VAEX VCAC VFTSE VXJ VSMI VSTOXX VDAX-NEW
The underline index and market
S&P 500 (U.S.)
AEX (Netherlands)
CAC 40 (France)
FTSE 100 (U.K.)
Nikkei 225 (Japan)
SMI (Switzerland)
EURO STOXX 50 (Eurozone)
DAX 30 (Germany)
Calculated by CBOE Euronext Euronext Euronext CSFI – Osaka U. SIX Swiss STOXX Ltd. Deutsche
Börse
Methodology New VIX (model-free) Deutsche Börse methodology - based on the New VIX methodology
Options used for calculations
The two nearest-term to 30-day expiration series, wide range of strike prices.
The two sub-indices closest to the 30-day expiration (based on nearest-term to 30-day expiration series).
Volatility period Fixed 30 calendar days
Last trading day Third Friday of the month Second
Friday of the month
Third Friday of the month
Starting year 1990 2000 2000 2000 1998 1999 1999 1992 Number of observations 5,295 2,814 2,811 2,801 3,194 3,041 3,069 4,808 Average 20.40 25.49 24.42 21.80 26.81 20.83 26.07 23.44 Standard deviation 8.24 11.68 9.84 9.57 9.35 8.99 10.47 10.16 Maximum 80.86 81.22 78.05 78.69 91.45 84.90 87.51 85.12 Minimum 9.31 10.12 9.24 9.10 11.52 9.24 11.6 9.35
B. Alternative volatility indices
U.S. indexes
(for stock index tests) Alternative indexes
(for methodology tests) Long-term indexes
(for time interpolation tests)
Index Name: VXD VXN RVX VDAX VXJ-CSFI VXV/VSMI6M/VSTOXX6M
/ VDAX-NEW6M
The underline index and market
Dow Jones Industrial Average
NASDAQ 100 Russell 2000
DAX 30 (Germany)
Nikkei 225 (Japan)
Same as VIX /
VSMI / VSTOXX /
VDAX-NEW
Calculated by CBOE Deutsche Börse
CSFI - Osaka University
Methodology Same as VIX Black-Scholes model
CSFI methodology
Options used for calculations Same as VIX
8 series near-the-money
Same as VXJ Nearest-term to 6 months (3 months for VXV), wide
range of strike prices
Volatility period Same as VIX Fixed 45 calendar
days Same as VXJ Floating 6 months (Fixed 3
months for VXV)
Last trading day Same as VIX Same as VXJ Same as VIX Starting year 1998 2001 2004 1998 1998 VXV from 2008, the others
same as VSMI, VSTOXX, and VDAX-NEW (the latter without 5/2005–10/2006)
Number of observations 3,171 2,495 1,763 3,303 3,194 Average 21.35 30.19 27.34 24.41 26.52 Standard deviation 8.50 14.78 11.24 9.41 9.78 Maximum 74.60 74.60 87.62 74.00 97.23 Minimum 9.28 15.00 14.44 10.98 10.97
34
Table 2: Test for the trading-break implied volatility hypothesis
The table reports the results of the following EGARCH-t model: ti
itii
itii
ititi
itit RRVTBREAKDAYV
22
0
2,5
22
0,4
12
1,32
5
1,,1 ,
ttt σzε ,
)log(')log( 2111
2 tttt σβγzzαωσ ,
where tV is the opening or closing VIX, or the change in the VIX overnight or during trading hours on day t (Tests 7 and 8); )5...1(, iDAY it are dummies corresponding to the weekdays; tTBREAK is a dummy corresponding to days other than Monday after non-trading days (or alternatively dummies corresponding to
one- two- and more than two-day trading breaks); tR is the percentage rate of return on the relevant stock index; t , tz and t are the innovation, standardized innovation, and the conditional standard deviation; and 11' tzEαωω is the conditional standard deviation constant term. The innovations follow the Student-t distribution. The closing and opening VIX data cover the period 1990–2010 and 1992–2010, respectively. Each line in the table reports the regression coefficients, while the t-values are reported in the line below (in brackets). Robust standard errors are obtained by Bollerslev-Wooldridge Quasi-Maximum Likelihood Estimates (QMLE). One and two asterisks indicate a two-tail test significance level of 5% and 1%, respectively. Day of the week variables Trading break variables EGARCH variables
Table 3: Test for the trading-break implied volatility hypothesis with non-overlapping daily VIX The table reports the results of the following EGARCH-t model:
ti
itii
ititi
itit εRRTBREAKDAYV
22
0
2,4
22
0,32
5
1,,1 ,
ttt σzε ,
)log(')log( 2111
2 tttt σβγzzαωσ ,
where Njt
Nt t VIXVIXV open open is the opening VIX on day t normalized by the weekly mean less the opening VIX on day t+j (j=1,…,4) normalized by the weekly mean;
)5...1(, iDAY it are dummies corresponding to the weekdays; tR is the percentage rate of return on the relevant stock index; t , tz and t are the innovation, standardized
innovation, and the conditional standard deviation; and 11' tzEαωω is the conditional standard deviation constant term. The innovations follow the Student-t distribution. Each The data covers the period of 1992–2010. line in the table reports the regression coefficients, while the t-values are reported in the line below (in brackets). Robust standard errors are obtained by Bollerslev-Wooldridge Quasi-Maximum Likelihood Estimates (QMLE). One and two asterisks indicate a two-tail test significance level of 5% and 1%, respectively.
Table 4: Tests for the trading-break volume and the quality of public information hypotheses The table reports the GMM estimate coefficients of the following system of equations:
jti
Njitjitj
iitji
Njt VOLUMETBREAKDAYVOLUME ,
12
1,,,3,2
5
1,,,1,
,
where )4...1(, jVOLUME Njt is the daily number of traded call or put indices’ options or equity (individual
stocks) options in the CBOE, normalized by all-day volume mean on day t; tTBREAK is a dummy corresponding to days other than Monday after non-trading days; and )5...1(, iDAY it are dummies corresponding to the weekdays. The data covers the period from November 2003 to 2010. The GMM is run with Bartlett kernel and Newey and West (1987, 1994) heteroskedasticity and autocorrelation (HAC) consistent standard errors with 7 lags, which corresponds to the automatic bandwidth parameter. Each line in the table reports the regression coefficients, while the t-values are reported in the line below (in brackets). One and two asterisks indicate a two-tail test significance level of 5% and 1%, respectively.
TBREAK and Monday coefficients are equal within two types of options
13.2 p=0.0105
37
Table 5: Tests for the market efficiency hypothesis The table reports the results of the following EGARCH-t model:
ti
itii
itii
itittti
ititi
itit RRVIXFUTTBREAKFUTDAYTBREAKDAYVIX
22
0
2,7
22
0,6
12
1,54
5
1,,32
5
1,,1 )()( ,
ttt σzε ,
)log(')log( 2111
2 tttt σβγzzαωσ ,
where tVIX is the VIX on day t; )5...1(, iDAY it are dummies corresponding to the weekdays; tTBREAK is a dummy corresponding to days other than Monday after non-trading days; tFUT is a dummy corresponding to the time period during which futures on the VIX were traded (April 2004 – 2010); tR is the percentage
rate of return on the relevant stock index; t , tz and t are the innovation, standardized innovation, and the conditional standard deviation; and 11' tzEαωω is the conditional standard deviation constant term. The innovations follow the Student-t distribution. Each line in the table reports the regression coefficients, while the t-values are reported in the line below (in brackets). Robust standard errors are obtained by Bollerslev-Wooldridge Quasi-Maximum Likelihood Estimates (QMLE). One and two asterisks indicate a two-tail test significance level of 5% and 1%, respectively.
Dependent variable and model
Day of the week variables Post-non-Monday trading-
break
Since the introduction of futures in 4/2004 EGARCH variables Log- likelihood
Table 6: The international evidence The table reports the GMM estimate coefficients of the following system of equations:
jti
jitjijtjji
itjijt VTBREAKDAYV ,12
1,,,3,,,2
5
1,,,1,
,
where jtV , in Panel A is the volatility index and in Panel B the GARCH(1,2) price volatility in market j
( 8...1j ) on day t; )5...1(, iDAY it are dummies corresponding to the weekdays; and jtTBREAK , are dummies corresponding to days other than Monday after non-trading days. The data covers the period from 2000 to 2010. The GMM is run with Bartlett kernel and Newey and West (1987, 1994) heteroskedasticity and autocorrelation (HAC) consistent standard errors with 9 lags, which corresponds to the automatic bandwidth parameter. Each line in the table reports the regression coefficients, while the t-values are reported in the line below (in brackets).
Table 7: Tests for the options’ expiration day The table reports the GMM estimate coefficients of the following system of equations:
jti
jitjii
itjii
ittjijtji
itjijt VWEEKWEEKMONDAYTBREAKDAYV ,12
1,,,5
4
1,,,4
4
1,,,3,,2
5
1,,,1, ))((
,
where jtV , is the volatility index in market j ( 8...1j ) on day t; )5...1(, iDAY it are dummies corresponding to the weekdays; jtTBREAK , are dummies
corresponding to days other than Monday after non-trading days; ))(( ,itt WEEKMONDAY are dummies corresponding to Mondays as a function of the week of the month excluding the fifth Monday (the first trading day in the fourth week is the first trading day after the expiration date except for Japan); and )4...1(, iWEEK it are week of the month dummies excluding the fifth week. The data covers the period from 2000 to 2010. The GMM is run with Bartlett kernel and Newey and West (1987, 1994) heteroskedasticity and autocorrelation (HAC) consistent standard errors with 9 lags, which corresponds to the automatic bandwidth parameter. Each line in the table reports the regression coefficients, while the t-values are reported in the line below (in brackets). One and two asterisks indicate a two-tail test significance level of 5% and 1%, respectively.
Day of the week dummy variables Post-non-Monday trading-
Table 8: Tests for technical and methodological biases The table reports the results of the following EGARCH-t model running with alternate control variables:
ti
itii
itii
itii
tiii
ititi
itit RRVVOLHOLIDAYTBREAKDAYV
22 2,7
22
0,6
12
1,5
2
1,,4
2
1,,32
5
1,,1 ,
ttt σzε ,
)log(')log( 2111
2 tttt σβγzzαωσ ,
where tV is the volatility index on day t; )5...1(, iDAY it are dummies corresponding to the weekdays; tTBREAK is a dummy corresponding to days other than Monday after non-trading days; )2,1(, iHOLIDAYS it are dummies corresponding to one-day and a full month before holidays; )2,1(, iVOL it is the price volatility
estimated by a GARCH(1,2) model, or directly from either past or future realized returns; tR is the percentage rate of return on the relevant stock index; t , tz and
t are the innovation, standardized innovation, and the conditional standard deviation; and 11' tzEαωω is the conditional standard deviation constant term. The innovations follow the Student-t distribution. Each line in the table reports the regression coefficients, while the t-values are reported in the line below (in brackets). Robust standard errors are obtained by Bollerslev-Wooldridge Quasi-Maximum Likelihood Estimates (QMLE). One and two asterisks indicate a two-tail test significance level of 5% and 1%, respectively.
41
Day of the week variables Other explanatory variables EGARCH variables
Dependent variable Mon. Tues. Wed. Thu. Fri.
Post-non-Monday
trading-break
One-day before holiday
All-month before holiday ω' α β γ
Log- likelihood ratio
(equal days)
A. Tests for time interpolation and option expiration date biases 1. VSMI6M (Switzerland)
In this Appendix, we determine the model that best fits the VIs’ daily data.
First, we use the Augmented Dickey–Fuller (ADF) test (Dickey and Fuller, 1979; and
Said and Dickey, 1984) to check for the existence of a unit root in the VIs. The test is
conducted with a constant and alternating number of lags, from zero to 22 lags, which
corresponds to a full month. Taking the number of lags which reveals the smallest test
statistic in absolute terms, the hypothesis of an existing unit root corresponding to six
VIs is rejected at a 1% significance level, and corresponding to two VIs at a 5%-level.
Having rejected the unit root hypothesis, we next search the model among
Engle’s (1982) ARCH, Bollerslevs’ (1986) GARCH and Nelson’s (1991) EGARCH
models that best fits the data. Focusing on the VIX, Table A1 compares the results of
the following alternate models,
ti
itit VIXVIX
12
1,10 ,
ttt /σεz ,
ARCH: 211
2 tt εαωσ ,
GARCH : 212
211
212
211
2 ttttt σβσβεαεαωσ ,
EGARCH: )log()()(log 2111111
2 ttttt σβγzzEzαωσ , (A1)
where tVIX is the VIX closing values on days t, and t , tz and t are the innovation,
standardized innovation and the conditional standard deviation, on day t.
<< Insert Table A1 >>
To deal with serial correlation, the models include the time-lag VIX series
( itVIX ). Specifically, we test for the first 22 days’ lags, which cover a full month of
trading days. However, as in all the models the coefficients corresponding to lags 13
to 22 are found to be insignificant, the models reported include only the first 12 lags.
Test 1 reports the results corresponding to the most parsimonious ARCH(1)
model. The conditional volatility coefficients (ω and α1) are highly significant but,
according to the Ljung–Box portmanteau test, the standardized residuals as well as the
squared standardized residuals are significantly autocorrelated at various lags. Indeed,
in Test 2, which corresponds to the GARCH(1,1) model, the GARCH coefficient (β1)
is highly significant. In addition, the Ljung–Box statistics show no significant
autocorrelations in the residuals. However, the residuals’ empirical distribution
43
reveals extremely high leptokurtosis. Indeed, the Jarque-Bera test rejects the
hypothesis that the residuals follow the normal distribution.
To deal with the high leptokurtosis, Bollerslev (1987), Baillie and Bollerslev
(1989) and Baillie and DeGennaro (1990) propose the GARCH-t model, which
assumes that the residuals follow a Student’s t-distribution. Test 3 reports the results
corresponding to the GARCH-t(1,1) model with Student’s t-distribution. The
conditional volatility coefficients are highly significant and the BIC and AIC
information criteria are substantially smaller than those obtained from the normal
GARCH model, which suggests that this model better fits the data. Indeed, the
distribution’s degrees of freedom is highly significant. In addition, in Tests 4 and 5,
which correspond to the GARCH-t(2,1) and GARCH-t(1,2) models, respectively, the
additional conditional volatility coefficients (α2 and β2) are both insignificant, the
Lagrange Multiplier test statistic for additional GARCH term is close to zero, and
including these variables increases the information criteria. Thus, adding additional
variables into the conditional volatility equation reduces model performance.
While the GARCH models assume a symmetric effect of positive and negative
shocks to volatility, many studies empirically find that a negative shock leads to a
higher conditional variance in the subsequent period than a positive shock (see, for
example, Christie, 1982; Pagan and Schwert, 1990 Nelson 1991; and Engle and Ng,
1993). To avoid the symmetrical assumption, which does not conform to the empirical
evidence, we employ Nelson’s (1991) EGARCH model. Apart from allowing for an
asymmetrical effect of positive and negative shocks, the EGARCH model also avoids
the GARCH restrictions on the autoregressive coefficients. Indeed, Pagan and
Schwert (1990), Nelson (1991), Hentschel (1995) and others find that the EGARCH
model better forecasts volatility than other models.
Test 6 employs the EGARCH-t(1,1) model. Like with the GARCH model, the
conditional volatility coefficients are highly significant and, according to the Ljung–
Box statistics, there are no significant autocorrelations in the residuals. Moreover, the
information criteria are much smaller than those corresponding to the GARCH model.
Thus, we find that the EGARCH-t(1,1) model with Student’s t-distribution
best fits the VIX time series as it handles all statistical issues which may bias the
results.21 In unreported tests, we find very similar results with all other VIs. The only
21The standardized residuals in Table 1 reveal some skewness. In unreported tests, we repeat the main tests of this study while assuming an EGARCH-t model with asymmetric Student’s t-distribution. As
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difference between the VIs is the number of significant autoregressive lag variables,
which is always smaller than 13. Therefore, in the analysis of this study, we employ
the EGARCH-t(1,1) model with 12 autoregressive lag variables.
the results in those tests are very similar to those corresponding to the Student’s t-distribution they are not reported, but are available upon request from the authors.
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Table A1: Model specification
The table reports the results of the following alternate regressions: t
iitit VIXVIX
12
1,10 ,
tttz / ,
ARCH: 211
2 tt εαωσ ,
GARCH : 212
211
212
211
2 ttttt ω ,
EGARCH: )log(')log( 211111
2 tttt σβγzzαωσ ,
where tVIX and itVIX are the VIX values on day t and it ; i ranges from 1 to 12, where in all tests
no lag larger than 12 is found to be significant; tz is the standardized innovation, t is the innovation
and t is the conditional standard deviation, all on day t; and 11' tzEαωω is the conditional
standard deviation constant term corresponding to the EGARCH model. The invocations are assumed
to follow either the standardized normal or Student-t distribution (GARCH-t model). Each line in the
table reports the regression coefficients, while the t-values are reported in the line below (in brackets).
Robust standard errors are obtained from the Bollerslev-Wooldridge Quasi-Maximum Likelihood
Estimates (QMLE). The BIC and AIC denote the Schwarz’s (Bayesian) and Akaike’s Information
Criteria. The Ljung–Box portmanteau test statistics Q and Q2 test for the hypothesis that the first 1, 10
and 20 autocorrelation coefficients corresponding to the standardized residuals and squared residuals,
respectively, are simultaneously equal to zero, where the p-values are reported in the line below (in
brackets). The last six lines report the descriptive statistic corresponding to the standardized residuals
and the Jarque-Bera statistic, which tests the hypothesis that the residuals’ skewness and the kurtosis in
excess to that corresponding to the standard normal distribution both equal zero. Finally, one and two
asterisks indicate a two-tail test significance level of 5% and 1%, respectively.