How does implied volatility differ from model based volatility forecasts?

Transcription

1 How does implied volailiy differ from model based volailiy forecass? # * Ralf Becker, Adam E. Clemens and James Curchin # Economic Sudies, School of Social Sciences, Universiy of Mancheser, School of Economics and Finance, Queensland Universiy of Technology Corresponding auhor Adam Clemens School of Economics and Finance Queensland Universiy of Technology GPO Box 2434, Brisbane Q, Ausralia a.clemens@qu.edu.au Ph , Fax Absrac Much research has addressed he relaive performance of opion implied volailiies and economeric model based forecass in erms of forecasing asse reurn volailiy. The general paern is ha implied volailiy is a superior forecas. Some auhors aribue his o he fac ha opion markes use a wider informaion se when forming heir forecass of volailiy. An alernaive reason may be ha he way in which hisorical daa is used differs across he forecasing approaches. This aricle considers hese issues and deermines wheher S&P 500 implied volailiy reflecs a se of economic informaion beyond is impac on he prevailing level of volailiy and wheher he mapping of hisorical daa varies widely across he approaches. I is found, ha while he implied volailiy subsumes his informaion, as do model based forecass, his is only due o is impac on he curren, or prevailing level of volailiy. Therefore, i appears as hough implied volailiy does no reflec a wider informaion se han model based forecass, meaning ha implied volailiy forecass simply reflec volailiy persisence in much he same way of as do economeric models. The manner in which implied volailiy maps hisorical daa ino a forecas differs from how model based forecass do so. Keywords: Implied volailiy, VIX, volailiy forecass, informaional efficiency. JEL Classificaion: C12, C22, G00, G14 Acknowledgemens: The auhors hank Adrian Pagan whose commens on relaed work moivaed he curren research, San Hurn for consrucive feedback on earlier versions of he aricle and seminar paricipans a he Queensland Universiy of Technology

2 1 Inroducion The behaviour of opion implied volailiies have araced a grea deal of research aenion. Boh he relaive forecas accuracy and informaional efficiency of implied volailiies (IV) have been considered by numerous auhors. Fleming (1998), Jiang and Tian (2003) and Becker, Clemens and Whie (2006, 2007), amongs ohers have examined wheher various IV measures subsume hisorical informaion (predominanly reurn daa) commonly used when forecasing volailiy. While Fleming (1998) and Jiang and Tian (2003) find ha IV is efficien wih respec o such informaion, Becker, Clemens and Whie (2006) find ha S&P 500 IV does no compleely subsume a diverse se of informaion including model based forecass (MBF). Becker, Clemens and Whie (2007) find ha IV conains no informaion beyond volailiy persisence as capured by MBF. While hese resuls are imporan, we sill do no ruly undersand he fundamenal differences beween IV and MBF. This paper seeks o redress his. Poon and Granger (2003, 2005) provide wide ranging surveys of aricles comparing various forecasing approaches. The general paern revealed by Poon and Granger (2003, 2005) is ha opion based IV produce superior forecass of volailiy relaive o compeing MBF. Two plausible explanaions for his superioriy can be proposed. Firs, Poon and Granger (2003) sae ha i is of lile surprise ha IV forecass are superior as hey are based on a larger and imelier informaion se. Second, i could be argued ha he superior forecas performance of IV can be aribued o opion markes uilising more complex funcions of hisorical daa when forming forecass, as hey are no consrained by a paricular funcion form ha maps pas volailiy informaion ino volailiy forecass. This paper will invesigae wheher any empirical evidence exiss o suppor hese conjecures. We consider ineres rae, commodiy price and exchange rae daa o esablish wheher IV incorporaes such informaion ha is no rouinely 1 included in volailiy models. This informaion has been seleced as i could poenially influence opion marke paricipans expecaions of fuure equiy volailiy. Volailiy forecass may reflec such informaion in one of wo ways. Firs, economic informaion may be refleced in he curren level of volailiy, and hus boh classes of volailiy forecass, MBF and IV may be relaed o such informaion. Second, i may be possible ha such informaion informs forecass beyond is influence on he curren level of volailiy, an effec only relevan o IV. An approach similar o ha used in he informaional efficiency sudies discussed above is aken o esablish in he manner in which such economic informaion is incorporaed ino he differen volailiy forecass. We esablish ha he addiional economic informaion considered here is relaed o he curren level of volailiy and hence also o all volailiy forecass, be hey MBF or IV. More ineresingly we esablish ha some bu no all volailiy forecass produce expeced volailiy changes ha are correlaed o economic informaion available a he ime of he forecass. This indicaes ha only a subgroup of forecass reflec addiional economic informaion. I is, however, no clearly obvious ha IV forecass are superior in capuring such informaion. While all volailiy forecass reflec proxies for hisorical volailiy, MBF do so in a very srucured way. Whereas his need no be he case wih IV forecass. Here a simple mapping of pas volailiy informaion ino he differen volailiy forecass reveals ha IV does so in a more flexible manner, highlighing a poenial advanage of he model free IV approach. This paper proceeds as follows. Secion 2 presens he daa relevan o his sudy. Secion 3 oulines he mehodology uilized o address he research quesions a hand. Secions 4 and 5 presen he empirical resuls and concluding commens 1 An excepion is Glosen, Jagannahan and Runkle (1993), who include ineres rae daa.

3 respecively. 2 Daa To address he research quesion a hand four differen ses of daa are required. Equiy reurns, an esimae of IV, realisaions of equiy volailiy and he economic, non-reurn daa, specifically erm srucure, commodiy prices and exchange rae informaion are uilized here. Each se of daa will now be discussed in urn. The sudy is based on daily S&P 500 index reurns, from 2 January 1990 o 17 Ocober 2003 (3481 daily observaions). The implied volailiy measure uilized here is ha provided by he Chicago Board of Opions Exchange, he VIX 2. The VIX is an implied volailiy index derived from a number of pu and call opions on he S&P 500 index, which generally have srike prices close o he curren index value wih mauriies close o he arge of 22 rading days 3. I is derived wihou reference o a resricive opion pricing model. For echnical deails relaing o he consrucion of he VIX index, see Chicago Board of Opions Exchange (2003). Afer allowing for a poenial volailiy risk premium, he VIX is consruced o be a general measure of he marke's esimae of average S&P 500 volailiy over he subsequen 22 rading days (Blair, Poon and Taylor 2001, and Chrisensen and Prabhala, 1998) 4. As highlighed by Jiang and Tian (2003), he advanages of such a model-free approach o compuing implied volailiy are wo-fold 5. Relaive o a model-based esimae such as Black-Scholes, a model-free esimae incorporaes more informaion from a range of observed opion prices. The measure of acual volailiy used here is realised volailiy (RV), consruced from inra-day S&P 500 index daa (see Andersen, Bollerlsev, Diebold and Labys 2001, 2003 for a discussion of RV) 6. In dealing wih pracical issues such as inra-day seasonaliy and sampling frequency when consrucing daily RV, he signaure plo mehodology of Andersen e al. (1999) is followed. Given his approach, daily RV esimaes are consruced using 30 minue S&P 500 index reurns. The se of economic (non-reurn) informaion comprises of variables ha can be reasonably assumed o reflec general economic condiions and hence influence equiy marke performance and volailiy. The variables are he slope of he erm srucure represened by he difference beween one and en year US Treasury bond yields (slope), he credi spread beween BBB raed commercial paper and US Treasury bills (cspr), absolue daily oil price change (oil) as a measure of volailiy in he oil marke, and an indicaor variable (doil) which is uniy when he change in oil price is posiive and zero oherwise. While his is by no means an exhausive lis of economic variables, hey relae o changes in he level of economic aciviy (cspr), inflaionary expecaions (slope) and he headline commodiy price in erms of oil prices, which impacs on he cos incurred by many firms and individuals, and hus inflaion. These are variables ha are available a he daily frequency. The informaion considered is of a wider naure han ha radiionally incorporaed in MBF. 2 The VIX index used here is he mos recen version of he index, inroduced on Sepember 22, VIX daa for his sudy was downloaded from he CBOE websie. 3 The daily volailiy implied by he VIX can be calculaed when recognising ha he VIX quoe is equivalen o 100 imes he annualised reurn sandard deviaion. Hence (VIX /( )) represens he daily volailiy measure (see CBOE, 2003). 4 Quoing from he CBOE Whie paper (2003) on he VIX, "VIX [...] provide[s] a minue-by-minue snapsho of expeced sock marke volailiy over he nex 30 calendar days." 5 They uilise a differen approach o ha embodied ino he calculaion of he VIX. 6 Inraday S&P 500 index daa were purchased from Tick Daa, Inc.

4 3 Mehodology 3.1 Model-based volailiy forecass The MBF considered here are seleced from a range of differen model classes frequenly applied in he financial economerics lieraure. Seleced MBF are he GARCH(1,1) (gar), an asymmeric GARCH-ype GJR hreshold model (gjr), a sochasic volailiy (sv) model as well as a shor memory (arma) and long memory (arfima) ime-series model of RV. These models are also exended by he inclusion of RV as an addiional explanaory variable (garrv, gjrrv, and svrv) 7. As he VIX is designed as a fixed 22 day ahead forecas, each of he models are used o produce forecass of average 22 day ahead volailiy. Forecass are based on parameers esimaed recursively from a rolling window of 1000 observaions. This procedure resuls in day-ahead forecass. 3.2 Economic informaion and forecas efficiency In order o invesigae if and how economic informaion eners volailiy forecass in general, and he VIX in paricular, we will invesigae a number of quesions. Firs, is he curren level of volailiy relaed o he seleced economic informaion? If his was no he case, one could no reasonably expec volailiy forecass o reflec any such informaion. Second, i will be esed wheher volailiy forecass are relaed o economic informaion. This is a sraighforward corollary o he firs research quesion. Such a relaion could merely be due o he fac ha volailiy forecass use curren and pas volailiy informaion. The VIX has a concepual advanage in ha i may exrac informaion from economic daa ha is relevan for fuure realisaions of volailiy, informaion no normally included in models for volailiy. In order o examine his, wo furher hypoheses are esed. Third, i is esablished wheher he prediced changes in he level of volailiy (forecas curren level of volailiy) is relaed o economic informaion. Only if his was he case could one reasonably argue ha a volailiy forecas allows for curren economic informaion o drive is forecas beyond is effec on he curren level of volailiy. Finally, i will be esed wheher he volailiy forecass are efficien, esing wheher forecas errors made, are correlaed wih economic informaion available a he ime he forecas was formed. Two economeric ools will be used o es hese hypoheses. Firs, following Fleming (1998) and Becker, Clemens and Whie (2006), is he generalized mehod of momens (GMM) as i enables us o es wheher a series ε ( y, x ; φ) (obained afer esimaing a se of parameers φ ) and a se of k2 economic variables, q are orhogonal or no. Parameer esimaes of φ are obained by minimising where V = g( y, x, z ; φ)' H g( y, x, z ; φ), (1) 7 See Becker, Clemens and Whie (2007) for exac specificaions of hese models.

5 1 g( y, x, z, φ) = T z = ' ' ( x q ) ' 1 = T T = 1 T = 1 ε ( y, x ; φ) z ( y x φ) z ' (2) The definiion of he scalar series and he k 1 1 vecor x depend on he paricular quesion a hand. The weighing marix H is chosen o be he variancecovariance marix of he momen condiions in g ( y, x, z; φ), where allowance is made for residual correlaion (see Hansen and Hodrick, 1980). In his conex, he es for k2 overidenifying resricions (as z produces k 1 + k2 momen condiions and φ is a ( k 1 1) parameer vecor o be esimaed) is used o es he null hypohesis ha ε ( y, x ; φ) and q are uncorrelaed. The second esing procedure is Harvey and Newbold (2000) generalizaion of Hoelling s es for a zero mean in a vecor-valued random variable. In his paper s conex he ( k 2 1) random variable is h = ( ~ y q~ ). Variables wih ~ are de-meaned and if y and q are uncorrelaed hen E ( ~ y ~ ) = q 0. The null hypohesis ha y and every elemen in q are uncorrelaed is esed using y ( ) T k2 1 MS hvˆ = h (3) k 2 ( T 1) where Vˆ is he sample variance covariance marix of h as defined in Harvey and Newbold s (2000) allowing for auocorrelaion due o he overlapping naure of he 22 day ahead forecass. The calculaed es saisic is compared agains criical values coming of he disribuion. Fk 2, T k2 The firs hree of he hypoheses posed above can be invesigaed wih boh mehodologies. This is useful as here is only limied evidence on he empirical properies of he above esing procedures especially when using variance-covariance marices which allow for auocorrelaion of a relaively high order. The firs sep is o invesigae wheher he economic informaion is relaed o he curren level of volailiy. If his was no he case, one would no expec a volailiy forecas o incorporae such informaion. Here y is he prevailing level of volailiy, is a consan and comprises he five economic variables slope, cspr, oil, doil x q and wi. As he volailiy models uilise differen proxies for volailiy, he quesion of wheher he choice of proxy, realized volailiy, squared, or absolue daily reurns, is crucial when evaluaing his hypohesis. I is well known ha he daily series of volailiy proxies (in paricular he squared and absolue daily reurns) are noisy proxies of he laen volailiy process, and hence i is examined wheher hese volailiy proxies averaged over a number of days are correlaed o he average of he seleced economic variables over hese days. The second hypohesis o be esed is wheher he volailiy forecass hemselves are correlaed wih he seleced economic informaion. For his purpose y is he volailiy forecas, y = f i,, where f i, is he volailiy forecas of he ih model made a period for he average volailiy over he period +1 o +22, x is a consan and remains q = slope, cspr, oil, doil. q ( ) Tesing wheher prediced volailiy changes are correlaed wih economic informaion will reveal wheher volailiy forecass do use economic daa beyond is

6 influence on he curren level of volailiy. For ha purpose he hird es ses y = fi, RV, x is a consan and q is as before. Lasly, i will be invesigaed wheher he differen forecass are efficien wih respec o he economic informaion available a he ime he forecass are formed. The GMM mehodology will be used o esimae Mincer-Zarnowiz ype regressions by seing y = RV , he average realized volailiy over he nex 22 business MBF MBF days, x is x = ( 1 f i, ) or x = ( 1 f i, ), where f i, is a vecor of all model based volailiy forecass and is defined above. q 3.3 Informaion Mapping In order o examine he second poenial difference beween MBF and IV, we relae he differen forecass o pas proxies of volailiy. This will esablish wheher IV volailiy forecass are more flexible in he way hey map pas volailiy informaion ino volailiy forecass. This may explain why a) IV rouinely ouperform single model based volailiy forecass bu b) a combinaion of model-based volailiy forecass appears o perform as well as he IV forecas. The volailiy proxies used here is he realized volailiy RV. To compare he mappings of pas volailiy implied by he various forecass i is necessary o use a flexible approach wihou assuming a specific funcional form. In a relaed conex Ghysels, Sana-Clara and Valkanov (2005, 2006) proposed he Mixed Daa Sampling (MIDAS) approach. The MIDAS regression of he forecass on pas volailiy proxies is achieved by esimaing volailiy proxy) f f i, α, φ and θ = ( θ 1, θ 2 ) in (exemplary for RV as k max i, ) k= 0 = α + φ b ( k, θ RV k + ε, (4) where b( k, θ ) is he weighing funcion applied o pas volailiy k β, θ1, θ 2 k max b ( k, θ ) = (5) k max j = β, θ 1 1, θ j 2 k max β (., θ 1, θ 2 ) is he Bea probabiliy disribuion and kmax is he maximum number of lags. The shape of he esimaed weigh funcions may change across differen subsample wih such variaion revealing a degree of flexibiliy in erms of mapping pas volailiy informaion ino volailiy forecass. 4 Resuls 4.1 Economic Informaion We firs consider he resuls ha reveal how economic informaion eners volailiy forecass and in paricular wheher he VIX does display any significan advanage compared o MBF. INSERT TABLE 1 ABOUT HERE From Table 1 we can see ha he null hypohesis of orhogonaliy beween he chosen economic variables and he curren level of volailiy as proxied by realized y o

7 volailiy, squared daily reurns and absolue reurns is clearly rejeced by boh he GMM es for overidenifying resricions (GMM) and he Harvey-Newbold-Hoelling (HNH) es. This is rue for daily observaions bu also for averages over longer ime periods. Only for averages over 30 days do he rejecions become marginal. This is likely he resul of he es s reduced power when dealing wih srongly overlapping daa. INSERT TABLE 2 ABOUT HERE This resul foreshadows he resuls shown in Table 2. Here i is demonsraed, again using he GMM and HNH es, ha he volailiy forecass are significanly correlaed wih economic informaion no direcly incorporaed ino he volailiy models bu available a he ime he forecass are formed. This resul is no surprising given he clear correlaion beween he curren level of volailiy and he economic informaion. I is well known ha volailiy forecass are driven by he curren level of volailiy. Volailiy forecass, in paricular hose derived from volailiy models, will depend on he esimaed long-run mean for volailiy and how quickly he volailiy process is expeced o rever o his level. Here we will invesigae wheher he change in volailiy prediced by he various volailiy forecass, f i, RV, is correlaed o he seleced economic informaion. INSERT TABLE 3 ABOUT HERE Table 3 displays he resuls of he es for overidenifying resricions and he HNH es when y = fi, RV, x is a consan and q = ( slope, cspr, oil, doil ). Ineresingly, he VIX does no produce forecass which predic volailiy changes ha are correlaed o economic informaion available a he ime,, a which he forecass are formed. Predicably, mos MBF display he same paern. I is, however, ineresing o noe ha garrv and gjrrv do behave differenly as heir prediced volailiy changes are correlaed o available economic informaion. The reason for his resul is he naure in which RV eners he volailiy model. RV eners he condiional volailiy equaion as an exra variable. If i wasn included, he GARCH model would, as discussed above, esimae a long-run volailiy and mean reversion o ha long-run mean from he daa, in paricular he hisory of squared reurns. When including RV ino his model one allows non-smoohed curren volailiy o ener he volailiy forecas. As i was esablished above, his curren level of volailiy is indeed relaed o he economic informaion available a ime and hence i is naural ha he expeced volailiy change will be relaed o he economic informaion. The same effec can be seen for he gjrrv model. This hen raises he quesion why he arma and arfima models which are soley based on curren and passed RV do no display he same paern. In hese models he hisory of curren and pas realized volailiies is used o exrac informaion on he long-run average volailiy and he speed of mean reversion from he daa. RV does no serve o incorporae immediae volailiy informaion ino he model in he same way as i does for he garrv and he gjrrv. The arma and arfima models, herefore, do no produce forecass of volailiy changes ha are relaed o economic informaion. A his sage i is worh noing ha his does no auomaically imply ha forecass from he garrv and he gjrrv models are superior o ha of VIX volailiy forecass. Here i is merely invesigaed how hese forecass are relaed o economic informaion. INSERT TABLE 4 ABOUT HERE

8 Lasly, we will esablish wheher he forecass are efficien wih respec o he economic informaion available a ime. If a forecas is efficien wih respec o a cerain informaion se, is forecas errors should be uncorrelaed wih ha informaion. Table 4 displays he resuls of he J-es for overidenifying resricions which ess he null hypohesis ha he forecas errors are indeed uncorrelaed wih he informaion in q = ( slope, cspr, oil, doil ). The resuls indicae ha in general he volailiy forecass are efficien. Surprisingly, he VIX volailiy forecas is he only one ha exhibis a marginal rejecion of his null hypohesis. Alhough a he level of rejecion and aking ino accoun he number of hypohesis ess underaken here, no oo much weigh ough o be pu on his single resul. In summarising he resuls so far, a number of ineresing findings arise. All volailiy forecass reflec he seleced economic informaion. Such informaion is refleced in he forecass hrough heir link o he curren level of volailiy. As such, ha informaion seems of lile value in erms of forecasing volailiy beyond his relaionship. As a consequence i appears as if volailiy forecass are indeed efficien wih respec o he seleced0 economic informaion, indicaing ha here is no scope for beer use of ha informaion in he conex of forecasing volailiy. These resuls leave i open why he VIX is commonly found o be a superior volailiy forecas compared o individual model based forecass. An alernaive explanaion is invesigaed in he following Secion. 4.2 Informaion Mapping In he previous secion i was impossible o esablish ha he apparen superioriy of he VIX is due o is abiliy o incorporae informaion ha is no direcly represened in he majoriy of volailiy models. The second concepual advanageof implied volailiy is ha i is no a formal model ha generaes he volailiy forecas. In conras, demand and supply condiions in opion markes produce his forecas. This implies ha he relaionship beween volailiy forecass and he available hisoric volailiy informaion is unresriced, and i may be such added flexibiliy ha delivers he imporan advanage o implied volailiy based forecass. In oher words, any model based volailiy model may be oo resricive in how pas volailiy informaion is mapped ino volailiy forecass. While i is no direcly obvious how i could be proven ha i is his concepual difference ha is responsible for he improved forecas accuracy of he VIX when compared agains individual MBF, we will esablish wheher or no he relaionship beween VIX and pas volailiy proxies is significanly more flexible han ha refleced in MBF. The model in equaions (4) and (5) is used o map volailiy forecass o volailiy proxies available a he ime of he forecas (kmax = 500). In order o evaluae wheher his mapping changes hrough ime, his esimaion is done for rolling subsamples of lengh 1000 (2460 forecass are available) wih sep size INSERT Figures 1 o 3 ABOUT HERE In order o illusrae he findings he mappings for he four volailiy forecass, vix, gar, garrv and arfima are shown. The weigh funcions b( k, θ ) are displayed in Figure 1 wih he volailiy proxy being RV 9. For ease of exposiion, only he weighs for he firs 50 (ou of a maximum of 500) lags are displayed. Figure 1 reveals a 8 The firs subsample is for forecass 1 o 1000, he second subsample for forecass 51 o 1050, he hird for 101 o 1100, ec. 9 Qualiaively similar resuls are found when pas squared or absolue daily reurns are used.

9 number of ineresing resuls. In general he vix and arfima forecass appears o pu more weigh on he disan volailiy hisory. This is o be expeced, as he arfima forecas is designed o capure long-memory in he volailiy process, bu i is ineresing o see ha he vix also appears o reflec his feaure. The corollary of ha is ha he gar and garrv forecass, on average, pu more emphasis on he more recen informaion. The weigh funcions are almos exclusively monoonically decreasing. When mapping he pas realized volailiies ino he differen volailiy forecass (Figure 3) arfima, gar and garrv display very lile variaion across he differen subsamples. Ineresingly, he garrv forecass show very large weigh on he very shor lag realized volailiies, which wihou doub, is a resul of he inclusion of realized volailiy as an addiional explanaory variable ino he GARCH volailiy equaion. Lasly, and mos imporanly for he conjecure evaluaed here, he mappings of he vix forecass display he larges amoun of variaion in heir mappings. Thus appears as if he relaive weigh given by opion markes o he immediae and he longer volailiy hisory varies significanly hrough ime. This variaion, while presen in all volailiy models, is much more marked han in he MBF. This finding lends suppor o he conjecure ha forecass derived from implied volailiies may be superior o MBF as hey allow more flexibiliy in erms of how pas volailiy informaion is used. In paricular opion markes may decide from period o period, wha amoun of pas informaion is relevan for he immediae (here 22 day ahead) fuure, whereas he MBF used here (and in all relaed sudies) are esimaed on he basis of a fixed esimaion window lengh. 5 Concluding remarks The behavior of opion implied volailiy and economeric model based forecass has araced a grea deal of research aenion. Much of his has focused on relaive forecas accuracy and he informaional efficiency of implied volailiy. Generally, i has been found ha implied volailiy provides a more accurae volailiy forecas relaive o hose generaed from economeric models. A commonly held view is ha his resul is due o he fac ha opion implied volailiies capure a wider range of informaion han forecass based on hisorical reurn daa. An alernaive conjecure is ha he models used impose a cerain amoun of rigidiy in he way in which he volailiy hisory is used o produce volailiy forecass. This paper demonsraed how boh claims can be examined. In paricular i was considered wheher boh implied volailiy and model based forecass reflec a se of economic informaion. The seleced se of informaion relaes o he erm srucure of ineres raes and commodiy prices. I was found ha boh implied volailiy and model-based forecass do subsume he seleced se of economic informaion. I was furher esablished ha his informaion eners hrough is impac on he prevailing level of volailiy and i is no apparen ha his informaion eners VIX forecass in any wider sense. Therefore i seems as hough implied volailiy does no use he seleced economic informaion in a fundamenally differen way han he model based volailiy forecass. The work presened here is, of course, no sufficien o conclude decisively ha implied volailiies will never capure such informaion. I is possible ha forecass of such economic variables are indeed aken ino accoun. As beer ime series forecass become available, fuure research may show ha implied volailiies do make use of ha informaion. Lasly, he analysis presened clearly demonsraes ha he manner in which implied volailiies makes use of hisorical volailiy informaion changes across. This clearly disinguishes implied volailiy from model based volailiy forecass and may explain or a leas conribue o explain why previous research has esablished ha

10 volailiy forecass implied from opion markes appear o be superior o model based volailiy forecass. References Andersen, T.G., and Bollerslev, T., and Diebold, F.X., and Labys, P., (Undersanding, opimizing, using and forecasing) realized volailiy and correlaion, Working Paper, Universiy of Pennsylvania. Andersen T.G., Bollerslev T., Diebold F.X. and Labys P., The disribuion of exchange rae volailiy. Journal of he American Saisical Associaion 96, Andersen T.G., Bollerslev T., Diebold F.X. and Labys P., Modeling and forecasing realized volailiy. Economerica 71, Becker R. and Clemens A. and Whie S., On he informaional efficiency of S&P500 implied volailiy, Norh American Journal of Economics and Finance, 17, Becker R. and Clemens A. and Whie S., Does implied volailiy provide any informaion beyond ha capured in model-based volailiy forecass?, Journal of Banking and Finance, 31, Blair B.J., Poon S-H. and Taylor S.J., 2001, Forecasing S&P 100 volailiy: he incremenal informaion conen of implied volailiies and high-frequency index reurns. Journal of Economerics, 105, Chicago Board of Opions Exchange, 2003, VIX, CBOE Volailiy Index. Chrisensen B.J. and Prabhala N.R., The relaion beween implied and realized volailiy. Journal of Financial Economics 50, Fleming J., The qualiy of marke volailiy forecass implied by S&P 100 index opion prices. Journal of Empirical Finance 5, Ghysels, E., Sana-Clara, P. and Valkanov, R., 2005, There is a risk-reurn radeoff afer all, Journal of Financial Economics, 76, Ghysels, E., Sana-Clara, P. and Valkanov, R., 2006, Predicing volailiy: geing mos ou of reurn daa sampled a differen frequencies, Journal of Economerics, 131, Glosen L.R., Jagannahan R. and Runkle D.E., On he relaion beween he expeced value and he volailiy of he nominal excess reurn on socks. Journal of Finance 48, Hansen L.P. and Hodrick R.J., Forward exchange raes as opimal predicors of fuure spo raes: an economeric analysis. Journal of Poliical Economy 88, Harvey D. and Newbold, P., ess for muliple forecas encompassing. Journal of Applied Economerics 15, Jiang, G.J. and Tian, Y.S., Model-free implied volailiy and is

11 informaion conen. Unpublished manuscrip. Mincer, J. and Zarnowiz, V., The evaluaion of economic forecass. In: Mincer, J. (ed.), Economic Forecass and Expecaions. New York: Naional Bureau of Economic Research. Poon S-H. and Granger C.W.J., Forecasing volailiy in financial markes: a review. Journal of Economic Lieraure. 41, Poon S-H. and Granger C.W.J., Pracical issues in forecasing volailiy. Financial Analyss Journal, 61,

12 TABLE SECTION RV r 2 r LAGS J HNH J HNH J HNH Table 1: Correlaion beween volailiy proxies and economic informaion. J: p- value for he es for overidenifying resricions in equaions (1) and (2) wih 2 y = RV, r or r, = 1 and q = slope, cspr, oil, doil. For LAGS = 2, 5, 10, 20 x ( ) and 30, y and q are he averages of he respecive variables over LAGS periods. The variance-covariance marix of he momen condiions, H in equaion (1), allows for correlaion of order (LAGS - 1). HNH: p-values of he Harvey-Newbold-Hoelling es wih y and q defined as for GMM. The variance-covariance marix Vˆ in equaion (3), allows for correlaion of order (LAGS - 1). Forecas J-saisic p-values (J) HNH VIX gar gjr sv arma arfima garrv gjrrv svrv Table 2: Correlaion beween volailiy forecass and economic informaion. J: p- value of he es for overidenifying resricions from equaions (1) and (2) wih y = f i,, x = 1 and q = ( slope, cspr, oil, doil ). The variance-covariance marix of he momen condiions, H in equaion (1), allows for correlaion of order 21. HNH: p- values of he Harvey-Newbold-Hoelling es wih y and q defined as for J. The variance-covariance marix Vˆ in equaion (3), allows for correlaion of order 21.

13 Forecas J-saisic p-values (J) HNH VIX gar gjr sv arma arfima garrv gjrrv svrv i x and ( ) Table 3: Correlaion beween expeced volailiy changes and economic informaion. J: p-value of he es for overidenifying resricions from equaions (1) and (2) wih y = f, RV, = 1 q = slope, cspr, oil, doil. The variancecovariance marix of he momen condiions, H in equaion (1), allows for correlaion of order 21. HNH: p-values of he Harvey-Newbold-Hoelling es wih y and q defined as for J. The variance-covariance marix Vˆ in equaion (3), allows for correlaion of order 21.

14 Forecas Error R 2 p-values(j) VIX gar gjr sv arma arfima garrv gjrrv svrv All MBF Table 4: Forecas efficiency. R 2 : R 2 of he Mincer-Zarnowiz regressions esimaed MBF y, = ( 1 ) or ( 1 ) = RV x f i, from equaions (1) and (2) wih x = f i,, where MBF f, is a vecor of all model based volailiy forecass. q = ( slope, cspr, oil, doil ). i p-values(j): p-value of he es for overidenifying resricions from equaions (1) and (2) wih y, x and q as above. The variance-covariance marix of he momen condiions, H in equaion (1), allows for correlaion of order 21.

15 0.2 Bea weighs VIX 0.2 Bea weighs ARFIMA Weighs 0.1 Weighs Lag Lag 0.2 Bea weighs GARCH 0.2 Bea weighs GARCHRV Weighs 0.1 Weighs Lag Lag Figure 1: Informaion Mapping Absolue Reurns. Weigh funcions b ( k, θ ) (equaion (5)) for VIX, ARFIMA, GARCH and GARCHRV volailiy forecass using daily realized volailiy, RV, as volailiy proxies.