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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

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

Keywords: options market microstructure, asymmetric information, implied volatility,

market efficiency

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1. Introduction

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.

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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

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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);

U.K. VFTSE (FTSE 100); Japanese VXJ (Nikkei 225); Swiss VSMI (SMI);

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.

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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

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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.

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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.


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.

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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.

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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


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).


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.


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.


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.


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.


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

Log- likelihood ratio Dependent variable Intercept Mon. Tues. Wed. Thurs. Fri. Post-non-Monday

trading-break 1-day break

2-day break

>2-day break

ω' α β γ

Weekend and holiday trading breaks Equal weekdays

1. opent t VIXV 0.5910 0.2521 0.1997 0.2055 0.0907 0.4371 -0.1473 0.1868 0.9778 0.0804 275.2 (13.41**) (5.59**) (4.42**) (4.56**) (2.01*) (7.62**) (-11.05**) (10.54**) (267.00**) (6.55**) p<0.0001

2. close tt VIXV 0.5372 0.1037 0.1354 0.1804 -0.0034 0.6068 -0.2120 0.2493 0.9656 0.1127 603.8 (16.43**) (3.06**) (4.14**) (5.45**) (-0.11) (14.04**) (-13.62**) (12.73**) (200.40**) (8.29**) p<0.0001

Trading break by duration Equal trading breaks 3. opent t VIXV 0.1788 0.3468 0.3980 0.5387 -0.1483 0.1878 0.9775 0.0810 5.2

(4.46**) (3.24**) (15.40**) (9.79**) (-11.08**) (10.56**) (263.80**) (6.55**) P=0.2628 4. close tt VIXV 0.1054 0.2172 0.4228 0.7209 -0.1991 0.2319 0.9662 0.1209 46.7

(3.61**) (2.09*) (21.14**) (17.67**) (-13.22**) (12.31**) (207.80**) (9.01**) p<0.0001 Weekend versus non-weekend trading break 5. opent t VIXV 0.3365 0.2490 0.1976 0.2038 0.0903 0.4052 0.2451 0.4541 -0.1483 0.1878 0.9775 0.4052 2.8

(2.62**) (5.48**) (4.38**) (4.52**) (2.00**) (3.67**) (1.98*) (6.78**) (-11.08**) (10.56**) (263.80**) (3.67**) P=0.5920 6. close tt VIXV

-0.0409 0.0835 0.1349 0.1789 -0.0007 0.2490 0.5723 0.7823 -0.2068 0.2421 0.9654 0.1167 33.3 (-0.43) (2.44*) (4.16**) (5.43**) (-0.02) (2.41*) (6.17**) (15.32**) (-13.40**) (12.49**) (201.90**) (8.61**) p<0.0001

Overnight trading break and reversal Equal weekdays 7. close 1-open ttt VIXVIXV

0.4236 0.0066 -0.0042 -0.0111 -0.1353 0.0402 -0.1405 0.1877 0.9799 -0.0117 6.0 (25.89**) (0.40) (-0.26) (-0.66) (-8.40**) (0.99) (-11.19**) (11.03**) (252.30**) (-1.23**) p=0.1978

8. openclose t tt VIXVIXV -0.0336 -0.0429 -0.0227 0.0120 -0.0162 0.0682 -0.1868 0.2391 0.9678 0.0399 615.7 (-1.60) (-2.05*) (-1.11) (0.60) (-0.82) (1.37) (-12.31**) (12.16**) (172.90**) (3.50) p<0.0001

35

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


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.

Post-non-Monday

trading-break

Dependent variable Mon.

less Wed. Tue.

less Thu. Wed.

less Fri. Thu.

less Sat. Fri.

less Sun. ω' α β γ Log- likelihood

ratio

1. open 1open tt t VIXVIXV

0.1629 0.1177 0.0785 0.1543 -0.5226 0.0341 -0.3417 0.5249 0.7704 0.0035 261.0 (4.29**) (3.30**) (2.36*) (4.96**) (-17.11**) (0.38) (-14.43**) (15.33**) (30.98**) (0.14) p<0.0001

Mon.-Tue.

less Wed.-Thu. Tue.-Wed.

less Thu.-Fri. Wed.-Thu.

less Fri.-Sat. Thu.-Fri.

less Sat.-Sun. Fri.-Sat.

less Sun.-Mon.

2. open 2open tt t VIXVIXV

0.2884 0.2268 0.2683 -0.3418 -0.3944 0.0812 -0.2567 0.4156 0.8875 -0.0118 278.1 (6.98**) (5.53**) (7.02**) (-8.47**) (-9.01**) (0.80) (-13.58**) (15.44**) (59.36**) (-0.58) P<0.0001

Mon.-Tue.

less Thu.-Fri. Tue.-Wed.

less Fri.-Sat. Wed.-Thu.

less Sat.-Sun Thu.-Fri.

less Sun.-Mon. Fri.-Sat.

less Mon.-Tue.

3. close 3open ttt VIXVIXV

0.3991 0.3968 -0.2210 -0.1851 -0.2818 0.1841 -0.2215 0.3606 0.9099 -0.0007 256.4 (10.24**) (9.96**) (-5.45**) (-3.85**) (-6.35**) (1.70) (-12.59**) (13.74**) (72.81**) (-0.04) p<0.001

Mon.-Tue.

less Fri.-Sat. Tue.-Wed.

less Sat.-Sun. Wed.-Thu.

less Sun.-Mon. Thu.-Fri.

less Mon.-Tue. Fri.-Sat.

less Tue.-Wed.

4. close 4open ttt VIXVIXV

0.5441 -0.0896 -0.0771 -0.0767 -0.1278 0.0181 -0.1960 0.3111 0.9223 0.0111 181.9 (14.51**) (-2.20*) (-1.69) (-1.71) (-3.19**) (0.20) (-10.86**) (11.90**) (71.46**) (0.59) p<0.0001

36

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.

Monday Tuesday Wednesday Thursday Friday

Post-non-Monday

trading-break R2 Wald χ2

(equal days)

1. Index call options -0.0592 0.1009 0.1133 0.1438 0.0152 -0.1492 0.72 160.3 (-4.59**) (6.98**) (7.58**) (8.94**) (1.18) (-5.17**) p<0.0001

2. Index put options -0.0677 0.0895 0.1061 0.1080 0.0708 -0.1046 0.72 141.8 (-5.12**) (5.97**) (7.30**) (8.14**) (4.76**) (-3.23**) p<0.0001

3. Equity call options 0.0084 0.0844 0.0795 0.0966 0.0351 -0.0620 0.76 105.3 (0.75) (7.87**) (7.63**) (9.10**) (2.88**) (-2.61**) p<0.0001

4. Equity put options -0.0142 0.1163 0.0739 0.0990 0.0325 -0.0747 0.76 190.3 (-1.30) (10.29**) (6.96**) (9.30**) (2.69**) (-3.00**) p<0.0001

Wald χ2 Days coefficients are equal across all types of options

41.1 18.3 9.2 12.6 28.8 11.9 p<0.0001 p<0.0001 p=0.0265 p=0.0057 p<0.0001 p=0.0078

Days coefficients are equal within the put and call options

7.7 17.7 0.69 9.0 28.6 4.15 p=0.0218 p<0.0001 p=0.7074 p=0.0114 p<0.0001 p=0.1254

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


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

ratio (equal days) Mon. Tues. Wed. Thurs. Fri. Mon. Tues. Wed. Thurs. Fri.

Post-non-Monday trading-

break ω' α β γ

1. tVIX with additional dummies from 4/2004

0.6838 0.1493 0.1706 0.1893 -0.0230 0.6532 -0.2764 -0.0434 0.0361 0.0361 0.1001 -0.0675 -0.2160 0.2555 0.9657 0.1120 532.2 (18.46**) (3.86**) (4.55**) (5.04**) (-0.63) (12.01**) (-8.08**) (-1.23) (-0.83) (1.12) (3.22**) (-0.78) (-13.77**) (12.94**)(199.90**) (8.27**) p<0.0001

2. tVIX pre-future introduction (1990 -

3/2004)

0.7466 0.2087 0.2282 0.2479 0.0334 0.6659 -0.1702 0.1913 0.9609 0.1150 495.6 (15.65**) (4.20**) (4.70**) (5.06**) (0.70) (11.70**) (-9.40**) (8.68**) (140.00) (7.36**) p<0.0001

3. tVIX post-future introduction (4/2004 -

12/2010)

0.3814 0.0751 0.1333 0.1856 0.0409 0.5632 -0.2969 0.3401 0.9563 0.1421 77.5 (8.17**) (1.59) (2.91**) (4.03**) (0.91) (8.93**) (-9.05**) (8.17**) (109.60**) (5.17**) p<0.0001

38

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).

A. Volatility Index

Monday Tuesday Wednesday Thursday Friday Post-non-Monday

trading-break Wald χ2

(equal days)

VIX (U.S.) 0.4409 -0.1199 0.1174 0.0753 0.0544 1.1504 540.3 (16.40**) (-4.14**) (4.55**) (2.86**) (1.97*) (19.16**) p<0.0001

VSMI (Switzerland) 0.5560 -0.0071 0.1043 0.1465 -0.0330 0.3462 850.1 (28.95**) (-0.41) (5.93**) (8.19**) (-1.96) (8.77**) p<0.0001

VAEX (Netherlands) 0.7045 0.1459 0.2371 0.0315 -0.0878 0.2870 743.0 (24.69**) (6.58**) (10.09**) (1.14) (-3.79**) (5.41**) p<0.0001

VDAX-NEW (Germany) 0.6589 0.0679 0.1526 0.1478 -0.1156 0.7925 765.8 (24.26**) (2.89**) (6.05**) (5.75**) (-4.73**) (15.04**) p<0.0001

VXJ (Japan) 0.7268 0.3109 0.2947 0.2352 0.0677 0.2847 313.6 (18.12**) (9.11**) (9.45**) (7.08** (2.09*) (4.74**) p<0.0001

VSTOXX (Eurozone) 0.5888 0.0446 0.2058 0.0977 -0.0121 0.5534 396.5 (20.80**) (1.77) (7.84**) (3.55**) (-0.45) (10.72**) p<0.0001

VCAC (France) 0.7045 0.1443 0.1449 0.2266 0.0083 0.3356 405.9 (24.69**) (5.97**) (5.49**) (7.92**) (0.31) (8.34**) p<0.0001

VFTSE (U.K.) 0.6037 0.0289 0.2017 0.0976 -0.0354 0.6322 580.4 (22.91**) (1.27) (8.18**) (3.96**) (-1.49) (12.85**) p<0.0001

B. GARCH volatility

Monday Tuesday Wednesday Thursday Friday Post-non-Monday

trading-break Wald χ2

(equal days)

S&P 500 (U.S.) 0.1081 0.1040 0.1638 0.1044 0.0976 -0.0640 99.4 (8.34**) (9.57**) (13.71**) (9.02**) (8.46**) (-2.87**) p<0.0001

SMI (Switzerland) 0.2376 0.3118 0.2854 0.1145 0.2813 -0.3616 411.7 (13.01**) (17.42**) (17.10**) (6.68**) (15.75** (-12.58**) p<0.0001

AEX (Netherlands) 0.1289 0.2524 0.2374 0.0068 0.2505 -0.6259 277.1 (5.77**) (12.59**) (12.36**) (0.34) (12.12**) (-10.78**) p<0.0001

DAX 30 (Germany) 0.2016 0.2480 0.2850 0.0916 0.2670 -0.6817 461.4 (8.66**) (13.47**) (14.61**) (4.64**) (13.49**) (-15.72**) p<0.0001

Nikkei 225 (Japan) 0.4265 0.3790 0.5490 0.2644 0.3297 -0.0141 137.6 (13.92**) (13.57**) (18.15**) (9.28**) (12.19**) (-0.27) p<0.0001

Europe STOXX 50 (Eurozone)

0.2022 0.2232 0.2952 0.0525 0.2566 -0.5493 310.3 (8.40**) (11.56**) (14.67**) (2.59**) (12.53**) (-10.93**) p<0.0001

CAC 40 (France) 0.2397 0.1759 0.3119 0.0769 0.2289 -0.5994 351.6 (9.57** (8.29**) (14.67**) (3.67**) (10.91**) (-12.83**) p<0.0001

FTSE 100 (U.K.) 0.1177 0.2487 0.1171 0.1468 0.1393 -0.1582 135.4 (8.41**) (16.57**) (8.56**) (10.29**) (9.63) (-5.45**) p<0.0001

39

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-

break

Monday by week of the month

The week of the month

Wald χ2

Mon. Tues. Wed. Thu. Fri.

1st 2nd 3rd 4th (post-ex. day)

1st 2nd 3rd 4th (post-ex. week)

Equal days

Equal Mondays

Equal weeks

VIX (U.S.) 0.6166 -0.0469 0.1940 0.1472 0.1351 1.1628 -0.0430 0.0048 -0.1573 -0.3184 -0.0280 -0.0383 -0.1082 -0.1661 275.2 22.1 29.0 (12.37**) (-1.16) (5.29**) (3.84**) (3.43**) (19.40**) (-0.47) (0.07) (-1.93) (-4.13**) (-0.71) (-1.11) (-2.88**) (-4.71**) p<0.0001 p<0.0001 p<0.0001 VSMI (Switzerland)

0.6327 0.0231 0.1367 0.1796 0.0042 0.3489 -0.2271 -0.0568 0.0824 -0.1429 -0.0285 -0.0014 -0.0131 -0.1221 315.7 23.9 50.0 (19.30**) (0.97) (5.63**) (6.95**) (0.17) (7.77**) (-3.69**) (-1.11) (1.41) (-2.58**) (-1.09) (-0.06) (-0.51) (-5.26**) p<0.0001 p<0.0001 p<0.0001

VAEX (Netherlands)

0.5504 0.1870 0.2734 0.0675 -0.0530 0.2480 0.1088 0.1986 0.3138 0.2755 0.0451 -0.0022 -0.0981 -0.0797 267.2 5.77 25.4 (11.55**) (6.03**) (8.71**) (1.93) (-1.62) (5.19**) (1.28) (2.75**) (3.66**) (3.27**) (1.30) (-0.07) (-3.01**) (-2.58**) p<0.0001 p=0.1236 p<0.0001

VDAX-NEW (Germany)

0.6372 0.1195 0.2032 0.2026 -0.0565 0.7785 -0.0875 0.0191 0.1761 0.1482 -0.0452 0.0025 -0.0917 -0.1280 271.2 12.5 30.3 (13.40**) (3.59**) (6.18**) (5.54**) (-1.64) (14.13**) (-1.05) (0.28) (2.26*) (1.76) (-1.25) (0.08) (-2.78**) (-4.15**) p<0.0001 p=0.0058 p<0.0001

VXJ (Japan) 0.8684 0.3787 0.3659 0.3037 0.1446 0.3036 -0.4681 -0.1323 0.1043 -0.1125 -0.0181 0.0127 -0.1284 -0.1988 138.6 23.4 50.9 (14.36**) (9.80**) (9.57**) (7.66) (3.62**) (5.03**) (-4.06**) (-1.32) (1.13) (-1.23) (-0.44) (0.39) (-3.80) (-5.88**) p<0.0001 p<0.0001 p<0.0001

VSTOXX (Eurozone)

0.5297 0.0787 0.2375 0.1314 0.0278 0.5385 -0.0031 0.0319 0.2409 0.1142 -0.0162 0.0437 -0.0338 -0.1684 122.8 7.64 61.6 (10.90**) (2.16*) (6.64**) (3.37) (0.73) (10.97**) (-0.03) (0.44) (2.72**) (1.32) (-0.40) (1.31) (-0.91) (-4.95**) p<0.0001 p=0.0541 p<0.0001

VCAC (France)

0.6414 0.1877 0.1900 0.2785 0.0637 0.3396 0.0506 0.0259 0.0796 -0.0957 -0.0828 0.0099 -0.0974 -0.0660 154.4 4.65 12.3 (13.56**) (5.36**) (5.27**) (6.97) (1.72) (6.12**) (0.64) (0.36) (0.99) (-1.16) (-2.17*) (0.28) (-2.82**) (-1.82) p<0.0001 p=0.1990 p=0.0066

VFTSE (U.K.) 0.5549 0.0810 0.2581 0.1592 0.0308 0.6224 -0.0501 0.1358 0.2371 0.0785 -0.0840 -0.0286 -0.0442 -0.1426 172.8 9.49 21.4 (13.99**) (2.39*) (7.48**) (4.56) (0.88) (12.13**) (-0.62) (2.04*) (3.01**) (1.07) (-2.40*) (-0.87) (-1.29) (-4.41**) p<0.0001 p=0.0234 p<0.0001

40

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


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)

0.2119 0.1248 0.1537 0.1549 0.1011 0.2078 -0.0742 0.2857 0.9854 0.0789 70.0 (8.37**) (4.94**) (6.12**) (6.16**) (4.01**) (4.74**) (-6.42**) (3.27**) (302.30**) (2.14*) p<0.0001

2. VSTOXX6M (Eurozone)

0.2410 0.1446 0.1474 0.1344 0.0973 -0.0263 -0.2305 0.2714 0.9714 0.0425 68.5 (8.81**) (5.36**) (5.46**) (4.93**) (3.60**) (-0.66) (-10.95**) (10.62**) (177.80**) (2.53*) p<0.0001

3. VDAX-NEW6M (Germany)

0.1084 0.0521 0.0581 0.0473 -0.0167 -0.0018 -0.1864 0.1871 0.9528 0.0438 66.7 (7.03**) (3.34**) (3.82**) (3.10**) (-1.14) (-1.20) (-13.33**) (13.21**) (194.50**) (4.66**) p<0.0001

4. VXV (U.S.) 0.3495 0.1846 0.2821 0.2677 0.1869 0.4435 -0.2682 0.3114 0.9619 0.0549 25.0 (3.39**) (1.74) (2.73**) (2.66**) (1.83) (3.67**) (-4.69**) (5.11**) (67.14**) (1.28) p<0.0001

B. Test for holiday bias

5. VIX (U.S.) 0.5827 0.1487 0.1775 0.2252 0.0443 0.5905 0.0079 -0.0665 -0.2098 0.2461 0.9653 0.1163 586.9 (17.39**) (4.29**) (5.29**) (6.64**) (1.32) (13.21**) (0.21) (-4.55**) (-13.54**) (12.62**) (201.10**) (8.44**) p<0.0001

C. Tests while controlling past and future price volatility

GARCH

volatility Past

volatility Future

volatility

6. VIX (U.S.)

0.5163 0.0821 0.1148 0.1600 -0.0250 0.6051 0.0064 -0.2104 0.2467 0.9655 0.1143 602.5 (14.15**) (2.18*) (3.15**) (4.32**) (-0.69) (13.99**) (1.23) (-13.59**) (12.68**) (200.90**) (8.43**) p<0.0001

7. VIX (U.S.) 0.5331 0.1082 0.1352 0.1778 -0.0141 0.6012 0.0090 0.0026 -0.2102 0.2454 0.9617 0.1201 600.8 (16.70**) (3.25**) (4.22**) (5.46**) (-0.44) (14.04**) (2.41*) (1.82) (-13.33**) (12.56**) (187.10**) (8.56**) p<0.0001

D. Tests for methodological biases 8. VXJ-CF (Japan) 0.8138 0.3038 0.3503 0.2934 0.0751 0.4345 -0.1974 0.2533 0.9554 0.1432 303.7 (13.30**) (4.81**) (5.56**) (4.77**) (1.22) (6.59**) (-10.00**) (9.48**) (144.60**) (8.02**) p<0.0001 9. VDAX (Germany) 0.5189 0.2119 0.1699 0.1714 0.0520 0.4699 -0.2037 0.2460 0.9828 0.0729 337.5 (12.58**) (5.03**) (4.11**) (4.10**) (1.26) (6.78**) (-11.85**) (11.76**) (274.80**) (5.05**) p<0.0001

E. Tests for stock indices’ biases

10. VXD (Dow Jones) 0.4587 0.0798 0.1630 0.1477 -0.0287 0.6106 -0.2170 0.2643 0.9683 0.0786 343.7

(11.69**) (2.03*) (4.16**) (3.85**) (-0.74) (10.07**) (-11.03**) (10.85**) (162.20**) (4.58**) p<0.0001 11. VXN (NASDAQ 100) 0.6913 0.1694 0.2502 0.2441 0.0417 0.8137 -0.2193 0.2919 0.9586 0.1058 225.1 (10.97**) (2.58**) (3.94**) (3.84**) (0.66) (8.64**) (-11.02**) (10.68**) (125.00**) (5.40**) p<0.0001 12. RVX (Russell 2000) 0.6739 0.1387 0.2301 0.2550 0.0959 0.9736 0.6739 0.1387 0.2301 0.2550 302.4

(9.07**) (1.84) (3.09**) (3.48**) (1.32) (13.52**) (9.07**) (1.84) (3.09**) (3.48**) p<0.0001

42

Appendix A: Model specification

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

44

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.

45

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


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.

46

Model: 1.

ARCH(1) (Normal dist.)

2. GARCH(1,1)

(Normal dist.)

3. GARCH(1,1) (Student-t)

4. GARCH(2,1)

(Student-t)

5. GARCH(1,2) ( Student-t)

6. EGARCH(1,1)

(Student-t.) Constant 0.3337 0.2320 0.1989 0.1989 0.1987 0.2617 (5.63**) (4.61**) (5.11**) (5.11**) (5.11) (8.46**) AR1 1.0325 0.8910 0.88873 0.8877 0.8881 0.8998 (30.87**) (51.80**) (63.09**) (62.95**) (62.76) (68.39**) AR2 -0.1270 0.0104 -0.0061 -0.0065 -0.0069 -0.0046 (-3.09**) (0.47) (-0.34) (-0.36) (-0.38) (-0.26) AR3 0.0116 -0.0037 0.0189 0.0188 0.0188 0.0211 (0.38) (-0.17) (1.06) (1.06) (1.05) (1.20) AR4 -0.0270 0.0194 0.0043 0.0045 0.0047 0.0050 (-0.76) (0.91) (0.24) (0.25) (0.27) (0.28) AR5 0.0516 0.0349 0.0485 0.0483 0.0480 0.0419 (1.36) (1.69) (2.67**) (2.67**) (2.65) (2.45*) AR6 -0.0459 -0.0481 -0.0431 -0.0430 -0.0429 -0.0393 (-1.43) (-2.47*) (-2.48**) (-2.48*) (-2.47) (-2.29*) AR7 0.0396 0.0379 0.0220 0.0220 0.0220 0.0225 (1.41) (1.87) (1.31) (1.31) (1.31) (1.31) AR8 -0.0424 -0.0085 -0.0049 -0.0048 -0.0047 -0.0087 (-1.34) (-0.41) (-0.297) (-0.29) (-0.28) (-0.52) AR9 0.0490 0.0283 0.0358 0.0354 0.0351 0.0383 (1.25) (1.23) (2.07**) (2.05) (2.03) (2.31*) AR10 0.1227 0.0497 0.0368 0.0370 0.0372 0.035 (3.59**) (2.37*) (2.17*) (2.18*) (2.19) (2.12*) AR11 -0.1475 -0.0672 -0.0518 -0.0517 -0.0516 -0.0537 (-4.98**) (-3.39**) (-3.07**) (-3.07**) (-3.07) (-3.28**) AR12 0.0627 0.0410 0.0359 0.0359 0.0359 0.0242 (2.96**) (2.85**) (3.94**) (2.95**) (2.95) (1.98*)

GARCH variables ω 0.9810 0.0485 0.0273 0.0292 0.0251 (16.16**) (3.05**) (4.51**) (4.49**) (4.00**) ω' -0.0725 (-7.15**) α1 0.6935 0.1380 0.1504 0.1631 0.1738 0.0831 (8.86**) (6.13**) (8.29**) (7.45**) (6.04**) (5.85**) α2 0.7417 -0.0317 (6.58**) (-0.96) β1 0.8397 0.8508 0.0967 0.8593 0.9779 (34.50**) (53.31**) (0.95) (46.00**) (384.4**) β 2 0.1615 γ (14.44**)

Student-t distribution parameter Degrees of freedom 4.3789 4.3800 4.3822 4.7581

(15.65**) (15.64**) (15.64**) (17.25**)

Information criteria BIC 17680.9 16381.0 15668.3 15676.3 15675.8 15509.1 AIC 17575.7 16275.8 15556.6 15558.0 15557.5 15390.8

Residual diagnostics Ljung-Box Q(1,10,20) 15**, 72**, 87** 0.6, 3.3, 11 0.7, 3.2, 12 0.6, 3.2, 12 0.6, 3.26, 12 0.17, 2.2, 11 p-value (0.00, 0.00, 0.00) (0.44, 0.97, 0.95) (0.41, 0.98, 0.91) (0.43, 0.98, 0.91) (0.44, 0.98, 0.91) (0.69, 1.00, 0.93) Ljung-Box Q2(1,10,20) 7.6**, 422**, 667** 0.0, 7.0, 10 0.1, 4.6, 9.2 0.1, 4.5, 9.1 0.2, 4.4, 9.0 0,1, 4.2, 8.5 p-value (0.00, 0.00, 0.00) (0.95, 0.73, 0.95) (0.81, 0.91, 0.98) (0.71, 0.92, 0.98) (0.63, 0.93, 0.98) (0.77, 0.94, 0.99) Mean 0.0546 0.0497 0.1080 0.1082 0.1083 0.0938 Standard Deviation 0.9986 0.9992 1.0029 1.0028 1.0028 1.0188 Skewness 1.3147 1.2930 1.4945 1.4928 1.4917 1.6805 Excess Kurtosis 6.4052 7.0313 9.6930 9.6692 9.6556 11.619 Jarque-Bera test 10552.8 12354.7 22648.4 22542.4 22481.4 32205.3 p-value (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)

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