Volatility in Equilibrium: Asymmetries and Dynamic Dependencies*

Transcription

1 Review of Finance (2012) 16: doi: /of/f005 Advance Access publicaion: Mach 23, 2011 Volailiy in Equilibium: Asymmeies and Dynamic Dependencies* TIM BOLLERSLEV 1, NATALIA SIZOVA 2 and GEORGE TAUCHEN 3 1 Duke Univesiy, NBER, and CREATES, 2 Rice Univesiy, and 3 Duke Univesiy Absac. Sock make volailiy cluses in ime, appeas facionally inegaed, caies a isk pemium, and exhibis asymmeic leveage effecs. A he same ime, he volailiy isk pemium, defined by he diffeence beween he isk-neual and objecive expecaions of he volailiy, feaues sho memoy. This pape develops he fis inenally consisen equilibium-based explanaion fo all hese empiical facs. Using newly available high-fequency inaday daa fo he S&P 500 and he VIX volailiy index, he auhos show ha he qualiaive implicaions fom he new heoeical coninuous-ime model mach emakably well wih he disinc shapes and paens in he sample auocoelaions and dynamic coss-coelaions acually obseved in he daa. JEL Classificaion: C22, C51, C52, G12, G13, G14 1. Inoducion Modeling and foecasing of sock make eun volailiy has eceived unpecedened aenion in he academic lieaue ove he pas wo decades. The hee mos siking empiical egulaiies o emege fom his bugeoning lieaue aguably concen: (i) he highly pesisen own dynamic dependencies in he volailiy, 1 (ii) he exisence of a ypically posiive volailiy isk pemium as manifesed by he vaiance swap ae exceeding he coesponding expeced fuue volailiy, 2 (iii) he appaen asymmey in he lead lag elaionship beween euns * We would like o hank Benad Dumas (he Edio), an anonymous efeee, Toben G. Andesen, Lauen Calve, Niklas Wagne, and paicipans in he 2008 Volailiy Symposium a Cene fo Reseach in Economeic Analysis of Time Seies (CREATES), Univesiy of Aahus, Denmak, he 2009 Financial Economeics Confeence a Toulouse Univesiy, he 2010 SoFiE Confeence in Melboune, along wih seveal univesiy seminas fo helpful commens and discussion. Bolleslev s wok was suppoed by a gan fom he Naional Science Foundaion (NSF) o he Naional Bueau of Economic Reseach (NBER) and Cene fo Reseach in Economeic Analysis of Time Seies (CREATES) funded by he Danish Naional Reseach Foundaion. 1 The hisoically low volailiy in he yeas peceding he Fall 2008 financial cises and he subsequen susained heighened volailiy povide anecdoal evidence fo his idea. 2 The pepondeance of opions ades selling volailiy o gain he pemium indiecly suppos he noion of volailiy caying a isk pemium. Ó The Auhos Published by Oxfod Univesiy Pess [on behalf of he Euopean Finance Associaion]. All ighs eseved. Fo Pemissions, please jounals.pemissions@oup.com

2 32 T. BOLLERSLEV ET AL. and volailiy. 3 Despie hese now well-documened and geneally acceped empiical facs, no fomal heoeical model ye exiss fo explaining all hese feaues wihin a coheen economic famewok. This pape fills ha void by developing an eniely self-conained equilibium-based explanaion fo he obseved asymmey and volailiy isk pemium. The model is based on he Epsein Zin Weil ecusive pefeence sucue and is cas in coninuous ime, heeby allowing fo a diec assessmen of is abiliy o mach he qualiaive feaues of he daa acoss diffeen sampling fequencies, including inaday coss-coelaion paens as well as longe un dynamic dependencies. The specific conibuions of he pape ae wo-fold. Fis, fom an empiical pespecive, we use ula high-fequency 5-min daa on boh euns and volailiy. Ou use of he Chicago Boad Opions Exchange (CBOE) VIX volailiy index, consuced o mach o he isk-neual expecaion of he fowad inegaed vaiance, ogehe wih he acual S&P 500 index affods a much shape view of he dynamic asymmeies and coss-coelaions beween euns and volailiy han hiheo available in he lieaue. 4 Inuiively, he highe esoluion aces diecly o he fac ha coelaions ae second momen saisics, which ae well known o be much moe accuaely deemined fom inaday high-fequency, as opposed o say daily, daa. Second, fom a heoeical pespecive, we se foh a coninuous-ime model, albei highly sylized, ha uses he shadow pices implied by an opimizing economic agen opeaing wihin an endowmen envionmen o help undesand he hee siking egulaiies noed above hence, he em equilibium. The model fills he above-menioned void in he lieaue and sands in diec conas o he less fomal educed-fom explanaions poffeed fo some of he obseved egulaiies, mos noably hose associaed wih he so-called leveage effec fis obseved by Black (1976) and Chisie (1982). Woking in coninuous ime pesens a numbe of new heoeical challenges bu is essenial o avoid inenal iming inconsisencies in egad o he dynamic dependencies in he VIX, which is fomally defined as squae oo of he expecaion of he coninuously inegaed fowad vaiance. Even hough ou economic, o equilibium, model is oo sylized o be diecly esimable, is geneal qualiaive 3 Again, he heighened volailiy following Russia s defaul and he LTCM debacle in Sepembe 1998, he elaively low volailiy accompanying he apid un-up in pices duing he ech bubble, as well as he ecen shap incease in volailiy accompanying he Fall 2008 financial cises and shap make declines ae all in line wih his asymmey. 4 To he bes of ou knowledge, no clea depicions of he dynamic coss-coelaions, such hose in Figue 2 below, cuenly exiss in he lieaue. Using 5-min euns alone, Bolleslev, Livinova, and Tauchen (2006) have peviously adduced a negaive elaionship beween he magniude and he sign of conempoaneous and lagged euns. Unlike he VIX 2, howeve, he absolue euns povide exemely noisy measues of he local volailiy and, in un, do no affod a clea picue of he fowadposiive elaionship essenial o he isk-based explanaion of he pesen pape.

3 VOLATILITY IN EQUILIBRIUM 33 pedicions ae ich enough o be compaed wih he documened empiical paens, hus making he model and he basic undelying economic mechanisms efuable. Befoe discussing he model any fuhe, i is insucive o illusae he new empiical egulaiies ha we seek o explain. To ha end, he op mos solid line in Figue 1 shows he sample auocoelaions fo he aggegae make volailiy ou o a lag lengh of 90 days. The calculaions ae based on daily daa fo he squaed opion implied volailiy index VIX ove he pas wo decades; fuhe deails concening he daa and diffeen volailiy measues ae given in Secion 4. The auocoelaions in Figue 1 decay a a vey slow ae and emain numeically lage and saisically significan fo all lags. Consisen wih hese highly pesisen own dynamic dependencies in he volailiy, i is now widely acceped ha he ypical ae of decay is so slow as o be bes descibed by a facionally inegaed long memoy ype pocess; fo some of he ealies empiical evidence along hese lines, see, fo example, Robinson (1991), Ding, Gange, and Engle (1993), and Baillie, Bolleslev, and Mikkelsen (1996). The VIX index in effec epesens he make s expecaion of he cumulaive vaiaion of he S&P 500 index ove he nex monh plus any pemium fo beaing he coesponding volailiy isk. 5 Isolaing he vaiance isk pemium, he second line in Figue 1 shows he daily auocoelaions fo he diffeence beween he squaed VIX index and he 1 monh ahead foecass fom a simple educed-fom ime seies model fo he acually obseved daily ealized vaiaion in he S&P 500 index; fuhe deails concening he high fequency based ealized volailiy seies and he consucion of he model foecass ae again defeed o Secion 4. Alhough he auocoelaions sill indicae posiive own dynamic dependencies fo up o seveal weeks, he pemium is clealy no as pesisen as he volailiy pocess iself. Again, his is no a new empiical esul pe se. Fo insance, he analysis in Bolleslev, Bolleslev, and Mikkelsen (2011) also suppos he idea of elaively fas mean evesion in he volailiy isk pemium, as does he empiical evidence of facional coinegaion beween implied and ealized volailiy pesened by, fo example, Bandi and Peon (2006) and Nielsen (2007). 6 Nex, in ode o highligh he afoemenioned eun volailiy asymmey, he fis panel in Figue 2 plos he coss-coelaions beween leads and lags of he S&P 5 The vaiance isk pemium is fomally defined as he diffeence beween he expeced fuue vaiaion unde he isk-neual and acual pobabiliy measues. 6 As noed by a efeee, he lowe pesisence in he vaiance pemium could a leas in pa be due o an eos-in-vaiables-ype poblem ceaed by he use of an esimaed foecas poxy in place of he ue populaion condiional expecaion fo he squaed VIX. Howeve, on implemening he insumenal vaiables echnique ecenly developed by Hansen and Lunde (2010) o accoun fo his poblem, we find ha he obus o measuemen eos auocoelaions diffe lile fom hose shown in Figue 1. Fuhe deails concening hese esuls ae available in supplemenay maeial.

4 34 T. BOLLERSLEV ET AL. Figue 1. Sample auocoelaions. The op mos solid line shows he sample auocoelaions fo he VIX2 volailiy index o a lag lengh of 90 days. The lowe line shows he sample auocoelaions fo he vaiance isk pemium. The calculaions ae based on daily daa and vaiable definiions as descibed in moe deail in Secion euns and he squaed opion implied VIX volailiy index. 7 Bolleslev, Livinova, and Tauchen (2006) have peviously demonsaed he advanage of using high-fequency inaday euns fo moe effecively esimaing and analyzing he lead lag elaionship beween euns and volailiy using he absolue euns as a poxy fo he laen spo volailiy. We follow hei lead in he use of highfequency 5-min obsevaions. Howeve, insead of poxying he volailiy by he absolue euns, we ely on acual obsevaions on he S&P 500 euns and he VIX volailiy index ecoded a a 5-min sampling fequency in calculaing he sample coss-coelaions fo leads and lags anging up o 22 days o 1,716 leads 7 Noe ha Figue 2 shows coss-coelaions beween he levels of he vaiance-elaed vaiables because he expeced pa of fuue euns, efleced in he igh-hand side of he plos, depends on hei levels, no he fis diffeences o innovaions. Boh vaiance-elaed vaiables ae saionay in levels as seen fom Figue 1.

5 VOLATILITY IN EQUILIBRIUM 35 Figue 2. Sample coss-coelaions. The op panel shows he sample coss-auocoelaions beween he VIX 2 and lags and leads of he euns anging fom 22 o 22 days. The boom panel shows he sample coss-auocoelaions beween he vaiance isk pemium and he euns. The calculaions ae based on high-fequency 5-min daa and vaiable definiions as fuhe deailed in Secion 4.1. and lags a he 5-min sampling. High-fequency daa fo he VIX have only ecenly become available, so ha he coss-auocoelaions depiced in Figue 2 ae necessaily based on a shoe 5-yea calenda-ime span compaed wih he longe 18- yea sample of daily obsevaions used fo illusaing he own dynamic dependencies in he pevious Figue 1. Noneheless, he use of high-fequency daa ove his shoe sample sill eveals a siking negaive paen in he coelaions beween he volailiy and he lagged euns, lasing fo seveal days. On he ohe hand, he coelaions beween he volailiy and he fuue euns ae all posiive, albei close o zeo. This sysemaic paen in he high fequency based coss-coelaions beween euns and volailiy is diecly in line wih he empiical evidence fom numeous sudies based on coase lowe fequency daily daa and specific paameic models, including he ealy influenial wok by Fench, Schwe, and Sambaugh (1987), Schwe (1990), Nelson (1991), Glosen, Jagannahan, and Runkle (1993), and

6 36 T. BOLLERSLEV ET AL. Figue 3. Calibaed auocoelaions and coss-coelaions. The plos ae based on he model calibaed in Appendix C. The wo lef-hand panels show he model-implied auocoelaions fo he VIX 2 volailiy index and he vaiance isk pemium o a lag lengh of 90 days. The igh-hand panels show he model-implied coss-coelaions beween he VIX 2 volailiy index and he vaiance isk pemium o a lag lengh of 22 days. Campbell and Henschell (1992). Also, following Black (1976), he lef pa of Figue 2 and he negaive coelaions beween lagged euns and cuen volailiy has now commonly efeed o in he lieaue as a leveage effec, while he igh pa of he figue and he posiive coelaions beween cuen volailiy and fuue euns have been emed a volailiy feedback effec. 8 Fuhe elaing ou empiical findings o he exising lieaue, i is woh noing ha while numeically small compaed o he esimaes epoed in some of he 8 I is now widely ageed ha he negaive coelaions beween lagged euns and cuen volailiy have lile if anyhing o do wih changes in financial leveage (see, e.g., Figlewski and Wang, 2002). In fac, he wo effecs may be viewed as flip sides of he same coin. Quoing fom Campbell, Lo, and MacKinlay (1997, chape 12): If expeced sock euns inceases when volailiy inceases, and if expeced dividends ae unchanged, hen sock pices should fall when volailiy inceases.

7 VOLATILITY IN EQUILIBRIUM 37 Figue 4. Model-implied auocoelaions. The op panel shows he auocoelaions fo he VIX 2 volailiy index o a lag lengh of 90 days. The solid line gives he model-implied auocoelaions unde he assumpion of long memoy in he undelying fundamenal volailiy pocess. The boom panel shows he auocoelaions fo he vaiance isk pemium p. The solid line gives he modelimplied auocoelaions. The pai of dashed lines included in boh panels epesens 95% confidence inevals fo he coesponding sample auocoelaions based on daily daa fom 1990 hough above-menioned sudies based on coase daily daa, he magniudes of he cosscoelaions in Figue 2 ae fequency dependen. As such, hey only appea misleadingly low. Fo insance, aking ino accoun he diffeences in sampling fequency, he coelaion of he squaed VIX wih he conempoaneous 5-min eun anslaes ino a coelaion of oughly 0.20 wih daily euns. Taking he analysis one sep fuhe, he boom panel in Figue 2 shows he coss-coelaions beween he 5-min S&P 500 euns and a moe moden volailiy-ype measue, he vaiance isk pemium, whee as befoe he vaiance isk pemium is defined as he diffeence beween he squaed VIX index and he coesponding foecas consuced fom a simple educed-fom ime seies model fo he daily ealized volailiies. Compaing his plo o he eun volailiy

8 38 T. BOLLERSLEV ET AL. Figue 5. Model-implied coss-coelaions. The op panel shows he sample coss-auocoelaions beween he VIX 2 volailiy index and lags and leads of he euns anging fom 22 o 22 days. The boom panel shows he coss-coelaions beween he vaiance isk pemium p and he euns. The solid lines give he coss-coelaions implied by he heoeical model. The pai of dashed lines epesens 95% confidence inevals fo he coesponding sample coss-coelaions based on high-fequency 5-min obsevaions fom 2003 hough dependencies in he op panel, he signs of he coss-coelaions geneally coincide. Howeve, hee is a noiceably fase decay owad zeo in he magniude of he coss-coelaions beween he vaiance isk pemium and he lagged euns compaed o he decay in he coss-coelaions beween he squaed VIX index iself and he lagged euns. 9 This diffeence closely mios he diffeence in he shape and he ae of decay in he sandad sample auocoelaions fo he wo daily volailiy seies depiced in Figue 1. The key empiical eun volailiy paens and dynamic dependencies illusaed in he wo figues ae consisen wih he idea ha volailiy caies a isk 9 Bolleslev and Zhou (2006) have peviously noed ha he eun volailiy asymmey end o be songe fo implied han fo ealized volailiies.

9 VOLATILITY IN EQUILIBRIUM 39 Table I. Summay Saisics The able epos summay saisics fo coninuously compounded euns, implied vaiances VIX 2, monhly ealized vaiances RV,þ 22, and he vaiance isk pemiumˆp ¼ VIX 2 berv ;þ22. The ealized vaiances ae consuced fom he summaion of high-fequency 5-min squaed euns. The expecaions fo he fuue vaiances _ E RV ;þ22 ae based on he HAR-RV foecasing model discussed in he ex. All he vaiables ae in pecenage fom. The daily daa exend fom Januay o Ocobe The 5-min sample spans Sepembe o Augus VIX 2 RV ; +22 bp Daily sampling ( ) Mean Sandad deviaion Skewness Excess kuosis Five-min sampling ( ) Mean Sandad deviaion Skewness Excess kuosis pemium. Sandad equilibium-based models build aound a epesenaive agen wih ime-sepaable uiliy ules ou piced volailiy isk. Insead, following he lieaue on so-called long-un isk models pioneeed by Bansal and Yaon (2004), we will hee assume a epesenaive agen wih Epsein Zin Weil pefeences, anamoun o a pefeence fo ealy esoluion of unceainy. Ou model is cas in coninuous ime, heeby avoiding any assumpions abou he decision ineval of he agen. The Epsein Zin Weil pefeence sucue was fis employed in a coninuous-ime asse picing seing by Duffie and Epsein (1992a). In his siuaion he sochasic discoun faco (SDF) will depend no only on he consumpion gowh ae bu also on he fuue invesmen oppouniies. 10 Consequenly, he aggegae make eun will be a funcion of he expeced gowh in he economy, as in he adiional ime-sepaable uiliy case, as well as he unceainy abou he fuue economic gowh (see, e.g., Campbell, 1996). Inuiively, his explains why invesos may be willing o pay an unceainy pemium, and in un why he VIX may diffe fom he coesponding acual eun volailiy, and why he coesponding vaiance isk pemium may ac as a sepaaely piced isk faco. The same mechanism involving ime-vaying economic unceainy and a pefeence fo ealy esoluion of unceainy also undelies he model of Bolleslev, Tauchen, and Zhou (2009), which paallels he pesen sudy in allowing he volailiy-of-volailiy in he economy, o he economic unceainy, o be 10 In conas o he expession fo he SDF involving he compensao funcion deived in Duffie and Epsein (1992a), we find i moe convenien o wok wih he discoun faco expessed in ems of he consumpion gowh ae and he make eun.

10 40 T. BOLLERSLEV ET AL. Table II. Calibaion Paamees The able epos he paamee values used in he calibaion of he heoeical model discussed in Secion B. BYa and BY denoe small adjusmens o, o values aken diecly fom, Bansal and Yaon (2004), BST denoes values calibaed by he auhos, and K denoes values fom Kiku (2008). Paamee Souce Pefeences q log(0.999) BYa c 7.50 BYa w 2.50 BYa Consumpion gowh l g BY j x log(0.960) BYa p x 0:044 ffiffiffiffiffi l BY Dividend gowh l d BYa / x / p3.0 BY 4:0 ffiffiffiffiffiffiffiffiffi 0:55 BYa, K / x p0.0 K d 4:0 ffiffiffiffiffiffiffiffiffi 0:45l BYa, K Volailiy l BY j log(0.760) BST l q BST j q log(0.0001) BST u q 0.07 BST deemined by is own sepaae sochasic pocess. The new equilibium model developed hee is also elaed o he long-un isk model of Dechsle and Yaon (2011), in which he expeced gowh ae in consumpion and he volailiy of consumpion gowh ae boh allowed o jump. 11 The pesen pape exends boh hese sudies by consideing he coss-coelaion beween volailiy and euns a all leads and lags. Of couse, as noed above, i has long been ecognized fom educed-fom analysis (e.g., Campbell, Lo, and MacKinlay, 1997, chape 12) ha he pice of volailiy isk mus be negaive, in un implying a negaive conempoaneous coelaion beween eun and volailiy. This negaive coelaion was, o he bes of ou knowledge, fis placed wihin a sucual equilibium famewok based on Epsein Zin Weil pefeences by Bansal and Yaon (2004) (see hei Equaion (12) and suounding discussion). Wih he noiceable excepions of Bolleslev, Tauchen, and Zhou (2009) and Dechsle and Yaon (2011), howeve, ohe sucual fomulaions geneally pesume ovely esicive dynamics fo he evoluion of economic unceainy, 11 A elaed long-un isk model in which he economic unceainy, o he volailiy of consumpion gowh, is allowed o jump in coninuous ime has also ecenly been exploed by Eake (2008), in an aemp o explain he exisence of a (on aveage) posiive volailiy isk pemium.

11 VOLATILITY IN EQUILIBRIUM 41 Table III. Calibaion Resuls The column denoed Obseved efes o he sample values fo he elevan momens in he sudies indicaed in he Souce column. The coesponding populaion momens implied by he heoeical model discussed in Secion B evaluaed a he paamee pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi values given in Table II ae epoed in he las column denoed Calibaion. E( m, f, ) and Vað m; Þ efe o he equiy pemium and he equiy eun sandad deviaion, especively. Fo souces, BST denoes values epoed in Table I above, BY efes o values aken fom Bansal and Yaon (2004), and DY denoes values fom Dechsle and Yaon (2011). Obseved Souce Calibaion Sysem dynamics Consumpion a E(g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1.80 BY 1.80 Vaðg Þ 2.93 BY 2.70 Dividends a E(d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 1.54 DY 1.54 Vaðd Þ DY Coelaions a Co(g, g 1 ) 0.43 DY 0.18 Co(d, d 1 ) 0.14 DY 0.10 Co(d, g ) 0.59 DY 0.74 Model-implied momens Reuns a pe( ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f, ) 0.82 DY 1.19 Vað f ; Þ 1.89 DY 4.43 E( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m, f, ) 6.23 DY 5.95 Vað m; Þ DY Volailiy and pemium E(VIX 2 ) BST E(p ) 8.96 BST 7.25 a The epoed value is appopiaely annualized o ease inepeaion. which imply a counefacual consan vaiance isk pemium ha simply canno explain he ich coss-coelaion paens seen in Figue 2. Also, in conas o he discee-ime fomulaions employed in mos pevious sudies, Bolleslev, Tauchen, and Zhou (2009) and Dechsle and Yaon (2011) boh included, he coninuous-ime fomulaion adoped hee has he disinc advanage of allowing fo he calculaion of inenally consisen model implicaions acoss all sampling fequencies and eun hoizons. Of couse, as aleady noed, he new model developed hee also accommodaes much iche and empiically ealisic longe un volailiy dependencies, including he possibiliy of facional inegaion. Moeove, ou coninuous-ime seup pemis an inenally consisen definiion of he isk-neual expecaions and he VIX volailiy index, heeby avoiding he inheen poblem in discee-ime asse picing models wih Genealized Auoegessive Condiional Heeoskedasic-ype eos ha he (condiional) vaiance is known one peiod in advance and heefoe fomally canno geneae a vaiance pemium. The new model developed below is also elaed o he mulifacal appoach pu foh in he seies of papes by Calve and Fishe (2007, 2008). In paicula, on

12 42 T. BOLLERSLEV ET AL. assuming ha he dividend gowh volailiy follows a mulifacal pocess, as in Calve and Fishe (2002), along wih an Epsein Zin Weil-ype epesenaive agen, as in he model developed hee, he equilibium models in Calve and Fishe (2007, 2008) ae also able o geneae endogenous volailiy feedback effecs and long memoy ype feaues in he volailiy, along wih negaive skewness in he euns due o he impac of leaning. None of hese fome sudies, howeve, has consideed he implicaions of he mulifacal seup and assumpions fo he isk-neual expecaion of he volailiy as embedded wihin he VIX, no he dynamic dependencies in he coesponding volailiy isk pemium. Ohe ecen sudies concened wih he equilibium picing of volailiy isk include Gabaix (2010) and Wache (2010), boh of whom analyze he implicaions of ae disases, and Leau, Ludvigson, and Wache (2009), who emphasize he ole of low-fequency movemens in macoeconomic unceainy fo explaining lowfequency muliyea movemens in sock make valuaions. Seveal sudies moe squaely ooed in he opion-picing lieaue have also exploed he equilibium implicaions of allowing fo iche volailiy dynamics and nonsandad pefeence sucues; see, fo example, he ecen papes by Benzoni, Collin-Dufesne, and Goldsein (2006), Eake and Shaliasovich (2008), Sana-Claa and Yan (2010), and he efeences heein. The empiical focus of he pesen pape and he use of high-fequency inaday daa fo he S&P 500 euns and he VIX ae disincly diffeen fom all hese pevious sudies, and o he bes of ou knowledge, no ohe coheen economic mechanism fo explaining all he dynamic dependencies and asymmeies in he volailiy and volailiy isk pemium depiced in Figues 1 and 2 is ye available in he lieaue. In ode o focus on he volailiy channels ha we seek o illuminae, he model is delibeaely kep as simple as possible and hus would no be expeced o mach all asse picing momens. Noneheless, in keeping wih he basic seup in Bansal and Yaon (2004), he geneal modeling famewok is flexible enough o allow fo a easonable mach wih many of he moe sandad momens as well. The plan fo he es of he pape is as follows. The new heoeical model is fomally defined and solved in Secion 2. This secion also biefly discusses a simple calibaion fo he model designed o mach some of he moe sandad asse picing momens. The equilibium implicaions fom an exended long-memoy vesion of he model in egad o he key eun volailiy asymmeies and own dynamic volailiy dependencies ha we seek o explain ae pesened in Secion 3. The daa used in he consucion of he figues discussed above and he model s abiliy o epoduce hese basic empiical feaues ae he subjec of Secion 4. Secion 5 concludes. Mos of he mahemaical poofs ae defeed o wo appendixes, as ae fuhe deails concening he sylized calibaion alluded o in Secion 2.

13 VOLATILITY IN EQUILIBRIUM Volailiy in Equilibium The classic coninuous-ime ineempoal capial asse picing model (CAPM) of Meon (1973) is ofen used o jusify he exisence of a volailiy isk pemium. Howeve, his model is inconsisen wih obseved long-ange volailiy dependence as seen in Figue 1 and is incapable of explaining he dynamic leveage effec and asymmeic eun volailiy dependencies shown in Figue 2. The coninuous-ime endowmen economy developed hee insead builds on he discee-ime long-un isk model pioneeed by Bansal and Yaon (2004). We begin by descibing an iniial coninuous-ime model seup and soluion unde sho-memoy Makov dynamics. We nex validae is pedicions fo sandad vaiables via a simple calibaion. We also show how i can explain he dynamic asymmeies bu fails on maching longange volailiy dependence. We subsequenly show how o adjus he model o incopoae empiically elevan long-memoy dependencies while peseving he key eun volailiy implicaions of he iniial model. 2.1 INITIAL MODEL SETUP AND ASSUMPTIONS Le he local geomeic gowh ae of consumpion C in he economy be denoed by g [ dc C, which we assume follows he coninuous-ime pocess g ¼ðl g þ x Þd þ g; dw c : ð1þ Hee l g denoes he consan long-un mean gowh ae, x is he mean-zeo sochasic componen of consumpion gowh, g, efes o economic unceainy, ha is, he condiional volailiy of he gowh ae, and W c is a sandad Wiene pocess. The sochasic gowh componen follows he sandad dynamics dx ¼ j x x d þ x dw x ; ð2þ whee j x > 0, and W x is a sandad Wiene pocess independen of W c. Fo small j x, his is a long-un isk-ype specificaion, bu we absac fom sochasic volailiy of consumpion gowh iself. We also assume a dividend asse wih dividend gowh dynamics (d ¼ dd /D ), d ¼ðl d þ / x x Þd þ / g; dw c þ / x dw x þ d dw d ; whee l d efes o he uncondiional mean dividend gowh ae, he /s eflec he dividend s exposues o he consumpion isk facos, and he dividend gowh innovaion volailiy d is assumed o be consan (nonsochasic) fo simpliciy. Impoanly, we assume ha he volailiy dynamics in he economy ae govened by he coninuous-ime affine pocesses, ð3þ

14 44 T. BOLLERSLEV ET AL. d 2 g; ¼ j ðl 2 Þd þ p ffiffiffiffi dw q ; pffiffiffiffi q dq ¼ j q ðl q q Þd þ u q q dw ; ð4þ ð5þ whee he wo Wiene pocesses W and W q ae independen and joinly independen of W c and W x, and he paamees saisfy he nonnegaiviy esicions l > 0, l q > 0, j > 0, j q > 0, and u q > The sochasic volailiy pocess 2 g; epesens ime-vaying economic unceainy in consumpion gowh, wih he volailiy-of-volailiy pocess q in effec inducing an addiional souce of empoal vaiaion in ha same pocess. 13 We assume ha he epesenaive agen s consumpion and invesmen decisions ae based on he maximizaion of Epsein Zin Weil ecusive pefeences. As fomally shown in Appendix A, his implies he following equilibium elaionship beween he ineempoal maginal ae of subsiuion, M, consumpion, C, and he cumulaed eun on he aggegae wealh pofolio, R, d logm þ h w d logc þð1 hþd logr ¼ qh d; whee q denoes he insananeous subjecive discoun faco, w equals he ineempoal elasiciy of subsiuion, and he paamee h is defined by h [ ð1 cþ 1 w 1 1 ; whee c efes o he coefficien of isk avesion. The expession in Equaion (6) is naually inepeed as he coninuous-ime vesion of he discee-ime equilibium elaionship deived in Epsein and Zin (1991). In he following, we will mainain he assumpions ha c > 1andw > 1, which eadily implies ha h < Consisen wih he empiical egulaiies discussed in Secion 1, hese ð6þ ð7þ 12 We also assume ha l q j q > 0:50u 2 q, which ensues posiiviy of q, and ha l is sufficienly lage elaive o j, so ha negaiviy of 2 g; is highly unlikely and he subsequen appoximaions easonable. 13 Empiical evidence in suppo of ime-vaying volailiy-of-volailiy in consumpion gowh has ecenly been pesened in Bolleslev, Tauchen, and Zhou (2009). Alenaively, Dechsle and Yaon (2011) conside a discee-ime model wih jumps in volailiy (and expeced gowh aes), in pa moivaed by esimaes epoed in he opion-picing lieaue. 14 The assumpion ha c > 1 is geneally ageed upon. Ealy esimaes by, fo example, Hall (1988) and Campbell and Mankiw (1989), pu w < 1, bu hese esuls have subsequenly been called ino quesion by Bansal and Yaon (2004) among many ohes, and we follow he moe ecen lieaue in assuming ha w > 1.

15 VOLATILITY IN EQUILIBRIUM 45 specific paamee esicions ensue, among ohe hings, ha asse pices fall on news of posiive volailiy shocks and ha volailiy caies a posiive isk pemium. 2.2 INITIAL MODEL SOLUTION Le W ¼ Wð 2 g; ; q ; x Þ denoe he pice-dividend aio, o equivalenly he piceconsumpion o he wealh-consumpion aio, of he asse ha pay he consumpion endowmen fc þs g s2½0;nþ. The equilibium sochasic diffeenial equaion fo log(w ) involves he ecipocal of W(z ), which we appoximae via WðzÞ 1 expð WÞ expð log WÞðlogWðzÞ log WÞ ¼j 0 j 1 logwðzþ, whee j 1 > Now, conjecuing a soluion fo log(w ) as an affine funcion of he hee sae vaiables, 2 g;, q, and x, logðw Þ¼A 0 þ A 2 g; þ A q q þ A x x ; and solving fo he A coefficiens, i follows fom Appendix B ha A 0 ¼ j 0 q þð1 w 1Þl g þ A l j þ A q l q j q þ ha2 x 2 x 2 ; j 1 A ¼ cð1 w 1Þ 2ðj þ j 1 Þ ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jq A q ¼ j 2 h q þ j 1 þ j 2 1 u 2 q A2 ; A x ¼ hu 2 q 1 1 w j x þ j 1 : ð8þ The esicions ha c > 1 and w > 1 eadily imply ha he impac coefficien associaed wih boh he volailiy sae vaiables ae negaive; ha is, A < 0 and A q < O, pu diffeenly ha he make falls on posiive volailiy news. Fom 15 This appoximaion plays a simila ole o ha of he sandad Campbell Shille discee-ime appoximaion, and simila expessions have been used in a coninuous-ime seing by, fo example, Campbell and Viceia (2002). 16 The soluion fo A q epesens one of a pai of oos o a quadaic equaion. Howeve, i is he economically meaningful oo as i implies ha he pemium disappeas fo u q / 0, o when q is consan, as would be equied by he lack of abiage.

16 46 T. BOLLERSLEV ET AL. hese explici soluions fo he fou coefficiens, i is now possible o deduce he educed-fom expessions fo ohe vaiables of inees. In paicula, as shown in Equaion (B.13) in Appendix B, he equilibium dynamics of he logaihmic cumulaive eun pocess is given by d logr d ¼ q þ l g w þ hðh 1ÞA2 x 2 x h 1 Ax x þ A d x 2 x þ / x 2 þ x! w Ad q ðj q þ j d 1 Þq A d ðj þ j d 1 Þþ/2 2 g; d 2 2 d 2 þ / g; dw c þ A d q u pffiffiffiffi q q dw q þða d x x þ / x ÞdW x þ d dw d : þ A d pffiffiffiffi q dw The diecional effecs of inceases in he endowmen volailiy, 2 g;, on he local expeced eun ae geneally ambiguous. Howeve, fo sufficienly high levels of isk avesion c and ineempoal subsiuion w, endowmen volailiy posiively affecs he local expeced eun. Meanwhile, inceases in he volailiy-ofvolailiy, q, unambiguously, incease he local expeced eun, eflecing he compensaion fo beaing volailiy isk. On he ohe hand, diffusive-ype innovaions in he volailiy and he volailiy-of-volailiy, dw q and dw, boh have a negaive impac on he local euns, consisen wih a leveage-ype effec. To fuhe appeciae he implicaions of he model, i is insucive o conside he equiy pemium deived in Equaion (B.12), p ; [ 1 d d½r d ; MŠ R d M ¼ c/ 2 g; ðh 1ÞðA A d þ A qa d q u2 q Þq ðh 1ÞðA x A d x x þ / x Þ x : ð9þ ð10þ The fis em, c 2 g;, is akin o a classic isk-eun ade-off elaionship. I does no epesen a volailiy isk pemium pe se, howeve, bu ahe changing pices of consumpion isk induced by he pesence of sochasic volailiy. Insead, he second em, ð1 hþj 2 1 ða A d þ A qa d q u2 q Þq, has he inepeaion of a ue volailiy isk pemium, epesening he confounding impac of he wo diffusive-ype innovaions, dw q and dw. The exisence of his ue volailiy isk pemium depends cucially on he dual assumpions of ecusive uiliy, o h 6¼ 1, as volailiy

17 VOLATILITY IN EQUILIBRIUM 47 would no ohewise be a piced faco, and ime-vaying volailiy-of-volailiy, in he fom of he q pocess. 2.3 CALIBRATION As a check on he iniial model discussed above, we undeook a simple calibaion o documen ha i is able o mach he aggegae se of momens geneally ageed upon in he macofinance lieaue. The deails ae lef o Appendix C. The wo main messages of he calibaion ae as follows. Fis, ou iniial model does indeed give a consisen equilibium isk-based explanaion of he dynamic coss-coelaions beween sock make eun and volailiy. This is eviden by compaing he model-implied dynamic coss-dependencies displayed in he igh-hand panel of Figue 3 o he obseved dependencies seen in Figue 2. Evidenly, he ageemen is quie close. 17 Second, as seen by conasing he model-based auocoelaions in he lef-hand panels of Figue 3 o he obseved auocoelaions in Figue 1, he iniial model is unable o accoun fo he vey slow decay in hese ha is so widely documened in he lieaue. 18 This model failue can, of couse, be aced diecly o he pesumed Makov specificaions fo he undelying dynamics. The challenge, hen, is o exend he undelying dynamics o incopoae longange dependencies while a he same ime peseving he equilibium isk-based explanaion fo he dynamic leveage and volailiy feedback effecs seen in Figue This exension enails some ahe deailed echnical analysis. Since we know fom he sylized calibaion in Appendix C ha he iniial model is indeed capable of maching he geneally ageed upon se of aggegae macofinance momens, we subsequenly simplify somewha he analysis by assuming away he dynamic dependencies in he consumpion endowmen and dividend gowh pocesses, while adaping he coninuous-ime seup and coesponding model soluion o accommodae vey flexible dynamic dependencies in he economic unceainy, including long memoy ype effecs. 17 Also, he insananeous coelaion beween he changes in he VIX 2 and he euns implied by he model calibaed in he appendix equals and as such is eniely consisen wih he coesponding numeically lage empiical esimaes epoed in he exan lieaue. 18 The empiical and model-implied counepas of he auo- and coss-coelaions fo he expeced ealized volailiy, as fomally defined below, closely paallel o hose fo he VIX 2 depiced in Figues 1 3. These addiional esuls and gaphs ae available upon eques. 19 To he bes of ou knowledge, wih he excepion of he disincly diffeen mulifacal appoach in Calve and Fishe (2007, 2008), he exan equilibium models all enail Makov dynamics and heeby canno accoun fo he long-ange dependence eviden in Figue 1.

18 48 T. BOLLERSLEV ET AL. 2.4 GENERAL MODEL SOLUTION Numeous compeing coninuous-ime sochasic volailiy models have been poposed in he lieaue. We hee build on he famewok of Come and Renaul (1996) in assuming ha 2 g; may be descibed by he geneal epesenaion, 2 g; ¼ 2 þ ð N a s pffiffiffiffi q sdw s : ð11þ By appopiae choice of he moving aveage weighs faðsþg s2½0;nþ, his epesenaion obviously includes he affine pocess in Equaion (4) as a special case. Impoanly, by suiable choice of he mapping s / a(s), he pocess fo 2 g; may also exhibi vaious foms of long-ange dependence. In paicula, seing 0 1 ð s aðsþ a k e ks e ku u a dua ð12þ Cð1 þ aþ 0 esuls in he classic facionally inegaed pocess, whee a denoes he longmemoy paamee. To complee he specificaion of he model and sill allow fo acable closedfom soluions, we will assume away he pedicabiliy in consumpion gowh in Equaion (2), ha is, x [ 0, while mainaining he idenical law of moion fo he volailiy-of-volailiy in Equaion (5). The acual soluion saegy, which is new and echnically demanding, diffes somewha fom ha fo he iniial model. The full deails ae given in Appendix D; a pecis follows. In paallel o he soluion fo he sho-memoy model discussion above, we sa by conjecuing a soluion fo he logaihmic pice-consumpion aio now of he fom logðw Þ¼A 0 þ A q q þ ð N A s pffiffiffiffi q sdw s ; ð13þ whee A 0, A q, and faðsþg s2½0;nþ ae o be deemined. Some cae is needed because of subleies elaed o possible abiage oppouniies unde long memoy ype dependencies (Roges, 1997). The saegy ha we use elies on he fac ha in he absence of abiage, he eun on a aded secuiy mus follow a semimaingale. This allows us o spli up he euns ino a dif and a local maingale componen. This decomposiion is possible when A() exiss and is diffeeniable a zeo. Subsiuing he conjecued soluion ino he picing Equaion (6), he following odinay diffeenial equaion fo > s,

19 VOLATILITY IN EQUILIBRIUM and wo egula equaions, c 1 A 0 w 1 ð sþ j 1 Að sþ ¼ 2 að sþ; 49 ð14þ A 0 ¼ h 2 u2 q A2 q ðj q þ j 1 ÞA q þ hað0þ2 ¼ 0; ð15þ 2 A q j q h q j 0 þ 1 w 1 l g q c 2 1 w 1 2 : ð16þ j 1 Fom he appendix, he soluions o his sysem of equaions ae þð N c 1 w 1 AðsÞ ¼ e j1ð sþ aðsþds; 2 A q ¼ j q þ j 1 A q j q h q j 0 þ A 0 ¼ s ð17þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jq 2 h þ j 2 1 u 2 q Að0Þ2 ; ð18þ hu 2 q 1 w 1 l g q c 2 1 w 1 2 ; ð19þ j 1 which exiss and is well-defined subjec o a eminal condiion uling ou explosive bubble soluions and ohe mild egulaiy condiions. As befoe, fom his se of soluions, i is possible o deduce he educed-fom expessions fo all ohe vaiables of inees. In paicula, in paallel o he expession fo he euns in he sho-memoy model in Equaion (9) above, i follows fom Appendix D ha he educed-fom expession fo he euns in he geneal model may be expessed as, d logðr Þ¼l R; d þ g; dw c pffiffiffiffi q pffiffiffiffi þ A q u q q dw þ Að0Þ q dw ; ð20þ whee he dif is defined by, l R; ¼ q þ l g w þ 1 2 þ c g; w ðj q þ j 1 ÞA q q : ð21þ Similaly, fom Equaion (D.7) in Appendix C, he equilibium equiy pemium akes he fom,

20 50 T. BOLLERSLEV ET AL. 1 p ; ¼ c 2 þð1 hþ½a 2 q u2 q þ Að0Þ2 Šq ¼ c 2 þ 2 h 1 ðj q þ j 1 ÞA q q : ð22þ Unde he peviously discussed paamee esicions c > 1 and w > 1, implying ha h < 0, he equiy pemium emains posiive. Moe geneally, as long as c > 1 w, o h < 1, i emains he case ha sochasic volailiy caies a posiive isk pemium. Noe also ha he insananeous equiy pemium only depends on he faðsþg s2½0;nþ weighs and he possible long-un dependencies in he volailiy hough he cumulaive impac deemined by he inegal soluion fo A(0) in Equaion (17). 3. Dynamic Equilibium Dependencies The equilibium expessions discussed in he pevious secion chaaceize how he equiy pemium depends on he insananeous volailiy and how he insananeous eun esponds o conempoaneous volailiy innovaions wihin he model. This secion fuhe deails he model s implicaions in egad o he dynamic dependencies in he volailiy and he volailiy isk pemium and how hese volailiy measues covay wih leads and lags of he euns a diffeen hoizons. We will subsequenly confon hese heoeical pedicions wih he key empiical egulaiies discussed in Secion VIX AND THE VOLATILITY RISK PREMIUM One of he key feaues of he model is ha he economic unceainy efleced in 2 g; may exhibi long-ange dependence, while he volailiy of he unceainy, q, is a sho-memoy pocess. This, in un, has impoan implicaions fo he seial coelaion popeies of he equivalen o he VIX volailiy index implied by he model and he coesponding volailiy isk pemium and how hese measues coelae wih he euns. To begin, conside he (fowad) inegaed vaiance, o quadaic vaiaion, of he asse pice S, IV ; þ N [ þð N s ¼ d ½logS; logsš s ; ð23þ whee he squae backe epesens he sandad quadaic vaiaion pocess. Fom Equaion (D.8) in Appendix D, he educed-fom expession fo he inegaed vaiance may be convenienly wien as follows:

21 VOLATILITY IN EQUILIBRIUM 51 IV ; þ N ¼ þð N 2 g;s ds þða2 q u2 q þ Að0Þ2 Þ ð þ N q s ds: ð24þ The inegaed vaiance is, of couse, andom and no obseved unil ime þ N. The coesponding vaiance swap ae is defined as he ime isk-neualized expecaion of IV,þN, say E Q ðiv ;þn Þ. This isk-neual expecaion may in heoy be calculaed in a compleely model-fee fashion fom a coss secion of opion pices (see, e.g., Ca and Madan, 1998; Bien-Jones and Neubege, 2000; Jiang and Tian, 2005). This way of calculaing he vaiance swap ae diecly mios he definiion of he (squaed) VIX volailiy index fo he S&P 500, VIX 2 [ EQ ðiv ; þ N Þ; ð25þ whee he hoizon N is se o 1 monh o 22 days. 20 This same isk-neual expeced vaiaion may alenaively be calculaed wihin he specific equilibium model seing. In paicula, i follows fom Equaion (D.9) in Appendix D ha VIX 2 ¼ b x;0 þ ð N pffiffiffiffi h x s q sdw s þ b x;q q ; ð26þ whee he dependence on N has been suppessed fo noaional convenience. The fh x ðsþg s2½0;nþ weighs depend on he faðsþg s2½0;nþ moving aveage coefficiens and impoanly inhei any long-memoy decay in hose coefficiens. As such, an evenual slow hypebolic decay in he auocoelaions fo VIX 2 would heefoe be eniely consisen wih he implicaions fom he geneal heoeical model; ha is, CoðVIX 2 ; VIX2 þ s Þ¼c hs b h s > S; ð27þ whee c h > 0 and b h < 0 ae consans and S denoes a sufficienly long lag so ha he sho-un dependencies have dissipaed. Nex, conside he vaiance isk pemium, as fomally defined by he diffeence beween he isk-neual and objecive expecaions of IV,þN, p [ E Q ðiv ; þ N Þ E P ðiv ; þ N Þ: ð28þ Wheeas E Q ðiv ;þn Þ and E P ðiv ;þn Þ boh depend on he consumpion gowh volailiy and he volailiy-of-volailiy of ha pocess, he vaiance isk pemium 20 A moe deailed descipion of he mechanical calculaion of he VIX index is available in he whie pape on he CBOE Web sie; see also he discussion in Jiang and Tian (2007).

22 52 T. BOLLERSLEV ET AL. is simply an affine funcion of he volailiy-of-volailiy o q. Specifically, fom Equaion (D.10) in Appendix D, p ¼ b p;0 þ b p;1 q ; ð29þ whee b p,0 > 0 and b p,1 > 0, eflecing he posiive pemium fo beaing volailiy isk. Inuiively, fo h < 1, invesos have a pefeence fo ealy esoluion of unceainy, while w > 1 implies ha hee is a negaive link beween he volailiy and he P/D aio. Meanwhile, he SDF only depends indiecly on shocks o he volailiy hough (h 1)R. Thus, any asse ha is posiively coelaed wih volailiy will be beaing a negaive isk pemium. As such, he pemium fo he vaiance isk exposue naually inceases if he unceainy abou volailiy inceases, ha is, he volailiy-of-volailiy, as chaaceized by he q pocess. Since he exposue of he vaiance swap o volailiy shocks diecly mios he exposue of he volailiy, he vaiance pemium ha esuls fom he covaiance beween he SDF and he vaiance swap heefoe only depends on q. Even hough he vaiance isk pemium will geneally be posiive, only if q is ime vaying will he pemium also be ime vaying. Moeove, fom Equaion (29) above, he p pocess simply inheis he dynamic dependencies in he q pocess and should exhibi a elaively fas exponenial decay in is auocoelaion sucue. Tha is, Coðp ; p þ s Þ¼c q e j qs whee c q > 0 denoes a posiive consan. 3.2 RETURN VOLATILITY CORRELATIONS s > 0; ð30þ The equilibium expessions fo he vaiance swap ae and he pemium discussed above also have some impoan and diecly esable implicaions fo he dynamic coss-coelaions fo he VIX 2 and p wih he euns. To help elucidae he economic mechanisms undelying hese dependencies, i is insucive o fis eview he pedicions unde sho-memoy dynamics. We subsequenly discuss he geneal case, explicily allowing fo long-memoy dynamics. The coss-coelaions beween he vaiance pemium and he euns ae easie o calculae han hose fo he VIX, and we begin by consideing hese. Le [ d log(r ) denoe he insananeous eun. We will efe o he cosscoelaions beween he ime pemium p and he fuue euns, þs fo s > 0, as he fowad coelaions. The fowad coelaions epesen he exen o which he pemium is able o foecas he euns. The coelaions beween he pemium p and he lagged euns, þs fo s < 0, on he ohe hand, epesen

23 VOLATILITY IN EQUILIBRIUM 53 he impac of movemens in he pas euns on he cuen vaiance pemium. Given he well known nea unpedicabiliy of euns, we would expec he fowad coelaions o be posiive, eflecing he pemium fo beaing volailiy isk, bu small and quickly declining o zeo fo longe inedaily eun hoizons. We would expec he lagged coelaions o be negaive, bu inceasing o zeo fo longe daily lags, consisen wih he exisence of a dynamic leveage-ype effec. The fomal heoeical pedicions fom he model confim his inuiion. Specifically, fom he esuls fo he sho-memoy model deived in Appendix B, i follows ha fo s > 0, Coðp ; þ s Þ¼b R;q Vaðq ÞK q e j qs ; whee b R,q epesens he sensiiviy of he insananeous euns o he q pocess. Since b R,q > 0 and K q > 0, he fowad coelaions ae all posiive. Similaly, i follows fom he appendix ha he coss-coelaions fo s < 0 saisfy, Coðp ; s Þ¼ðb R;q Vaðq ÞþA q / 2 q l qþk q e j qs : Since he high-fequency euns ae close o unpedicable, he value fo b R,q is likely o be small. Hence, we would expec he second em involving A q < 0o dominae he expession in paenhesis, and consequenly, all he backwad coelaions o be negaive. In summay, he model pedics, a e Coðp ; þ s Þ¼ k qjsj S < 0; a þ e k ð31þ qs S 0; whee a < 0 and a þ < 0. As discussed fuhe below, his pedicion does indeed adhee vey closely wih he paen in he empiical coelaions. The dynamic coss-coelaions beween he VIX 2 and he eun ae a bi moe involved han hose fo he vaiance pemium. Sill, he basic inuiion is essenially he same, excep ha he acual fomulas now also depend on he volailiy pocess 2 g; iself and is coelaion wih he euns. In paicula, efeing o Appendix B, he fowad coelaions fo s > 0 akes he fom, CoðVIX 2 ; þ sþ ¼b R; Vað 2 g; ÞK e j s þ b R;q Vaðq ÞK q e j qs : The sign of b R, will depend upon he pefeence paamees w and c. Howeve, i may easonably be expeced o be posiive, 21 so ha he fowad coss-coelaions will again be posiive, wih he decay owad zeo ulimaely deemined by he dominan value of j o j q. As fo he pemium, he backwad coelaions fo s < 0 ae slighly moe complicaed, aking he fom, 21 The pooypical values w ¼ 2.5 and c ¼ 7.5 used in ou calibaion imply ha b R, ¼ 12.6.

24 54 T. BOLLERSLEV ET AL. CoðVIX 2 ; sþ ¼ðb R; Vað 2 g; ÞþA l ÞK e j s þðb R;q Vaðq Þ þ A q / 2 q l qþk q e j qs : As discussed above, given he difficulies in pedicing euns, we would expec he b R, and b R,q ems o be elaively small and dominaed by he ems involving he A < 0 and A q < 0 coefficiens ha deemine he insananeous esponse of he euns o volailiy innovaions. Consequenly, he backwad coelaions ae naually expeced o be all negaive. In summay, CoðVIX 2 ; aq; e þ sþ k qjsj þ a ; e k jsj s < 0; a q; þ e kqs þ a ; e k s s 0; ð32þ whee a q,, a, < 0 and a q,þ, a,þ < 0. Again, hese heoeical model pedicions closely mach wha we see in he daa. The geneal model allowing fo long memoy in he economic unceainy essenially gives ise o he same basic paens and pedicions. The fomal deivaions ae somewha moe complicaed; howeve, he acual values of he coss-coelaions will ulimaely depend upon he specific pocess fo 2 g; and he coesponding moving aveage coefficiens faðsþg s2½0;nþ. We biefly skech he elevan ools and ideas equied o evaluae he coelaions. The economics of he poblem emain exacly he same. The main ineacions beween he eun and he volailiy ae wo-fold: one consiss in he fowad effec of volailiy innovaions on fuue expeced euns and he ohe involves he feedback effec of lagged eun innovaions, o he diffusive pa of he euns, on cuen volailiy. To elucidae hese sepaae effecs wihin he geneal model seing, i is useful o define he auxiliay vaiable g; ;s [ þ s dw c þ s þ A pffiffiffiffiffiffiffiffiffi qu q q þ s dw q pffiffiffiffiffiffiffiffiffi þ s þ Að0Þ q þ s dw þ s s < 0; l R; þ s s 0; which equals he local diffusive pa of he equilibium eun pocess fo s < 0 and he local mean of he equilibium eun pocess fo s 0, especively. As such, he basic shape of he coss-covaiances beween he vaiance isk pemium p and,s diecly mios ha of he coss-covaiances wih he euns, þ s. In paicula, i follows diecly fom he expession fo p in Equaion (29) ha he fowad coelaions wih,s mus be popoional o he auocovaiances of he q pocess. Tha is, fo s > 0, Covðp ; ;s Þ¼K e j qs ; whee K > 0 denoes a posiive consan of popoionaliy. To deive he backwad coelaions, wie q in inegal fom,

25 VOLATILITY IN EQUILIBRIUM 55 q ¼ u q ð u ¼ N e j qðu Þ pffiffiffiffiffi q udw q u : Fom his expession, i follows ha fo s < 0, Covðp ; ;s Þ¼e jqjsj pffiffiffiffiffiffiffiffiffi Eðb p;1 u q q þ s dw q þ s ; A pffiffiffiffiffiffiffiffiffi qu q q þ s dw q ¼ A q b p;1 u 2 q l qe j qjsj ; þ s Þ so ha all he backwad auocovaiances ae again negaive and decay a an exponenial ae, povided ha A q < 0 as again implied by c > 1 and w > 1. These dynamic paens in he coss-covaiances fo,s diecly anslae ino he cosscoelaions fo,s, and in un he coss-coelaions fo he euns þ s, mioing he implicaions fom he sho-memoy model summaized in Equaion (31) above. The heoeical pedicions fo he dynamic coss-coelaions beween he VIX and he euns wihin he geneal model seing ae no quie as clea-cu as hose fo he vaiance pemium. The q pocess deemining he vaiance isk pemium essenially ges confounded wih he classical consumpion isk pemium, and hee ae also poenial side effecs fom long-ange dependence. Howeve, he undelying economic mechanisms emain he same as fo he sho-memoy model, esuling in a simila paen of mixed negaive backwad coelaions and posiive fowad coelaions. 4. Empiical Resuls The equilibium famewok developed above compleely chaaceizes he dynamic dependencies in he euns and he volailiy. Of couse, he specific soluion of he model will invaiably depend upon he choice of pefeence paamees and he values of he paamees fo he undelying consumpion gowh ae and volailiy dynamics. Meanwhile, he model is obviously somewha sylized, and diec esimaion based on acual consumpion daa would be challenging a bes. Insead, we nex illusae he model s qualiaive implicaions in egad o he auocoelaions and coss-coelaions deived in he pevious secion and, in paicula, how well he basic paens implied by he model mach hose of he acual daa depiced in Figues 1 and 2. We begin wih a discussion of he daa and peinen summay saisics undelying he figues. 4.1 DATA DESCRIPTION Ou ick-by-ick daa fo he S&P 500 fuues conac wee obained fom Tick Daa Inc. To alleviae he impac of make micosucue noise in he