Asymmetric Information, Perceived Risk and Trading Patterns: The Options Market

Transcription

1 Asymmeric Informaion, Perceived Risk and Trading Paerns: The Opions Marke Guy Kaplanski * Haim Levy** March 01 * Bar-Ilan Universiy, Israel, Tel: , Fax: , [email protected]. ** The Hebrew Universiy of Jerusalem, 91905, and he Academic Cener of Law and Business, Israel. Tel: Fax: [email protected] (Corresponding auhor). 0

2 Asymmeric Informaion, Perceived Risk and Trading Paerns: The Opions Marke Absrac Asymmeric informaion models are esed using opions implied volailiy and volume of rade in eigh inernaional markes. We explore he relaions beween he rading break ime duraion, he qualiy of public informaion, he discreion of opions liquidiy raders o pospone heir rades, and he inerday and inraday implied volailiy and volume of rade in opions. Alhough asymmeric informaion is generally relaed o he underline asse, we find ha i srongly affecs he invesmen sraegies adoped by he various opions raders which, in urn, affec implied volailiy and opions volume of rade. The curren analysis sheds new ligh on hose sraegies and heir inerrelaions wih he sock marke. The inroducion of fuures on implied volailiy in 004 is also explored. JEL Classificaion Numbers: D8, G1, G14 Keywords: opions marke microsrucure, asymmeric informaion, implied volailiy, marke efficiency 0

3 1. Inroducion Sock marke sudies provide compelling empirical evidence for sysemaic inerday and inraday paerns in sock price volailiy and volume of rade. Several heoreical adverse selecion models wih asymmeric informaion have been employed o explain hese phenomena; each enails differen predicions corresponding o he ineremporal sock price and volume of rade behavior, depending on he underline assumpions. In his sudy, we focus on he opions marke by sudying he ineremporal rading paerns in eigh inernaional opions markes. While he exising heoreical models and he relevan empirical sudies mainly focus on he effec of informaion asymmery corresponding o he underline asse on he asse iself, we sudy his effec on he opions wrien on his asse. The effec of asymmeric informaion corresponding o he underline asse on opions is no rivial, as asymmeric informaion is expeced o simulaneously increase he risk and decrease he price of he underline asse. These effecs have a conradicing influence on he price of call opions, bu enhance effecs in he same direcion in regard o he price of pu opions. Employing daa on opions wrien on various asses, we es several hypoheses ha shed ligh on he alernae asymmeric informaion models suggesed in he lieraure for he sock marke, and on he implied invesmen sraegies adoped by he various opions raders. By incorporaing implied volailiy ino he analysis, we add anoher dimension o he exising models: ha of invesor perceived risk (for, say, he nex 30 calendar days) which, o he bes of our knowledge, has no been previously explored in his conex. We analyze he perceived risk relaions of uninformed opions liquidiy raders, he flow of public informaion ha resolves he informaion asymmery, and he invesmen sraegies employed by he various paries which, in urn, sysemaically affec he inerday and inraday opions volume of rade. Does he opions marke reveal inerday and inraday rade paerns similar o hose observed in he sock marke? How do uninformed opions raders proec hemselves agains raders who possess privae informaion? Are he rading sraegies adoped by various raders affeced by he qualiy of public informaion? Do he fuures on he U.S. volailiy index (he VIX), inroduced in 004, miigae he risk induced by informaion asymmery? Are he empirical resuls unique o he U.S. marke? The aim of his sudy is o answer hese and oher relaed quesions. To achieve his goal, we use Foser and Viswanahan s (1990) heoreical model as a 1

4 springboard for posulaing he hypoheses regarding he opions marke. They sugges a rich heoreical model wih several predicions as regards sock marke behavior. As heir alernae se of assumpions implies differen predicions, we empirically examine heir (and oher suggesed models ) various ses of assumpions and infer which se of assumpions (namely, he heoreical model) bes conforms o he opions marke. As a rading break is a major cause of asymmeric informaion, we explore he relaions beween implied volailiy and volume of rade in opions, and he radingbreak ime duraion during which privae informaion is accumulaed. Thus, we go beyond he weekend and also es holiday and overnigh rading breaks, as well as he reversal during rading hours a reversal which occurs due o he revealing process of privae informaion hrough rade. These relaions shed ligh on he qualiy of public informaion, he way uninformed opions raders proec hemselves agains privae informaion, and wheher hey have he discreion o pospone heir rade aciviies an acion which depends on he qualiy of he public informaion. This analysis also shows when, and how quickly, privae informaion is revealed. As implied volailiies corresponding o subsequen days include overlapping days, we sugges ess which measure he daily differences in implied volailiy, ne of he overlapping days effec. Finally, he inroducion of fuures on implied volailiy in 004 enables us o separaely explore he role of privae informaion before and afer 004. This analysis indicaes ha he various opions raders use he fuures marke o eiher proec hemselves or exploi privae informaion, hereby miigaing he asymmeric informaion perceived risk which, in urn, improves marke efficiency. In a non-rigorous manner, Figure 1 illusraes he highligh of his sudy wih he U.S. VIX, which corresponds o he S&P 500 Index s implied volailiy (a similar figure is obained wih oher markes indices). The figure presens he average VIX a marke opening and marke closing imes, he average rading volume in he CBOE corresponding o index opions, and he acual price volailiy calculaed from realized reurns on he S&P 500 Index, as a funcion of he day of he week. << Inser Figure 1 >> As can be seen from he figure, he average VIX and volume reveal sysemaic paerns across he weekdays. The average VIX in Figure 1a is highes on Monday; i decreases during he week, where he opening VIX is higher han he closing VIX, especially a he beginning of he week. In conras, he average rading volume,

5 presened in Figure 1b, is a is lowes on Monday; i increases unil Thursday and hen decreases on Friday. Finally, he average realized price volailiy (as measured by he GARCH model), presened in Figure 1c, is almos he same, wih only minor nonmonoonic changes across he days. Figure 1 reveals inverse paerns in implied volailiy and rading volume of opions across he weekdays. These paerns are no induced by acual price volailiy, as no paricular paern is observed in his variable. The more rigorous saisical analysis reveals ha hese paerns are no relaed o he day of he week, bu raher o he weekend rading-break. This rading-break effec is neiher due o changes in economic fundamenals nor o mechanical and saisical biases relaed o he volailiy index calculaion mehod. I is raher nicely explained by he exising heoreical models dealing wih privae informaion accumulaed during rading breaks, and he invesmen sraegies employed by he various opions raders in he presence of asymmeric informaion. The privae informaion accumulaed during he rading break is anoher risk componen ha uninformed raders face; hence, his facor is also aken ino accoun when esablishing heir invesmen sraegies. Moreover, we find ha he rading-break effec is a global phenomenon which is no unique o he U.S. marke. Finally, we show ha he opions marke resuls are consisen wih he resuls repored by French and Roll (1986), corresponding o acual sock price volailiy during rading and non-rading days. The srucure of his paper is as follows: Secion presens he exising heoreical models and he empirical evidence regarding inemporal sock price volailiy and rading volume, and posis he hypoheses ha are relevan o he opions marke. Secion 3 presens he daa and mehodology. Secion 4 repors he empirical resuls. Secion 5 repors he resuls corresponding o alernaive models and robusness checks, while Secion 6 concludes. Some echnical, albei imporan, ess are relegaed o he Appendix.. Exising heory, he empirical evidence and hypoheses of his sudy The discovery of inemporal sysemaic paerns in socks realized price volailiy goes back o Fama (1965), Granger and Morgensern (1970), Chrisie (1981) and French and Roll (1986), all of whom find ha sock price volailiy is significanly higher during rading days han during non-rading days. French and Roll (1986) provide compelling evidence showing ha his phenomenon is due o privae 3

6 informaion ha affecs prices when informed raders rade (see also Barclay, Lizenberger and Warner, 1990; and Soll and Whaley, 1990). Sock price volailiy also reveals an inraday U-shape. Wood, McInish and Ord (1985) and Harris (1986) find ha volailiy is higher a marke opening and closing imes han during he middle of he day. Amihud and Mendelson (1987), Lockwood and Linn (1990), Foser and Viswanahan (1993), and Soll and Whaley (1990) show ha his U-shape is no symmeric, as volailiy is larger a marke opening imes han a marke closing imes. Sock rading volume also reveals sysemaic inerday and inraday paerns; however, here are conflicing views and empirical evidence regarding he correlaion beween sock price volailiy and volume. Jain and Joh (1988) find ha sock rading volume is lower on Mondays and Fridays han on oher days, which implies a negaive correlaion beween volume and volailiy. On he oher hand, during rading hours, Soll and Whaley (1990) find ha higher volailiy is accompanied by high volume, indicaing a posiive correlaion. Foser and Viswanahan (1993) find ha for he more acively raded socks, volume and volailiy are posiively correlaed as regards inraday aciviy, bu negaively correlaed as regards inerday aciviy. Several heoreical models are employed o explain hese sock marke empirical resuls. Kyle (1985) shows ha in a marke wih hree ypes of raders informed raders, noise raders, and compeiive marke makers privae informaion is gradually incorporaed ino prices. Glosen and Milgrom (1985) show ha adverse selecion can accoun for he exisence of he bid-ask spread and ha ransacion prices are informaive in he presence of adverse selecion; hus, spreads end o decline wih rade. Admai and Pfleiderer (1988) expand he model o include discreionary liquidiy raders, who can ime heir rade aciviies. These raders lead o rading concenraions during he day, which can explain he volailiy inraday asymmeric U-shape. Foser and Viswanahan (1990) sugges ha in he presence of privae informaion uniformed discreionally liquidiy raders have an incenive o pospone heir rade aciviies o oher days, while waiing for public informaion. This model explains boh he inraday and inerday paerns in sock price volailiy and volume, and he correlaions in hese paerns. The main predicions of Foser and Viswanahan s model as regards he sock marke, which also have implicaions ha relae o he opions marke, are as follows: 4

7 1. As privae informaion is received during all imes, bu revealed only during rading hours, sock price volailiy is expeced o be higher afer rading breaks, in paricular when he marke is open and privae informaion is a is highes level.. Uninformed discreionary liquidiy raders will avoid rading on days following non-rading days, in order o seer clear of he adverse high coss implied by privae informaion. Thus, sock rading volume is expeced o be lower on Mondays and afer holidays when privae informaion is high. 3. The incenive o pospone rading depends on he process by which privae informaion is revealed. I is prediced ha wih he regular release of high qualiy public informaion, here will be wo days before Friday wih concenraed rading. In conras, poor public informaion is expeced o lead o only one day (Friday) of concenraed rading each week. Of course, he marke aggregae resuls also depend on he proporion of discreionary liquidiy raders in he marke. Based on hese predicions abou he sock marke, below we posi and es several hypoheses regarding implied volailiy and volume of rade in opions. Generally, an increase in uncerainy of he uniformed liquidiy rades regarding he value of he underline asse is expeced o decrease is price, due o he increase in he required risk premium. Therefore, here are wo effecs on he opion price: The increase in uncerainy (due o he asymmeric informaion risk) increases he price of all opions, and he decrease in he underline asse price decreases he price of call opions and increases he price of pu opions. 1 Thus, while he oal effec on he opion price depends on he opion ype and he relaive magniude of he wo effecs, in boh cases he increased uncerainy regarding he underline asse is expeced o 1 Jones and Shemesh (010) show ha he rae of reurn on opions is relaively low over he weekend, a phenomenon ha is no relaed o he change in he price of he underline asse. They also show ha he oal implied volailiy decreases over he weekend in conradicion o wha is repored by French and Roll (1986), he prediced resuls given by he privae informaion and asymmeric informaion models, and he resuls repored here. There are several possible reasons for he differen resuls in he wo sudies. Firs, Jones and Shemesh focus on opions of individual socks, while we focus on opions of sock indices, e.g., he S&P 500 Index. Suppor for his possible reason for he differences is ha when hey repor some resuls on indices opions, hey obain inconclusive resuls. Oher possible sources for he differences are he differen periods covered, he differen implied volailiy measures employed (calendar versus oal implied volailiy, Model-free versus Black- Scholes), he differen mehodologies employed o measure he implied volailiy on differen days of he week, and finally, heir use of opions closing values, which overshadows he higher implied volailiy a marke open. 5

8 increase he opion implied volailiy. To explore his predicion as regards implied volailiy, we es he following hypohesis wih opions daa: H1. The rading-break implied volailiy (TBIV) hypohesis: Implied volailiy afer rading breaks is no significanly differen from ha during rading hours. The alernaive hypohesis assers ha implied volailiy is relaively higher afer rading breaks, due o higher risk as perceived by uninformed raders, a risk which decreases when public signals are received. The TBIV hypohesis has several spinoffs. Firs, i is separaely esed for weekend, holiday and overnigh rading-breaks. According o he TBIV hypohesis, implied volailiy is expeced o be higher, albei no wih he same magniude, afer all ypes of rading breaks. Second, he longer he rading break he greaer he expeced amoun of privae informaion; hence, he larger he risk perceived by he uninformed raders. Therefore, we also es wheher he higher implied volailiy is correlaed wih he rading-break ime duraion. To es wheher privae informaion is gradually revealed during rading hours, we es wheher implied volailiy decreases during rading hours. Finally, o es wheher discreionary liquidiy opions raders pospone heir rades o oher days as suggesed by Foser and Viswanahan (1990) in regard o raders in underline asses we es for possible paerns in implied volailiy across all weekdays. We now urn o he hypohesis regarding he inerday paern in rade volume. If, indeed, discreionary liquidiy raders in he opions marke pospone heir rades o oher days, hen according o Foser and Viswanahan s model, hey will decrease heir rading on Mondays and afer holidays. This leads o he following hypohesis: H. The rading-break volume (TBV) hypohesis: The opions rading volume afer non-rading days is no significanly differen from ha on oher days. The alernaive hypohesis assers ha he volume is lower afer non-rading days due o uninformed discreionary liquidiy opions raders who pospone heir rades. While he volume on Mondays is expeced o be relaively low, he exac rading paern on he oher days of he week depends on he qualiy of public informaion. In he case of regular release of accurae and high qualiy public informaion, discreionary liquidiy raders will pool heir rades ino wo days before Since he well-known Monday effec in reurns has significanly aenuaed over he las decades (Schwer, 003), he decline in he underline asse price during he period covered in his sudy probably does no reflec he increase in uncerainy. 6

9 Friday, whereas in he case of poor public informaion hey will pool heir rades on Friday. To explore his issue and he role of he qualiy of public informaion, we es he following hypohesis: H3. The qualiy of public informaion (QPI) hypohesis: The opions rading volume paern over he weekdays does no depend on wheher he underline asse is an individual sock or an index. The alernaive hypohesis assers ha he rading paern is differen for opions wrien on individual socks and indices, as accurae public informaion corresponding o indices is released more regularly, on average, han ha on individual socks. If he higher implied volailiy afer rading breaks is due o higher risk induced by privae informaion, he inroducion of fuures on he VIX in 004 has possibly served o miigae his phenomenon. This is because new insrumens are generally expeced o improve marke efficiency. In paricular, hese fuures enable raders o hedge agains privae informaion risk and also provide anoher relaively low-cos channel for informed raders o exploi heir privae informaion which, in urn, expedies he flow of privae informaion o he marke. To es wheher he fuures on implied volailiy have indeed miigaed he effec of privae informaion, we es he following hypohesis: H4.The marke efficiency (ME) hypohesis: The abiliy o rade implied volailiy in he fuures marke did no significanly change he inerday paern in implied volailiy. The alernaive hypohesis assers ha he abiliy o rade implied volailiy miigaed he inerday paerns in implied volailiy. Alhough he empirical resuls in his sudy rejec he null hypoheses presened above, here is always a possibiliy ha he observed significan phenomena are caused by economic facors or echnical biases, which are correlaed wih he predicions of he heoreical models. Therefore, we conduc several robusness ess. These ess rejec he hypoheses assering ha he paerns in he opions marke are due o he following facors: economic fundamenals which are incorporaed in acual price volailiy (where price volailiy is measured by various mehods) and in he underline asse price reurns; saisical and mehodological biases including he disincion beween rading days and calendar days corresponding o he calculaion of implied volailiy, and various numbers of rading days due o holidays; implied volailiy calculaion mehods (in paricular, he volailiy index ime inerpolaion and mehodology); he ype of opions underline asses and, mos imporanly, he opions 7

10 expiraion day. Finally, we es wheher he resuls are affeced by specific characerisics of he local markes and cross-border inefficiencies like he differen rading hours, currency effecs and biases relaed o rading mehods and he marke s various selemen procedures. 3. Daa and mehodology To measure implied volailiy, we employ he well-known volailiy indices (VIs). The VI measures he volailiy expecaion as implied by he opion prices. The daily daa of he following eigh primaries VIs and heir underline sock indexes are employed: The U.S. VIX (S&P 500); Duch VAEX (AEX); French VCAC (CAC 40); U.K. VFTSE (FTSE 100); Japanese VXJ (Nikkei 5); Swiss VSMI (SMI); Eurozone s VSTOXX (EURO STOXX 50); and he German VDAX-NEW (DAX 30). 3 Panel A in Table 1 presens he main characerisics of he eigh VIs. << Inser Table 1 >> All of he VIs were calculaed backwards ino he pas, providing us wih a leas 10 years of daily daa, wih 1 and 19 years of daa in he case of he VIX and he VDAX-NEW, respecively. The VIX, VAEX, VCAC, VFTSE and VXJ employ he New-VIX mehodology, and he VSMI, VSTOXX and VDAX-NEW are also based on his mehodology wih some modificaions. This mehodology firs adoped by he CBOE in 003, when he VIX was recalculaed backward ino he pas is based on Brien-Jones and Neuberger s (000) model-free mehodology, which esimaes volailiy expecaions by averaging he weighed prices of pu and call opions over a wide range of srike prices. To explore wheher he resuls repored in his sudy are affeced by he VIs calculaion mehod, he underline sock index or he opions ime o expiraion, we also sudy he following alernaive VIs, presened in Panel B: The VXD (Dow Jones Indusrial), VXN (NASDAQ 100), and RVX (Russell 000) are used o verify ha he resuls are general, raher han confined o a specific underline sock index. The VDAX (DAX 30) and CSFI-VXJ (Nikkei 5) are used o verify ha he resuls are 3 The daa on he U.S. VIs, he S&P 500 Index, and he opions rading volume are provided by he CBOE. The daa on he VSMI, VSTOXX, and VDAX-NEW, as well as heir alernaive indexes, are provided by he SIX Swiss Exchange, STOXX Limied Company and he Deusche Börse exchange, respecively. The daa on he Japanese indexes is provided by The Cener for he Sudy of Finance and Insurance (CSFI), Osaka Universiy. Finally, he daa on he VAEX, VCAC and he VFTSE, as well as heir alernaive indexes, are provided by he NYSE Euronex Group. 8

11 no echnically induced by he New-VIX mehodology. 4 The VSMI6M (SMI), VSTOXX6M (EURO STOXX 50) and VDAX-NEW6M (DAX 30) measure he floaing six-monh expecaion volailiy from one opions series whose expiraion day is he closes o six monhs, wihou ime inerpolaion. These VIs are used o verify ha resuls are neiher induced by he opions expiraion day nor by he VI s ime inerpolaion procedure. As for he U.S. marke, here is no six-monh floaing VI. For comparison purposes, we also repor he VXV (S&P 500), which measures he fixed hree-monh expecaion volailiy. As implied volailiy is affeced by economic fundamenals, i is also imporan o measure he VI relaive o he acual price volailiy o verify ha he observed resuls are no induced by economic fundamenals, which are accouned for in price volailiy. Therefore, we employ he daily ime series of he VI as well as he daily price volailiy. To conduc an analysis of hese ime series, one firs needs o choose he appropriae economeric model. The choice of he model is imporan because he VI s ime series incorporae several well-known economeric issues ha may bias he resuls. Firs, like he volailiy ime series, which may have a uni roo (Pagan and Schwer, 1990), he VI may also have a uni roo. Second, volailiy is serially correlaed and he VI is inherenly serially correlaed. 5 Finally, like acual volailiy, he VI may also reveal memory in response o shocks, and a correspondingly high degree of heeroskedasiciy (for he exisence of hese phenomena in volailiy, see e.g. Poerba and Summers, 1986, French, Schwer, and Sambaugh, 1987 and Schwer, 1990). To handle hese issues, our firs ask is o choose he appropriae ime series saisical model, which akes ino accoun all hese problemaic issues. Comparing he various alernae models, presened in more deail in Appendix A, we find ha he Exponenial Generalized Auoregressive Condiional Heeroskedasic(1,1) model wih Suden s -disribuion (EGARCH-) and 1 auoregressive lag variables bes handles he saisical issues menioned above. Therefore, his model is employed in he main analysis. 6 4 The VDAX employs he Deusche Börse s old mehodology, which is based on he Black-Scholes opion pricing model, near-he-money opions and corresponds o 45 calendar days, while he CSFI- VXJ employs he Cener for he Sudy of Finance and Insurance novel model-free mehodology. 5 On each day, he VI measures he expeced volailiy for he nex 30 calendar days; hence, he index values corresponding o day and day -1 include 9 common days. 6 For he advanage of he EGARCH model as regards volailiy ime series see, for example, Nelson (1991), Pagan and Schwer (1990) and Henschel (1995). Alernaively, we also employed a GARCH 9

12 To esimae acual price volailiy, like many oher sudies we use a GARCH model (for a review, see Poon and Granger, 003). We use he GARCH(1,) model which, unlike he GARCH(1,1), eliminaes significan auocorrelaions corresponding o all lags. Furhermore, comparing oher models we find ha in seven markes he GARCH(1,) is he bes fiing model as measured by Schwarz s (1978) BIC and Akaike s (1974) AIC crieria. 7 Finally, in he robusness ess, price volailiy is also direcly esimaed from realized reurns, where he analysis incorporaes boh he expos and ex-ane price volailiy corresponding o he VI period. 4. Empirical resuls In his secion, we repor on several significan rading paerns in he opions marke. The possibiliy ha he resuls are arifacs induced by echnical biases is explored in Secion The rading-break implied volailiy (TBIV) hypohesis Based on he resuls repored in Appendix A, o analyze he VIs we employ he following EGARCH-(1,1) model, while assuming ha he residuals follow he Suden- disribuion. Specifically, we employ he following model: V 5 1 1, iday, i TBREAK 3, iv i 4, ir i 5, ir i, i 0 i 0 ε z σ, log( σ ) ω α( z E z ) γz β log( σ ), (1) where V is he volailiy index (or a funcion of i) on day ; DAY, i ( i 1...5) are dummies corresponding o he weekdays; days oher han Mondays afer non-rading days; he opions underline sock index on day, and TBREAK is a dummy corresponding o R is he percenage rae of reurn on, z and are he innovaion, sandardized innovaion and he condiional sandard deviaion, respecively. Marke volailiy and reurns are correlaed in a complex manner (e.g., Glosen, Jagannahan, and Runkle, 1993; French, Schwer, and Sambaugh, 1987; model in which he innovaions follow eiher he Suden- or he normal disribuion as well as he Auoregressive Inegraed Moving Average (ARIMA) (3,0,3) model which, according o he BIC and AIC informaion crieria, is he bes fi ARIMA model. As he resuls wih hese models are very similar o hose repored in his sudy, for breviy s sake hey are no repored, bu are available upon reques. 7 In he U.S., he GARCH(,) model reveals slighly beer resuls. As he differences are small, for he sake of consisency, o calculae he U.S marke volailiy we also employ he GARCH(1,) model. 10

13 Campbell and Henschel 199; Brand and Kang, 004; and Avramov, Chordia and Goyal, 006). Specifically, as he paerns in he VI coincide wih he well-known weekend effec in reurns, he resuls corresponding o he VIs may be induced by he effec in reurns. To accoun for his possibiliy and o conrol for any oher bias induced by reurns, he regressions also include he reurns variable ( R ) and is lags over a full monh ( rading days) as explanaory variables. As he dependen variable is volailiy, in he main ess we also include he squared reurns ( R ) and is lags as explanaory variables. 8 Table repors Eq. (1) resuls, wih he U.S. VIX. << Inser Table >> Tes 1 examines he VIX opening values, where he Monday coefficien corresponds o days subsequen o he weekend rading break, and he TBREAK coefficien corresponds o non-monday days subsequen o he rading break. Tes 1 reveals ha boh he Monday and TBREAK coefficiens are several imes larger han he oher days coefficiens. The Friday coefficien, on he oher hand, is subsanially smaller han he oher coefficiens. Finally, he Log-likelihood saisic for equal days indicaes ha he differences across he days are highly significan ( p ). The resuls wih he VIX closing values in Tes are very similar. The Monday and TBREAK coefficiens are, once again, several imes larger han he oher days coefficiens, while he Friday coefficien is smaller han he oher coefficiens, and he differences across he days are highly significan. Can he high VIX on Monday be aribued o specific characerisics of he Monday or wo-day weekend rading break? To answer his quesion, Tess 3 6 include dummies ha correspond o days subsequen o one-, wo- and more han woday rading breaks. Tess 3 and 4, which do no include he weekdays dummies, examine he effec of he duraion of he rading break on he regression coefficien. For boh opening and closing VIX, he hree rading-break coefficiens are significanly posiive. Moreover, he coefficiens increase wih he rading break ime duraion, and he hypohesisha he rading break coefficiens are equal is rejeced as regards he VIX closing values ( p ). As he wo-day rading break 8 In unrepored ess, we verified ha excluding he reurns and squared reurns variables do no change he main resuls. In separae ess, we also include yearly dummy variables which conrol for oulier years wih paricularly high and low VIs. As hese variables are found o be insignifican, hese ess are no repored. 11

14 observaions mainly consis of weekends, heir number is much larger han he number of one- and more han wo-day observaions, which explains he relaively high -value corresponding o he wo-day rading break. Tess 5 and 6 also include he weekdays dummies. Thus, he wo-day rading break variable includes all of he weekend effecs, as measured by he Monday variable, plus oher wo-day rading breaks ha do no end on Mondays. Hence, hese wo variables are highly correlaed, which decreases he -value of hese wo variables due o mulicollineariy. Indeed, we find ha he -value corresponding o hese wo variables subsanially decreases in comparison o he -values repored in he previous ess. This phenomenon is mos profound in Tes 6, where he Monday coefficien urns ou o be insignifican. This resul indicaes ha he rading breaks affec he increase in he VIX, raher han various Monday-specific facors. Finally, in Tess 5 and 6 all he coefficiens corresponding o rading breaks are larger han he coefficiens corresponding o weekdays; his suppors he TBIV hypohesis. Afer non-rading days, uninformed opions raders face addiional risk, due o privae informaion accumulaed during he rading break refleced in he higher VIX. As he longer he rading break he more privae informaion is expeced o be accumulaed, his risk and correspondingly, he VIX increase wih he rading break ime duraion. Two resuls repored in Tess 1 6 require some furher explanaion. Firs, he VIX is significanly lower on Fridays han on oher days. Second, alhough he resuls are similar wih boh he opening and closing VIX, hey differ in magniude. The lower VIX on Fridays can be explained by means of he TBIV hypohesis, as well as by a mechanical bias relaed o calendar days, and which conforms o he findings of French and Roll (1986). According o Foser and Viswanahan s model, when high qualiy public informaion is regularly released, discreionary liquidiy raders pool heir rade ino wo days before Friday. Dealing wih opions wrien on he S&P 500 Index, he regularly released public informaion is probably of high qualiy (relaive o informaion on individual socks). Therefore, if a large porion of raders pool heir rade, say, on Wednesday and Thursday, hen all privae informaion is revealed in Thursday s closing prices. Hence, on Friday, all raders are informed, uncerainy due o privae informaion vanishes, and he VIX is relaively low. Alhough he lower VIX on Friday conforms, under reasonable assumpions, o he TBIV hypohesis i may also be induced by a mechanical bias. 1

15 The VIX reflecs he implied volailiy corresponding o he nex 30 calendar days. As a resul, he VIX on Friday relaes o a smaller number of rading days. 9 As according o French and Roll (1986) price volailiy over non-rading days is lower han ha on rading days, he smaller number of rading days corresponding o he VIX on Friday may accoun for he lower VIX on Fridays. Of course, i is also possible ha boh he release of public informaion and he mechanical bias, which operae in he same direcion, may accoun for he lower VIX on Friday. Le us now address he differences beween he opening and closing VIX. The wo hypoheses below es wheher privae informaion is also accumulaed overnigh; hence, he VIX increases, and wheher during rading hours privae informaion is, a leas parially, revealed, leading o a decline in he VIX. As he ime periods corresponding o overnigh and rading hours are relaively shor, he effecs, if hey exis, are expeced o be less profound in comparison o hose corresponding o weekends and holidays. To es for he exisence of an overnigh rading break effec, he dependen variable in Tes 7 is he overnigh change in he VIX, which is calculaed as he opening VIX less he previous day s closing VIX. As previously, he Monday and TBREAK coefficiens are posiive and highly significan. However, he oher coefficiens are relaively small, and he Friday coefficien is significanly negaive. Thus, he VIX increases afer weekends and holidays, decreases on Friday mornings, and does no significanly change over he oher nighs. The increase in he VIX afer weekends and holidays conforms o he TBIV hypohesis. The decrease in he VIX on Friday mornings is also in line wih his hypohesis. As wih high qualiy public informaion which is more relevan for he VIX and he underline S&P 500 Index virually all privae informaion is revealed by he end of Thursday. Hence, on Friday mornings all raders are informed, no risk premium is required for privae informaion, and he VIX decreases. Finally, he oher weekdays insignifican coefficiens sugges ha here is no significan overnigh rading-day effec. This is probably because no much informaion is received over he relaively shor overnigh rading break, which is also 9 A 30-calendar-day window, saring on Friday, includes he subsequen four weeks plus wo nonrading days: Saurday and Sunday. In conras, a 30-calendar-day window saring on he oher days includes he nex four weeks plus eiher one non-rading (Thursday) or wo rading days (Monday- Wednesday). 13

16 in line wih he general resul of French and Roll (1986): ha during rading breaks informaion is received a a slower pace han during rading hours. As privae informaion is revealed during rading hours, according o he TBIV hypohesis he risk induced by privae informaion is expeced o diminish during rading hours; hence, he closing VIX is expeced o be lower han he opening VIX. Indeed, Figure 1 shows ha he average closing VIX is lower han he average opening VIX, in paricular on Mondays and Tuesdays, where a simple -es rejecs he hypohesis of equal means ( p ). To furher es his predicion, he dependen variable in Tes 8 is he change in he VIX during rading hours, which is calculaed as he closing VIX less he opening VIX on he same day. In line wih he TBIV hypohesis, apar from Thursdays he days coefficiens are negaive and on Mondays and Tuesdays hey are relaively large, where he laer is also significan. Thus, i seems ha he VIX decreases during rading hours, in paricular on Mondays and Tuesdays when a greaer amoun of privae informaion accumulaed during he weekend is revealed. Ye, he significance of his resul depends on wheher he squared reurns conrol variables are included or no in he regression. 10 To complee he descripion of he ess in Table, noe ha in line wih he resuls repored in Appendix A regarding he VIs ime series, in all he ess he EGARCH coefficiens (α, β and γ) are highly significan. As expeced, he reurn and, o some exen also he squared reurn variables, are significanly negaively and posiively correlaed, respecively, a various lags (o avoid a complex able hese coefficiens are no repored in he able). However, he effecs in he VIX are highly significan afer conrolling for reurns. 4..Overlapping period in he VIX calculaion The significan inraday and inerday paerns repored so far are found in he VIX values, which include overlapping days. For example, he opening VIX on Monday and he subsequen Tuesday, which corresponds o 30 calendar days, i.e. o he period ha ends on he fifh Tuesday and Wednesday, respecively, include 9 overlapping days. These overlapping days are no expeced o sysemaically bias he resuls because hey are common o boh VIX values and have a similar effec or, more precisely, a random effec raher han a sysemaic one, which is expeced o be 10 In unrepored ess, we found ha wihou he squared reurn variables he Monday and Tuesday coefficiens are highly significanly negaive. Thus, he relaively small -values in Tes 8 are probably due o he correlaion beween he daily difference in he VIX and squared reurns variable and is lags. 14

17 canceled ou on average. Therefore, if he opening VIX on Mondays is higher han ha on Tuesdays, i implies ha he perceived volailiy corresponding o Mondays is higher han he perceived volailiy corresponding o Wednesdays. To supplemen he overlapping analysis, we also measure he pairwise differences in he VIX, afer deducing he overlapping days volailiy. For example, he opening VIX on Mondays and Tuesdays correspond o he periods ending on he fifh Tuesday and Wednesday, respecively. Hence, he opening VIX on Monday less Tuesday measures he difference in daily perceived volailiy corresponding o Monday and Wednesday, where here is a ime period of 30 calendar days beween hese wo days. 11 Similarly, he opening VIX on Monday less Wednesday corresponds o he perceived volailiy on Monday-Tuesday less ha on Wednesday-Thursday, which comes 30 calendar days laer. Finally, he opening VIX on Monday less Thursday corresponds o Monday-Wednesday less Wednesday-Friday, which comes 30 calendar days laer. Ignoring he common Wednesday, i acually measures he difference in he perceived volailiy corresponding o Monday-Tuesday and Thursday-Friday. By he same logic, he opening VIX on Monday less ha on Friday measures he difference in he perceived volailiy corresponding o (afer ignoring he common days) Monday-Tuesday less Friday-Saurday. Finally, as we are ineresed in he inerday effec across he weekdays o conrol for he long-erm rend across he weeks, we normalize he VIX values according o he weekly mean. Thus, all observaions for each week are divided by he relevan weekly mean, which reduces he possibiliy ha he resuls are biased by he long-erm rend and oulier periods during which he VIX was very high or very low. This also reduces he possibiliy ha he 30-day ime period beween he daily VIX values biases he resuls The perceived volailiy corresponding o he eliminaed days is no necessarily he same on each day. Therefore, alhough here is no reason o believe ha here are more han random changes across he many years covered in his sudy, we look a boh he coefficiens corresponding o Monday less oher days (e.g., Wednesday) and he coefficiens corresponding o oher days less Monday, where in he firs case Monday precedes he oher days by 30 days and in he laer case he oher days precede Monday by 30 days. Thus, if he resuls are biased, due o he overlapping days in favor of he TBIV hypohesis in one case, hey are expeced o be biased agains i in he oher case. This is because in boh cases almos he same days are eliminaed. For example, when comparing Monday less Wednesday in 30 days and Wednesday of he same week less Monday in 30 days here are 8 common days among he 30 eliminaed days. Hence, obaining he same resuls in boh cases reduces he possibiliy ha he resuls are spurious. 1 The resuls wih he raw VIX (i.e. wihou normalizaion) are generally similar wih only slighly smaller -values probably due o rend biases. 15

18 difference in one of he VIX pairs. 13 << Inser Table 3 >> Table 3 repors he resuls of Eq. (1), where he dependen variable is he The firs column in he able repors he coefficien corresponding o he VIX on Mondays or he VIX on Mondays and Tuesdays less he VIX on oher days. In all he ess, he coefficien is significanly posiive, indicaing ha he perceived volailiy on Mondays is significanly higher han ha on Wednesdays (Tes 1), and he perceived combined volailiy on Mondays and Tuesdays is significanly higher han ha on he combined Wednesdays and Thursdays (Tes ), Thursdays and Fridays (Tes 3) and Fridays and Saurdays (Tes 4). The resul in Tes 1 ha he daily perceived volailiy on Mondays is higher han ha on Wednesdays is paricularly imporan, as comparing he perceived volailiy corresponding o Mondays and Wednesdays compleely bypasses he mechanical bias, due o a varied number of rading days. This is because Mondays and Wednesdays have he same number of rading days in a forward-looking 30-calendar-day window; hence, hey are no exposed o he lower volailiy on non-rading days repored by French and Roll (1986). Consisen wih he resuls repored above, all he coefficiens which correspond o non-monday days less Monday are negaive and mos of hem are also significan. For example, he fifh coefficien in Tes indicaes ha he combined perceived volailiy on Friday and Saurday is significanly lower han ha on Sunday and Monday (a -value of 9. 01). Thus, in line wih he TBIV hypohesis, he daily perceived volailiy on Monday, and possibly also on Tuesday, is higher han ha on oher days and hese resuls are inac wheher Monday precedes he oher day or vice versa which, as previously explained, reduces he possibiliy ha he resuls are biased by he eliminaed overlapping days. Higher perceived volailiy a he beginning of he week is a general phenomenon, which is no confined o Mondays. For example, he second and hird coefficiens in Tes 1 show ha he perceived volailiy on Tuesdays and Wednesdays are significanly higher han ha on Thursdays and Fridays, respecively. Consisen resuls are obained in all oher ess. Thus, in line wih he TBIV hypohesis and he previous resuls wih overlapping periods, he daily perceived volailiy is a is 13 Alhough he EGARCH coefficien, γ, is no significan, for he sake of consisency Table 3 repors he resuls corresponding o he EGARCH model (he GARCH model resuls are very similar). The 16

19 highes level on Mondays, when privae informaion is high; i hen decreases over he week as privae informaion is revealed hrough rade. Two oher resuls emerge from Table 3, which are consisen across he various ess. Firs, he daily perceived volailiy on Saurdays is lower han ha on oher days (e.g. he fourh coefficien in Tes 1), which conforms o French and Roll s resul ha volailiy on Saurdays is lower han ha on he weekdays. In conras, he daily perceived volailiy on Sundays is higher han ha on oher days (e.g. he fifh coefficien in Tes 1). Alhough his las resul seems o conradic he fac ha volailiy on non-rading days is smaller han on rading days, i probably reflecs he highes level of accumulaed privae informaion before he Monday rading and he high volailiy as recorded on Monday morning. To summarize, he resuls repored so far reveal ha in line wih he TBIV hypohesis, he VIX is significanly higher on days afer non-rading days han on oher days, in paricular when he marke opens and privae informaion is a is highes level. These resuls are robus o serial correlaion, overlapping days in he VIX, mechanical bias due o a varied number of rading days, and sock marke reurns. Finally, he VIX is significanly lower on Fridays han on oher weekdays. This resul can be explained by boh a mechanical bias due o he number of rading days (which conforms o he findings of French and Roll, 1986), and by Foser and Viswanahan s (1990) model The rading break volume (TBV) hypohesis In his secion, we es he TBV hypohesis for various ypes of opions. As he disribuion of he volume daa is unknown, we employ Hansen s (198) Generalized Mehod of Momens (GMM) analysis o esimae he following sysem of equaions: 5 1 N N, j 1, i, jday, i, jtbreak 3, i, jvolume i, j j VOLUME,, (3) where VOLUME N, j ( j 1...4) is he normalized daily volume of raded opions in he CBOE corresponding o index call opions ( j 1), index pu opions ( j ), individual sock call opions ( j 3), and individual sock pu opions ( j 4 ) on day ; TBREAK is a dummy corresponding o days oher han Monday afer non-rading days; and DAY, i ( i 1...5) are dummies corresponding o he weekdays. To be able o model in Table 3 does no include auoregressive variables o avoid mulicollineariy, due o he correlaion beween he daily volailiy and and is lags. R 17

20 compare he coefficiens across he equaions, he daily volume corresponding o each ype of opions is normalized by he relevan all-day mean. The daa on volume is provided by he CBOE and covers he period from 003 o 010. Table 4 repors he resuls of he regression corresponding o Eq. (3). << Inser Table 4 >> Le us firs discuss he resuls corresponding o he index opions. The Monday and TBREAK coefficiens in Tess 1 and are significanly negaive, whereas he oher days coefficiens are posiive and mos of hem are highly significan. Indeed he Wald saisics in boh ess, repored in he las column of he able, rejec he hypohesis of equal weekdays coefficiens (p<0.0001). Moreover, he TBREAK coefficiens are larger in absolue erms han he Monday coefficiens and he hypohesis of equal Monday and TBREAK coefficiens is rejeced a p= (see he las row in he able). Finally, he Friday coefficiens are smaller han hose corresponding o oher non-monday weekdays. The lower volume afer weekend and holiday rading breaks conforms o Foser and Viswanahan s model and is consisen wih he TBV and TBIV hypoheses. As afer rading breaks privae informaion is a is highes level, discreionary opions liquidiy raders pospone heir rade o oher days; hence, a relaively low volume is recorded. Moreover, he higher TBREAK coefficiens (in absolue erms) in comparison o he Monday coefficiens suggess ha, as wih implied volailiy, so, oo wih volume he inensiy of he effec increases wih he rading break s ime duraion. This is because 68% of TBREAK observaions (which do no include regular weekends) correspond o more han wo-day rading breaks. 14 Furher in line wih his model, as accurae public informaion corresponding o indices is probably regularly released, discreionary liquidiy opions raders are expeced o pool heir rades ino wo days, prior o Friday. Hence, he volume rade on Friday is also expeced o be lower han ha on he oher weekdays, which is precisely wha we obain. The resuls in Tess 3 and 4, which correspond o individual sock opions, are similar o hose wih indices, bu less profound. The mos imporan difference which emerges from he comparison of index and individual sock opions is ha wih individual sock opions he Monday coefficiens are low, bu no as low as wih he 14 In he U.S., mos holidays fall on Mondays. Therefore, mos of he TBREAK observaions correspond o Tuesdays afer hree-day rading breaks. 18

21 index opions. This resul is significan as he hypohesis ha he days coefficiens across he four ypes of opions are equal is rejeced for all weekdays. The relaively weaker resuls corresponding o opions on individual socks conform o he qualiy of informaion hypohesis. As he qualiy of regularly released public informaion is expeced o be higher in regard o indices compared o individual socks, i is likely ha discreionary liquidiy raders in index opions pospone heir rades more ofen han hose rading individual sock opions. Hence, a larger decline is expeced on Mondays in he volume rade corresponding o index opions. Finally, obaining similar paerns in he volume of rade corresponding o boh pu and call opions reduces he possibiliy ha he resuls are echnical in naure, induced by he expeced decline in he price of he underline asse, due o asymmeric informaion risk. This is because he decline in he underline asse price is no expeced o have a symmerical effec on pu and call opions. However, he increase in uncerainy due o asymmeric informaion, which is our main explanaion for he resuls, is expeced o have a symmerical effec on boh ypes of opions The marke efficiency (ME) hypohesis: The fuures rade on he VIX In April 004, CBOE inroduced fuures on he VIX. 15 As previously explained, according o he EM hypohesis, i is expeced ha he fuures on he VIX miigae marke inefficiencies due o asymmeric informaion. To es he EM hypohesis, Tes 1 in Table 5 repeas he Eq. (1) analysis, while including addiional weekdays dummies corresponding o he period during which he fuures on he VIX were raded. This procedure covers he longes possible ime period for which daa is available; hence, i conains a relaively large number of observaions. For breviy s sake, in Table 5 and in he remainder of he sudy he VI daa only corresponds o closing values. << Inser Table 5 >> The days coefficiens corresponding o he whole period reveal a paern ha is very similar o he one obained so far. However, he Monday and Friday coefficiens corresponding o he period during which he fuures were raded are significanly negaive and posiive, respecively. Thus, he oal effec on Mondays 15 In February 006, he CBOE inroduced opions on he VIX. As his period is already incorporaed in he period corresponding o he fuures, and as he opions on he VIX are expeced o furher miigae he effec in implied volailiy, we focus on he fuures marke and he period saring from

22 and Fridays during he period he fuures were raded, which is equal o he sum of he wo Monday and wo Friday coefficiens, respecively, has significanly aenuaed. 16 Tess and 3, which separaely es he wo sub-periods (wih a lower number of observaions in comparison o Tes 1), reveal similar resuls. As he number of observaions in each sub-period is differen, le us focus on he regression coefficiens raher han on he -values. While he coefficiens corresponding o he more recen period are generally smaller, due o a lower VIX on average, he Monday coefficien decreased by 0.365, whereas he Friday coefficien increased by (he oher days coefficiens decreased by , and 0.063, respecively). Thus, consisen wih he resuls of Tes 1, he decrease on Monday is he highes, while on Friday he coefficien increased. The resuls repored in Table 5 sugges ha he inerday paern in implied volailiy has significanly aenuaed since he inroducion of fuures on he VIX, bu i sill remains highly significan. These resuls demonsrae how derivaive insrumens improve marke efficiency, presumably by reducing he asymmeric informaion risk. Obviously, causaliy is no proven and furher research is required o deermine he exac relaions beween he rade in opions and fuures on implied volailiy, which is beyond he scope of his sudy Alernaive economic explanaions: The inernaional evidence Figure presens he average VIs and price volailiies corresponding o he eigh markes covered in his sudy. All he VIs, presened in Figure a, are highes on Mondays and lowes on Fridays, whereas he acual price volailiies, presened in Figure b, are very similar across he days wih he excepion ha hey are only slighly higher on Tuesdays. Thus, a similar paern in implied volailiy is observed in all markes, while no such phenomenon is observed in regard o price volailiy. << Inser Figure >> To es wheher he inerday paern is significan and similar across markes, we employ a GMM analysis o esimae he following sysem of equaions: 5 1 V, j 1, i, jday, i, jtbreak, j 3, i, jv i, j, j, (4) 16 For example, while he Monday coefficien corresponding o he whole period is equal o , he combined Monday coefficien corresponding o he fuures period is equal o

23 where V, j is eiher he volailiy index (Panel A) or he GARCH(1,) price volailiy (Panel B) in marke j ( j ) on day ; TBREAK, j are dummies corresponding o days oher han Monday afer non-rading days in marke j; and DAY, i ( i 1...5) are dummies corresponding o he weekdays. Table 6 repors he resuls of his analysis. << Inser Table 6 >> Like wih he U.S. VIX, he Monday coefficiens corresponding o all he VIs in Panel A are several imes larger, and he Friday coefficiens are smaller, han he ohers days coefficiens. The TBREAK coefficiens are also larger han he ohers days coefficiens, apar from he one corresponding o Japan. Finally, in all he markes he hypohesis assering ha he coefficiens are equal is significanly rejeced (see he las column in he able). In sharp conras o he resuls in Panel A, he Monday coefficiens in Panel B, which correspond o he GARCH price volailiy, are he same size order as he oher days coefficiens, smaller han he Tuesday coefficiens, and five of hem are even smaller han he Friday coefficiens. The TBREAK coefficiens are all negaive and mos of hem are highly significan. The resuls in Table 6 show ha he higher implied volailiy afer rading breaks and he lower implied volailiy on Fridays is a global phenomenon ha exiss in all eigh markes. The coexisence of his phenomenon in all he markes suggess ha his phenomenon canno be explained by specific characerisics of he local markes. This resul also eliminaes he possibiliy ha he inerday paern is induced, for example, by he differen rading hours across he markes, any currency effecs and biases relaed o rading mehods and marke selemen procedures, all of which have been proposed in he pas as poenial explanaions for he weekend effec in reurns. Finally, his phenomenon does no exis in regard o sock price volailiy, which reduces he possibiliy ha i is induced by economic fundamenals incorporaed in sock prices. 5. Rejecing echnical and mehodological explanaions In his secion, we show ha he resuls repored above are no merely arifacs induced by some echnical biases. To avoid unnecessary repeiions, when he ess are sraighforward we analyze he U.S. VIX, which is he mos maure index wih he longes daa hisory. 1

24 5.1. Rejecing he opions expiraion day as a poenial explanaion As can be seen from Table 1, all he VIs underline opions expire on Fridays. Thus, he VI inerday paerns may be induced by echnical biases in he VI calculaions when shifing from one opion series ha has expired o anoher series, or due o unique rading paerns around he expiraion day. Noe, however, ha he opions expiraion day canno explain he inerday paerns in volume, as he volume daa incorporaes he ransacions corresponding o all opions series. Wang, Li and Erickson (1997) explore he possibiliy ha he opions expiraion day induces he weekend effec in reurns. Following heir mehodology, we break he monh ino five weeks 17 and es wheher he inerday paern in he VIs differs in regard o he remaining ime o opions expiraion. Thus, we employ a GMM analysis o esimae he following sysem of equaions: V, j 4 4, i, j 5 1, i, j WEEK DAY, i, i 1, j 5, i, j V TBREAK i, j, j, j, 4 3, i, j ( MONDAY )( WEEK, i ) (5) where V, j is he volailiy index in marke j ( j ) on day ; DAY, i ( i 1...5) are dummies corresponding o he weekdays; TBREAK, are dummies corresponding o j days oher han Monday afer non-rading days; ( MONDAY )( WEEK, i ),( i 1...4) are dummies corresponding o Mondays wihin he paricular weeks of he monh excluding he fifh Monday, if i exiss; and WEEK, i ( i 1...4) are dummies corresponding o week of he monh excluding he fifh week. The hypohesis esed by Eq. (5) assers ha he paerns in he VIs are relaed o he opions expiraion day; herefore, hey differ across he weeks of he monh, depending upon he remaining ime period unil expiraion. Presumably, he closer he underline opions o expiraion, he greaer/lesser he inerday paern. Focusing on he higher VIs on Mondays, o es his hypohesis we add four Monday dummies, ( MONDAY )( WEEK,i ), which allow he Monday coefficien o vary depending on he remaining ime period unil expiraion. As he opions expiraion day may induce a 17 The firs week of he monh is defined as he week ha conains he firs rading day of he monh. If he firs rading day of he monh is a Monday, hen i will be he Monday in he firs week of he monh; oherwise, here is no Monday observaion for he firs week of he monh. As Wang, Li and Erickson (1997) noe, his definiion ensures ha he Monday of he fourh week of he monh always follows he opions expiraion day (where in Japan, he Monday of he hird week of he monh always follows he opions expiraion day).

25 sysemaic paern across he weeks, we conrol for his possibiliy by adding four dummies, WEEK, i, which capure any weekly paern over he monh ha is no unique o he days wihin he week. Table 7 repors he resuls of he GMM coefficiens esimaed from Eq. (5). In all he markes, he inerday paern is robus o he remaining ime unil expiraion. The Monday coefficiens are significanly larger han he oher days coefficiens, he TBREAK coefficiens are also larger han he oher days coefficiens (apar from Japan) and he Friday coefficiens are smaller han he oher days coefficiens. << Inser Table 7 >> In addiion, he four Mondays week coefficiens, ( MONDAY )( WEEK,i), are generally insignifican and in four markes he hypohesis of equal Monday coefficiens wihin he monh is no rejeced a a 1% significance level. Thus, higher VIs on Mondays is common o all weeks and no significan paern as regards Mondays across he weeks is found, which indicaes ha his phenomenon is no induced by he opions expiraion day on a paricular week. Ineresingly, he hird and he fourh week coefficiens are negaive and mos of hem are significan, where he differences across he week coefficiens are highly significan (see he las column in he able). Thus, i seems ha he shif o a new opions series and he expiraion of he opions sysemaically affec he VIs. However, his week-of-he-monh bias does no change he main resuls regarding he inerday paern in he VIs. The resuls in Table 7 show ha he inerday paern in implied volailiy is robus o he opions expiraion day. However, he VIs are imely inerpolaed o reflec he volailiy over a ime period of 30 calendar days, 18 a procedure which may induce hidden biases ha affec he inerday resuls. To univocally deermine ha he resuls are robus o he opions expiraion day, as well as o he VIs ime inerpolaion, Tess 1, and 3 in Table 8 repor he resuls of Eq. (1), where he dependen variable is one of he Vis, which measures he floaing six-monh implied 18 The VIX ime inerpolaion formula, for example, is as follows: T ( N N ) /( N N ) T ( N N ) /( N N ) ( N / N ) VIX T 30 T T 30 T T T where T 1 and T are he remaining ime periods o expiraion (in annual erms calculaed in resoluion of minues) corresponding o he wo opions series, whose expiraion ime period is closer o 30 days; 1 and are he volailiies of he wo opions series as derived from heir prices; and N T is he 3,

26 volailiy: he VSMI6M, VSTOXX6M and VDAX-NEW6M; floaing means ha he VI corresponds o a floaing period wihou ime inerpolaion. << Inser Table 8 >> Like wih he regular, fixed 30-day VIs, he Monday and TBREAK coefficiens corresponding o he floaing six-monh VIs are larger han he oher coefficiens and he Friday coefficiens are smaller, where he differences are highly significan (see he las column). Thus, a similar inerday paern also exiss in regard o six-monh VIs, which are no imely inerpolaed; in addiion, hey are subsanially less sensiive o he remaining ime unil expiraion and o a one-week ime shif from Monday o Friday, which is only a fracion of he remaining ime unil expiraion. As expeced, he differences across he days coefficiens in his case are smaller han hose corresponding o he 30-day VIs because he risk implied by privae informaion accumulaed over he wo-day rading break is expeced o be smaller, on average, for six-monh expeced volailiy han for 30-day expeced volailiy. To show ha he smaller coefficiens are no relaed o he ime inerpolaion, bu raher o he longer expeced volailiy period, in Tes 4 he dependen variable is he VXV, which measures he fixed, raher han he floaing, hree-monh expecaion volailiy in he U.S. marke. Like wih he oher VIs, he inerday paern is significan and he coefficiens are also smaller han hose corresponding o he 30-days VIs. Thus, alhough his index is imely inerpolaed, like he oher long-erm VIs, i reveals a smaller in magniude, ye significan, inerday paern. 5.. Rejecing holidays as a poenial explanaion As he VIX reflecs he volailiy over 30 calendar days and, as according o French and Roll (1986), price volailiy is lower during non-rading days, any sysemaic paern in he number of rading days wihin a rolling 30-calendar-day window may induce a sysemaic paern in he VIX. Holidays creae sysemaic paerns in he number of rading days; his is paricularly imporan because many U.S. holidays fall on Monday. Therefore, Tes 5 includes wo holiday variables ha explore wheher he reduced number of rading days due o holidays affecs he VI. number of minues in ime period T. Noe ha when one series expires, his formula is used o exrapolae he VIX from wo series whose expiraion ime period is longer han 30 days. 4

27 The firs variable is a dummy for he las rading day before holidays, which ess wheher here is a holiday effec similar o he holiday effec in reurns, in which reurns are relaively high on he las rading day before he holiday (Lakonishok and Smid, 1988; and Kim and Park, 1994). The second variable is a dummy for all of he days wihin he monh before he holiday, i.e. he days followed by an unusually small number of rading days. As can be seen, he inerday paern in he VIX is robus o holidays, as he resuls are similar o hose obained in he previous ess. Thus, a varying number of rading days due o holidays does no accoun for he inerday paern. This resul, ogeher wih he fac ha he inerday paern is also observed in oher markes characerized by differen holidays, rule ou holidays as a possible explanaion for he inerday paern in implied volailiy. Ineresingly, he coefficien corresponding o he pre-holiday monh is negaive and highly significan. This resul suggess ha he marke recognizes he lower volailiy during non-rading days in comparison o ha during rading days (French and Roll, 1986). Hence, a smaller number of rading days wihin a forwardlooking 30-calendar-day induces a lower expeced volailiy Rejecing acual risk and economic fundamenals as a possible explanaion So far, we have found ha he inerday paern in implied volailiy does no exis in he GARCH price volailiy and is also robus o marke reurns. Thus, if one assumes ha price volailiy and marke reurns accoun for economic fundamenals relevan o implied volailiy, hen economic fundamenals do no accoun for he observed inerday paern in he VI. However, while he VI reflecs he volailiy expecaion for he nex 30 calendar days, which may include a varied number of rading days, price volailiy is calculaed from rading-day realized reurns. This disincion beween calendar and rading days may reduce he abiliy of price volailiy o imely reflec a daily resoluion of he economic fundamenals relevan o he VI. Moreover, so far we have employed price volailiy from realized reurns known on day (he GARCH mehod). One may reasonably argue ha because he VI measures volailiy expecaion, he fuure acual price volailiy is more relevan in accouning for he economic fundamenals ha affec he VI. To es his argumen, as well as he possible bias due o he disincion beween calendar and rading days, we also measure acual volailiy direcly from realized reurns while relying on boh pas reurns (he ex-pos mehod) and fuure reurns (he ex-ane mehod). 5

28 To calculae he forward-looking 30-calendar-day fuure volailiy direcly from realized reurns, we use he following well-known formula (e.g., Bakshi and Kapadia, 003): VOL 5 N 100 ( R R), (6) N 1 where R is he reurn on he S&P 500 Index on day ; R is he mean reurn; and N is he precise number of rading days in a forward-looking 30-calendar-day window. Similarly, o calculae he pas volailiy, he parameer in Eq. (6) runs from N o 0, where N is he precise number of rading days in a backward-looking 30- calendar-day window. The division by N should, in principle, correc he bias due o he changing number of rading days wihin he 30-calendar-day window. 19 Tess 6 and 7 in Panel C of Table 8 repor he resuls of Eq. (1) where eiher he GARCH volailiy, or he pas and fuure price volailiy calculaed by Eq. (6), are also included as conrol variables, respecively. In line wih he resuls in Table, he inerday paern in he VI is robus o pas and fuure price volailiies; hence, his paern is no explained by economic fundamenals which are realized in pas or fuure acual price volailiies. 5.4 Oher robusness ess The New-VIX mehodology may suffer from hidden biases of which we are no aware. A firs indicaion ha such possible biases do no accoun for he inerday paerns is he fac ha he paerns also exiss in he VSMI, VSTOXX and VDAX- NEW, whose mehodologies differ from hose of he oher indexes (see Table 1). To furher refine he analysis, he dependen variable in Tess 8 and 9 is one of he VIs whose mehodology is differen from he NEW-VIX mehodology: he CSFI s novel mehodology CSFI-VXJ (Japan), and he Deusche Börse s old mehodology VDAX (Germany). 0 As can be seen, he wo VIs reveal significan inerday paerns. Thus, he New-VIX mehodology does no induce he inerday paern, as hese 19 In unrepored ess, we also calculaed he forward-looking 1-rading-day pas and fuure volailiies, by forcing N 1 in Eq. (6). As he resuls in his case are very similar o hose obained wih he forward-30-calendar-day volailiies, hese ess are no repored. Noe ha o obain he annualized volailiy, he daily volailiy is muliplied by 5, which is consisen wih he definiion over rading days. Muliplying he daily sandard deviaion by 365 calendar days (which is common in he indusry) has only a consan affec, which does no change he inerday paern resuls. 0 The CBOE s old mehodology VXO (U.S.) is no included in he analysis as i is calculaed for 1 rading days; hence, each observaion corresponds o a varied number of calendar days. Therefore, a comparison across he weekdays is biased. This is one of he reasons why he CBOE replaced i wih he new VIX. 6

29 indices are based on differen mehodologies. The VDAX resuls also reinforce he resuls in Table 7, which show ha he inerday paern is no induced eiher by he VIs ime inerpolaion or by he opions expiraion day, as he VDAX employs a differen ime inerpolaion procedure corresponding o marke volailiy over 45 calendar days. While we find inerday paerns in eigh VIs ha correspond o differen sock indices, hese indices share one hing in common: hey are all major indices ha include socks from large firms and diversified indusries. To univocally deermine ha he paerns are no unique o major sock indices, in Tess 10, 11 and 1 he dependen variable is one of he VIs whose opion underline sock index eiher belongs o paricular indusries (Dow Jones Average Indusrial and NASDAQ 100) or includes socks from relaively small firms (Russell 000). As can be seen, he inerday paerns are highly significan in all ess, suggesing ha hey are no relaed o he sock index characerisics. 6. Concluding Remarks Several heoreical models sugges ha various disinc groups of invesors are involved in rading risky asses. The main disincion beween he various raders corresponds o he available level of informaion. Informed raders have he advanage over uninformed liquidiy raders; hence, uninformed liquidiy raders perceived risk increases wih he accumulaion of privae informaion by he informed raders. These asymmeric informaion models predic several resuls regarding ineremporal behavior of sock price volailiy and volume, depending on he invesmen sraegies adoped by he various paries. Generally speaking, hese heoreical predicions have been empirically confirmed by employing daa corresponding o he sock marke. In his paper, we analyze he ineremporal behavior of he implied volailiy and volume of rade in opions, covering eigh inernaional markes. The implied volailiy reflecs he perceived risk by raders, including he increase in risk, due o he knowledge ha here are informed invesors who can ake advanage of heir privae informaion. We find ha here are sysemaic inraday and inerday paerns in volume of rade in opions and in he opions implied volailiy. The main resul is ha implied volailiy significanly increases, while he volume of rade significanly decreases afer weekend and holiday rading breaks. Furhermore, he longer he 7

30 rading break he sharper hese phenomena, presumably because more privae informaion is accumulaed, which creaes a rade wih asymmerical informaion. Uninformed liquidiy raders who have he discreion o pospone heir rade aciviies are he main reason for he observed decrease in volume of rade in opions afer a rading break. Some privae informaion is revealed during rading hours, which induces a decrease in he closing implied volailiy, relaive o he opening implied volailiy as measured on he firs wo days afer he rading break. We find ha he invesmen sraegies adoped by he various paries are associaed wih he qualiy of he revealed public informaion during rading hours. The higher he qualiy of informaion revealed during he rade he larger he moivaion of uninformed raders o pospone heir rade while waiing for his informaion. The inroducion of rade in fuures on he VIX in 004 has miigaed he observed phenomena, probably because liquidiy raders use his insrumen o hedge heir risk, while informed raders use i o expedie he exploiaion process of heir privae informaion. Finally, he resuls are robus o economic fundamenals, which may accoun for he observed phenomena, he fac ha he implied volailiy in boh of he wo days under comparison include many overlapping days, he various mehods of calculaing he implied volailiy, he opions expiraion day, he various underline asses of he opions, he various inernaional opion markes covered in his sudy, and oher possible mechanical biases. 8

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32 Glosen, L.R. and Milgrom P.R Bid, Ask, and Transacion Prices in a Specialis Marke wih Heerogeneously Informed Agens. Journal of Financial Economics 14, , Glosen, L.R. Jagannahan, R. and Runkle, D.E, On he Relaion beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks. Journal of Finance. 48, Granger, C.W.J. and Morgensem, O Predicabiliy of Sock Marke Prices. Heah-Lexingon, Lexingon, MA. Hansen, L Large Sample Properies of Generalized Mehod of Momen Esimaors. Economerica 50, Harris, L A Transacion Daa Sudy of Weekly and Inradaily Paerns in Sock Reurns. Journal of Financial Economics 16, Henschel, L All in he Family: Nesing Symmeric and Asymmeric GARCH Models. Journal of Financial Economics 39, Jain, P. and Joh, G The Dependence beween Hourly Prices and Trading Volume. Journal of Financial and Quaniaive Analysis 3, Jones, C. and Shemsh, J The Weekend Effec in Equiy Opion Reurns. Working Paper, Universiy of Souhern California. Kim, C.W. Park, J Holiday Effecs and Sock Reurns: Furher Evidence. Journal of Financial and Quaniaive Analysis 9, Kyle, A Coninuous Aucions and Insider Trading, Economerica 53, Lakonishok, J. and Smid, S Are Seasonal Anomalies Real? A Niney Year Perspecive. Review of Financial Sudies 1, Lockwood, L.J. and Linn, S.C An Examinaion of Sock Marke Reurn Volailiy during Overnigh and Inraday Periods, Journal of Finance 45, Nelson, D.B Condiional Heeroskedasiciy in Asse Reurns: A New Approach, Economerica 59, Newey, W.K. and Wes, K.D A Simple Posiive Semi-Definie, Heeroskedasiciy and Auocorrelaion Consisen Covariance Marix. Economerica 55, Newey, W.K. and Wes, K.D Auomaic Lag Selecion in Covariance Marix Esimaion. Review of Economic Sudies 61, Pagan, A.R. and Schwer, G.W Alernaive Models for Condiional Sock Volailiy. Journal of Economerics 45, Poon, S. and Granger, C Forecasing Volailiy in Financial Markes: A Review. Journal of Economic Lieraure 41, Poerba, J. and Summers, L The Persisence of Volailiy and Sock Marke Flucuaions. American Economic Review 76, Said, E. and Dickey, D.A Tesing for Uni Roos in Auoregressive Moving Average Models of Unknown Order. Biomerika 71, Schwarz, G.E Esimaing he Dimension of a Model. Annals of Saisics 6, Schwer, G.W Sock Volailiy and he Crash of 87. Review of Financial Sudies 3, Soll, H.R. and Whaley, R.E Sock Marke Srucure and Volailiy. Review of Financial Sudies 3, Schwer, G.W Anomalies and Marke Efficiency. In Consaninides, G. Harris, M. Sulz, R.M. Handbook of he Economics of Finance, Norh-Holland, Wang, K. Yuming, L. and Erickson, J A New Look a he Monday Effec. Journal of Finance 5, Wood, R. McInish, T. and Ord, J An Invesigaion of Transacion Daa for NYSE Socks. Journal of Finance 40,

33 Figure 1a Figure 1b Figure 1c Figure 1. Average VIX, volume and acual price volailiy as a funcion of he day of he week The figures presen he average VIX a marke opening and marke closing (Figure 1a), volume of raded index opions (Figure 1b), and index price volailiy (Figure 1c) as a funcion of he day of he week. The VIX is he CBOE implied volailiy index corresponding o opions wrien on he S&P 500 Index. Volume is he number of index opions raded in he CBOE. Index price volailiy is a GARCH(1,) model sandard deviaion corresponding o he S&P 500 Index. The daa on he VIX and volailiy covers he period of , while he daa on volume covers he period of Ocober For comparison purposes, he lef-hand y-axis in Figures 1a and 1c is scaled o be he same (60 basis poins). The y-axis on he righ-hand side ranslaes he values on he lef-hand side ino a percenage deviaion from he all-day mean. 31

34 Figure a Figure b Figure. Average volailiy index and price volailiy in eigh markes The figure presens he average volailiy index (Figure 1a) and he average index price volailiy (Figure b) in eigh markes as a funcion of he day of he week. The volailiy indices are he VIX (U.S.), VAEX (Neherlands), VCAC (France), VFTSE (U.K.), VXJ (Japan), VSMI (Swizerland), VSTOXX (Eurozone), and he VDAX-New (Germany). Index price volailiy is calculaed as a GARCH (1,) model sandard deviaion on he relevan sock index reurns. The firs year s daa ranges from 1990 o 000, depending on he index (see Table 1), and he las year repored is 010. For illusraion purposes, he y-axis values are cenered by subracing he relevan index all-day mean from all values. 3

35 Table 1: The various volailiy indices The able repors he descripive saisic of he volailiy indices employed in his sudy. Panel A corresponds o he eigh markes main volailiy indices, while Panel B corresponds o he alernaive indices which employ differen mehodologies. A. Inernaional volailiy indices Index Name: VIX VAEX VCAC VFTSE VXJ VSMI VSTOXX EURO The underline index and S&P 500 AEX CAC 40 FTSE 100 Nikkei 5 SMI STOXX 50 marke (U.S.) (Neherlands) (France) (U.K.) (Japan) (Swizerland) (Eurozone) VDAX- NEW DAX 30 (Germany) Calculaed by CBOE Euronex Euronex Euronex CSFI Osaka U. SIX Swiss STOXX Ld. Deusche Börse Mehodology Opions used for calculaions Volailiy period New VIX (model-free) The wo neares-erm o 30-day expiraion series, wide range of srike prices. Fixed 30 calendar days Deusche Börse mehodology - based on he New VIX mehodology The wo sub-indices closes o he 30- day expiraion (based on neares-erm o 30-day expiraion series). Las rading day Third Friday of he monh Second Friday of he monh Third Friday of he monh Saring year Number of observaions 5,95,814,811,801 3,194 3,041 3,069 4,808 Average Sandard deviaion Maximum Minimum U.S. indexes (for sock index ess) B. Alernaive volailiy indices Alernaive indexes (for mehodology ess) Index Name: VXD VXN RVX VDAX VXJ-CSFI Dow Jones The underline index and Russell DAX 30 Nikkei 5 Indusrial NASDAQ 100 marke 000 (Germany) (Japan) Average Calculaed by Mehodology CBOE Same as VIX Deusche Börse Black- Scholes model CSFI - Osaka Universiy CSFI mehodology Long-erm indexes (for ime inerpolaion ess) VXV/VSMI6M/VSTOXX6M / VDAX-NEW6M Same as VIX / VSMI / VSTOXX / VDAX-NEW Opions used for calculaions Same as VIX 8 series near-hemoney Same as VXJ Neares-erm o 6 monhs (3 monhs for VXV), wide range of srike prices Volailiy period Same as VIX Fixed 45 calendar days Same as VXJ Floaing 6 monhs (Fixed 3 monhs for VXV) Las rading day Same as VIX Same as VXJ Same as VIX Saring year VXV from 008, he ohers Number of observaions 3,171,495 1,763 3,303 3,194 same as VSMI, VSTOXX, Average and VDAX-NEW (he laer wihou 5/005 10/006) Sandard deviaion Maximum Minimum

36 Table : Tes for he rading-break implied volailiy hypohesis The able repors he resuls of he following EGARCH- model: V 5 1 1, iday, i TBREAK 3, iv i 4, ir i 5, ir i, i 0 i 0 ε z σ, ) 1 log( σ ω' α z γz β log( σ ), 1 where V is he opening or closing VIX, or he change in he VIX overnigh or during rading hours on day (Tess 7 and 8); DAY, i ( i 1...5) are dummies corresponding o he weekdays; TBREAK is a dummy corresponding o days oher han Monday afer non-rading days (or alernaively dummies corresponding o one- wo- and more han wo-day rading breaks); R is he percenage rae of reurn on he relevan sock index;, z and are he innovaion, sandardized innovaion, and he condiional sandard deviaion; and ω ' ω α1e z 1 is he condiional sandard deviaion consan erm. The innovaions follow he Suden- disribuion. The closing and opening VIX daa cover he period and , respecively. Each line in he able repors he regression coefficiens, while he -values are repored in he line below (in brackes). Robus sandard errors are obained by Bollerslev-Wooldridge Quasi-Maximum Likelihood Esimaes (QMLE). One and wo aserisks indicae a wo-ail es significance level of 5% and 1%, respecively. Day of he week variables Trading break variables EGARCH variables Dependen variable Inercep Mon. Tues. Wed. Thurs. Fri. Pos-non-Monday rading-break 1-day break -day break >-day break ω' α β γ Log- likelihood raio Weekend and holiday rading breaks Equal weekdays 1. V VIX open (13.41 ** ) (5.59 ** ) (4.4 ** ) (4.56 ** ) (.01 * ) (7.6 ** ) ( ** ) (10.54 ** ) (67.00 ** ) (6.55 ** ) p< V VIX close (16.43 ** ) (3.06 ** ) (4.14 ** ) (5.45 ** ) (-0.11) (14.04 ** ) (-13.6 ** ) (1.73 ** ) (00.40 ** ) (8.9 ** ) p< Trading break by duraion Equal rading breaks 3. V VIX open (4.46 ** ) (3.4 ** ) (15.40 ** ) (9.79 ** ) ( ** ) (10.56 ** ) (63.80 ** ) (6.55 ** ) P= V VIX close (3.61 ** ) (.09 * ) (1.14 ** ) (17.67 ** ) (-13. ** ) (1.31 ** ) (07.80 ** ) (9.01 ** ) p< Weekend versus non-weekend rading break 5. V VIX open (.6 ** ) (5.48 ** ) (4.38 ** ) (4.5 ** ) (.00 ** ) (3.67 ** ) (1.98 * ) (6.78 ** ) ( ** ) (10.56 ** ) (63.80 ** ) (3.67 ** ) P= V VIX close (-0.43) (.44 * ) (4.16 ** ) (5.43 ** ) (-0.0) (.41 * ) (6.17 ** ) (15.3 ** ) ( ** ) (1.49 ** ) (01.90 ** ) (8.61 ** ) p< Overnigh rading break and reversal Equal weekdays 7. V VIX open VIX -1close (5.89 ** ) (0.40) (-0.6) (-0.66) (-8.40 ** ) (0.99) ( ** ) (11.03 ** ) (5.30 ** ) (-1.3 ** ) p= V VIX close VIX open (-1.60) (-.05 * ) (-1.11) (0.60) (-0.8) (1.37) (-1.31 ** ) (1.16 ** ) (17.90 ** ) (3.50) p<

37 Table 3: Tes for he rading-break implied volailiy hypohesis wih non-overlapping daily VIX The able repors he resuls of he following EGARCH- model: where V VIX N open VIX N j open V 5 1, iday, i TBREAK 3, ir i 4, i i 0 i 0 ε z σ, i R ε, 1 log( σ ) ω' α z γz β log( σ ), 1 1 is he opening VIX on day normalized by he weekly mean less he opening VIX on day +j (j=1,,4) normalized by he weekly mean; DAY, i ( i 1...5) are dummies corresponding o he weekdays; R is he percenage rae of reurn on he relevan sock index;, z and are he innovaion, sandardized innovaion, and he condiional sandard deviaion; and ω ' ω α1e z 1 is he condiional sandard deviaion consan erm. The innovaions follow he Suden- disribuion. Each The daa covers he period of line in he able repors he regression coefficiens, while he -values are repored in he line below (in brackes). Robus sandard errors are obained by Bollerslev-Wooldridge Quasi-Maximum Likelihood Esimaes (QMLE). One and wo aserisks indicae a wo-ail es significance level of 5% and 1%, respecively. Dependen variable 1. V. V 3. V 4. V VIX open VIX 1open VIX open VIX open VIX open VIX 3 close VIX open VIX 4 close Mon. less Wed. Tue. less Thu. Wed. less Fri. Thu. less Sa. Fri. less Sun. Pos-non- Monday rading-break ω' α β γ Log- likelihood raio (4.9 ** ) (3.30 ** ) (.36 * ) (4.96 ** ) ( ** ) (0.38) ( ** ) (15.33 ** ) (30.98 ** ) (0.14) p< Mon.-Tue. less Wed.-Thu. Tue.-Wed. less Thu.-Fri. Wed.-Thu. less Fri.-Sa. Thu.-Fri. less Sa.-Sun. Fri.-Sa. less Sun.-Mon (6.98 ** ) (5.53 ** ) (7.0 ** ) (-8.47 ** ) (-9.01 ** ) (0.80) ( ** ) (15.44 ** ) (59.36 ** ) (-0.58) P< Mon.-Tue. less Thu.-Fri. Tue.-Wed. less Fri.-Sa. Wed.-Thu. less Sa.-Sun Thu.-Fri. less Sun.-Mon. Fri.-Sa. less Mon.-Tue (10.4 ** ) (9.96 ** ) (-5.45 ** ) (-3.85 ** ) (-6.35 ** ) (1.70) (-1.59 ** ) (13.74 ** ) (7.81 ** ) (-0.04) p<0.001 Mon.-Tue. less Fri.-Sa. Tue.-Wed. less Sa.-Sun. Wed.-Thu. less Sun.-Mon. Thu.-Fri. less Mon.-Tue. Fri.-Sa. less Tue.-Wed (14.51 ** ) (-.0 * ) (-1.69) (-1.71) (-3.19 ** ) (0.0) ( ** ) (11.90 ** ) (71.46 ** ) (0.59) p<

38 Table 4: Tess for he rading-break volume and he qualiy of public informaion hypoheses The able repors he GMM esimae coefficiens of he following sysem of equaions: 5 1 N N, j 1, i, jday, i, jtbreak 3, i, jvolume i, j j VOLUME, where VOLUME N, j ( j 1...4) is he daily number of raded call or pu indices opions or equiy (individual socks) opions in he CBOE, normalized by all-day volume mean on day ; TBREAK is a dummy corresponding o days oher han Monday afer non-rading days; and DAY, i ( i 1...5) are dummies corresponding o he weekdays. The daa covers he period from November 003 o 010. The GMM is run wih Barle kernel and Newey and Wes (1987, 1994) heeroskedasiciy and auocorrelaion (HAC) consisen sandard errors wih 7 lags, which corresponds o he auomaic bandwidh parameer. Each line in he able repors he regression coefficiens, while he -values are repored in he line below (in brackes). One and wo aserisks indicae a wo-ail es significance level of 5% and 1%, respecively. 1. Index call opions. Index pu opions 3. Equiy call opions 4. Equiy pu opions Monday Tuesday Wednesday Thursday Friday Pos-non- Monday rading-break R Wald χ (equal days) (-4.59 ** ) (6.98 ** ) (7.58 ** ) (8.94 ** ) (1.18) (-5.17 ** ) p< (-5.1 ** ) (5.97 ** ) (7.30 ** ) (8.14 ** ) (4.76 ** ) (-3.3 ** ) p< (0.75) (7.87 ** ) (7.63 ** ) (9.10 ** ) (.88 ** ) (-.61 ** ) p< (-1.30) (10.9 ** ) (6.96 ** ) (9.30 ** ) (.69 ** ) (-3.00 ** ) p< Wald χ Days coefficiens are equal across all ypes of opions p< p< p=0.065 p= p< p= Days coefficiens are equal wihin he pu and call opions p=0.018 p< p= p= p< p=0.154, TBREAK and Monday coefficiens are equal wihin wo ypes of opions 13. p=

39 Table 5: Tess for he marke efficiency hypohesis The able repors he resuls of he following EGARCH- model: VIX ε z σ, 5 5 1, iday, i TBREAK 3, iday, i FUT ) 4TBREAK ( FUT ) 1 log( σ ) ω' α z γz β log( σ ), ( VIX R R, where VIX is he VIX on day ; DAY, i ( i 1...5) are dummies corresponding o he weekdays; TBREAK is a dummy corresponding o days oher han Monday afer non-rading days; FUT is a dummy corresponding o he ime period during which fuures on he VIX were raded (April ); R is he percenage rae of reurn on he relevan sock index;, z and are he innovaion, sandardized innovaion, and he condiional sandard deviaion; and ω ' ω α1e z 1 is he condiional sandard deviaion consan erm. The innovaions follow he Suden- disribuion. Each line in he able repors he regression coefficiens, while he -values are repored in he line below (in brackes). Robus sandard errors are obained by Bollerslev-Wooldridge Quasi-Maximum Likelihood Esimaes (QMLE). One and wo aserisks indicae a wo-ail es significance level of 5% and 1%, respecively. 5, i i i 0 6, i i i 0 7, i i Dependen variable and model 1. VIX wih addiional dummies from 4/004. VIX pre-fuure inroducion (1990-3/004) Day of he week variables Pos-non- Mon. Tues. Wed. Thurs. Fri. Monday radingbreak Mon. Tues. Wed. Thurs. Fri. Since he inroducion of fuures in 4/004 EGARCH variables Log- Pos-non- Monday rading- ω' α β γ likelihood raio (equal days) break (18.46 ** ) (3.86 ** ) (4.55 ** ) (5.04 ** ) (-0.63) (1.01 ** ) (-8.08 ** ) (-1.3) (-0.83) (1.1) (3. ** ) (-0.78) ( ** ) (1.94 ** )( ** ) (8.7 ** ) p< (15.65 ** ) (4.0 ** ) (4.70 ** ) (5.06 ** ) (0.70) (11.70 ** ) (-9.40 ** ) (8.68 ** ) (140.00) (7.36 ** ) p< VIX pos-fuure inroducion (4/ /010) (8.17 ** ) (1.59) (.91 ** ) (4.03 ** ) (0.91) (8.93 ** ) (-9.05 ** ) (8.17 ** ) ( ** ) (5.17 ** ) p<

40 Table 6: The inernaional evidence The able repors he GMM esimae coefficiens of he following sysem of equaions: where V, j 5 1 V, j 1, i, jday, i, j, jtbreak, j 3, i, jv i, j, j in Panel A is he volailiy index and in Panel B he GARCH(1,) price volailiy in marke j ( j ) on day ; DAY, i ( i 1...5) are dummies corresponding o he weekdays; and TBREAK, j are dummies corresponding o days oher han Monday afer non-rading days. The daa covers he period from 000 o 010. The GMM is run wih Barle kernel and Newey and Wes (1987, 1994) heeroskedasiciy and auocorrelaion (HAC) consisen sandard errors wih 9 lags, which corresponds o he auomaic bandwidh parameer. Each line in he able repors he regression coefficiens, while he -values are repored in he line below (in brackes)., VIX (U.S.) VSMI (Swizerland) VAEX (Neherlands) VDAX-NEW (Germany) VXJ (Japan) VSTOXX (Eurozone) VCAC (France) VFTSE (U.K.) S&P 500 (U.S.) SMI (Swizerland) AEX (Neherlands) DAX 30 (Germany) Nikkei 5 (Japan) Europe STOXX 50 (Eurozone) CAC 40 (France) FTSE 100 (U.K.) A. Volailiy Index Monday Tuesday Wednesday Thursday Friday Pos-non-Monday rading-break Wald χ (equal days) (16.40 ** ) (-4.14 ** ) (4.55 ** ) (.86 ** ) (1.97 * ) (19.16 ** ) p< (8.95 ** ) (-0.41) (5.93 ** ) (8.19 ** ) (-1.96) (8.77 ** ) p< (4.69 ** ) (6.58 ** ) (10.09 ** ) (1.14) (-3.79 ** ) (5.41 ** ) p< (4.6 ** ) (.89 ** ) (6.05 ** ) (5.75 ** ) (-4.73 ** ) (15.04 ** ) p< (18.1 ** ) (9.11 ** ) (9.45 ** ) (7.08 ** (.09 * ) (4.74 ** ) p< (0.80 ** ) (1.77) (7.84 ** ) (3.55 ** ) (-0.45) (10.7 ** ) p< (4.69 ** ) (5.97 ** ) (5.49 ** ) (7.9 ** ) (0.31) (8.34 ** ) p< (.91 ** ) (1.7) (8.18 ** ) (3.96 ** ) (-1.49) (1.85 ** ) p< B. GARCH volailiy Monday Tuesday Wednesday Thursday Friday Pos-non-Monday rading-break Wald χ (equal days) (8.34 ** ) (9.57 ** ) (13.71 ** ) (9.0 ** ) (8.46 ** ) (-.87 ** ) p< (13.01 ** ) (17.4 ** ) (17.10 ** ) (6.68 ** ) (15.75 ** (-1.58 ** ) p< (5.77 ** ) (1.59 ** ) (1.36 ** ) (0.34) (1.1 ** ) ( ** ) p< (8.66 ** ) (13.47 ** ) (14.61 ** ) (4.64 ** ) (13.49 ** ) (-15.7 ** ) p< (13.9 ** ) (13.57 ** ) (18.15 ** ) (9.8 ** ) (1.19 ** ) (-0.7) p< (8.40 ** ) (11.56 ** ) (14.67 ** ) (.59 ** ) (1.53 ** ) ( ** ) p< (9.57 ** (8.9 ** ) (14.67 ** ) (3.67 ** ) (10.91 ** ) (-1.83 ** ) p< (8.41 ** ) (16.57 ** ) (8.56 ** ) (10.9 ** ) (9.63) (-5.45 ** ) p<

41 Table 7: Tess for he opions expiraion day The able repors he GMM esimae coefficiens of he following sysem of equaions: where V, j 1, i, jday, i, jtbreak, j 3, i, j ( MONDAY )( WEEK, i ) 4, i, jweek, i 5, i, jv i, j, j V, j is he volailiy index in marke j ( j ) on day ; DAY, i ( i 1...5) are dummies corresponding o he weekdays; TBREAK, j are dummies corresponding o days oher han Monday afer non-rading days; ( MONDAY )( WEEK,i ) are dummies corresponding o Mondays as a funcion of he week of he monh excluding he fifh Monday (he firs rading day in he fourh week is he firs rading day afer he expiraion dae excep for Japan); and WEEK, i ( i 1...4) are week of he monh dummies excluding he fifh week. The daa covers he period from 000 o 010. The GMM is run wih Barle kernel and Newey and Wes (1987, 1994) heeroskedasiciy and auocorrelaion (HAC) consisen sandard errors wih 9 lags, which corresponds o he auomaic bandwidh parameer. Each line in he able repors he regression coefficiens, while he -values are repored in he line below (in brackes). One and wo aserisks indicae a wo-ail es significance level of 5% and 1%, respecively., Day of he week dummy variables Pos-non- Monday by week of he monh The week of he monh Wald χ Monday Mon. Tues. Wed. Thu. Fri. radingbreak 1 s nd 3 rd 4 h (posex. day) 1 s nd 3 rd 4 h (posex. week) Equal days Equal Mondays Equal weeks VIX (U.S.) (1.37 ** ) (-1.16) (5.9 ** ) (3.84 ** ) (3.43 ** ) (19.40 ** ) (-0.47) (0.07) (-1.93) (-4.13 ** ) (-0.71) (-1.11) (-.88 ** ) (-4.71 ** ) p< p< p< VSMI (Swizerland) (19.30 ** ) (0.97) (5.63 ** ) (6.95 ** ) (0.17) (7.77 ** ) (-3.69 ** ) (-1.11) (1.41) (-.58 ** ) (-1.09) (-0.06) (-0.51) (-5.6 ** ) p< p< p< VAEX (Neherlands) (11.55 ** ) (6.03 ** ) (8.71 ** ) (1.93) (-1.6) (5.19 ** ) (1.8) (.75 ** ) (3.66 ** ) (3.7 ** ) (1.30) (-0.07) (-3.01 ** ) (-.58 ** ) p< p=0.136 p< VDAX-NEW (Germany) (13.40 ** ) (3.59 ** ) (6.18 ** ) (5.54 ** ) (-1.64) (14.13 ** ) (-1.05) (0.8) (.6 * ) (1.76) (-1.5) (0.08) (-.78 ** ) (-4.15 ** ) p< p= p< VXJ (Japan) (14.36 ** ) (9.80 ** ) (9.57 ** ) (7.66) (3.6 ** ) (5.03 ** ) (-4.06 ** ) (-1.3) (1.13) (-1.3) (-0.44) (0.39) (-3.80) (-5.88 ** ) p< p< p< VSTOXX (Eurozone) (10.90 ** ) (.16 * ) (6.64 ** ) (3.37) (0.73) (10.97 ** ) (-0.03) (0.44) (.7 ** ) (1.3) (-0.40) (1.31) (-0.91) (-4.95 ** ) p< p= p< VCAC (France) (13.56 ** ) (5.36 ** ) (5.7 ** ) (6.97) (1.7) (6.1 ** ) (0.64) (0.36) (0.99) (-1.16) (-.17 * ) (0.8) (-.8 ** ) (-1.8) p< p= p= VFTSE (U.K.) (13.99 ** ) (.39 * ) (7.48 ** ) (4.56) (0.88) (1.13 ** ) (-0.6) (.04 * ) (3.01 ** ) (1.07) (-.40 * ) (-0.87) (-1.9) (-4.41 ** ) p< p=0.034 p<

42 Table 8: Tess for echnical and mehodological biases The able repors he resuls of he following EGARCH- model running wih alernae conrol variables: V ε z σ, 5 1 1, iday, i TBREAK 3, iholiday, i 4, ivoli, 5, iv i 6, ir i 7, ir i, i 0 i 1 log( σ ) ω' α z γz β log( σ ), 1 1 where V is he volailiy index on day ; DAY, i ( i 1...5) are dummies corresponding o he weekdays; TBREAK is a dummy corresponding o days oher han Monday afer non-rading days; HOLIDAYS, i ( i 1,) are dummies corresponding o one-day and a full monh before holidays; VOL, i ( i 1,) is he price volailiy esimaed by a GARCH(1,) model, or direcly from eiher pas or fuure realized reurns; R is he percenage rae of reurn on he relevan sock index;, z and are he innovaion, sandardized innovaion, and he condiional sandard deviaion; and ω ' ω α1e z 1 is he condiional sandard deviaion consan erm. The innovaions follow he Suden- disribuion. Each line in he able repors he regression coefficiens, while he -values are repored in he line below (in brackes). Robus sandard errors are obained by Bollerslev-Wooldridge Quasi-Maximum Likelihood Esimaes (QMLE). One and wo aserisks indicae a wo-ail es significance level of 5% and 1%, respecively. 40

43 Day of he week variables Oher explanaory variables EGARCH variables Dependen variable Mon. Tues. Wed. Thu. Fri. Pos-non- Monday rading-break One-day before holiday All-monh before holiday ω' α β γ Log- likelihood raio (equal days) A. Tess for ime inerpolaion and opion expiraion dae biases 1. VSMI6M (Swizerland) (8.37 ** ) (4.94 ** ) (6.1 ** ) (6.16 ** ) (4.01 ** ) (4.74 ** ) (-6.4 ** ) (3.7 ** ) (30.30 ** ) (.14 * ) p< VSTOXX6M (Eurozone) (8.81 ** ) (5.36 ** ) (5.46 ** ) (4.93 ** ) (3.60 ** ) (-0.66) ( ** ) (10.6 ** ) ( ** ) (.53 * ) p< VDAX-NEW6M (Germany) (7.03 ** ) (3.34 ** ) (3.8 ** ) (3.10 ** ) (-1.14) (-1.0) ( ** ) (13.1 ** ) ( ** ) (4.66 ** ) p< VXV (U.S.) (3.39 ** ) (1.74) (.73 ** ) (.66 ** ) (1.83) (3.67 ** ) (-4.69 ** ) (5.11 ** ) (67.14 ** ) (1.8) p< VIX (U.S.) B. Tes for holiday bias (17.39 ** ) (4.9 ** ) (5.9 ** ) (6.64 ** ) (1.3) (13.1 ** ) (0.1) (-4.55 ** ) ( ** ) (1.6 ** )(01.10 ** ) (8.44 ** ) p< C. Tess while conrolling pas and fuure price volailiy GARCH Pas Fuure volailiy volailiy volailiy 6. VIX (U.S.) (14.15 ** ) (.18 * ) (3.15 ** ) (4.3 ** ) (-0.69) (13.99 ** ) (1.3) ( ** ) (1.68 ** )(00.90 ** ) (8.43 ** ) p< VIX (U.S.) (16.70 ** ) (3.5 ** ) (4. ** ) (5.46 ** ) (-0.44) (14.04 ** ) (.41 * ) (1.8) ( ** ) (1.56 ** ) ( ** ) (8.56 ** ) p< D. Tess for mehodological biases 8. VXJ-CF (Japan) (13.30 ** ) (4.81 ** ) (5.56 ** ) (4.77 ** ) (1.) (6.59 ** ) ( ** ) (9.48 ** ) ( ** ) (8.0 ** ) p< VDAX (Germany) (1.58 ** ) (5.03 ** ) (4.11 ** ) (4.10 ** ) (1.6) (6.78 ** ) ( ** ) (11.76 ** ) (74.80 ** ) (5.05 ** ) p< E. Tess for sock indices biases 10. VXD (Dow Jones) (11.69 ** ) (.03 * ) (4.16 ** ) (3.85 ** ) (-0.74) (10.07 ** ) ( ** ) (10.85 ** ) (16.0 ** ) (4.58 ** ) p< VXN (NASDAQ 100) (10.97 ** ) (.58 ** ) (3.94 ** ) (3.84 ** ) (0.66) (8.64 ** ) (-11.0 ** ) (10.68 ** ) (15.00 ** ) (5.40 ** ) p< RVX (Russell 000) (9.07 ** ) (1.84) (3.09 ** ) (3.48 ** ) (1.3) (13.5 ** ) (9.07 ** ) (1.84) (3.09 ** ) (3.48 ** ) p<

44 Appendix A: Model specificaion In his Appendix, we deermine he model ha bes fis he VIs daily daa. Firs, we use he Augmened Dickey Fuller (ADF) es (Dickey and Fuller, 1979; and Said and Dickey, 1984) o check for he exisence of a uni roo in he VIs. The es is conduced wih a consan and alernaing number of lags, from zero o lags, which corresponds o a full monh. Taking he number of lags which reveals he smalles es saisic in absolue erms, he hypohesis of an exising uni roo corresponding o six VIs is rejeced a a 1% significance level, and corresponding o wo VIs a a 5%-level. Having rejeced he uni roo hypohesis, we nex search he model among Engle s (198) ARCH, Bollerslevs (1986) GARCH and Nelson s (1991) EGARCH models ha bes fis he daa. Focusing on he VIX, Table A1 compares he resuls of he following alernae models, VIX 1 0 1, ivix i, z, ε /σ ARCH: σ ω α ε, 1 1 GARCH : σ ω α ε α ε β σ β σ, where EGARCH: log( σ ) ω α ( z E z ) γz β log( σ ), (A1) 1 1 VIX is he VIX closing values on days, and, z and are he innovaion, sandardized innovaion and he condiional sandard deviaion, on day. 1 << Inser Table A1 >> To deal wih serial correlaion, he models include he ime-lag VIX series ( VIX i ). Specifically, we es for he firs days lags, which cover a full monh of rading days. However, as in all he models he coefficiens corresponding o lags 13 o are found o be insignifican, he models repored include only he firs 1 lags. Tes 1 repors he resuls corresponding o he mos parsimonious ARCH(1) model. The condiional volailiy coefficiens (ω and α 1 ) are highly significan bu, according o he Ljung Box pormaneau es, he sandardized residuals as well as he squared sandardized residuals are significanly auocorrelaed a various lags. Indeed, in Tes, which corresponds o he GARCH(1,1) model, he GARCH coefficien (β 1 ) is highly significan. In addiion, he Ljung Box saisics show no significan auocorrelaions in he residuals. However, he residuals empirical disribuion

45 reveals exremely high lepokurosis. Indeed, he Jarque-Bera es rejecs he hypohesis ha he residuals follow he normal disribuion. To deal wih he high lepokurosis, Bollerslev (1987), Baillie and Bollerslev (1989) and Baillie and DeGennaro (1990) propose he GARCH- model, which assumes ha he residuals follow a Suden s -disribuion. Tes 3 repors he resuls corresponding o he GARCH-(1,1) model wih Suden s -disribuion. The condiional volailiy coefficiens are highly significan and he BIC and AIC informaion crieria are subsanially smaller han hose obained from he normal GARCH model, which suggess ha his model beer fis he daa. Indeed, he disribuion s degrees of freedom is highly significan. In addiion, in Tess 4 and 5, which correspond o he GARCH-(,1) and GARCH-(1,) models, respecively, he addiional condiional volailiy coefficiens (α and β ) are boh insignifican, he Lagrange Muliplier es saisic for addiional GARCH erm is close o zero, and including hese variables increases he informaion crieria. Thus, adding addiional variables ino he condiional volailiy equaion reduces model performance. While he GARCH models assume a symmeric effec of posiive and negaive shocks o volailiy, many sudies empirically find ha a negaive shock leads o a higher condiional variance in he subsequen period han a posiive shock (see, for example, Chrisie, 198; Pagan and Schwer, 1990 Nelson 1991; and Engle and Ng, 1993). To avoid he symmerical assumpion, which does no conform o he empirical evidence, we employ Nelson s (1991) EGARCH model. Apar from allowing for an asymmerical effec of posiive and negaive shocks, he EGARCH model also avoids he GARCH resricions on he auoregressive coefficiens. Indeed, Pagan and Schwer (1990), Nelson (1991), Henschel (1995) and ohers find ha he EGARCH model beer forecass volailiy han oher models. Tes 6 employs he EGARCH-(1,1) model. Like wih he GARCH model, he condiional volailiy coefficiens are highly significan and, according o he Ljung Box saisics, here are no significan auocorrelaions in he residuals. Moreover, he informaion crieria are much smaller han hose corresponding o he GARCH model. Thus, we find ha he EGARCH-(1,1) model wih Suden s -disribuion bes fis he VIX ime series as i handles all saisical issues which may bias he resuls. 1 In unrepored ess, we find very similar resuls wih all oher VIs. The only 1 The sandardized residuals in Table 1 reveal some skewness. In unrepored ess, we repea he main ess of his sudy while assuming an EGARCH- model wih asymmeric Suden s -disribuion. As 43

46 difference beween he VIs is he number of significan auoregressive lag variables, which is always smaller han 13. Therefore, in he analysis of his sudy, we employ he EGARCH-(1,1) model wih 1 auoregressive lag variables. he resuls in hose ess are very similar o hose corresponding o he Suden s -disribuion hey are no repored, bu are available upon reques from he auhors. 44

47 Table A1: Model specificaion The able repors he resuls of he following alernae regressions: where VIX and i VIX z /, 1 0 1, ivix i, ) ω' α1 ARCH: σ ω α ε, GARCH : ω , EGARCH: log( σ z γz β log( σ ), VIX are he VIX values on day and i ; i ranges from 1 o 1, where in all ess no lag larger han 1 is found o be significan; z is he sandardized innovaion, is he innovaion and is he condiional sandard deviaion, all on day ; and ω ' ω α1e z 1 is he condiional sandard deviaion consan erm corresponding o he EGARCH model. The invocaions are assumed o follow eiher he sandardized normal or Suden- disribuion (GARCH- model). Each line in he able repors he regression coefficiens, while he -values are repored in he line below (in brackes). Robus sandard errors are obained from he Bollerslev-Wooldridge Quasi-Maximum Likelihood Esimaes (QMLE). The BIC and AIC denoe he Schwarz s (Bayesian) and Akaike s Informaion Crieria. The Ljung Box pormaneau es saisics Q and Q es for he hypohesis ha he firs 1, 10 and 0 auocorrelaion coefficiens corresponding o he sandardized residuals and squared residuals, respecively, are simulaneously equal o zero, where he p-values are repored in he line below (in brackes). The las six lines repor he descripive saisic corresponding o he sandardized residuals and he Jarque-Bera saisic, which ess he hypohesis ha he residuals skewness and he kurosis in excess o ha corresponding o he sandard normal disribuion boh equal zero. Finally, one and wo aserisks indicae a wo-ail es significance level of 5% and 1%, respecively. 45

48 Model: 1. ARCH(1) (Normal dis.). GARCH(1,1) (Normal dis.) 3. GARCH(1,1) (Suden-) 4. GARCH(,1) (Suden-) 5. GARCH(1,) ( Suden-) 6. EGARCH(1,1) (Suden-.) Consan (5.63 ** ) (4.61 ** ) (5.11 ** ) (5.11 ** ) (5.11) (8.46 ** ) AR (30.87 ** ) (51.80 ** ) (63.09 ** ) (6.95 ** ) (6.76) (68.39 ** ) AR (-3.09 ** ) (0.47) (-0.34) (-0.36) (-0.38) (-0.6) AR (0.38) (-0.17) (1.06) (1.06) (1.05) (1.0) AR (-0.76) (0.91) (0.4) (0.5) (0.7) (0.8) AR (1.36) (1.69) (.67**) (.67 ** ) (.65) (.45 * ) AR (-1.43) (-.47 * ) (-.48**) (-.48 * ) (-.47) (-.9*) AR (1.41) (1.87) (1.31) (1.31) (1.31) (1.31) AR (-1.34) (-0.41) (-0.97) (-0.9) (-0.8) (-0.5) AR (1.5) (1.3) (.07 ** ) (.05) (.03) (.31 * ) AR (3.59 ** ) (.37 * ) (.17 * ) (.18 * ) (.19) (.1 * ) AR (-4.98 ** ) (-3.39 ** ) (-3.07 ** ) (-3.07 ** ) (-3.07) (-3.8 ** ) AR (.96 ** ) (.85 ** ) (3.94 ** ) (.95 ** ) (.95) (1.98 * ) GARCH variables ω (16.16 ** ) (3.05 ** ) (4.51 ** ) (4.49 ** ) (4.00 ** ) ω' (-7.15 ** ) α (8.86 ** ) (6.13 ** ) (8.9 ** ) (7.45 ** ) (6.04 ** ) (5.85 ** ) α (6.58 ** ) (-0.96) β (34.50 ** ) (53.31 ** ) (0.95) (46.00 ** ) (384.4 ** ) β γ (14.44 ** ) Suden- disribuion parameer Degrees of freedom (15.65 ** ) (15.64 ** ) (15.64 ** ) (17.5 ** ) Informaion crieria BIC AIC Residual diagnosics Ljung-Box Q(1,10,0) 15 **, 7 **, 87 ** 0.6, 3.3, , 3., 1 0.6, 3., 1 0.6, 3.6, ,., 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 Q (1,10,0) 7.6 **, 4 **, 667 ** 0.0, 7.0, , 4.6, , 4.5, , 4.4, 9.0 0,1, 4., 8.5 p-value (0.00, 0.00, 0.00) (0.95, 0.73, 0.95) (0.81, 0.91, 0.98) (0.71, 0.9, 0.98) (0.63, 0.93, 0.98) (0.77, 0.94, 0.99) Mean Sandard Deviaion Skewness Excess Kurosis Jarque-Bera es p-value (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) 46