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

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

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