The predictive power of volatility models: evidence from the ETF market

Transcription

1 The predicive power of volailiy models: evidence from he ETF marke AUTHORS ARTICLE INFO JOURNAL FOUNDER Chang-Wen Duan Jung-Chu Lin Chang-Wen Duan and Jung-Chu Lin (4). The predicive power of volailiy models: evidence from he ETF marke. Invesmen Managemen and Financial Innovaions, () "Invesmen Managemen and Financial Innovaions" LLC Consuling Publishing Company Business Perspecives NUMBER OF REFERENCES NUMBER OF FIGURES NUMBER OF TABLES The auhor(s) 8. This publicaion is an open access aricle. businessperspecives.org

2 Invesmen Managemen and Financial Innovaions, Volume, Issue, 4 Chang-Wen Duan (Taiwan), Jung-Chu Lin (Taiwan) The predicive power of volailiy models: evidence from he ETF marke Absrac This sudy uses exchange-raded fund (ETF) daa o invesigae he abiliy of he ime-series volailiy model, he implied volailiy model, and he inraday reurn volailiy model o forecas reurn volailiy. Among various ETFs, we adop NASDAQ Index Tracking Sock (QQQ) as he sample because i has corresponding volailiy index (VIX) issued which is necessary. The resuls show ha all volailiy models applied in his sudy can reliably forecas volailiy. The Glosen-Jagannahan-Runkle GARCH model is superior o he GARCH model, implying ha he reurn volailiy of QQQ is asymmeric. Among he added incremenal informaion, QQQ Volailiy Index (QQV) of he American Sock Exchange has beer abiliy in forecasing he reurn volailiy of QQQ, followed by he NASDAQ Volailiy Index (VXN) of he Chicago Board Opions Exchange, and hen by he inraday reurn volailiy. The probable reason is ha he urnover of QQQ opions is higher han ha of he NASDAQ Index Opions (NDX) and causes QQV o conain subsanially more informaion han VXN and o predic volailiy beer. We also find he predicive power of he ime-series GARCH model is weaker han ha of he volailiy model wih QQV embedded as incremenal informaion. Since QQQ, as an ETF, has diversified is non-sysemaic risks, he GARCH model using non-sysemaic risk informaion o predic volailiy is ineviably worse han ha using implied volailiy. Idenical resuls are achieved when examining ou-of-sample forecasing performance. Keywords: volailiy model, implied volailiy, volailiy index, incremenal informaion. JEL Classificaion: G4, G7. Inroducion Developing a feasible volailiy model o assis wih describing and predicing he volailiy of reurns on financial asses has long been a focus of research. This is because volailiy underpins he risks, pricing, and allocaion of asses. Among he various incremenal informaional variables embedded in volailiy models, i is widely believed ha if he opions marke is informaionally efficien, implied volailiy exraced from opions prices is he opimal predicor of fuure volailiy (implied volailiy hypohesis). However, he empirical evidence on wheher opions prices or hisorical daa in ime-series models conains much more informaion abou fuure volailiy is mixed. Lamoureux and Lasrapes (993) rack individual sock opions o es several volailiy models. Their resuls rejec he orhogonaliy resricion ha he forecas from ime-series models should no have predicive power on op of implied volailiy and are hus inconsisen wih he implied volailiy hypohesis. Such conradicion moivaes a lo of subsequen sudies o explore he performance of implied volailiy using differen daa and more complee mehods. According o Rubensein (994), fundamenal srucural change occurred in opions markes afer he US marke crash of Ocober 987. Opions prices conain much more valuable informaion han oher asse prices since hen. Relevan sudies, e.g. Day and Lewis (99), Chrisensen and Prabhala Chang-Wen Duan, Jung-Chu Lin, 4. (998), Fleming (998), and Mayhew and Sivers (3), among ohers, find ha opions wih higher rading volume provide more informaion on fuure volailiy. To improve he predicive abiliy of imeseries volailiy models, sudies also include highfrequency daa in he models. A key finding is ha if he research samples consis of individual socks, indices, or opions wih high rading volume or high frequency daa, he volailiy models using hese samples gain informaional superioriy over hose models ha use only hisorical reurn daa (Andersen and Bollerslev, 998; Andersen, Bollerslev, Diebold and Ebens, ; Mayhew and Sivers, 3). Pas sudies focus heir aenion on individual socks or composie indices while esing he implied volailiy hypohesis. Exchange-raded fund (ETF), as a highly innovaive and convenien insrumen for spo index rading, has become he highligh of marke ransacions. Ye few sudies arge he ETF marke o invesigae he predicabiliy of volailiies exraced from various models. To address he lieraure gap, his paper aims o explore he performance of various volailiy models using ETF daa. Among various ETFs, we adop an ETF ha has corresponding opions issued as sample in order ha he implied volailiy can be calculaed. Moreover, wo ypes of opions correspond o he ETF: one is based on he underlying index he ETF rack, and he oher is based on he ETF iself. This sudy hus chooses an ETF ha has boh ypes of opions issued in order o explore implied volailiy of which ype can beer predic he reurn volailiy. Finally, since ETF, as a

3 Invesmen Managemen and Financial Innovaions, Volume, Issue, 4 fund racking some index, has diversified is nonsysemaic risks, we are ineresed in invesigaing wheher ime-series volailiy models which include only non-sysemaic risk informaion, e.g. he generalized auoregressive condiional heeroscedasiciy (GARCH) model, can provide all relevan informaion on fuure volailiy of ETFs. American Sock Exchange (AMEX) issued a volailiy index (VIX) based on NASDAQ Index Tracking Sock (QQQ) opions on January 7,. This VIX, known as he AMEX QQQ Volailiy Index (QQV), is hus an index for he implied volailiy of QQQ opions. On he oher hand, he underlying index ha he QQQ racks, i.e. he NASDAQ Index, has corresponding NASDAQ Index Opions (NDX) issued. Chicago Board Opions Exchange (CBOE) also compiles a volailiy index, known as he CBOE NASDAQ Volailiy Index (VXN), o represen he implied volailiy of NDX. Therefore, he racker fund, QQQ, conform o he sample selecion sandard of his sudy ha an ETF has wo ypes of opions issued a he same ime. Moreover, according o he evidence of pas lieraure, sudies using acively raded commodiies as samples can achieve consisen and meaningful resuls. The ETF QQQ, on his poin, has exremely high rading volumes compared o any oher sock, e.g. on Sepember 4, 7, he rading volume of QQQ on he NASDAQ reached US$99,8,65, ranking he second highes. We hus adop QQQ as he sample o invesigae he predicive power of volailiy models and o explore implied volailiy exraced from which ype of opions can beer predic he reurn volailiy. The resuls conribue o he horough undersanding of he predicive power of various volailiy indices. We begin he analysis by comparing he abiliy of wo ime series models, he GARCH and Glosen- Jagannahan-Runkle GARCH (GJR-GARCH) models, o explain reurn volailiy. Nex, we inroduce differen volailiy indices ino models and compare heir incremenal informaion effecs. Furher, we observe wheher 5-minue inraday reurns provide beer informaion for he volailiy of he underlying asse reurns han he lags of daily reurn volailiy. The resuls show ha all volailiy models applied in his paper have forecasing abiliy. The GJR-GARCH model is superior o he GARCH model, which implies ha he reurn volailiy of QQQ is asymmeric. In erms of incremenal informaion from volailiy indices, he AMEX was merged wih NASDAQ in 998 and all is hree earlies ETFs were moved o NASDAQ o rade. Glosen, Jagannahan and Runkle (993). model embedding QQV is beer han ha incorporaing VXN. Finally, we find ha since ETF has diversified is non-sysemaic risks, he ime series model using he lag of he error erm o predic ETF volailiy is ineviably worse han ha using implied volailiy. This aricle proceeds as follows. Secion reviews he lieraure. Secions and 3 describe he research models and mehodology, respecively. Secion 4 presens he empirical resuls. The las secion presens he conclusions.. Lieraure review The lieraure regarding he volailiy of sock reurns usually assumes sock reurns sochasic and normally disribued and assumes he variance of sock reurns consan. The assumpion of consan variance, however, is called ino quesion by many researchers. Fama (965) considers he disribuion of sock prices lepokuric and faailed and he changes in prices of sock no independen. If a larger volailiy appears in a paricular period, anoher larger volailiy will follow in a subsequen period, known as volailiy clusering. Therefore, he reurn volailiy should no be a consan. Morgan (976) finds he variance of sock reurns varying over ime and demonsraes he heeroscedasiciy of reurn volailiy on sock ime series. To consider he heeroscedasiciy of volailiy, Engle (98) develops he auoregressive condiional heeroscedasiciy (ARCH) model. The model defines he disribuion of condiional error erms as a normal disribuion and les he condiional variance have a linear relaionship wih he square of pas error erms. Engle (98) finds ha he ARCH model no only improves he predicive performance of he ordinary leas square mehod, bu also acquires a more accurae forecas of variance. Bollerslev (986) adds condiional variance o he ARCH model, exending he ARCH model o he GARCH model. The GARCH model makes he lag srucure of he condiional variance more flexible and reasonable. Considering ha he reurn volailiy varies over ime, Domowiz and Hakkio (985) use he ARCH model o fi he ime-varying variance and esed he exisence of ime-varying risk premium. Such a model is he so-called ARCH-Mean model. Bollerslev, Engle and Wooldridge (988) exend he ARCH-Mean model o he GARCH-Mean model. Black (976), Chrisie (98), and Schwer (99) indicae ha he disribuion of reurn volailiies was asymmeric. They demonsrae ha negaive reurn shocks have larger effecs on reurn volailiies han posiive reurn shocks. They also poin ou ha he asymmery of volailiy could be

4 Invesmen Managemen and Financial Innovaions, Volume, Issue, 4 explained by he financial leverage effec. In oher words, negaive reurn shocks make sock prices decline, making he deb/equiy raio rise and resuling in increased fuure reurn volailiy on socks. On he conrary, a posiive reurn shock will decrease he volailiy. Therefore, in order o describe precisely he asymmery of reurn volailiy, asymmeric models based on he GARCH model are subsequenly proposed, e.g. asymmeric GARCH (AGARCH) of Engle (99), exponenial GARCH (EGARCH) of Nelson (99), hreshold GARCH (TGARCH) of Zakoian (994), and GJR model of Glosen, Jagannahan and Runkle (993). Some sudies have demonsraed he implied volailiy hypohesis o be rue. There is a school of opinion aribues he resul o he fac ha while esimaing ime-series volailiy models, hese sudies use only closing price daa of socks in heir compuaion. For insance, Laane and Rendleman (976) and Schmalensee and Trippi (978) find ha he power of implied volailiy is superior o ha of hisorical volailiy on forecasing realized volailiy. Chiras and Manaser (978) show ha if dividend yields are incorporaed, he predicive power of implied volailiy is no longer significanly beer han ha of hisorical volailiy. The evidence is mixed regarding he implied volailiy hypohesis. Day and Lewis (99) use daa of opions on S&P index fuures o compare he predicive powers of implied volailiy, hisorical volailiy, GARCH model, and EGARCH model. They demonsrae ha he ime series models migh provide subsanially more informaion han he implied volailiy model. Lamoureux and Lasrapes (993) exploi daa of sock opions on CBOE and obain a conclusion idenical o ha of Day and Lewis (99). Canina and Figlewski (993) adop a daa sample drawn from he se of weekly selemen prices of all call opions on he S&P Index from 983 o 987. They find implied volailiy o be a biased and inefficien esimaor and incapable of gahering he informaion he hisorical volailiy conains. Becker, Clemens and Whie (7) and Becker and Clemens (8) also indicae ha hisorical daa subsume imporan informaion ha is no incorporaed in opion prices, suggesing ha implied volailiy has poor performance on volailiy forecasing. However, he implied volailiy from he index opion has been widely sudied and oally differen resuls are obained. Jorion (995) repors ha implied volailiy is superior o hisorical reurn volailiy in erms of boh predicive power and he exen of informaion conen using he daa of opions on foreign currency fuures. Chrisensen and Prabhala (998) adop he same S&P Index opions as hose of Canina and Figlewski (993) as heir sample and acquire exacly he opposie resuls. They find ha no only he predicive power of implied volailiy is superior o ha of hisorical volailiy, bu also he implied volailiy incorporaes subsanially more informaion on fuure volailiy. Chrisensen and Hansen () furher include boh in-he-money and ou-of-he-money opions on he S&P Index o consruc a rade weighed average of implied volailiies. They also incorporae he daa of he pu opion. Their resuls are idenical o hose of Chrisensen and Prabhala (998). In recen sudies, he empirical evidence from Becker, Clemens and McClelland (9) and Frijns, Tallau and Tourani- Rad () documens ha he implied volailiies from index opions can capure mos of he relevan informaion in he hisorical daa. Some sudies aemp o uncover he predicive informaion from inraday daa o forecas reurn volailiy. Andersen and Bollerslev (998) use ick daa o compare he predicive powers of ARCH and sochasic volailiy models over volailiy. They find ha boh models provide superior volailiy forecass and he use of he high-frequency daa conribues o he accuracy of volailiy measuremens. Andersen, Bollerslev, Diebold and Labys () believe ha he five-minue horizon of inraday daa is shor enough o make he esimaed realized volailiies free from measuremen error. Moreover, Andersen, Bollerslev, Diebold and Ebens () focus on inraday daa for 3 socks in he Dow Jones Indusrial Average o observe he disribuions of realized volailiies. They find ha he disribuions of realized volailiies are highly righskewed, implying asymmery of reurn volailiies. Blair, Poon and Taylor () choose he S&P Index as he underlying o evaluae he predicive power of reurn volailiies. They incorporae implied volailiies and 5-minue high frequency daa o compare heir informaion conen in he GJR model. They find ha high frequency reurn daa are able o enhance he adapive and predicive abiliies of models, bu ha he implied VIX of he S&P Index opions provides he mos accurae forecass. Engle and Ng (993) using he Japanese sock reurn o compare he EGARCH and GJR models find ha he GJR is he bes a parsimoniously capuring he asymmeric effec. Mayhew and Sivers (3) expand heir sample o cover 5 individual socks, which are divided ino hree sub-samples by rading volume. They employ GARCH and GJR ime series models, implied volailiy model, and highfrequency inraday reurn daa o compare he forecasing power of incremenal informaion over A-he-money opions.

5 Invesmen Managemen and Financial Innovaions, Volume, Issue, 4 reurn volailiy. Their resuls indicae ha he implied volailiy reliably ouperforms he GARCH model for boh high and low rading volume socks. In addiion, he implied volailiy of index opions subsumes reliable incremenal volailiy informaion and he GARCH model can explain he reurns of individual socks wihou corresponding opions issued. For hose socks ha have insufficien volailiy informaion conen because of he lower liquidiy of he corresponding opions, he implied volailiy of index opions also provides superior incremenal informaion abou fuure firm-level volailiy. Liu and Hung () also invesigae he performance of various volailiy forecass for he S&P sock index series. They compare he symmeric GARCH model wih hree differen ypes of disribuions agains GJR-GARCH and EGARCH models. Their empirical resuls indicae ha he GJR-GARCH model achieves he mos accurae volailiy forecass.. Empirical models The sudy uses he ime-series volailiy model, he implied volailiy model, and he high-frequency inraday reurn volailiy model o examine he predicive power of various volailiies. Among imeseries models, we adop GARCH, a widely acceped propery of volailiy, as he basis of he model. According o he empirical evidence of Baillier and Bollerslev (989), Bollerslev (987), Engle and Bollerslev (986), and Hsieh (989), he exisence of volailiy clusering in speculaive reurns is ubiquious. Many of hese sudies find ha he simple GARCH (, l) model provides a decen firs approximaion of he observed emporal dependencies in daily daa. Thus, we adop he GARCH (, ) model as our ime-series volailiy model. The model for reurn dynamics is se and esimaed as follows: R R () ~N(, h ) () h i ( + D ) L 5R5min, + 4 IVindex,, L IV 3 ETF, + (3) where R = ln(p /p - ), p is he spo price of underlying asse,, and are parameers of he model, is he error erm, h is he funcion of generalized condiional reurn heeroscedasiciy; D is a dummy variable ha akes a value of if - <, oherwise a value of zero; L indicaes he lag operaor, IV is implied volailiy, R 5min is he sum of squared 5-minue inraday reurns of he ETF and a proxy for realized volailiy. By resricing, 3, 4 and 5 in equaion (3) o a value of zero, his specificaion ness he GARCH (, ) model: h. L (4) Allowing for he asymmeric phenomenon of financial asse reurns ha GARCH (, ) is unable o describe, we resric 3, 4 and 5 in equaion (3) o a value of zero and hus urn his equaion ino he sandard GJR-GARCH (, ) model: ( + D) h. L (5) This equaion is divided ino wo subses by he posiive/negaive of reurn errors. We can gauge, hrough he significance of, wheher he explanaory abiliy of each error erm of he wo subses is significanly differen from each oher. The significance of indicaes he asymmery of volailiies. For purposes of discussing he predicive effecs of implied volailiy on sock-reurn volailiy, we resric,, 4 and 5 in equaion (3) o a value of zero and hus urn he condiional heeroscedasiciy model ino a single-facor volailiy model ha considers only one facor, he implied volailiy of he underlying: (6) 3 h, 3 IVETF, where IV ETF is he implied variance from opions on he ETF; he saisical significance of 3 reveals ha he implied volailiy has sufficien informaion conen o predic reurn volailiy. Resricing, 5 and in equaion (3) o a value of zero, we ge: h IV IV 3 3 ETF, 4 index, L (7) which is an equaion for modeling GARCH, bu incorporaing wo oher variaions o compare he effecs of incremenal informaion: IV ETF is he implied volailiy from opions on he ETF, and IV index is he implied volailiy from opions on he index. Mayhew and Sivers (3) indicae ha comparing he values of 3 and 4 can help assess which has beer predicive power. Tha is o say, he significance of eiher coefficien implies ha he respecive variaion has sufficien informaion conen for volailiy forecasing. 3

6 Invesmen Managemen and Financial Innovaions, Volume, Issue, 4 Andersen and Bollerslev (998) believe ha he incorporaion of high frequency inraday daa in a volailiy model can enhance he model s abiliy o explain reurn volailiy. Therefore, we subsume boh he inraday reurns and implied volailiies o find ou which has he more powerful incremenal informaion. In accordance wih Blair e al. (), we choose a 5-minue frequency o sieve reurns from everyday rading daa for he period 8:3 o 5: CST, hen square and sum hese 5-minue reurns o proxy for inraday reurn volailiy; furhermore, in consideraion of he overnigh effec, he rading daa afer 5: of he previous day are also included o compue he inraday volailiy of each day. This volailiy model is as follows: R h min, 3 IVETF, L L, (8) where R 5min is he inraday volailiy based on 5- minue reurns. If 5 is significan, hen he informaion ha inraday daa conain has forecasing power. Mayhew and Sivers (3) also apply he above model, bu allow for lagged daily reurn shocks ( ) and lagged 5-minue sum of squared reurns ( R 5min, ) o have a differen lagged decay srucure. Thus, his specificaion enables one o compare volailiy informaion from daily reurn shocks versus he inraday reurn volailiy of differen lags. According o evidence from Mayhew and Sivers (3), reurn-shocks and inraday reurn volailiy of older lags add essenially no explanaory power. Therefore, we resric o a value of zero and discuss only he power of inraday reurn volailiy of lag o explain daily reurn volailiy while verifying he GARCH+V5 model, a GARCH (, ) model wih inraday reurn volailiy as is incremenal informaion; ha is, and older reurn-shock and inraday reurn volailiy are no considered. Such an empirical mehod does no model he GJR asymmery (Mayhew and Sivers, 3). 3. Mehodology AMEX and CBOE began issuing QQQ opions on March 3, 999 and February 7,, respecively. AMEX sared o issue QQV on January 3,. To ease he shocks ha he issuances of QQQ opions and QQV have creaed in he marke, he daa period begins a March 3,, wo monhs afer he launch of QQV, and ends a June 3, 3. A oal of 7 rading monhs of he daa period provide 563 daily observaions. Daily daa are used and high-frequency inraday price daa of QQQ are US Cenral Sandard Time. gahered o compue inraday 5-minue reurn volailiy. In addiion, due o he measuremen errors of implied volailiy based on opion pricing heory and he smile effec exhibied by he implied volailiy, we use wo VIXs, QQV of QQQ opions and VXN of NASDAQ Index opions, compiled wih he Whaley (993) mehod and by CBOE, o avoid he measuremen errors. Opions daa are from he Prophne Company in he Unied Saes and inraday daa are from he Tickdaa Company. Whaley (993) uses he implied volailiies of eigh near-he-money opions o calculae an implied volailiy index. There are four calls and four pus in his sample and pairs of neares-he-money exercise prices are chosen o calculae a weighed average of eigh implied volailiies. As such, he implied volailiy of an a-he-money opion wih a consan rading days o expiry could be consruced. We also have an ou-of-sample predicion period for comparing he forecasing abiliy of he empirical models. Since volailiy index, however, is based on he a-he-money opions wih rading days o expiry, here we allow an ou-of-sample predicion period o have rading days o expiry also o faciliae he comparison of he predicive performance. The numerical mehod of Bernd, Hall, Hall, and Hausman (974) is applied o esimae he model parameers. The log-likelihood funcion value (Log-L) of Bollerslev and Wooldridge (99) is used in order o compare he fi abiliy of he models. As regards he issue of comparing he incremenal informaion of various volailiies, he likelihood raio (LR) ess are employed o execue ess. Finally, hree es saisics, mean absolue error (MAE), roo mean square error (RMSE), and mean absolue percenage error (MAPE), are compued o examine he predicive power of he models while comparing he ou-of-sample forecas performance. 4. Empirical resuls If a ime series is non-saionary, he execuion of a regression will cause spurious regression. We use an augmened Dickey-Fuller (ADF) es o es he saionary of he daa series. In addiion, we perform normaliy, sabiliy, auocorrelaion, and heeroscedasiciy ess on variaions. Table presens he descripive saisics for QQQ. Observing he skewness and kurosis coefficiens, we find ha he disribuion of QQQ price reurns is boh righ-skewed and lepokuric. The Jarque-Bera es also indicaes ha he disribuion is no normal. The resuls evidence he fa-ailed characerisic of QQQ price reurns and hus using ARCH or GARCH o describe he heeroscedasiciy of price reurns appears correc. 4

7 Table. Basic saisics of QQQ reurns Saisic variables Saisic values Sample size 584 Mean.8 Sandard deviaion.73 Maximum.63 Minimum Skewness coefficien.9 Kurosis coefficien Jarque-Bera es value 3.93** Noe: *, **, and *** indicae significance a he %, 5%, and % levels, respecively. The resuls of he sabiliy, heeroscedasiciy, and auocorrelaion ess are repored in Table. The resul of he uni roo es shows ha boh daily price reurns and inraday price reurns rejec he exisence of uni roo, indicaing heir saionary. The Q saisic es of Ljung and Box (978) shows ha he Q saisic is saisically significan in lag 6,, 8, 4, and 3 of he square of residual erms, indicaing he exisence of heeroscedasiciy in residual variance. Furhermore, he resuls of he ARCH-LM es (Engle, 98) show he significance of he LM saisic and hus demonsrae he heeroscedasiciy of price reurn residuals. Observing he auocorrelaion of residuals, while seing he number of lag erm 4, he correlaion coefficiens of lag,, 3, 9, and 33 are saisically significan, demonsraing he auocorrelaion of reurn residuals. Table. Saisical es of QQQ Inercep and rend included ADF s uni roo es Only inercep included Daa ype Lag Tes saisic Lag Tes saisic Price reurn -8.67*** -8.65*** Inraday reurn *** *** Model: R R, Invesmen Managemen and Financial Innovaions, Volume, Issue, 4 Ljung and Box Q es for heeroscedasiciy Auocorrelaion es Q(lag) Saisic Lag Correlaion Q(6) 77.39*** ** Q() 4.3*** ** Q(8) 74.8*** 3.893** Q(4) 96.59*** ** Q(3) 37.4*** ** ARCH-LM es T R saisic 5.69*** Noe: ADF uni roo es uses AIC rule o choose he bes lag erm. ARCH-LM is he ARCH (6) saisic; *, **, and *** indicae significance a he %, 5%, and % levels, respecively. Table 3 presens he empirical resuls of he GARCH, GJR-GARCH and QQV volailiy models. The coefficien of GJR-GARCH is saisically significan, revealing he asymmery of QQQ reurn volailiy. Comparing he GARCH, GJR-GARCH, and QQV models, he Log-L value of he QQV model is he highes, showing ha his model has he bes predicive power over oher reurn volailiy models. Furher, we use LR saisics o compare he incremenal informaion effecs of GARCH and GJR-GARCH models wih QQV added and find ha boh LR values of he wo models are saisically significan, demonsraing ha adding QQV ino he models can enhance he abiliy o forecas fuure reurn volailiy. Therefore, QQV has incremenal informaion for predicing he reurn volailiy of QQQ. Comparing he Log-L values of hese wo models, he GJR-GARCH model wih QQV embedded provides beer predicive power han he GARCH model wih QQV embedded. Neverheless, he Log-L values of hese wo models are abou he same. The coefficien of QQV, 3, is saisically significan only when i is embedded in a GARCH model. Table 3. GARCH, GJR-GARCH and he condiional variance models wih incremenal informaion QQV ~ N(, h), ( + D ) R min h 3 IVETF, 4 IVindex, L L 5 5, where R = ln(p /p - ), p is he spo price of an underlying asse,, and are parameers of he model, is he error erm, h is he funcion of generalized condiional reurn heeroscedasiciy; D is a dummy variable ha akes on a value of if - <, oherwise i akes on a value of ; L indicaes he lag operaor; IV ETF is QQV, an implied volailiy for QQQ opions; IV index is VXN, an implied volailiy for NASDAQ Index opions; R is he sum of squared 5-minue inraday reurns and proxy for inraday volailiy. 5min, Coefficien -4 3 Model GARCH GJR-GARCH QQV GARCH + QQV GJR-GARCH + QQV.538 (.97).48*** (.8).346 (-.8) -.7* (-.48).84*** (3.49) *** (-6.).56*** (.9) -.85*** (-.78) -. (-.66).34* (.6) -.34* (-.) -.5 (-.93).97** (.89).3 (.3) We use he Akaike s Informaion Crierion (AIC) rule, recommended by Engle and Yoo (987), o choose he bes lag erm. 5

8 Invesmen Managemen and Financial Innovaions, Volume, Issue, 4 Table 3 (con.). GARCH, GJR-GARCH and he condiional variance models wih incremenal informaion QQV Coefficien Model GARCH GJR-GARCH QQV GARCH + QQV GJR-GARCH + QQV *** (44.5) 968*** (7.5).449* (.8).98*** (.5) Log-L LR 7.5*** 3.55*** Noe: Values in parenheses are saisics calculaed wih Bollerslov-Wooldrige robus sandard errors; *, **, and *** indicae significance a he %, 5%, and % levels, respecively. QQQ is an ETF racking NASDAQ- Index. Table 4 shows a very high correlaion beween QQV and VXN. The correlaion coefficien of he wo VIX is as high as Figure displays he movemens of he QQV, VXN, and QQQ prices. Also, eviden from his figure is he high correlaion beween QQV and VXN. In ligh of Mayhew and Sivers (3), he implied volailiy subsumes reliable informaion conen abou he reurn volailiy of he underlying. Therefore, we furher observe wheher VXN, a VIX based on he NASDAQ Index, also subsumes informaion abou he price reurn volailiy of QQQ and compare VXN wih QQV. Table 4. Correlaion coefficiens marix of QQQ price reurns, QQV and VXN QQQ QQV VXN QQQ.8.65 QQV VXN Noe: The daa period sars from /3/3 o 3/6/3. Implied volailiy QQV VXN QQQ Price /3/ /4/ /5/ /6/ /7/ /8/ /9/ // // // // // /3/ /4/ /5/ /6/ /7/ /8/ /9/ // // // 3// 3// 3/3/ 3/4/ 3/5/ 3/6/ Table 5 presens he resuls of execuing volailiy models wih VXN. All he coefficien 4 s are saisically significan, indicaing ha he VIX of he NASDAQ- Index has predicive power over QQQ reurn volailiy. This oucome is idenical o ha of Mayhew and Sivers (3). Compared o Table 3, he Log-L value of he VXN model is 84.93, higher han ha of he GARCH (, ) model, indicaing ha he power o predic reurn volailiy in he VXN model is beer han ha of GARCH. Comparing he power o predic reurn 6 day Fig.. The dynamics of QQQ prices, QQV and VXN volailiy of he four models in Table 5, he GARCH model wih boh QQV and VXN has he sronges power and he mos incremenal informaion. If only comparing he exen o which wo variaions, QQV and VXN, affec reurn volailiy, we find ha he coefficiens in model QQV + VXN are all saisically significan, 3 equals.3, 4 equals.3, and 3 > 4, indicaing ha QQV s marginal explanaory power for QQQ reurn volailiy is higher han ha of VXN. This oucome is idenical o ha in model GARCH + QQV + VXN.

9 Model: R Invesmen Managemen and Financial Innovaions, Volume, Issue, 4 Table 5. GARCH and he condiional variance model wih incremenal informaion QQV and VXN ~ N(, h), ( + D ) R min h 3 IVETF, 4 IVindex, L L R, 5 5, where R = ln(p /p - ), p is he spo price of an underlying asse,, and are parameers of he model, is he error erm, h is he funcion of generalized condiional reurn heeroscedasiciy; D is a dummy variable ha akes on a value of if - <, oherwise i akes on a value of ; L indicaes he lag operaor; IV ETF is QQV, an implied volailiy for QQQ opions; IV index is VXN, an implied volailiy for NASDAQ index opions; R is he sum of squared 5-minue inraday reurns and proxy for inraday volailiy. 5min, -4 Coefficiens Models VXN GARCH + VXN QQV + VXN GARCH + QQV + VXN 5.93*** (9.6) -7.6** (-.) (-.7) *** (-5.9) -8.75*** (-6.) -.49** (-.67) 3 4.5*** (.6).67** (.8).3*** (.83).3* (.58).48*** (4.33).9*** (.96) 5.3 (.38).69 (.9) Log-L LR 8.5*** *** Noe: Values in parenheses are saisics calculaed wih Bollerslov-Wooldrige robus sandard errors; *, **, and *** indicae significance a he %, 5%, and % levels, respecively. Finally, we incorporae inraday 5-minue reurn volailiy daa ino he models o observe wheher inraday informaion enhances he modeling and forecasing of he QQQ reurn volailiy. Table 6 shows ha he coefficien 5 of he inraday 5- minue reurn volailiy (V5) model is saisically significan, revealing ha inraday reurn volailiy can predic he volailiy of QQQ reurns. Compared o ha of he GARCH (, ) model in Table 3, he Log-L value of he V5 model is and is larger han ha of he GARCH (, ) model, which uses daily daa, indicaing ha using high-frequency daa could indeed reduce noise from price reurns and enhance he abiliy o forecas he volailiy of reurns. Compared o he QQV model in Table 3 and he VXN model in Table 5, he Log-L value of he V5 model is he smalles, showing ha he V5 model does no provide beer informaion conen han he QQV or VXN models. The model ha has he highes Log-L value is he QQV model. Furhermore, we add he inraday reurn volailiy variable informaion o he GARCH model. The coefficien 5 of GARCH + V5 in Table 6 is saisically significan he coefficien of he residual erm is smaller han ha of GARCH, and drops from.943 in GARCH o.645 in GARCH + V5, indicaing ha he model wih exra volailiy informaion on QQQ inraday reurns has a beer Referring o Andersen and Bollerslev (998). abiliy o forecas reurn volailiy han GARCH. In erms of he incremenal informaion effecs of inraday reurns, he LR saisic of he GARCH + V5 model in Table 6 is 9.6 and is saisically significan, showing ha inraday reurn volailiy can indeed provide more informaion, increasing he abiliy of GARCH (, ) o forecas volailiy. This oucome is idenical o hose of Blair e al. (), and Mayhew and Sivers (3). Table 6. GARCH, ARCH and he condiional variance model wih inraday incremenal informaion Model: R 4 index, R, ~ N (, h ), ( + D) h 3 IVETF, L IV R L 5 5 min,, where R = ln(p /p - ), p is he spo price of an underlying asse,, and are parameers of he model, is he error erm, h is he funcion of generalized condiional reurn heeroscedasiciy; D is a dummy variable ha akes on a value of if - <, oherwise i akes on a value of ; L indicaes he lag operaor; IV ETF is QQV, an implied volailiy for QQQ opions; IV index is VXN, an implied volailiy for NASDAQ index opions; R 5min is he sum of squared 5-minue inraday reurns and proxy for inraday volailiy. Coefficiens -4 Models V5 GARCH + V5 ARCH + QQV + V5.53 (.93).59** (.9) *** (-6.9) 7

10 Invesmen Managemen and Financial Innovaions, Volume, Issue, 4 Table 6 (con.). GARCH, ARCH and he condiional variance model wih inraday incremenal informaion Coefficiens Models V5 GARCH + V5 ARCH + QQV + V5.44** (.3).946*** (46.).*** (.36).55** (.48).645*** (38.3).5** (.7).6*** (.).9** (.6) Log-L LR 9.6*** Noe: This paper discusses only he predicive power of lag reurn residuals and inraday reurns (V5) over volailiy of QQQ reurns, and hence we resric o a value of zero. Values in parenheses are -saisics calculaed wih Bollerslov- Wooldrige robus sandard errors; *, **, and *** indicae significance a he %, 5%, and % levels, respecively. Since Andersen and Bollerslev (998) use inraday daa o forecas fuure volailiy and find ha he ARCH model performs beer, we incorporae QQV and V5 ino he ARCH model and find ha all coefficiens are saisically significan. 3 is greaer han 5, showing again ha he abiliy of QQV o forecas he reurn volailiy of QQQ is superior o ha of inraday reurn volailiy. Table 7 presens he comparison of he performance of various volailiy models in ou-of-sample forecass. The resuls indicae ha he error is smalles for he QQV model, demonsraing ha using QQV o forecas QQQ reurn volailiy has a smaller predicion error. Performance index Table 7. Predicive errors Models GARCH GJR-GARCH QQV VXN V5 RMSE MAE MAPE Conclusion This paper invesigaes he abiliy of various reurn volailiy models o forecas fuure reurn volailiy of QQQ, an ETF wih diversificaion advanages. We compare he ime series volailiy model, he implied volailiy model, and he inraday reurn References volailiy model, in an effor o deermine which has he bes predicive power for he volailiy in he ETF marke. The evidence shows ha all empirical volailiy models considered in his sudy have predicive power o forecas volailiy, bu employing only a GARCH model o forecas volailiy canno subsume all informaion conen. Previous sudies demonsrae he asymmery of reurn volailiy. Thus, when describing he imevarying process of reurn volailiy, i is beer o use models ha consider his propery. The empirical resul ha he GJR-GARCH model is superior o he GARCH model suppors his viewpoin. Moreover, incremenal informaion incorporaed ino he models all enhances he abiliy o forecas reurn volailiy. QQV has he bes power o predic he volailiy of QQQ reurns, VXN is in he second place, and inraday reurn volailiy has he lowes predicive power. Since he rading volume of QQQ is far more han ha of NDX, such resuls are idenical o he conclusion of prior sudies ha he implied volailiy of opion wih higher liquidiy would have beer predicive power or more informaion over reurn volailiy han opion wih lower liquidiy. Since ETFs are index funds ha diversify almos all non-sysemaic risks, using lagged error erms as proxy variables ha represen non-sysemaic risks o predic reurn volailiy of ETFs would no subsume enough informaion conen. The empirical resuls demonsrae his poin of view. Incorporaing oher incremenal informaion ino a GARCH model could increase is abiliy o forecas fuure reurn volailiy, indicaing ha here exis incremenal informaion effecs wihin a GARCH model. QQV is he mos valuable among all such incremenal informaion. One of he major shorcomings of his sudy is he usage of no so curren daa se. As a resul of using old daa se, he findings may be no so robus. Consequenly, given ha he auhors of his sudy have limiaions o more recen daa and o enhance he robusness of he findings, a few direcions for fuure research are recommended. Firs of all, since he curren sudy analyzes old daa se, fuure research can be se up o exend or invesigae he opic of his sudy by using a more recen daa se or daa ses belonging o oher counries. In addiion, fuure research can look ino incorporaing oher economeric echniques such as EGARCH or oher asymmeric GARCH models (or regime-swiching models) in order o provide more robus findings.. Andersen, T.G. and T. Bollerslev (998). Answering he Skepics: Yes, Sandard Volailiy Models do Provide Accurae Forecass, Inernaional Economic Review, 49, pp Andersen, T.G., T. Bollerslev, F. Diebold and H. Ebens (). The Disribuion of Realized Sock Reurn Volailiy, Journal of Financial Economics, 6, pp

11 Invesmen Managemen and Financial Innovaions, Volume, Issue, 4 3. Andersen, T.G., T. Bollerslev, F.X. Diebold and P. Labys (). The Disribuion of Realized Exchange Rae Volailiy, Journal of he American Saisical Associaion, 96, pp Baillier, T. and T. Bollerslev (989). The Message in Daily Exchange Raes: A Condiional Variance Tale, Journal of Business and Economic Saisics, 7, pp Becker, R. and A.E. Clemens (8). Are Combinaion Forecass of S&P 5 Volailiy Saisically Superior? Inernaional Journal of Forecasing, 4, pp Becker, R., A.E. Clemens and A. McClelland (9). The Jump Componen of S&P 5 Volailiy and he VIX Index, Journal of Banking & Finance, 33, pp Becker, R., A.E. Clemens and S.I. Whie (7). Does Implied Volailiy Provide any Informaion beyond ha Capured in Model-based Volailiy Forecass? Journal of Banking & Finance, 3, pp Bernd, E., B. Hall, R. Hall and J. Hausman (974). Esimaion and Inference in Nonlinear Srucural Models, Annals of Economic and Social Measuremen, 3, pp Black, F. (976). Sudies in Sock Price Volailiy Changes, in Proceedings of he 976 Meeings of he Business and Economics Saisics Secion, American Saisical Associaion, pp Blair, B., S. Poon and S. Taylor (). Forecasing S&P Volailiy: The Incremenal Informaion Conen of Implied Volailiies and High-frequency Index Reurns, Journal of Economerics, 5, pp Bollerslev, T. (986). Generalized Auoregressive Condiional Heeroscedasiciy, Journal of Economerics, 3, pp Bollerslev, T. (987). A Condiional Heeroskedasic Time Series Model for Speculaive Prices and Raes of Reurn, Review of Economics and Saisics, 69, pp Bollerslev, T., R.F. Engle and J.M. Wooldridge (988). A Capial Asse Pricing Model wih Time-Varying Covariances, Journal of Poliical Economy, 96, pp Bollerslev, T. and J. Wooldridge (99). Quasi Maximum Likelihood Esimaion and Inference in Dynamic Models wih Time Varying Covariances, Economeric Reviews,, pp Canina, L. and S. Figlewski (993). The Informaional Conen of Implied Volailiy, Review of Financial Sudies, 6, pp Chiras, D.P. and S. Manaser (978). The Informaion Conen of Opion Prices and a Tes of Marke Efficiency, Journal of Financial Economics, 6, pp Chrisensen, B.J. and N.R. Prabhala (998). The Relaion beween Implied and Realized Volailiy, Journal of Financial Economics, 5, pp Chrisensen, B.J. and C.S. Hansen (). New Evidence on he Implied-realized Volailiy Relaion, European Journal of Finance, 8, pp Chrisie, A.A. (98). The Sochasic Behavior of Common Sock Variances: Value, Leverage, and Ineres Rae Effecs, Journal of Financial Economics,, pp Day, T. and C. Lewis (99). Sock Marke Volailiy and he Informaion Conen of Sock Index Opions, Journal of Economerics, 5, pp Domowiz, I. and C.S. Hakkio (985). Condiional Variance and he Risk Premium in he Foreign Exchange Marke, Journal of Inernaional Economics, 9, pp Engle, R.F. (98). Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of U.K. Inflaion, Economerica, 5, pp Engle, R.F. (99). Discussion: Sock Marke Volailiy and he Crash of 87, Review of Financial Sudies, 3, pp Engle, R.F. and T. Bollerslev (986). Modeling he Persisence of Condiional Variances, Economeric Reviews, 5, pp Engle, R.F. and V.M. Ng (993). Measuring and esing he impac of news on volailiy, Journal of Finance, 48, pp Engle, R.F. and B.S. Yoo (987). Forecasing and Tesing in Co-inegraed Sysems, Journal of Economerics, 35, pp Fama, E. (965). The Behavior of Sock Marke Prices, Journal of Business, 38, pp Fleming, J. (998). The Qualiy of Marke Volailiy Forecass Implied by S&P Index Opion Prices, Journal of Empirical Finance, 5, pp Frijns, B., Tallau, C. and A. Tourani-Rad (). Ausralian Implied Volailiy Index, Finsia Journal of Applied Finance,, pp Glosen, L., R. Jagannahan and D. Runkle (993). On he Relaion beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks, Journal of Finance, 48, pp Hsieh, D.A. (989). Modeling Heeroskedasiciy in Daily Foreign Exchange Raes, Journal of Business and Economic Saisics, 7, pp Jorion, P. (995). Predicing Volailiy in he Foreign Exchange Marke, Journal of Finance, 5, pp Lamoureux, C.G. and W.D. Lasrapes (993). Forecasing Sock-reurn Variances: Toward an Undersanding of Sochasic Implied Volailiies, Review of Financial Sudies, 6, pp Laane, H.A. and R.J. Rendleman (976). Sandard Deviaions of Sock Price Raios Implied in Opion Prices, Journal of Finance, 3, pp

12 Invesmen Managemen and Financial Innovaions, Volume, Issue, Liu, H.C. and J.C. Hung (). Forecasing S&P Sock Index Volailiy: The Role of Volailiy Asymmeric and Disribuion Assumpion in GARCH Models, Exper Sysems wih Applicaions, 37, pp Ljung, G.M. and G.E.P. Box (978). On a Measure of Lack of Fi in Time Series Models, Biomerika, 65, pp Mayhew, S. and S. Sivers (3). Sock Reurn Dynamics, Opion Volume, and he Informaion Conen of Implied Volailiy, Journal of Fuures Markes, 3, pp Morgan, I.G. (976). Sock Prices and Heeroscedasiciy, Journal of Business, 49, pp Nelson, D.B. (99). Condiional Heeroscedasiciy in Asse Reurns: A New Approach, Economerica, 59, pp Rubensein, M. (994). Implied Binomial Trees, Journal of Finance, 49, pp Schmalensee, R. and R. Trippi (978). Common Sock Volailiy Expecaions Implied by Opion Premia, Journal of Finance, 33, pp Schwer, G.W. (99). Sock Volailiy and he Crash of 87, Review of Financial Sudies, 3, pp Whaley, R.E. (993). Derivaives on Marke Volailiy: Hedging Tools Long Overdue, Journal of Derivaives,, pp Zakoian, J.M. (994). Threshold Heeroskedasic Models, Journal of Economic Dynamics and Conrol, 8, pp