A Tale of Two Indices


 Kenneth Ray
 1 years ago
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1 PEER CARR is he direcor of he Quaniaive Finance Research group a Bloomberg LP and he direcor of he Masers in Mahemaical Finance program a he Couran Insiue of New York Universiy NY. LIUREN WU is an associae professor of economics and finance a he Zicklin School of Business Baruch College Ciy Universiy of New York NY. A ale of wo Indices PEER CARR AND LIUREN WU In 1993 he Chicago Board of Opions Exchange (CBOE) inroduced he CBOE Volailiy Index. his index has become he de faco benchmark for sock marke volailiy. On Sepember 3 he CBOE revamped he definiion and calculaion of he volailiy index and backcalculaed he new index o 199 based on hisorical opion prices. On March 6 4 he CBOE launched a new exchange he Chicago Fuures Exchange and sared rading fuures on he new volailiy index. Opions on he new volailiy index are also planned. his aricle describes he major differences beween he old and he new volailiy indexes derives he heoreical underpinnings for he wo indexes and discusses he pracical moivaions behind he recen swich. I also looks a he hisorical behavior of he new volailiy index and discusses he pricing of VIX fuures and opions. In 1993 he Chicago Board of Opions Exchange (CBOE) inroduced he CBOE Volailiy Index (VIX). his index has become he de facor benchmark for sock marke volailiy. he original consrucion of his volailiy index uses opions daa on he S&P 1 index (OEX) o compue an average of he Black and Scholes [1973] opion implied volailiy wih srike prices close o he curren spo index level and mauriies inerpolaed a abou one monh. he marke ofen regards his implied volailiy measure as a forecas of subsequen realized volailiy and also as an indicaor of marke sress (Whaley []). On Sepember 3 he CBOE revamped he definiion and calculaion of he VIX and backcalculaed he new VIX o 199 based on hisorical opion prices. he new definiion uses he S&P 5 index (SPX) o replace he OEX as he underlying sock index. Furhermore he new index measures a weighed average of opion prices across all srikes a wo nearby mauriies. On March 6 4 he CBOE launched a new exchange he Chicago Fuures Exchange (CFE) o sar rading fuures on he new VIX. A he ime of wriing opions on he VIX are also planned. In his aricle we describe he major differences in he definiions and calculaions of he old and he new volailiy indexes. We derive he heoreical underpinnings of he wo indexes and discuss he pracical moivaions for he swich from he old o he new VIX. We also sudy he hisorical behavior of he new volailiy index and analyze how i ineracs wih sock index reurns and realized volailiies. Finally we discuss how o use opions on he underlying S&P 5 index o define valuaion bounds on VIX fuures and how o exploi informaion in he underlying opions marke and VIX fuures o price opions on he new VIX. DEFINIIONS AND CALCULAIONS he Old VXO I IS ILLEGAL O REPRODUCE HIS ARICLE IN ANY FORMA he CBOE renamed he old VIX he VXO and i coninues o provide quoes on his index. he VXO is based on opions on he OEX. I is an average of he BlackScholes implied volailiies on eigh nearhemoney SPRING 6 HE JOURNAL OF DERIVAIVES 13 Copyrigh 6
2 opions a he wo neares mauriies. When he ime o he neares mauriy is wihin eigh calendar days he nex wo neares mauriies are used insead. A each mauriy he CBOE chooses wo call and wo pu opions a he wo srike prices ha sraddle he spo level and are neares o i. he CBOE firs averages he wo implied volailiies from he pu and call a each srike price and hen linearly inerpolaes beween he wo average implied volailiies a he wo srike prices o obain he ahemoney spo implied volailiy. he inerpolaed ahemoney implied volailiies a he wo mauriies are furher inerpolaed along he mauriy dimension o creae a radingday volailiy which consiues he VXO. he BlackScholes implied volailiy is he annualized volailiy ha equaes he BlackScholes formula value o he opions marke quoe. he annualizaion is based on an acual/365 daycouning convenion. Insead of using his implied volailiy direcly he CBOE inroduced an arificial radingday conversion ino he calculaion of he VXO. Specifically le AMV ( ) denoe he ime BlackScholes ahemoney implied volailiy as an annualized percenage wih expiry dae. he CBOE convers his percenage o radingday volailiy V ( ) as V ( ) = AMV ( ) NC / N where NC and N are he number of acual calendar days and he number of rading days beween ime and he opion expiry dae respecively. he CBOE convers he number of calendar days ino he number of rading days according o he following formula: (1) N NC in(nc/7) () he VXO represens an inerpolaed radingday volailiy a rading days based on he wo radingday volailiies a he wo neares mauriies (V ( 1 ) and V ( )): VXO V N N1 = ( 1) + V ( ) N N N N 1 1 where N 1 and N denoe he number of rading days beween ime and he wo opion expiry daes 1 and respecively. Since each monh has around rading days he VXO represens a onemonh ahemoney implied volailiy esimae. Neverheless he radingday conversion in Equaion (1) raises he level of he VXO and makes (3) i no longer comparable o annualized realized volailiies compued from index reurns. hus he VXO compuaion mehodology has drawn criicism from boh academia and indusry for is arificially induced upward bias. he New VIX In conras o he old VXO which is based on nearhemoney BlackScholes implied volailiies of OEX opions he CBOE calculaes he new volailiy index VIX using marke prices insead of implied volailiies. I also uses SPX opions insead of OEX opions. he general formula for he new VIX calculaion a ime is Ki VS K e r O K F ( ) 1 ( )= ( i ) 1 i K i where is he common expiry dae for all of he opions involved in his calculaion F is he ime forward index level derived from coerminal index opion prices K i is he srike price of he ih ouofhemoney opion in he calculaion O (K i ) denoes he ime midquoe price of he ouofhemoney opion a srike K i K is he firs srike below he forward index level F r denoes he ime riskfree rae wih mauriy and K i denoes he inerval beween srike prices defined as K i (K i1 K i )/. For noaional clariy we suppress he dependence of r and F on he mauriy dae as no confusion will resul. Equaion (4) uses only ouofhemoney opions excep a K where O (K ) represens he average of he call and pu opion prices a his srike. Since K F he average a K implies ha he CBOE uses one uni of he inhemoney call a K. he las erm in Equaion (4) represens he adjusmen needed o conver his inhemoney call ino an ouofhemoney pu using pucall pariy. he calculaion involves all available call opions a srikes greaer han F and all pu opions a srikes lower han F. he bids of hese opions mus be sricly posiive o be included. A he exreme srikes of he available opions he definiion for he inerval K is modified as follows: K for he lowes srike is he difference beween he lowes srike and he nex lowes srike. Likewise K for he highes srike is he difference beween he highes srike and he nex highes srike. o deermine he forward index level F he CBOE chooses a pair of pu and call opions wih prices ha are closes o each oher. hen he forward price is derived via he pucall pariy relaion: (4) 14 A ALE OF WO INDICES SPRING 6 Copyrigh 6
3 F e r () (C (K) P (K)) K (5) he CBOE uses Equaion (4) o calculae VS( ) a wo of he neares mauriies of he available opions 1 and. hen he CBOE inerpolaes beween VS( 1 ) and VS( )o obain a VS() esimae a 3 days o mauriy. he VIX represens an annualized volailiy percenage of his 3day VS using an acual/365 daycouning convenion: NC NC VIX = ( 1 ) VS( 1) + ( VS ) ( ) 3 NC NC1 NC NC1 where NC 1 and NC denoe he number of acual days o expiraion for he wo mauriies. When he neares ime o mauriy is 8 days or fewer he CBOE swiches o he nexneares mauriy in order o avoid microsrucure effecs. he annualizaion in Equaion (6) follows he acual/365 daycouning convenion and does no suffer from he arificial upward bias incurred in he VXO calculaion. ECONOMIC AND HEOREICAL UNDERPINNINGS he Old VXO he VXO is essenially an average esimae of he onemonh ahemoney BlackScholes implied volailiy wih an arificial upward bias induced by he radingday conversion. Academics and praciioners ofen regard ahemoney implied volailiy as an approximae forecas for realized volailiy. However since he BlackScholes model assumes consan volailiy here is no direc economic moivaion for regarding he ahemoney implied volailiy as he realized volailiy forecas beyond he BlackScholes model conex. Neverheless a subsanial body of empirical work has found ha he ahemoney BlackScholes implied volailiy is an efficien alhough biased forecas of subsequen realized volailiy. Examples include Laane and Rendleman [1976] Chiras and Manaser [1978] Day and Lewis [1988] Lamoureux and Lasrapes [1993] Canina and Figlewski [1993] Fleming [1998] Chrisensen and Prabhala [1998] and Gwilym and Buckle [1999]. hus references o he VXO as a forecas of subsequen realized volailiy are based more on empirical evidence han on any heoreical linkages. Carr and Lee [3] idenify an economic inerpreaion for ahemoney implied volailiy in a heoreical framework ha goes beyond he BlackScholes (6) model. hey show under general marke seings ha he ime ahemoney implied volailiy wih expiry a ime represens an accurae approximaion of he condiional riskneural expecaion of he reurn volailiy during he ime period [ ]: AMV() [RV ol ] (7) where [.] denoes he expecaion operaor under he riskneural measure condiional on ime filraion F and RVol denoes he realized reurn volailiy in annualized percenages over he ime horizon [ ]. Appendix A deails he underlying assumpions and derivaions for his approximaion. he resul in Equaion (7) assigns new economic meanings o he VXO which approximaes he volailiy swap rae wih a onemonh mauriy if we readjus he upward bias induced by he radingday conversion. Volailiy swap conracs are raded acively over he couner on major currencies and some equiy indexes. A mauriy he long side of he volailiy swap conrac receives he realized reurn volailiy and pays a fixed volailiy rae which is he volailiy swap rae. A noional dollar amoun is applied o he volailiy difference o conver he payoff from volailiy percenage poins o dollar amouns. Since he conrac coss zero o ener he fixed volailiy swap rae equals he riskneural expeced value of he realized volailiy. I is worh noing ha alhough he ahemoney implied volailiy is a good approximaion of he volailiy swap rae he payoff on a volailiy swap is nooriously difficul o replicae. Carr and Lee [3] derive hedging sraegies for volailiy swap conracs ha involve dynamic rading of boh fuures and opions. he New VIX he new VIX squared approximaes he condiional riskneural expecaion of he annualized reurn variance over he nex 3 calendar days: VIX [RV 3 ] (8) wih RV 3 RVol denoing he annualized reurn 3 variance from ime o 3 calendar days laer. Hence VIX approximaes he 3day variance swap rae. Variance swap conracs are acively raded over he couner on major equiy indexes. A mauriy he long side of he variance swap conrac receives a realized variance and pays a fixed variance rae which is he variance swap rae. SPRING 6 HE JOURNAL OF DERIVAIVES 15 Copyrigh 6
4 he difference beween he wo raes is muliplied by a noional dollar amoun o conver he payoff ino dollar paymens. A he ime of enry he conrac has zero value. Hence by noarbirage he variance swap rae equals he riskneural expeced value of he realized variance. Alhough volailiy swap payoffs are difficul o replicae variance swap payoffs can be readily replicaed up o a higherorder erm. he rading sraegy combines a saic posiion in a coninuum of opions wih a dynamic posiion in fuures. he riskneural expeced value of he gains from dynamic fuures rading is zero. he square of he VIX is a discreized version of he iniial cos of he saic opion posiion required in he replicaion. he heoreical relaion holds under very general condiions. We can hink of he VIX as he variance swap rae quoed in volailiy percenage poins. o undersand he replicaion sraegy and appreciae he economic underpinnings of he new VIX we follow Carr and Wu [4] in decomposing he realized reurn variance ino hree componens: RV F 1 K K S + dk 1 K S K + = ( ) + ( ) dk F F F df s s x x e x ( dx ds) 1 where S denoes he ime spo index level denoes he real line excluding zero and (dx d)is a random measure ha couns he number of jumps of size (e x 1) in he index price a ime. he decomposiion in Equaion (9) shows ha we can replicae he reurn variance by he sum of 1) he payoff from a saic posiion in a coninuum of European ouofhemoney opions on he underlying spo across all srike prices bu a he same expiry (firs line) ) he payoff from a dynamic rading sraegy holding [e r ( s) /( )] [(1/F s ) (1/F )] fuures a ime s (second line) and 3) a higherorder erm induced by he disconinuiy in he index price dynamics (hird line). aking expecaions under he riskneural measure on boh sides we obain he riskneural expeced value of he reurn variance on he lefhand side. We also obain he forward value of he sum of he sarup cos of he replicaing sraegy and he replicaion error on he righhand side. By he maringale propery he expeced value of he gains from dynamic fuures rading is zero under he riskneural measure. Wih deermin (9) isic ineres raes we have [ RV ]= (1) where denoes he approximaion error which is zero when he index dynamics are purely coninuous and of order O [(df/f) 3 ] when he index can jump: = r e O K ( ) K e x 1 x vs( x) dxds (11) where (x)dxd is he compensaor of he jumpcouning measure (dx d). he VIX definiion in Equaion (4) represens a discreizaion of he inegral in he heoreical relaion in Equaion (1). he exra erm (F /K 1) in Equaion (4) is an adjusmen for using a porion of he inhemoney call opion a K F. Appendix B provides a proof for he decomposiion in Equaion (9) and a jusificaion for he adjusmen erm in Equaion (4). herefore he new VIX index squared has a very concree economic inerpreaion. I can be regarded eiher as he price of a porfolio of opions or as an approximaion of he variance swap rae up o he discreizaion error and he error induced by jumps. Pracical Moivaion for he Swich x ( ) dk + he CBOE s swich from he old VXO o he new VIX was moivaed by boh heoreical and pracical consideraions. Firs unil very recenly he exac economic meaning of he VXO or he ahemoney implied volailiy was no clear in any heoreical framework beyond he BlackScholes model. I merely represens a monoonic bu nonlinear ransformaion of ahemoney opion prices. In conras he new VIX is he price of a linear porfolio of opions. he economic meaning of he new VIX is much more concree. Second he radingday conversion in he VXO definiion induced an arificial upward bias ha has drawn criicism from boh academia and indusry. hird alhough he VXO approximaes he volailiy swap rae i remains rue ha volailiy swaps are very difficul o replicae. In conras Equaion (9) shows ha one can readily replicae he variance swap payoffs up o a higherorder error erm using a saic posiion in a coninuum of European opions and a dynamic posiion in fuures rading. herefore despie he populariy of he VXO as a general volailiy reference index no deriva 16 A ALE OF WO INDICES SPRING 6 Copyrigh 6
5 ive producs have been launched on he VXO index. In conras jus a few monhs afer he CBOE swiched o he new VIX definiion i sared planning o launch fuures and opions conracs on he new VIX. VIX fuures sared rading on March 6 4 on he CFE. HISORICAL BEHAVIORS Based on hisorical daa on daily closing opion prices on he S&P 5 index and he S&P 1 index he CBOE has backcalculaed he VIX o 199 and he VXO o For our empirical work we choose he common sample period from January 199 o Ocober 18 5 spanning 5769 calendar days. We analyze he hisorical behavior of he wo indexes during his sample period wih a focus on he new VIX. We also download he wo sock indexes OEX and SPX and compue he realized reurn volailiies over he same sample period. For each day we compue he ex pos realized volailiy during he nex 3 calendar days according o he following equaion: ( ) 3 = ( ) 365 RVol 1 ln S / S j + j 1 j= 1 (1) where S j denoes he index level j calendar days afer day. We follow he indusry sandard by compuing he reurn squared wihou demeaning he reurn and by annualizing he volailiy according o he acual/365 daycouning convenion. We analyze how he volailiy indexes correlae wih he index reurns and reurn volailiies. Summary Saisics Exhibi 1 repors summary saisics on he levels and daily differences of he wo volailiy indexes (VXO and VIX) and heir corresponding 3day realized volailiies RVol SPX and RVol OEX. Since he VXO has an arificial upward bias as a resul of he radingday conversion we also compue an adjused index (VXOA) ha scales back he conversion in he VXO : VXOA = / 3 VXO where we approximaely regard he rading days as coming from 3 acual calendar days. All he volailiy series are represened in percenage volailiy poins. Since he VIX squared approximaes he 3day variance swap rae on he SPX and he VXOA approximaes he 3day volailiy swap rae on he OEX Jensen s inequaliy dicaes ha he VIX should be higher han he VXOA if he riskneural expeced values of he realized volailiies on he wo underlying sock indexes (OEX and SPX) are similar in magniude: SPX SPX VIX RVol 3 RVol 3 RVol Var VXOA SPX + ( + ) + ( + 3) ( ) = [ ] OEX [ RVol + 3] ( ) [ ] [ ] SPX SPX VIX VXOA Var RVol if RVol RVol OEX (13) (14) (15) Exhibi 1 shows ha he sample mean of he realized volailiy on he OEX is sighly higher han ha on he SPX. Neverheless he sample average of he VIX is higher han he sample average of he VXOA owing o Jensen s inequaliy. he sample average of he original VXO series is he highes mainly because of he erroneous radingday conversion. E XHIBI 1 Summary Saisics of Volailiy Indexes and Realized Reurn Volailiies Noes: Enries repor he sample average (Mean) sandard deviaion (Sdev) skewness excess kurosis and firsorder auocorrelaion (Auo) on he levels and daily differences of he new volailiy index VIX he 3day realized volailiy on SPX reurn (RV ol SP X ) he old volailiy index VXO is biascorreced version VXOA and he 3day realized volailiy on OEX reurn (RV ol OEX ). Each series has 5769 daily observaions from January 199 o Ocober All series are represened in percenage volailiy poins. SPRING 6 HE JOURNAL OF DERIVAIVES 17 Copyrigh 6
6 Comparing he volailiy index wih he corresponding realized volailiy we find ha on average he VIX is approximaely 5 percenage poins higher han he realized volailiy on he SPX and he VXOA is approximaely percenage poins higher han he corresponding realized volailiy on he OEX. o es he saisical significance of he difference beween he volailiy index and he realized volailiy we consruc he following saisic: sa = N X (16) S X where N 5769 denoes he number of observaions X denoes he difference beween he volailiy index and he realized volailiy he overline denoes he sample average and S X denoes he Newey and Wes [1987] sandard deviaion of X ha accouns for overlapping daa and serial dependence wih he number of lags opimally chosen following Andrews [1991] and an AR(1) specificaion. We esimae he saisic for (VIX RVol SPX ) a 14.9 and for (VIX RVol OEX ) a 6.7 boh of which are highly significan. he volailiy levels show moderae posiive skewness and excess kurosis bu he excess kurosis for daily differences is much larger showing poenial disconinuous index reurn volailiy movemens. Eraker e al. [3] specify an index dynamics ha conains consanarrival finieaciviy jumps in boh he index reurn and he reurn variance rae. By esimaing he model on SPX reurn daa hey idenify a srongly significan jump componen in he variance rae process in addiion o a significan jump componen in he index reurn. Wu [5] direcly esimaes he variance rae dynamics wihou specifying he reurn dynamics by using he VIX and various realized variance esimaors consruced from ick daa on SPX index fuures. He also finds ha he variance rae conains a significan jump componen bu ha he jump arrival rae is no consan over ime; insead i is proporional o he variance rae level. Furhermore he finds ha jumps in he variance rae are no rare evens bu arrive frequenly and generae sample pahs ha display infinie variaion. Exhibi repors he crosscorrelaion beween he wo volailiy indexes (VIX and VXO ) and he SPX OEX subsequen realized volailiies (RVol 3 and RVol 3 ). Each volailiy index level is posiively correlaed wih is corresponding subsequen realized volailiy bu he correlaion esimaes become close o zero when measured in daily changes. Neverheless he wo volailiy indexes are highly correlaed in boh levels (.98) and daily differences (.86). he wo realized volailiy series are also highly correlaed in boh levels (.99) and daily changes (.98). herefore jus as boh sock indexes provide a general picure of he overall sock marke so boh volailiy indexes proxy he overall sock marke volailiy. Given he close correlaion beween he VIX and he VXO and he planned obsolescence of he VXO we henceforh focus our analysis on he behavior of he new VIX. he Leverage Effec Exhibi 3 plos he crosscorrelaions beween SPX index reurns a differen leads and lags and daily changes in he volailiy index VIX wih he wo dashdoed lines denoing he 95% confidence band. he insananeous correlaion esimae is srongly negaive a.78 bu he correlaion esimaes a oher leads and lags are much smaller. Careful inspecion shows ha lagged reurns (wihin a week) show marginally significan posiive correlaions wih daily changes in he volailiy index indicaing ha index reurns predic fuure movemens in he volailiy index. However index reurns wih E XHIBI CrossCorrelaions beween Volailiy Indexes and Subsequen Realized Reurn Volailiies SPX Noes: Enries repor he conemporaneous crosscorrelaion beween VIX SPX 3day realized volailiy (RVol 3 ) VXO OEX and OEX 3day realized volailiy (RVol 3 ) boh in levels and in daily differences. 18 A ALE OF WO INDICES SPRING 6 Copyrigh 6
7 E XHIBI 3 CrossCorrelaions beween Reurn and Volailiy Noes: he sem bars represen he crosscorrelaion esimaes beween SPX index reurns a he relevan number of lags (in days) and he corresponding daily changes in VIX. he wo dashdoed lines denoe he 95% confidence band. negaive lags are no significanly correlaed wih daily changes in he volailiy index. herefore volailiy index movemens do no predic index reurns. he negaive correlaions beween sock reurns and sock reurn volailiies have been well documened. Neverheless since reurn volailiy is no observable he correlaion can be esimaed only under a srucural model for reurn dynamics. In Exhibi 3 we use he VIX as an observable proxy for reurn volailiy and compue he correlaion across differen leads and lags wihou resoring o a model for reurn dynamics. he srongly negaive conemporaneous correlaion beween sock (index) reurns and reurn volailiies capures he leverage effec firs discussed by Black [1976]: given a fixed deb level a decline in he equiy level increases he leverage of he firm (marke) and hence he risk for he sock (index). Various oher explanaions for he negaive correlaion have been proposed in he lieraure for example Haugen e al. [1991] Campbell and Henschel [199] Campbell and Kyle [1993] and Bekaer and Wu []. he Federal Open Marke Commiee Meeing Day Effec Balduzzi e al. [1] find ha rading volume bidask spreads and volailiy on reasury bonds and bills increase dramaically around Federal Open Marke Commiee (FOMC) meeing daes. he Federal Reserve ofen announces changes in he Fed Funds arge Rae and is views on he overall economy during he FOMC meeings. he anicipaion and ex pos reacion o hese announcemens in moneary policy shifs and assessmens creae dramaic variaions in rading and pricing behavior in he reasury marke. In his secion we use he VIX as a proxy for sock marke volailiy and invesigae wheher sock marke volailiy also shows any apparen changes around FOMC meeing days. We download he FOMC meeing day log from Bloomberg. During our sample period here were 144 scheduled FOMC meeings or approximaely 1 meeings per year. Exhibi 4 plos he ime series of he Fed Funds arge Raes in he lef panel and he basis poin arge changes during he scheduled FOMC meeing days in he righ panel. Among he 144 meeings 6 announced a change in he Fed Funds arge Rae. Among he 6 arge moves he change is 5 bp 45 imes 5 bp 16 imes and 75 bp once. On 5 occasions he change is posiive represening a ighening of moneary policy and on 37 occasions he change represens a rae cu and hence an easing of moneary policy. Armed wih he lis of FOMC meeing days we sor he VIX around he FOMC meeing days and compue he average VIX level each day from 1 days before o 1 days afer each FOMC meeing day. he lef panel of Exhibi 5 plos sample averages of VIX around FOMC meeing days. We observe ha he average volailiy level builds up before he FOMC meeing dae and hen drops markedly aferward. he volailiy index reaches is highes level he day before he meeing and drops o he lowes level 4 days afer he meeing. o invesigae he significance of he drop we measure he difference beween he volailiy index 1 day before and 1 day afer he meeing. he mean difference is.6 percenage volailiy poins wih a saisic of 4.6. Before he FOMC meeing marke paricipans disagree on wheher he Fed will change he Fed Funds arge Rae in which direcion and by how much. he fac ha he opionimplied sock index volailiy increases prior o he meeing and drops aferward shows ha he uncerainy abou moneary policy has a definie impac on he volailiy of he sock marke. his uncerainy is resolved righ afer he meeing. Hence he volailiy index drops rapidly afer he FOMC meeing. Since he VIX squared can be regarded as he variance swap rae on he SPX we also sudy wheher he iming of a variance swap invesmen around FOMC SPRING 6 HE JOURNAL OF DERIVAIVES 19 Copyrigh 6
8 E XHIBI 4 he Fed Funds arge Rae Changes Noes: he solid line in he lef panel plos he ime series of he Fed Funds arge Rae over our sample period. he spikes in he righ panel represens he arge rae changes in basis poins. E XHIBI 5 VIX Flucuaion around FOMC Meeing Days Noes: Lines represen he sample averages of he VIX levels (lef panel) and he average payoffs o long variance swap conracs (RV 3 VIX ) (righ panel) a each day wihin 1 days before and afer he FOMC meeing days. meeing days generaes differen reurns. he righ panel of Exhibi 5 plos he average ex pos payoff from going long he swap conrac around FOMC meeing days and holding he conrac o mauriy. he payoff is defined as he difference beween he ex pos realized variance and he VIX squared: (RV 3 VIX ) We find ha he average payoffs are negaive by going long he swap on any day. herefore shoring he swap conrac generaes posiive payoffs on average. Comparing he magniude differences on differen days we also find ha shoring he swap conrac 4 days prior o he FOMC meeing day generaes he highes average payoff and ha shoring he variance A ALE OF WO INDICES SPRING 6 Copyrigh 6
9 swap 4 days afer he FOMC meeing day generaes he lowes average payoff. he difference in average payoff beween invesmens on hese days is saisically significan wih a saisic of 9.9. herefore he evidence suggess ha i is more profiable o shor he SPX variance swap conrac 4 days before an FOMC meeing han 4 days afer. Variance Risk Premia Up o a discreizaion error and a jumpinduced error erm he VIX squared is equal o he riskneural expeced value of he realized variance on he SPX reurn during he nex 3 days: VIX [RV 3 ] (17) We can also rewrie Equaion (17) under he saisical measure as VIX [ M RV ] + + M = [ RV + 3]+ Cov M + 3 M 3 3 [ ] (18) where M denoes a pricing kernel beween imes and. For raded asses noarbirage guaranees he exisence of a leas one such pricing kernel (Duffie [199]). Equaion (18) decomposes he VIX squared ino wo erms. he firs erm [RV 3 ] represens he saisical condiional mean of he realized variance and he second erm capures he condiional covariance beween he normalized pricing kernel and he realized variance. he negaive of his covariance defines he ime condiional variance risk premium (VRP ): M VRP Cov M + 3 [ + 3] + 3 [ + 3] RV + 3 RV = + 3 [ RV + 3] VIX (19) aking uncondiional expecaions on boh sides we have [VRP ] [RV 3 VIX ] () hus we can esimae he average variance risk premium as he sample average of he differences beween he realized reurn variance and he VIX squared. Over our sample period he mean variance risk premium is esimaed a bp wih a Newey and Wes [1987] serialdependenceadjused sandard error of 17.. Hence he mean variance risk premium is srongly negaive. Riskaverse invesors normally ask for a posiive risk premium for reurn risk. hey require sock prices o appreciae by a higher percenage on average if sock reurns are riskier. In conras he negaive variance risk premium indicaes ha invesors require he index reurn variance o say lower on average o compensae for higher variance risk. herefore whereas higher average reurn is regarded as compensaion for higher reurn risk lower average variance levels are regarded as compensaion for higher variance risk. Invesors are averse no only o increases in he reurn variance level bu also o increases in he variance of he reurn variance. From he perspecive of a variance swap invesmen he negaive variance risk premium also implies ha invesors are willing o pay a high premium or endure an average loss when hey are long variance swaps in order o receive compensaion when he realized variance is high. Dividing boh sides of Equaion (18) by VIX we can rewrie he decomposiion in excess reurns: 1 = (1) If we regard VIX as he forward cos of he invesmen in he saic opion posiion required o replicae he variance swap payoff (RV 3 /VIX 1) capures he excess reurn from going long he variance swap. he negaive of he covariance erm in Equaion (1) represens he condiional variance risk premium in excess reurn erms: VRPR RV VIX Cov M + 3 M Cov M + 3 [ + 3] RV VIX [ ] RV VIX + 3 = () We can esimae he mean variance risk premium in excess reurn form hrough he sample average of he realized excess reurns ER 3 (RV 3 /VIX 1) which is esimaed a 4.16% wih a Newey and Wes [1987] sandard error of.87%. Again he mean variance risk premium esimae is srongly negaive and highly significan. Invesors are willing o endure a highly negaive excess reurn for being long variance swaps in order o hedge away upward movemens in he reurn variance of he sock index. he average negaive variance risk premium also suggess ha shoring he 3day variance swap and holding i o mauriy generaes an average excess reurn of 4.16%. We compue he annualized informaion raio using 3 dayapar nonoverlapping daa IR = 1 ER/ S ER where ER denoes he ime series average of he excess reurn and S ER denoes he serialdependenceadjused sandard deviaion esimae of he excess reurn. he informaion raio esimaes average 3.5 indicaing ha shoring 3day variance swaps is very profiable on average. M RV VIX + 3 SPRING 6 HE JOURNAL OF DERIVAIVES 1 Copyrigh 6
10 E XHIBI 6 Excess Reurns from Shoring 3Day Variance Swaps Noes: he lef panel plos he ime series of excess reurns from shoring 3day variance swaps on SPX and holding he conrac o mauriy. he righ panel plos he hisogram of excess reurns. o furher check he hisorical behavior of excess reurns from his invesmen we plo he ime series of he excess reurns in he lef panel and he hisogram in he righ panel of Exhibi 6. he ime series plo shows ha shoring variance swaps provides a posiive reurn 89% of he ime (5137 ou of he 5769 daily invesmens). However alhough he hisorical maximum posiive reurn is 89.53% he occasionally negaive realizaions can be as large as 4.4%. he hisogram in he righ panel shows ha he excess reurn disribuion is heavily negaively skewed. he high average reurn and high informaion raio sugges ha invesors ask for a very high average premium o compensae for he heavily negaively skewed risk profile. he payoff from shoring variance swaps is similar o ha from selling insurance which generaes a regular sream of posiive premiums wih small variaion bu wih occasional exposures o large losses. o invesigae wheher he classic Capial Asse Pricing Model (CAPM) can explain he risk premium from invesing in variance swaps we regress he excess reurns from being long he variance swap on he excess reurns from being long he marke porfolio: ER 3 (R m 3 R f ) e (3) where (R m R ) denoes he coninuously compounded 3 f excess reurn o he marke porfolio. If he CAPM holds we will obain a highly negaive bea esimae for he long variance swap reurn. If he CAPM can fully accoun for he risk premium he esimae for he inercep which represens he average excess reurn o a markeneural invesmen will no be significanly differen from zero. We proxy he excess reurn o he marke porfolio using he valueweighed reurn on all NYSE AMEX and NASDAQ socks (from CRSP) minus he onemonh reasury bill rae (from Ibboson Associaes). Monhly daa on he excess reurns are publicly available a Kenneh French s online daa library from July 196 o Sepember 5. We mach he sample period wih our daa and run he regression on monhly reurns over nonoverlapping daa using he generalized mehod of momens wih he weighing marix compued according o Newey and Wes [1987]. he regression esimaes are as follows wih saisics repored in parenheses: m ER = ( R R f) + e R = % (4) ( 65. 3)( 3. 1) he bea esimae is highly negaive consisen wih he general observaion ha index reurns and volailiy are negaively correlaed. However his negaive bea canno fully explain he negaive premium for volailiy risk. he esimae for he inercep or he mean beaneural excess A ALE OF WO INDICES SPRING 6 Copyrigh 6
11 reurn remains srongly negaive. he magniude of is no much smaller han he sample average of he raw excess reurn a 38.36%. hus he CAPM ges only he sign righ; i canno fully accoun for he large negaive premium on index reurn variance risk. his resul suggess ha variabiliy in variance consiues a separae source of risk ha he marke prices heavily. o es wheher he variance risk premium is ime varying we run he following expecaionshypohesis regressions wih he saisics repored in parenheses: RV 3 = VIX + e ( 5. ) ( 479. ) ( RV 3/ VIX 1) = VIX + e ( 14. 8) ( 1. 61) (5) Under he null hypohesis of consan variance risk premium he firs regression should generae a slope of one and he second regression should generae a slope of zero. Zerovariance risk premium would furher imply zero inerceps for boh regressions. he saisics are compued agains hese null hypoheses. Since he daily series of he 3day realized variance consiues an overlapping series we esimae boh regressions using he generalized mehod of momens and consruc he weighing marix accouning for he serial dependence according o Newey and Wes [1987] wih 3 lags. When he regression is run on he variance level he slope esimae is significanly lower han he null value of E XHIBI 7 Informaion Conen in VIX and GARCH volailiies in predicing fuure realized reurn variances Noes: Enries repor he esimaion resuls on resriced and unresriced versions of he following relaion: RV = a + bvix + cgarch + e he relaion is esimaed using he generalized mehod of momens. he covariance marix is compued according o Newey and Wes [1987] wih 3 lags. he daa are daily from January 199 o Ocober 18 5 generaing 5769 observaions for each series. one providing evidence ha he variance risk premium VRP is ime varying and correlaed wih he VIX level. When he regression is run on excess reurns in he second equaion he slope esimae is no longer significanly differen from zero suggesing ha he variance risk premium defined in excess reurn erms (VRPR ) is no highly correlaed wih he VIX level. Predicabiliy of Realized Variance and Reurns o Variance Swap Invesmens We esimae GARCH(11) processes on he S&P 5 index reurn innovaion using an AR(1) assumpion on he reurn process. hen we compare he relaive informaion conen of he GARCH volailiy and he VIX index in predicing subsequen realized reurn variances: RV 3 a b VIX cgarch e 3 (6) where GARCH denoes he ime esimae of he GARCH reurn variance in annualized basis poins. Exhibi 7 repors he generalized mehod of momen esimaion resuls on resriced and unresriced versions of his regression. When we use eiher VIX or GARCH as he only predicor in he regression he volailiy index VIX generaes an Rsquared approximaely 1 percenage poins higher han he GARCH variance does. When we use boh VIX and GARCH as predicors he slope esimae on he GARCH variance is no longer saisically significan and he Rsquared is only marginally higher han using VIX alone as he regressor. hus he GARCH variance does no provide much exra informaion over he VIX index. he resuls in Exhibi 7 show ha we can predic he realized variance using he volailiy index VIX. By using variance swaps invesors can exploi such predicabiliy and direcly conver hem ino dollar reurns. We invesigae wheher he predicabiliy of reurn variance has been fully priced ino he variance swap rae by analyzing he predicabiliy of he excess reurns from invesing in a 3 day SPX variance swap and holding i o mauriy. Firs we measure he monhly auocorrelaion of he excess reurns ER 3 using nonoverlapping 3dayapar daa. he esimaes average.1. When we run an AR(1) SPRING 6 HE JOURNAL OF DERIVAIVES 3 Copyrigh 6
12 E XHIBI 8 CrossCorrelaion beween SPX Monhly Reurns and Excess Reurns on 3day Variance Swaps VIX DERIVAIVES Given he explici economic meaning of he new VIX and is direc link o a porfolio of opions he launch of derivaives on his index becomes he naural nex sep. On March 6 4 he CBOE launched a new exchange he Chicago Fuures Exchange and sared rading fuures on he VIX. A he ime of wriing opions on he VIX are also being planned. In his secion we derive some ineresing resuls regarding he pricing of VIX fuures and opions. VIX Fuures and Valuaion Bounds Under he assumpion of noarbirage and coninuous marking o marke he VIX fuures price F vix is a maringale under he riskneural probabiliy measure : vix vix F [F 1 ] [VIX 1 ] (7) Noes: he sem bars represen he crosscorrelaion esimaes beween SPX reurns a differen lags and excess reurns on invesing in a 3day variance swap and holding i o mauriy. he esimaes are based on monhly nonoverlapping daa. he wo dashed lines denoe he 95% confidence band. Posiive numbers on he xaxis represen lags in monhs for index reurns. regression on he nonoverlapping excess reurns he R squared esimaes average 1.58%. hus he predicabiliy of excess reurns hrough mean reversion is very low. Alhough he volailiy level is srongly predicable invesors have priced his predicabiliy ino variance swap conracs so ha he excess reurns on hese swaps are no srongly predicable. Exhibi 3 shows ha SPX reurns predic fuure movemens in he VIX. Now we invesigae wheher we can predic he excess reurn on a variance swap invesmen using index reurns. Exhibi 8 plos he crosscorrelaion beween he excess reurn o he variance swap and he monhly reurn on SPX based on monhly sampled and hence nonoverlapping daa. he sock index reurn and he reurn on he variance swap invesmens show srongly negaive conemporaneous correlaion bu he nonoverlapping series do no exhibi any significan leadlag effecs. Hence despie he predicabiliy in reurn volailiies excess reurns on variance swap invesmens are no srongly predicable. his resul shows ha he SPX opions marke is relaively efficien. We derive valuaion bounds on VIX fuures ha are observable from he underlying SPX opions marke under wo simplifying assumpions: 1) he VIX is calculaed using a single srip of opions mauring a 1 wih 1 3/365 insead of wo srips and on a coninuum of opions prices raher han a discree number of opions; ) he SPX index has coninuous dynamics and ineres raes are deerminisic. he firs assumpion implies ha he VIX is given by VIX (8) where B 1 ( ) denoes he ime 1 price of a zero bond mauring a. he second assumpion furher implies ha he equaliy beween he VIX squared and he riskneural expeced value of he reurn variance is exac. Alernaively we can wrie VIX (9) Subsiuing Equaion (9) in Equaion (7) we have he VIX fuures as vix F = RV < 1 = ( ) B ( ) 1 1 = RV (3) hen he concaviy of he square roo and Jensen s inequaliy generaes he following lower and upper bounds for he VIX fuures: O ( K ) dk K 1 4 A ALE OF WO INDICES SPRING 6 Copyrigh 6
13 (31) he lower bound is he forward volailiy swap rae L RV which can be approximaed by a 1 forwardsaring ahemoney opion. he proof is similar o ha in Appendix A for he approximaion of a spo volailiy swap rae using he spo ahemoney opion. he upper bound is he forwardsaring variance swap rae quoed in volailiy percenage poins U RV 1 which can be deermined from he prices on a coninuum of opions a wo mauriies 1 and : U (3) he widh of he bounds is deermined by he riskneural variance of he forwardsaring realized volailiy: U L RV RV Var RV (33) When he marke quoe on VIX fuures (F vix ) is available we can combine i wih forwardsaring variance swap raes (U ) o deermine he riskneural variance of he fuure VIX: Var vix RV F RV = RV = ( ) RV ( 1 ) V 1 O( K ) O( K 1) dk = 1 B ( ) B ( 1) K 1 1 = ( ) ( ) = ( ) ( VIX ) = Var [ RV ] (34) herefore VIX fuures provide economically relevan informaion no only abou he fuure VIX level bu also abou he riskneural variance of he fuure VIX. We can use his informaion for pricing VIX opions. VIX Opions vix [ RV ] RV U ( F ) [ ] = ( ) = he VIX fuures marke ogeher wih he SPX opions marke provides he informaion basis for launching VIX opions. o see his we consider a call opion on VIX wih he erminal payoff (VIX 1 ) (35) where K is he srike price and 1 denoes he expiry dae of he opion. We have shown ha we can learn he condiional riskneural mean (m 1 ) and variance (m ) of VIX 1 from informaion in he VIX fuures marke and he underlying SPX opions marke: m ( VIX )= F vix 1 1 vix 1 m Var VIX U F ( ) = ( ) = (36) hus under cerain disribuional assumpions we can derive he value of he VIX opion as a funcion of hese wo momens. As an example if we assume ha VIX 1 follows a lognormal disribuion under measure we can use he Black formula o price VIX opions wih he wo momens in Equaion (36) as inpus: where d vix C B F N d KN d 1 vix 1 ln F / K + s ( 1 ) = s and s is he condiional annualized volailiy of ln VIX 1 which can be represened as a funcion of he firs wo condiional momens of VIX 1 s = (37) As anoher example if we assume ha he riskneural disribuion of VIX 1 is normal raher han lognormal we can derive he Bachelier opion pricing formula as a funcion of he firs wo observable momens of VIX 1 : vix C = B ( ) m N ( d) + ( F K) N( d) 1 (38) vix wih d = ( F K)/ m. For ahemoney opions (K vix F ) he Bachelier opion pricing formula reduces o a very simple form A = B( ) m / 1 CONCLUSION = ( ) ( ) ( ) 1 [ ] 1 1 vix m + ( F ) ln vix ( F ) O K B ( ) [ ] (39) he new VIX differs from he old VXO in wo key aspecs. Firs he wo indexes use differen underlyings: he SPX for he new VIX versus he OEX for he old VXO. Second he wo indexes use differen formulae in exracing volailiy informaion from he opions marke. he new VIX is consruced from he price of a porfolio ( ) ( ) O K 1 dk B ( 1 ) K d = d s 1 1 F vix ( ). SPRING 6 HE JOURNAL OF DERIVAIVES 5 Copyrigh 6
14 of opions and represens a modelfree approximaion of he 3day reurn variance swap rae. he old VXO builds on he 1monh BlackScholes ahemoney implied volailiy and approximaes he volailiy swap rae under cerain assumpions. he CBOE decided o swich from he VXO o he VIX mainly because he new VIX has a beer known and more robus economic inerpreaion. In paricular he variance swap underlying he new VIX has a robus replicaing porfolio whose opion componen is saic. In conras robus replicaion of he volailiy swap underlying he VXO index requires dynamic opion rading. Furhermore he VXO includes an upward bias induced by an erroneous radingday conversion in is definiion. Analyzing approximaely 15 years of daily daa on he wo volailiy indices we obain several ineresing findings on he index behavior. We find ha he new VIX averages abou percenage poins higher han he biascorreced version of he old index alhough he sample average of he 3day realized volailiy on SPX is.66 percenage poins lower han ha of OEX. he difference beween he new and old volailiy indexes is mainly induced by Jensen s inequaliy and he riskneural variance of realized volailiy. he hisorical behaviors of he wo volailiy indexes are oherwise very similar and hey move closely wih each oher. We also find ha daily changes in he volailiy indexes show very large excess kurosis suggesing ha he volailiy indexes conain large disconinuous movemens. We idenify a srongly negaive conemporaneous correlaion beween VIX and SPX index reurns confirming he leverage effec firs documened by Black [1976]. Furhermore alhough lagged index reurns show marginal predicive power on he fuure movemens of he VIX lagged movemens in he volailiy index do no predic fuure index reurns. When we analyze VIX behavior around FOMC meeing days during which moneary policy decisions such as Fed Funds arge Rae changes are ofen announced we find ha he volailiy index increases prior o he FOMC meeing bu drops rapidly afer he meeing showing ha uncerainy abou moneary policy has a direc impac on volailiy in he sock marke. Since he VIX squared represens he variance swap rae on he SPX he sample average difference beween he 3day realized reurn variance on he SPX and he VIX squared measures he average variance risk premium which we esimae o be bp and highly significan. When we represen he variance risk premium in excess reurns form we obain a mean esimae of 4.16% for being long a 3day variance swap and holding i o mauriy. he highly negaive variance risk premium indicaes ha invesors are averse o variaions in reurn variance and he compensaion for bearing variance risk can come in he form of a lower mean variance level under he empirical disribuion han under he riskneural disribuion. From he perspecive of variance swap invesors he negaive variance risk premium indicaes ha invesors are willing o pay a high average premium o obain compensaion (insurance) when he variance level increases. herefore shoring variance swaps and hence receiving he fixed leg generaes posiive excess reurns on average. he annualized informaion raio for shoring a variance swap is approximaely 3.5 which is much higher han for radiional invesmens. Neverheless he excess reurn disribuion accessed by being shor variance swaps is heavily negaively skewed. Negaive reurn realizaions are few bu large. he high informaion raio indicaes ha invesors ask for a high average reurn in order o compensae for he heavily negaively skewed risk profile. When we regress he excess reurns from being long he variance swap on he sock marke porfolio we obain a highly negaive bea. However he inercep of he regression remains highly negaive indicaing ha he classic Capial Asse Pricing Model canno fully accoun for he negaive variance risk premium. Invesors regard variabiliy in variance as a separae source of risk and charge a separae price for bearing his risk. Expecaions hypohesis regressions furher show ha he variance risk premium in variance levels is ime varying and correlaed wih he VIX level bu he variance risk premium in excess reurns form is much less correlaed wih he VIX level. We find ha he VIX can predic movemens in fuure realized variance and ha GARCH volailiies do no provide exra informaion once he VIX is included as a regressor. Neverheless he srong predicabiliy of he realized variance does no ransfer o srong predicabiliy in excess reurns for invesing in variance swaps. Finally we show ha he SPX opions marke provides informaion on valuaion bounds for VIX fuures. he widh of he bounds are deermined by he riskneural variance for forwardsaring reurn volailiy. Furhermore VIX fuures quoes no only provide informaion abou he riskneural mean of fuure VIX levels bu also combine wih informaion from he SPX 6 A ALE OF WO INDICES SPRING 6 Copyrigh 6
15 opions marke o reveal he riskneural variance of he VIX. his informaion can be used o price VIX opions. APPENDIX A Approximaing Volailiy Swap Raes wih AheMoney Implied Volailiies Le ( F ) be a probabiliy space defined on a riskneural measure. As in Carr and Lee [3] we assume coninuous dynamics for he index fuures F under measure : df / F = dw (A1) where he diffusion volailiy can be sochasic bu is variaion is assumed o be independen of he Brownian moion W in he price. Under hese assumpions Hull and Whie [1987] show ha he value of a call opion can be wrien as he riskneural expeced value of he BlackScholes formula evaluaed a he realized volailiy. he ime value of he ahemoney forward (K F ) opion mauring a ime can be wrien as AMC F N RVol = N (A) where RVol is he annualized realized reurn volailiy over [ ] RVol N RVol 1 s ds (A3) (A6) Since an ahemoney call value is concave in volailiy ( / F AMC is a slighly downwardbiased ) approximaion of he volailiy swap rae. As a resul he error erm is posiive. However Brenner and Subrahmanyam show ha he ahemoney implied volailiy is also given by AMC = F AMV + O (( ) ) (A7) Once again ( / F ) AMC is a slighly downwardbiased approximaion of he ahemoney implied volailiy. Subracing Equaion (A7) from Equaion (A6) implies ha he volailiy swap rae is approximaed by he ahemoney implied volailiy: (A8) he leading source of error in Equaion (A6) is parially canceled by he leading source of error in Equaion (A7). As a resul his approximaion has been found o be very accurae. APPENDIX B [ RVol ]= AMV + O(( ) ) Replicaing Variance Swaps wih Opions he inerpreaion of he new VIX as an approximaion of he 3day variance swap rae can be derived under a much more general seing for he dynamics of SPX index fuures: x df / F = dw + ( e 1)[ ( dx d) v ( x) dxd] Brenner and Subrahmanyam [1988] show ha a aylor series expansion of each normal disribuion funcion abou zero where F implies denoes he fuures price a ime jus prior o a jump denoes he real line excluding zero and he random measure (dx d) couns he number of jumps of size (e x 1) in he N RVol N RVol RVol index fuures O a ime. he process {v (x) x } compensaes he jump process J (e x 1) (dx ds) so ha he 3 RVol (A4) las erm in Equaion (A9) is he incremen of a pure jump = + O(( ) ) maringale. o avoid noaional complexiy we assume ha he jump componen in he price process exhibis finie variaion: [ RVol]= 3 F AMC + O (( ) ) 3 3 (A9) Subsiuing Equaion (A4) ino Equaion (A) implies ha AMC F RVol and hence he volailiy swap rae is given by (A5) ( x 1) v( x) dx < By adding he ime subscrips o and (x) we allow boh o be sochasic and predicable wih respec o he filraion F. o saisfy limied liabiliy we furher assume he wo sochasic SPRING 6 HE JOURNAL OF DERIVAIVES 7 Copyrigh 6
16 processes o be such ha he fuures price F is always nonnegaive and absorbing a he origin. Finally wih lile loss of generaliy we assume deerminisic ineres raes and dividend yields. Under hese assumpions he annualized quadraic variaion on he fuures reurn over horizon [ ] can be wrien as RV = 1 d + x ( dx d) (A1) have Applying Iō s lemma o he funcion f(f) ln F we 1 1 x ln( F) = ln ( F)+ dfs s ds + x e + dx ds F 1 Adding and subracing [(F /F ) 1] x (dx d) and rearranging we obain a represenaion for he quadraic variaion: F F ( ) RV = F F F F df s ln + s x x e 1 x ( dx ds) (A11) A aylor expansion wih he remainder of ln F abou he poin F implies lnf 1 = lnf + ( F F F ) K F 1 K K F + dk 1 ( ) ( K F K + ) dk (A1) Plugging Equaion (A1) ino Equaion (A11) we have F ( ) RV = 1 K K F + dk 1 K F K + ( ) + ( ) dk F F F df s s x x + e 1 x ( dx ds) (A13) s which is he decomposiion in Equaion (9) ha also represens a replicaing sraegy for he reurn quadraic variaion. aking expecaions under measure we obain he riskneural expeced value of he reurn variance on he lefhand side and he cos of he replicaion sraegy on he righhand side: ( ) ( ) F r e O K x x [ RV ]= dk e x v s x dxds K 1 ( ) where he firs erm denoes he iniial cos of he saic porfolio of ouofhemoney opions and he second erm is a higherorder error erm induced by jumps. F [ ] ( ) he VIX definiion in Equaion (4) represens a discreizaion of he opion porfolio. he exra erm (F /K 1) in he VIX definiion adjuss for he inhemoney call opion used a K F. o conver he inhemoney call opion ino he ouofhemoney pu opion we use he pucall pariy: r e ( ) r C K K e ( ) (A14) P K F K If we plug his equaliy ino Equaion (4) o conver all opion prices ino ouofmoney opion prices we have Ki VS K e r O K K ( ( ) )= ( i )+ (A15) where he second erm on he righhand side of Equaion (A 15) is due o he subsiuion of he inhemoney call opion a K by he ouofhemoney pu opion a he same srike K. If we furher assume ha he forward level is in he middle of he wo adjacen srike prices and approximae he inerval K by F K he las wo erms in Equaion (A15) cancel ou o obain Ki VS K e r ( ) ( )= O K i (A16) hus he VIX definiion maches he heoreical relaion for he riskneural expeced value of he reurn quadraic variaion up o a jumpinduced error erm and errors induced by discreizaion of srikes. ENDNOE he auhors hank Sephen Figlewski (he edior) Rober Engle Harvey Sein Benjamin Wurzburger and seminar paricipans a New York Universiy and he h Annual Risk Managemen Conference in Florida for many insighful commens. All remaining errors are ours. REFERENCES ( ) = ( )+ i K K F K F 1 + ( ) 1 ( ) K i ( ) Andrews D. Heeroskedasiciy and Auocorrelaion Consisen Covariance Marix Esimaion. Economerica 59 (1991) pp Balduzzi P. E.J. Elon and.c. Green. Economic News and Bond Prices: Evidence from he U.S. reasury Marke. Journal of Financial and Quaniaive Analysis 36 (1) pp Bekaer G. and G. Wu. Asymmeric Volailiies and Risk in Equiy Markes. Review of Financial Sudies 13 () pp A ALE OF WO INDICES SPRING 6 Copyrigh 6
17 Black F. Sudies of Sock Price Volailiy Changes. In Proceedings of he 1976 American Saisical Associaion Business and Economical Saisics Secion. Alexandria VA: American Saisical Associaion 1976 pp Black F. and M. Scholes. he Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy 81 (1973) pp Brenner M. and M. Subrahmanyam. A Simple Formula o Compue he Implied Sandard Deviaion. Financial Analyss Journal 44 (1988) pp Campbell J.Y. and L. Henschel. No News Is Good News: An Asymmeric Model of Changing Volailiy in Sock Reurns. Review of Economic Sudies 31 (199) pp Campbell J.Y. and A.S. Kyle. Smar Money Noise rading and Sock Price Behavior. Review of Economic Sudies 6 (1993) pp Canina L. and S. Figlewski. he Informaion Conen of Implied Volailiy. Review of Financial Sudies 6 (1993) pp Carr P. and R. Lee. AheMoney Implied as a Robus Approximaion of he Volailiy Swap Rae. Working paper New York Universiy 3. Carr P. and L. Wu. Variance Risk Premia. Working paper New York Universiy and Baruch College 4. Chiras D. and S. Manaser. he Informaion Conen of Opion Prices and a es of Marke Efficiency. Journal of Financial Economics 6 (1978) pp Chrisensen B.J. and N.R. Prabhala. he Relaion beween Implied and Realized Volailiy. Journal of Financial Economics 5 (1998) pp Day.E. and C.M. Lewis. he Behavior of he Volailiy Implici in Opion Prices. Journal of Financial Economics (1988) pp Eraker B. M. Johannes and N. Polson. he Impac of Jumps in Equiy Index Volailiy and Reurns. Journal of Finance 58 (3) pp Fleming J. he Qualiy of Marke Volailiy Forecas Implied by S&P 5 Index Opion Prices. Journal of Empirical Finance 5 (1998) pp Gwilym O.A. and M. Buckle. Volailiy Forecasing in he Framework of he Opion Expiry Cycle. European Journal of Finance 5 (1999) pp Haugen R.A. E. almor and W.N. orous. he Effec of Volailiy Changes on he Level of Sock Prices and Subsequen Expeced Reurns. Journal of Finance 46 (1991) pp Hull J. and A. Whie. he Pricing of Opions on Asses wih Sochasic Volailiies. Journal of Finance 4 (1987) pp Lamoureux C.G. and W.D. Lasrapes. Forecasing Sock Reurn Variance: oward an Undersanding of Sochasic Implied Volailiies. Review of Financial Sudies 6 (1993) pp Laane H.A. and R.J. Rendleman. Sandard Deviaion of Sock Price Raios Implied in Opion Prices. Journal of Finance 31 (1976) pp Newey W.K. and K.D. Wes. A Simple Posiive SemiDefinie Heeroskedasiciy and Auocorrelaion Consisen Covariance Marix. Economerica 55 (1987) pp Whaley R.E. he Invesor Fear Gauge. Journal of Porfolio Managemen 6 () pp Wu L. Variance Dynamics: Join Evidence from Opions and HighFrequency Reurns. Working paper Baruch College 5. o order reprins of his aricle please conac Dewey Palmieri a or Duffe D. Dynamic Asse Pricing heory nd ed. Princeon NJ: Princeon Universiy Press 199. SPRING 6 HE JOURNAL OF DERIVAIVES 9 Copyrigh 6
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