On the Intraday Relation between the VIX and its Futures

Transcription

1 On he Inraday Relaion beween he VIX and is Fuures Bar Frijns a, *, Alireza Tourani-Rad a and Rober I. Webb b a Deparmen of Finance, Auckland Universiy of Technology, Auckland, New Zealand b Universiy of Virginia, US *Corresponding auhor. Deparmen of Finance, Auckland Universiy of Technology, Privae Bag 92006, 1142 Auckland, New Zealand. Tel (ex. 5706); Fax ; bfrijns@au.ac.nz. 1

2 On he Inraday Relaion beween he VIX and is Fuures Absrac We sudy he inraday dynamics of he VIX and VIX fuures for he period January 2, 2008 o December 31, Considering firs he resuls of a Vecor Auoregression (VAR) using daily daa, we observe ha here is some evidence of causaliy from he VIX fuures o he VIX. Esimaing a VAR using ulra-high frequency daa, we find srong evidence for bi-direcional Granger causaliy beween he VIX and he VIX fuures. Overall, his effec appears o be sronger from he VIX fuures o he VIX han he oher way around. Impulse response funcions and variance decomposiions confirm he dominance of he VIX fuures. We furher show ha he causaliy from he VIX fuures o he VIX has been increasing over our sample period, whereas he reverse causaliy has been decreasing. This suggess ha he VIX fuures are become more and more imporan in he pricing of volailiy. We furher documen ha he VIX fuures dominae he VIX more on days wih negaive reurns, and on days wih high values of he VIX, suggesing ha hose are he days when invesors use VIX fuures o hedge heir posiions raher han rading in he S&P 500 index opions. Keywords: VIX, Fuures, Vecor Auoregressions, Ulra-High Frequency Daa JEL Codes: C11, C13. 2

3 1. Inroducion In 1993, he Chicago Board Opions Exchange (CBOE) inroduced he CBOE Volailiy Index (VIX). 1 This index has become he leading benchmark for sock marke volailiy and, more generally, for invesor senimen, due o is negaive relaion wih he S&P 500 index. 2 The VIX has also proven o be very useful in forecasing fuure marke volailiy, where he forecasing qualiies of he VIX ouperform radiional volailiy measures based on realized volailiy and GARCH models (Corrado and Miller, 2005 and Carr and Wu, 2006). However, while he VIX could be used for hedging purposes, i could no easily be raded. Theoreically, i would be possible replicae a porfolio of he underlying opions in he VIX and mainain he 30-day inerpolaed mauriy, bu he coss of doing so would be exorbian. To expedie rading in volailiy, as well as increase hedging opporuniies, he CBOE inroduced fuures on he VIX on March 26, These VIX fuures conracs (henceforh VXF) have become very popular. Due o he exisence of a srong negaive correlaion beween S&P 500 index reurns and he VIX, hese Fuures have proven o be a far more convenien hedging ool han S&P index opions (Szado, 2010). In 2006, CBOE inroduced VIX opions. In addiion, from January 2009 onwards, various exchange raded produc were inroduced ha derive heir value from VXF. These feaures have made he rade in VXF grow exponenially, wih more han 150,000 conracs per day in The VIX was originally based on implied volailiies, wih 30 days o expiraion, of eigh S&P 100 a-he-money pu and call opions (Whaley, 1993). In 2003, he VIX was expanded o include opions based on a broader index, he S&P 500, reflecing a more accurae view of marke volailiy. The valuaion model was also changed o a model-free basis (Brien-Jones and Neuberger, 2003). 2 The VIX is ofen referred o as he invesor fear gauge (Whaley, 2000, 2009). Generally, when invesors expec he sock marke o fall, hey will buy S&P pu opions for porfolio insurance. By doing so, invesors push up he opion prices and ulimaely he level of VIX. 3

4 In his paper, we are paricularly ineresed in he dynamic relaion beween he VIX and is fuures. Wih he inroducion of he VIX fuures, invesors can hedge volailiy eiher using opions on he S&P500 index or by aking posiions in VXF. Boh he demands for opions in he index and VIX fuures provide useful informaion regarding he marke s expecaion of fuure volailiy. An imporan quesion hen arises, which is where he informaion on fuure volailiy eners he marke, and which marke would lead in erms of incorporaing his new informaion. Prior research has used daily daa o (parially) address his quesion (see e.g. Konsaninidi, Skiadopolous and Tzagkaraki, 2008; Konsaninidi and Skiadopolous, 2011; and Shu and Zhang, 2012). However, inferring informaional efficiency and leadership is difficul using daily daa as informaional asymmeries beween markes ge los in he daa aggregaion process (i.e. if one marke leads he oher by, say, 5 minues, hen daily daa will no reveal much of his leadership). In his sudy, we herefore examine he dynamic relaion beween he VIX and he near-erm VIX fuures using inraday daa, where we sample a a 15-second frequency (which is he highes frequency a which he VIX is calculaed). Sampling a his frequency eliminaes all issues relaed o daa aggregaion, and allows us o ge a clear picure on he informaional efficiency and leadership in he relaion beween he VIX and is fuures. To our knowledge, we are he firs o examine he relaion beween he VIX and is fuures using inraday daa. We sudy he inraday dynamics of he VIX and VXF for he period January 2, 2008 o December 31, Considering firs he resuls of a Vecor Auoregression (VAR) using daily daa, we observe ha here is some evidence of Granger causaliy from he VIX Fuures o he VIX, which is in line wih Shu and Zhang (2012). However, he fi of his model is poor and he model is lef wih a considerably high residual correlaion of approximaely 0.8, suggesing ha here is a high degree of conemporaneous comovemen ha canno be explained by a daily 4

5 VAR. Esimaing a VAR using inraday daa, we find ha virually all conemporaneous effecs disappear (he residual correlaion is negligible), and we find srong evidence for bi-direcional Granger causaliy beween he VIX and VXF. Overall, his effec appears o be sronger from VXF o he VIX han he oher way around. Impulse response funcions and variance decomposiions confirm he dominance of he VIX Fuures. In furher analysis, we demonsrae ha he causaliy from VXF o he VIX has been increasing over our sample period, whereas he reverse causaliy has been decreasing. This suggess ha he VIX fuures are becoming more imporan in he pricing of volailiy. We furher documen ha VXF dominaes he VIX more on days wih negaive reurns, and on days wih high values of he VIX, suggesing ha hose are he days when invesors use VIX Fuures o hedge heir posiions raher han rading in he S&P 500 index opions. The remainder of his paper is srucures as follows. In secion 2, we review some of he relevan lieraure. Secion 3 describes daa used in his paper and presens some summary saisics. In secion 4, we presen our resuls. Finally, secion 6 concludes. 2. Lieraure Apar from he lieraure focusing on he volailiy pricing models (e.g, Zhu and Lian, 2012; Lu and Zhu, 2010; Brenner, Shu and Zhang, 2008, Lin, 2007, Zhang and Zu, 2006) and he addiion of a long VIX fuures posiion o equiy porfolios (Szado, 2010 and Alexander and Korovilas, 2011) here is a limied number of empirical papers invesigaing he efficiency of he VIX and VIX fuures markes. As for he issue of price discovery and causaliy in he marke 5

6 for volailiy, here is only one sudy, i.e., Shu and Zhang (2012). In his secion, we briefly provide an overview of hese sudies on he VIX and is fuures. Konsaninidi, Skiaopoulos and Tzagkaraki (2008) and Konsaninidi and Skiaopoulos (2011) invesigae he behavior of he implied volailiy indices, for he US and Europe, o assess wheher hey are predicable. While he auhors observe significan predicable paerns in he fuures on implied volailiy indices, none of hese paerns can be exploied hrough rading sraegies ha yield economically significan profis. Hence, from an economic poin of view hey canno rejec he efficiency of he volailiy fuures markes. Nossman and Wilhelmson (2009) es he expecaion hypohesis, using informaion on he erm srucure of volailiy, o es he efficiency of he VIX fuures marke. When hey allow for he exisence of a volailiy risk premium in heir analysis, Nossman and Wilhelmsson canno rejec fuures marke could predic he fuure VIX levels correcly. The paper closes o our sudy is ha of Shu and Zhang (2012). In his paper, auhors explicily examine price discovery beween he VIX and he VIX fuures. They use daily prices for he period Shu and Zhang (2012) find ha he VIX and he fuures are indeed coinegraed and proceed by using a Vecor Error Correcion model o assess he lead-lag ineracion beween spo VIX and VIX fuures. They find ha VIX fuures are informaive abou spo VIX and lead he spo marke in a linear error correcion model. Overall, hey conclude ha he VIX fuures have some price discovery funcion. 6

7 3. Daa and Summary Saisics We obain inraday daa for he VIX and he fuures on VIX from he Thomson Reuers Tick Hisory daabase (TRTH) mainained by SIRCA. We collec daa for he period January 2, 2008 o December 31, All daa are colleced in ick ime, wih poenial millisecond precision. The CBOE compues inraday values for VIX a approximaely 15-second inervals from 8.30 a.m p.m. Chicago ime (noe ha his ime period reflecs he normal rading hours for he S&P 500 index opions). The ime inerval beween he calculaions of he VIX is no exacly 15 seconds, bu slighly more. This implies ha a he sar of he day VIX may be compued a 9:30:15.12, 9:30:30.24, ec. bu during he rading day may be repored a, say, 10:30:22.54, ec. The VIX fuures were firs lised on he CBOE fuures exchange on March 6, The conracs use he VIX as he underlying and use a muliplier of $1,000. The conracs rade on CBOE Direc, he elecronic rading plaform of he CBOE. The minimum ick size of hese conracs is 0.01 index poins. The CBOE liss up o 9 near-erm serial monhs and five monhs on he February quarerly cycle. The regular rading hours for he VXF are from 8:30 a.m. o 3:15 p.m., he same inraday period over which he VIX is compued. Trading of he conracs erminaes on he las day before he final selemen dae. We collec ick-by-ick rade and quoe daa for all conracs raded during our sample period. However, o consruc a coninuous series, we splice ogeher he neares-erm conracs, which are rolled over on he day when he rading volume of he second neares-erm conrac exceeds he rading volume of he neares 7

8 erm conrac. 3 We focus on he neares-erm conrac as hese conracs should relae closeso he VIX. Since he conracs rade on an elecronic marke we collec he bid and ask quoes for hese conracs and compue he midpoin from hese quoes. This frequency and sample period provides us wih a oal of 1,987,254 observaions for each series. As he VIX is compued approximaely every 15 seconds, we sample VIX and he VXF a a 15-second frequency. 4 In addiion, we exclude he firs and las five minues of he rading day, o avoid any noise ha may be due o opening and closing affecing our resuls. INSERT FIGURE 1 HERE In Figure 1, we provide a ime series plo of he daa. The plo shows ha he VIX was relaively low a he sar of our sample period bu swifly increased and doubled from early Augus 2007 onwards, which can be aribued o he European Sovereign Deb Crisis. For he remainder of he period he VIX sayed relaively high, due o he coninuaion of he global financial crisis, and was decreasing only a a very slow rae. The VXF resembles he paern of he VIX closely, 3 In some cases he neares-erm conrac remains he mos acive one unil he selemen dae of he conrac. In his case, we roll he conrac over on he day before he las rading day, o avoid any price disorions ha may be due o final selemen of he conracs. 4 Since VIX is compued a slighly more han 15 seconds, we consruc 3 series, saring on he whole minue and 5 and 10 seconds afer he minue. By consrucing hese hree series, we aim o minimize he impac of sale values of VIX ha could affec our resuls. Noe ha we only repor he resuls for he sampling inerval ha sars on he whole minue. Oher sampling inervals yield similar resuls and are available upon reques. 8

9 bu as expeced he VXF is generally below he VIX when he VIX is high and is above he VIX when he VIX is low (see also Shu and Zhang, 2012). In Table 1, we presen summary saisics on he VIX and he VXF, respecively. In he firs wo columns of Table 1, we repor he summary saisics for he levels of he VIX and he VXF. Over our sample period, he VIX was on average 25.83, while he VXF was slighly higher a As can be seen from maximum and minimum values, he VIX has more exreme values han he VXF. This is also refleced in he sandard deviaion, which is higher for he VIX han for he VXF. Boh series have posiive skewness, as can be expeced and have excess kurosis. The persisence, a he 15 second frequency, is exremely high and he firs order auocorrelaion is no discernibly differen from Finally, when conducing a uni roo es on boh series, we observe ha we can rejec he presence of a uni roo for he VIX a he 1% level, while for he VXF we can only rejec he uni roo a he 10% level. INSERT TABLE 1 HERE The las wo columns of Table 1 repor summary saisics for he firs difference of he (log) VIX and VXF. These firs differences only include changes during he rading day, and hence exclude he overnigh change. On average, he changes in he VIX and VXF are nearly zero as could be expeced a hese are ulra-high frequencies daa. Again, as we observed in he levels, ΔVIX has more exreme changes han ΔVXF as can be seen from he maximum and minimum values, and from he sandard deviaions. Ineresingly, he skewness of ΔVIX is negaive a , whereas he skewness of ΔVXF is nearly zero. Alhough boh series display excess 9

10 kurosis, he excess kurosis in ΔVIX is much higher han ha in ΔVXF. The firs order auocorrelaion in boh series is negaive, a a value of for ΔVIX and for ΔVXF, suggesing ha here is sronger negaive auocorrelaion in he VIX fuures. Finally, he ADF saisics, for he difference series, sugges ha we can srongly rejec he presence of a uni roo in boh series. 4. Resuls 4.1 Daily Analysis To assess he relaionship beween he VIX and he VIX Fuures, we sar by conducing our analysis a a daily frequency as in Shu and Zhang (2012). Since he summary saisics show ha here is weak evidence of a uni roo in he VIX fuures, we compue firs differences of he log of he volailiy series. We hen esimae he following VAR: VIX VXF 1 1 ( L) VIX 1 ( L) VIX ( L) VXF 1 ( L) VXF , (1) where ϕ1(l), φ1(l), ϕ2(l), and φ2(l), are polynomials in he lag operaor of idenical lengh. Daily daa suggess an opimal lag lengh of one using he Schwarz Informaion Crierion (SIC). Hence, we esimae Equaion (1) as a VAR(1). 10

11 In Table 2, we repor he regression resuls for he VAR(1) as well as Newey-Wes adjused - saisics in parenheses. For ΔVIX, we find a negaive and significan coefficien on ΔVIX-1 suggesing ha here is negaive auocorrelaion in changes in he VIX a he daily frequency. We also find a posiive coefficien for ΔVXF-1, significan a he 5% level, suggesing ha a he daily frequency he VIX fuures have some predicive value for he changes in he VIX. For ΔVXF, we find ha here is no saisical evidence for predicabiliy of he changes in he VIX fuures, neiher lagged changes in he VIX or he VXF can be used o predic hese changes. Alhough we find some evidence of predicabiliy for ΔVIX, we noe ha he R 2 s of he regressions are quie low a 1.82% and 0.18% for ΔVIX and ΔVXF, respecively. The conemporaneous correlaion beween he residuals in he VAR is quie high a 0.813, suggesing a srong correlaion beween he series. Granger causaliy ess, repored in Panel B, confirm he findings of he coefficiens, i.e., here is Granger Causaliy from ΔVXF o ΔVIX, bu no he reverse. Overall, he resuls of our daily analysis are in line wih Shu and Zhang (2012). 4.2 Inraday Analysis The daily analysis reveals some evidence for predicabiliy of changes in he VIX. However, his analysis also revealed a very high conemporaneous correlaion beween he changes in he VIX and he VXF. Par of his conemporaneous correlaion may be due o daa aggregaion. Sampling a higher frequencies may reduce he conemporaneous correlaion and reveal more lead-lag dynamics. We herefore esimae he VAR in Equaion (1) using inraday ulra-highfrequency daa sampling a a 15 second frequency. Insead of esimaing Equaion (1) as one 11

12 big VAR using 1,987,254 observaions, we esimae he model every day in he sample, so ha we obain a daily series of coefficiens and saisics. Firs, we deermine he opimal lag lengh of he VAR by compuing he SIC for 1 lag up o 10 lags every day. Then, we compue he average SIC over all days in he sample. We find ha he average SIC is lowes for a VAR wih hree lags. Hence, we esimae all coefficiens and compue all saisics based on daily models using hree lags. In Panel A of Table 3, we repor he resuls for he inraday VAR(3) model. We repor coefficiens and indicae significance using aserisks (based on Newey-Wes correced sandard errors obained from he ime series of coefficiens). In brackes, we repor he 2.5% and 97.5% percenile values from he ime series of coefficiens. When we consider he dynamics of ΔVIX, we find evidence of some negaive auocorrelaion, wih he firs and second lag significan a he 5% level and values of and , respecively. The coefficien for he hird auoregressive lag is posiive a and alhough very small, i is significan a he 10% level. For he coefficiens on he lagged values of ΔVXF, we find ha all hree coefficiens are posiive and significan a he 1% level, wih values of 0.172, 0.119, and 0.058, for lags one, wo and hree, respecively. This provides evidence ha here is some predicabiliy for he changes in he VIX based on lagged changes in he VXF. The R 2 of his regression, 9.26%, is considerably higher han ha for he regression using daily daa, suggesing ha much more of he variaion in he inraday changes in he VIX can be explained by pas informaion. For he changes in he VXF, we noe ha lagged values of ΔVIX 12

13 have a posiive and significan effec on changes in he VXF for all hree lags. This finding suggess ha here is some predicabiliy of changes in he VXF based on lagged values of ΔVIX. This conrass he findings ha we observed for he daily esimaion. We also find significan evidence for negaive auocorrelaion in he changes in he VIX fuures a his frequency, wih all hree lags yielding negaive and significan coefficiens. Again, compared wih he daily VAR, he R 2 of he inraday VAR is considerably higher a 4.12%, bu sill lower for he regression for ΔVIX. A his high level of daa frequency, we also observe ha he conemporaneous correlaion in he changes in VIX and VXF is close o zero (on average , wih 2.5% and 97.5% percenile values a and , respecively). This finding indicaes he presence of high posiive conemporaneous correlaion beween daily changes in VIX and VXF is driven by daa aggregaion. INSERT TABLE 3 HERE In Panel B of Table 3, we repor he resuls for Granger causaliy ess, where we repor he average Granger causaliy saisic over he sample period and repor he percenages of significan Granger causaliy saisics a convenional significance levels. When we consider he Granger causaliy from ΔVXF o ΔVIX, we find ha he average coefficien is equal o , hus giving very srong saisical evidence for a causal relaion running from ΔVXF o ΔVIX (noe ha he 1% criical value for his saisic is 11.30). When we consider he percenage of days ha we find a significan effec of ΔVXF on ΔVIX, we find ha beween 92.37% (85.45%) of he days here is significan causal effec measured a he 10% (1%) 13

14 significance level. When we invesigae he causal effec of ΔVIX on ΔVXF, we find an average es saisic of 14.31, which, alhough lower han he causaliy saisic of ΔVXF on ΔVIX, is sill highly significan. Nex, we compue he percenage of days ha here is a causal effec of ΔVIX on ΔVXF. I is observed ha on 63.99% (42.05%) of he days here is significan evidence for causaliy running from ΔVIX o ΔVXF measured a he 10% (1%) level. Overall, he Granger Causaliy ess reveal ha here is significan evidence for reverse causaliy, bu he effec appears o be sronger running from ΔVXF o ΔVIX han vice versa. Given ha he conemporaneous correlaion beween ΔVIX and ΔVXF is nearly zero, we can easily esimae impulse response funcions and compue variance decomposiion assuming ha boh series are conemporaneously uncorrelaed. 5 In Figure 2, we plo he impulse response funcions for 10 seps ahead, where we apply a uni shock o he residuals of each series. Panels A and B of Figure 2 show he responses o a uni shock in ΔVIX. As can be seen, his shock has some impac on changes in he VIX, bu dies ou afer abou four periods. Panel B shows ha his shock leads o a small increase in he VXF afer which i decreases and again dies ou afer abou four periods. Overall, a uni shock o ΔVIX does no lead o a change of more han 0.1 (in absolue erms) in he VXF. Panels C and D show he responses of a uni shock in ΔVXF. Considering Panel D firs, we noe ha a uni shocks o ΔVXF leads o a drop in he VXF afer abou wo periods, and dies ou afer abou 5 periods. The response of ΔVIX o a shock in ΔVXF (Panel C), shows a posiive reacion in ΔVIX afer wo periods and dies ou again afer abou 5 periods. Overall, he impulse response analysis shows ha here is 5 Noe ha his addiional analysis is difficul o conduc in a meaningful way using daily daa, due o he high conemporaneous correlaions observed in daily daa. 14

15 bidirecional spillover beween ΔVIX and ΔVXF, however, he response in he VIX o a shock in he VXF is subsanially larger han he response of he VXF o a shock in he VIX. The las rows of Table 3 repor he Variance Decomposiion, where we decompose he variance of ΔVIX (ΔVXF) ha is due o eiher ΔVIX or ΔVXF. When we decompose he variance of ΔVIX, we find ha 94.53% of he variance comes from ΔVIX, whereas 5.47% originaes from changes in he VXF. Vice versa, he variance of changes in he VXF is for 3.54% due o changes in he VIX and he remaining 96.46% is due o is own changes. In line wih he Granger causaliy ess and he impulse response funcions, his resul suggess ha he changes in he VXF have a greaer influence on he changes in he VIX hen he oher way around. 4.3 Time-variaion in Granger Causaliy The nex quesion we address is wheher we observe any ime variaion in he inraday causal relaion beween ΔVIX and ΔVXF. In Table 4, we repor Granger causaliy saisics per year during our sample period. The firs column of Table 4 repors he causaliy saisics from ΔVXF o ΔVIX. We noe ha since 2008, here is an upward rend in he causaliy saisics, indicaing ha causaliy from ΔVXF o ΔVIX has becoming sronger. However, we also observe a sligh drop off in he causaliy saisic in he las year in he sample in For he causaliy in he opposie direcion, we observe a downward rend in he saisic going from an 15

16 average of in 2008 o in 2012, which is jus below he 5% significance level. Hence he causal effec of ΔVIX on ΔVXF seems o have died off over ime. INSERT TABLE 4 HERE In he nex wo columns of Table 4, we repor he average saisics for he variance decomposiion per year over our sample period. The firs of hese columns repors he percenage of variance of ΔVIX ha is aribuable o ΔVXF. Again, we noe an upward rend over ime going from 2.62% in 2008 o 10.04% in 2011, afer which i declines o 5.86% in The percenage of variance of ΔVXF aribuable o ΔVIX shows a less clear picure, which sars a 4.66% in 2008 and declines o 3.06% in To provide a visual represenaion of he ime variaion in he Granger Causaliy beween ΔVIX and ΔVXF, we plo he 10-day moving average of he log of he raio of he Granger causaliy saisics in Figure 3, i.e. log(gcδvxf/gcδvix), where GCΔVXF is he Granger causaliy saisic of causaliy from ΔVXF o ΔVIX, and GCΔVIX is he Granger causaliy saisic of causaliy from ΔVIX o ΔVXF. From his plo, we observe ha here is an increase in his raio. A he sar of he sample period he raio is less han zero, suggesing ha he causaliy from ΔVIX o ΔVXF is sronger. However, his raio quickly becomes posiive. Overall, here is an upward rend in his raio. INSERT FIGURE 3 HERE 16

17 In order o explain wha drives he changes and increase in his raio, we conduc he following regression analysis, i.e. GC FuVol VXF log rend R SP VIX GC 1 _ 2 log( ) 3 log, (2) VIX OpVol where GC GC VXF log is he daily raio of he Granger causaliy saisics, rend capures he VIX ime rend ha we observed in he raio, R_SP is he reurn on he S&P 500 index on day, log(vix) is he log of he VIX on day, and FuVol log is he log of he raio of daily OpVol volume raded in he near-erm VIX fuures relaive o he daily volume raded in he near-erm S&P 500 index opions on day. 6 In Table 5, we repor he resuls for Equaion (2), where we include variables sep-by-sep. We repor all coefficiens and Newey-Wes correced -saisics in parenheses. Firs, we esimae he regression only including he ime rend. The resuls, in he firs column of Table 5, show ha his ime rend is highly significan, and his regression produces an adjused R 2 of 28.43%. 6 Noe ha we use a derended version of his raio as here is a srong posiive upward rend in his variable. We have also conduced he analysis wih he raio of daily volume raded in he near-erm VIX fuures relaive o he volume raded in he near-erm S&P500 index pu opions. The resuls for his analysis are nearly idenical o he ones repored in his paper. 17

18 This provides clear evidence of an upward rend in he informaiveness of he VIX fuures over he VIX. In he second column, we add he reurns on he S&P500 index. The coefficien on hese reurns is negaive and significan a he 1% level. This suggess ha VIX fuures are more informaive on days when reurns are negaive and could be due o he increased hedging using VIX fuures on days wih negaive reurns. The adjused R 2 increases slighly relaive o he model ha only includes he ime rend o 29.28%. INSERT TABLE 5 HERE Nex, we include he log of he VIX. We find ha he coefficien for his erm is posiive and significan a he 1% level. Hence his suggess ha when he VIX is high, he informaiveness of he VXF relaive o he VIX increases. Again, his could be due o he increased hedging aciviy in using VIX fuures, when uncerainy (VIX) in he marke is high. The adjused R 2 of his regression is 31.40%, suggesing ha he VIX is more informaive for he causaliy raio han he reurns on he S&P 500. In he nex column, we include he raio of volume raded in he VIX fuures relaive o he volume raded in he S&P 500 opions. One reason why we could expec an increase in he informaiveness of he VXF is because more people are using he VIX fuures o hedge heir posiions han he opions on he S&P500. We observe ha he raio is insignifican in his regression, showing ha an increase in aciviy in he VIX fuures relaive o he opions does no explain he raio of causaliy. Finally, we esimae a regression where we include boh he reurns on he S&P 500 and log of he VIX. Prior lieraure has shown ha here is a srong and negaive relaion beween hese wo variables (e.g. Whaley, 2000) and hence we need o include boh in a single regression o deermine wheher he 18

19 negaive relaion beween he reurns on he S&P500 and he relaive informaiveness of he VXF is driven by he VIX, and vice versa. When we include boh variables, we observe ha boh mainain heir sign and significance, suggesing ha boh reurns on he S&P500 and he VIX are informaive for he causaliy beween he VIX and is fuures. 5. Conclusion In his paper, we examine he inraday dynamics of he VIX and VXF for he period January 2, 2008 o December 31, In line wih Shu and Zhang (2012), we observe some evidence of Granger causaliy from he VXF o he VIX a a daily level. However, a he inraday level, we find srong evidence for bi-direcional Granger causaliy beween he VIX and he VXF. Overall, his effec appears o be sronger from he VXF o he VIX han he oher way around, which is confirmed by impulse response funcions and variance decomposiions. We documen ha he causaliy from he VXF o he VIX has been increasing over our sample period, whereas he reverse causaliy has been decreasing. This suggess ha he VIX fuures are become more and more imporan in he pricing of volailiy. We furher find ha he VIX fuures dominae he VIX more on days wih negaive reurns, and on days wih high values of he VIX, suggesing ha hose are he days when invesors use VIX fuures o hedge heir posiions raher han rading in he S&P 500 index opions. Overall, our resuls sugges ha he VIX fuures are informaionally dominan over he VIX in reflecing fuure volailiy. 19

20 References Alexander, C. and Korvilos, D. (2011). The Hazards of volailiy diversificaion, ICMA Cenre Discussion Paper in Finance, No. DP Black, F. (1975). Fac and Fanasy in he Use of Opions, Financial Analyss Journal, 3, Brenner, M., Shu, J., and Zhang, J. (2008). The marke for valiliy rading: VIX fuures, working paper, New York Universiy, Sern School of Business. Brien-Jones. M., and Neuberger, A. (2000). Opion Prices, Implied Price Processes, and Sochasic Volailiy, Journal of Finance, 55, CBOE (2009). VIX: The CBOE Volailiy Index, Whie Paper, Chicago Board Opions Exchange, (Available a Carr, P. and Wu, L. (2006). A ale of wo indices, Journal of Derivaives, 13, Corrado, C., and Miller, T. (2005). The Forecas Qualiy of CBOE Implied Volailiy Indexes. Journal of Fuures Markes, 25, Fernandes, M., Medeiros, and Scharh, M. (2007). Modeling and Predicing he CBOE Marke Volailiy, Working Paper, Queen Mary Universiy. Gran, M., M.. K. Gregory, and Lui, J. (2007). Volailiy as an Asse, Goldman Sachs Global Invesmen Research. November. Konsaninidi, E, Skiadopolous, G., and Tzagkaraki, E. (2008). Can he evoluion of implied volailiy be forecased? Evidence from European and US implied volailiy indices, Journal of Banking and Finance, 32, Konsaninidi, E, and Skiadopolous, G. (2011). Are VIX Fuures Prices Predicable? An Empirical Invesigaion, Inernaional Journal of Forecasing, 27,

21 Lin, Y. N. (2007). Pricing VIX Fuures: Evidence from Inegraed Physical and Risk Neural robabiliy Measures, Journal of Fuures Markes, 27, Lu, Z. and Zhu, Y. (2010). Volailiy Componens: The Term Srucure Dynamics of VIX Fuures. Journal of Fuures Markes, 30, Mixon, S. (2007). The Implied Volailiy Term Srucure of Sock Index Opions. Journal of Empirical Finance, 14, Nossman, M. and Wilhelmson, A. (2009). Is he VIX Fuures Marke Able o Predic he VIX Index? A Tes of he Expecaion Hypohesis, The Journal of Alernaive Invesmen, Fall, Shu, J. and Zhang, J. (2012). Causaliy in he VIX fuures marke. Journal of Fuures Markes 32, Szado, E., (2009). VIX fuures and opions A case sudy of porfolio diversificaion, Journal of Alernaive Invesmens, Fall, Toikka, M., E.K. Tom, S. Chadwick, and M. Bol-Chrismas (2004). Volailiy as an Asse? CSFB Equiy Derivaives Sraegy, February 26. Zhang, J., and Zhu, Y., (2006). VIX Fuures, Journal of Fuures Marke, 26, Whaley, R. (1993). Derivaives on Marke Volailiy: Hedging Tools Long Overdue, Journal of Derivaives, 1 (Fall): Whaley, R. (2000). The Invesor Fear Gauge: Explicaion of he CBOE VIX. Journal of Porfolio Managemen, 26, Whaley, R. (2009). Undersanding he VIX. Journal of Porfolio Managemen, 35,

22 Zhu, S.-P., and Lian, G.-H. (2012). An Analyical Formula for VIX Fuures and is Applicaions. Journal of Fuures Markes, 32,

23 Table 1. Descripive Saisics Levels Differences VIX VXF ΔVIX ΔVXF Mean e e-07 Median Max Min Sandard Deviaion Skewness Kurosis ρ ADF -3.64*** -2.78* *** *** 23

24 Table 2. Daily VAR Analysis Panel A: Esimaion Resuls ΔVIX ΔVXF C ΔVIX *** ΔVXF ** R Panel B: Granger Causaliy Tess Causaliy from ΔVXF o ΔVIX Causaliy from ΔVIX o ΔVXF 4.84**

25 consan Table 3. Inraday VAR Analysis Panel A: Esimaion Resuls ΔVIX -4.80e-06*** [-6.11e-05,5.82e-05] ΔVXF -1.07e-06 [-4.60e-05, 5.24e-05] ϕ *** [-0.486, 0.169] ϕ *** [-0.295, 0.124] ϕ * [-0.154, 0.128] 0.056*** [-0.036, 0.174] 0.033*** [-0.053, 0.127] 0.017*** [-0.074, 0.096] φ *** [0.015, 0.503] φ *** [-0.010, 0.391] φ *** [-0.033, 0.203] *** [-0.289, 0.003] *** [-0.166, 0.028] *** [-0.108, 0.036] R [0.0057,0.3147] [0.0075,0.0954] Panel B: Granger Causaliy and Variance Decomposiion Causaliy from ΔVXF o ΔVIX Causaliy from ΔVIX o ΔVXF Mean Perc. exceeding 10% Cri. Val % 63.99% Perc. exceeding 5% Cri. Val % 57.15% Perc. exceeding 1% Cri. Val % 42.05% Variance Decomposiion ΔVIX ΔVXF due o ΔVIX 94.53% 3.54% due o ΔVXF 5.47% 96.46% 25

26 Table 4. Causaliy and Variance Decomposiion by Year Causaliy from VD ΔVIX due ΔVIX o ΔVXF o ΔVXF Causaliy from ΔVXF o ΔVIX VD ΔVXF due o ΔVIX % 4.66% % 2.80% % 3.49% % 3.68% % 3.06% 26

27 Table 5. Regression Resuls for he Causaliy Raios GC log GC VXF VIX GC log GC VXF VIX GC log GC VXF VIX GC log GC VXF VIX GC log GC VXF VIX α (0.260) (0.18) *** (-3.43) rend *** *** *** (12.21) (12.28) (12.94) R_SP *** (-4.10) log(vix) 1.041*** (3.37) FuVol log OpVol (0.259) *** (12.16) (0.43) *** (-3.21) *** (12.82) *** (-03.18) 0.983*** (3.155) R 2 (adj)

28 Figure 1. Time Series Plo of VIX and VXF VIX VXF 28

29 Figure 2. Impulse Response Funcions 29

30 Figure 3. Moving Average of he Log Causaliy Raio 30