The implied convexity of VIX futures. Robert T. Daigler Brice Dupoyet * Fernando Patterson

Transcription

1 The implied convexiy of VIX fuures Rober T. Daigler Brice Dupoye * Fernando Paerson March 13, 2014 Keywords: Implied convexiy, VIX fuures variance, Volailiy JEL classificaion: G12; G13; G17 * Correspondence auhor, Chapman Graduae School of Business, Deparmen of Finance, RB 209A, Miami, Florida Tel: , Fax: , E- mail: dupoyeb@fiu.edu Rober T. Daigler is Knigh Ridder Research Professor of Finance, Florida Inernaional Universiy, RB 206B, Miami, Florida. daiglerr@fiu.edu Brice Dupoye is an Associae Professor of Finance, Florida Inernaional Universiy, RB 209A, Miami, Florida. dupoyeb@fiu.edu Fernando Paerson is a Docoral Candidae in Finance a Florida Inernaional Universiy, Miami, Florida. fpa001@fiu.edu

2 The implied convexiy of VIX fuures ABSTRACT We examine he convexiy adjusmen needed o value VIX fuures prices by exracing he implied convexiy componen from VIX fuures prices and he corresponding spo opion marke prices. The resulan implied convexiy value is he missing adjusmen beween he observed VIX fuures price and he forwardsaring variance swap rae. We hen sudy he properies of his convexiy adjusmen, boh as a ime series and wih respec o various marke facors. We find ha implied convexiy ofen violaes no-arbirage condiions since i is frequenly negaive, and ha implied convexiy is srongly saisically relaed o boh marke volailiy and VIX fuures ime o expiraion. We also invesigae he abiliy of implied convexiy o predic realized VIX fuures price variances, finding ha i underesimaes he realized VIX fuures variance 71% of he ime. 1. Inroducion VIX fuures pricing is an acive area of fuures markes research, boh because volailiy producs are considered a new asse class and because no VIX fuures pricing model is currenly acceped as he definiive model by he financial communiy. The difficulies in pricing VIX fuures arise from he need o accuraely esimae a number of complicaed facors under differen marke scenarios, he imporance of he sabiliy of hese facors, and he size of he mispricing for curren models. Consequenly, he limied marke-based empirical evidence does no suppor he long-erm pricing sabiliy for any of he exising models. As Poon and Granger (2003) poin ou, much remains o be done o undersand he process and characerisics of volailiy. The lack of susainable empirical suppor for sandard VIX fuures pricing models makes his sudy boh ineresing and necessary. In his paper we inroduce he concep of implied convexiy of 2

3 VIX fuures and hen examine is properies. 1 One way o hink abou he convexiy of VIX fuures pricing is ha i provides he missing piece needed o help deermine how VIX fuures prices behave. Alhough several researchers previously have modeled convexiy (discussed shorly), heir convexiy componen is ofen an approximaion value as well as being model-dependen. Therefore, developing and examining implied convexiy from marke-based values provides an alernaive o undersanding a key facor affecing he behavior of VIX fuures prices. Such an examinaion is similar o how raders employ implied volailiy and herefore i can provide new insighs ino how o creae a more accurae and realisic VIX fuures pricing model. Afer describing our approach o he VIX fuures convexiy issue, we examine he empirical properies of he implied convexiy series. We find ha implied convexiy ofen violaes he noarbirage condiion relaed o he forward-saring variance swap rae. These resuls show ha insiuional raders (or marke makers) can poenially earn arbirage-free profis. Addiionally, regressing implied convexiy on S&P 500 volailiy and on he spo VIX yields a saisically significan relaion. Noably, implied convexiy increases during highly volaile periods, such as he Fall of Alernaively, implied convexiy decreases as he VIX fuures reaches expiraion. Finally, we find ha implied convexiy is associaed wih fuure realized VIX fuures variance, alhough implied convexiy does consisenly underesimae he fuure variance of VIX fuures. This resul is similar o he phenomenon whereby he implied volailiy of an opion consisenly overesimaes he fuure realized volailiy of he underlying asse. This aricle is organized as follows: secion 2 examines curren VIX fuures models and he associaed convexiy adjusmen; secion 3 defines he naure of he problem and provides a 1 Whereas his convexiy adjusmen migh no be implied in he pure convenional model-dependen sense, i is neverheless implied by he difference beween he forward-saring variance swap rae and he square of he VIX fuures price. 3

4 summary of he heory; secion 4 describes our mehodology and approach o he problem; secion 5 deails he daa used in his sudy; secion 6 examines he resuls; and secion 7 concludes. 2. Curren models of VIX fuures pricing Since he underlying cash asse for VIX fuures is no raded as an index, he behavior of he VIX fuures price series differs from a ypical cos-of-carry fuures conrac. 2 Moreover, he sochasic process of volailiy over ime is no well undersood, leading o difficulies in designing an effecive model for VIX fuures pricing. Wang and Daigler (2013) examine par of his issue by underaking an analysis of he volailiy of volailiy and he behavior of he implied volailiy skew. However, hey do no apply hese facors o a pricing model. In addiion, he VIX fuures price is he marke s esimae of he fuure VIX level a he expiraion of he conrac. Therefore, he fuures price is affeced by he characerisics of he fuure VIX, as well as hose of he spo VIX, and he S&P 500 underlying he opions. In urn, hese insrumens are affeced by mean-reversion, jumps, he dynamic naure of he VIX and VIX fuures volailiy over ime, and he shifing erm srucure of volailiy. Some of he more complee endeavors o price VIX fuures are Lin (2007), Zhu and Lian (2012), and Mencia and Senana (2013). However, hese complex models conain a large number of parameers. Since he parameers change over ime in relaion o heir esimaion from differen sampling characerisics, he consisency of hese models is quesionable. For example, Zhu and Lian (2012) sae ha Jus as an issue raised in Zhang & Huang (2010), we also believe ha searching for a reliable esimaion mehod o deermine he model parameers from marke daa 2 Tradable ou-of-he-money S&P 500 opions do deermine he underlying cash VIX. However, several facors make his cash asse difficul o rade on a risk-free basis for arbirage purposes. One major facor is ha he porfolio of opions would need o be dynamically raded o make i always equivalen o he underlying cash index used for VIX fuures selemen. Such rading would be cosly; moreover, many of he needed opions have limied liquidiy. Anoher issue includes he uncerainy ha he selemen procedure creaes, i.e. acual opion prices raded a he opening of selemen day are employed o calculae he selemen price raher han he bid-ask average used o calculae he cash VIX. 4

5 remains a challenge. While our work presened in his paper has demonsraed anoher alernaive, complicaed sochasic models probably will no gain populariy among marke praciioners, unil a convincing approach can be acceped and agreed upon by a majoriy of researchers. As such, here are a handful of sudies, such as Huskaj and Nossman (2012), ha esimae he parameers exogenously wih an improved rae of success in capuring he erm srucure of VIX fuures compared o previous sudies. Oher sudies, for example Grunbichler and Longsaff (1996), Zhang and Zhu (2006), and Dupoye, Daigler, and Chen (2011), provide simpler, more racable models involving some combinaion of he Consan Elasiciy of Variance (CEV) and Poisson jumps (generally based on he Cox-Ingersoll-Ross (1985) approach). However, he sabiliy of he resuls for hese models over ime needs more empirical suppor. Zhang and Zhu (2006) develop a sochasic variance model for he evoluion of he VIX over ime, showing ha heir model overprices VIX fuures by 16-44%. When using only one year of daa, parameer fiing reduces he errors o 2-12%. However, he ime period in quesion hey employed was a very low volailiy period, unlike any ime period ha would include Zhu and Lian (2012) develop a VIX fuures pricing model displaying sochasic volailiy and simulaneous jumps for boh he underlying asse and he corresponding volailiy, wih he Heson (1993) sochasic volailiy model as he underlying volailiy process. They find ha jumps in he underlying asse improve he pricing of VIX fuures, whereas jumps in volailiy do no. Their empirical comparisons of he various models and heir individual facors are ineresing. However, he resuls of heir model wih differen volailiy processes ypically show errors in excess of 5%, and he daa series ends before he volaile period in he Fall of Anoher facor affecing VIX fuures pricing is he convexiy adjusmen. A few auhors do incorporae he convexiy adjusmen ino heir models, using an approximaion of he Taylor series 5

6 and a seleced volailiy process. For example, Lin (2007) employs a second-order correcion approximaion, Lu and Zhu (2010) use Kalman filer and maximum likelihood o esimae heir model, and Zhang e al. (2010) use a hird order approximaion based on he Heson sochasic volailiy model. Zhu and Lian (2012) compare heir exac formula o he oher approximaion mehods, finding ha hese approximaions can creae large percenage errors. Through he use of simulaions, Zhu and Lian also show ha he errors increase as volailiy increases. Raher han proposing ye anoher approximaion mehod for convexiy, we back ou convexiy from he acual VIX fuures prices, esing his implied convexiy for several periods of varying marke volailiy, including he Fall of This implied convexiy allows us o examine he behavior of he acual convexiy adjusmen needed o jusify he curren VIX fuures price. 3. Modeling VIX fuures pricing and he convexiy issue The pricing of physical commodiy fuures conracs is based on he cos-of-carry model. Violaions of he cos-of-carry model beyond he ransacion bounds creae arbirage opporuniies. Alhough he cash VIX is deermined from S&P 500 opion prices, he conversion from he implied fuure variance capured by a porfolio of ou-of-he-money S&P 500 opions o he implied volailiy represened by he cash VIX resuls in a convexiy issue. As such, he need arises o esimae he VIX fuures price variance (or volailiy) from he presen ime o he conrac s expiraion. Esimaing he fuures price variance poses a challenge for pricing purposes, since he fuure realized variance of VIX fuures is unknown by definiion and mus herefore be esimaed. One approach o dealing wih his convexiy issue is o develop a model o forecas he VIX fuures price volailiy from he presen ime unil he conrac s expiraion. However, such a model enails assumpions concerning he pah of fuure VIX fuures prices and herefore includes model risk. A 6

7 second approach o examining he convexiy of VIX fuures prices is o employ acual fuures and cash VIX prices and solve for he implied convexiy of VIX fuures. This paper inroduces he concep of implied convexiy and hen examines he associaed empirical properies of his new measure. In his secion we firs provide a brief summary of he main issues and associaed equaions relaed o VIX fuures pricing, and hen elaborae on our implied convexiy approach o VIX fuures pricing. 3 The firs sep in undersanding he VIX fuures convexiy is o define he associaed variance. For oday a ime, for any random variable x T whose value is revealed a ime T >, we know ha 2 Var x E x E x (1) 2 ( T ) ( T ) ( T ) Subsiuing he value of he VIX fuures price a expiraion, F T, for x T ino (1) we obain 2 Var F E F E F (2) 2 ( T ) ( T ) ( T ) Furhermore, since E ( F ) E ( VIX ) F he fair value of he VIX fuures price oday, and since T T E F 2 2 ( T ) E ( VIX T ) he forward-saring variance swap rae, we can express he fair value of he VIX fuures a ime oday, expiring a ime T as: F E VIX Var F 2 ( T ) ( T ) (3) As noed in he CBOE VIX whie paper (2003), he expression Var (F T ) is he variance of he VIX fuures prices from he presen o he fuures expiraion, which can be viewed as he daily cumulaive variance of he VIX fuures prices from he curren ime o expiraion ime T. Since F E VIX E VIX E VIX by reason of he convexiy of he square roo funcion, and 2 2 ( T ) ( T ) ( T ) since Var (F T ) is he elemen ha compensaes exacly for he difference found in he convexiy 3 In our developmen of he models we simplify he wide variey of symbols found in his srand of he lieraure. 7

8 inequaliy, he variance of he VIX fuures prices is ofen referred o as he concaviy/convexiy adjusmen. However, his adjusmen is no direcly observable. In order o be esimaed, he convexiy adjusmen requires he specificaion of boh a model and a corresponding se of assumpions: herefore, his correcing facor is model dependen. The use of differen VIX dynamic specificaions can consequenly lead o radically differen VIX fuures variance esimaes for such a model. The nex secion shows how we examine his issue by using an approach similar o he concep of implied volailiy for opions wrien on radiional securiies. 4. Mehodology One approach o he convexiy adjusmen for VIX fuures is o embark on a ques for he perfec model. However, his approach requires some arbirariness in is model specificaion, complex calibraions of he model iself, and ambiguous exercises in forecasing for he fuure variance relaed o he convexiy adjusmen. Insead, we compue he implied convexiy using acual VIX fuures prices and he S&P 500 near- and far-erm opion-implied volailiy values. We hen analyze he properies of he convexiy adjusmen facor over ime, as well as wih respec o several marke dynamics. This new approach of examining he convexiy adjusmen provides new informaion relaed o VIX fuures pricing and poenial arbirage opporuniies wih VIX fuures. VIX fuures conracs sared rading in March of 2004 and herefore provide observable prices and volailiies since ha dae. However, he forward-saring variance swap raes needed for he cash comparison are no raded on exchanges. Alernaively, forward-saring variance swap raes can be deermined from he value of a porfolio of S&P 500 opions. Therefore, we ake advanage of publicly repored volailiy indices ha reflec he prices of he S&P 500 opions porfolios. Forward-saring variance swap raes can hen be compued in a way ha is similar o inferring he 8

9 forward curve from a yield curve. As described below, we calculae hese values by aking advanage of he addiive naure of variances and of he aforemenioned index volailiy measures racked and repored by he CBOE. Here we deail he procedure used o exrac he implied convexiy value. Firs, on he dae he VIX fuures conrac expires, he 30-day implied volailiy forecas by he VIX fuures is simply he implied volailiy deermined from he S&P 500 opions series ha expires 30 days afer he VIX fuures expiraion. However, before he expiraion of he VIX fuures, he 30-day implied volailiy underlying he fuures conrac is a forward-saring implied volailiy, i.e. a 30-day volailiy saring a he fuures expiraion. Consequenly, we mus calculae he forward implied volailiy (variance) as of any specific day/ime before VIX fuures expiraion. In order o inroduce he concep of implied convexiy we firs examine he calculaion of variance in general. For hree ordered ime poins or markers idenified as 1, 2, and 3, such ha 1 < 2 < 3, he relaion among he variances associaed wih he various corresponding periods needed o deermine a oal variance over ime period 1 o 3 is as follows: ( ) ( ) ( ) (4) where represens he annualized variance for he period of ime beween markers i and j. 4 2 i j However, in he conex of a VIX fuures conrac priced a ime, expiring a ime T, and delivering he fuure implied marke volailiy (VIX) based on he S&P 500 opions expiring in 30 days from VIX fuures expiraion - and using calendar days in his descripion - we develop equaion (5) providing he oal variance beween he presen ime and 30 days afer he VIX fuures expires: ( T 30 ) ( T ) 30 (5) T 30 T T T 30 4 Noe ha he variances do no echnically need o be annualized, provided ha hey reflec he same frequency of reurns. For insance, he variances could reflec daily, weekly, or monhly reurns, as long as hey are all of he same frequency ype. Thus, one can easily annualize all he variances by simply muliplying boh sides of equaion (4) by he proper scaling facor. 9

10 2 where represens he variance beween he presen and 30 days afer expiraion of he fuures T 30 2 conrac, is he variance beween he presen ime and he VIX fuures expiraion, and T 2 T T 30 is he variance beween he VIX fuures expiraion and 30 days afer expiraion. The laer is also he forward-saring variance swap rae represening he implied variance embedded in VIX fuures prices, and we can herefore isolae his rae appearing in equaion (5) as follows: T 30 T T T 30 T 30 T (6) The CBOE racks wo indices relaing o he cash VIX ha also employ a porfolio of S&P 500 opions and are calculaed using he mehodology developed for he VIX, namely he VIN (he implied volailiy for he near-erm S&P 500 opion expiraion) and he VIF (he implied volailiy for he far-erm S&P 500 opion expiraion). These series are he individual componens ha deermine he cash VIX. However, insead of represening a consan n-day-ahead volailiy as he VIX does, he VIN and he VIF deermine he volailiy beween now and he firs-monh opion expiraion (VIN) and he volailiy beween now and he nex-monh opion expiraion (VIF). Therefore, he VIX can be compued as a weighed average beween he VIN and he VIF. We can employ he VIN and VIF direcly o compue he forward rae implied variance, if he mauriy of he VIN corresponds exacly o ha of he expiraion of he VIX fuures conrac and 2 he VIF corresponds exacly o 30 days afer he VIX expiraion. In ha case represens he T 30 square of he far-erm VIX (VIF) and represens he square of he near-erm VIX (VIN) in 2 T equaions (5) and (6). Consequenly, he forward-saring variance swap rae in ha siuaion is expressed as: T 30 T E ( VIX ) VIF VIN T T T 30 (7) 10

11 In realiy, he horizons of he VIN and he VIF do no perfecly mach he expiraion daes of he VIX fuures conrac for he VIN and he corresponding 30 days laer for he VIF; in fac, he difference in he number of days beween VIN and VIF is eiher 28 or 35 days, no 30 days. 5 For equaion (7) o be applicable, we need an adjused horizon for he VIN o mach ha of he expiraion of he VIX fuures conrac, and an adjused horizon for he VIF o mach 30 days afer he expiraion of he VIX fuures conrac, which would hen provide he 30-day forward-saring variance for he VIX fuures. This can be accomplished by judiciously weighing he VIN and he VIF indices. If he horizon of he VIN is n 1 calendar days afer he expiraion of he VIX fuures conrac and he mauriy of he VIF is n 2 days afer T+30, we can creae he needed adjused values for boh indices by compuing: 30 n n 2 1 Adjused _ VIN VIN VIF 30 n2 n1 30 n2 n1 (8) n 30 n 2 1 Adjused _ VIF VIN VIF 30 n2 n1 30 n2 n1 (9) We can hen apply equaion (7) o he new horizon-adjused volailiy indices, yielding an equaion for he forward-saring variance swap rae when ime adjusmens are needed o mach he VIX fuures expiraion: T 30 T E ( VIX ) Adjused _ VIF Adjused _ VIN T T T 30 (10) Finally, using he obained forward-saring variance swap rae along wih he curren (ime-) observed VIX fuures price, he marke-implied convexiy can be derived as: 5 The VIN and he VIF are based on he S&P 500 index opion conracs ha expire on he hird Friday of he expiraion monh. On he oher hand, VIX fuures conracs expire on he Wednesday ha is 30 days prior o he hird Friday of he calendar monh immediaely following he monh in which he fuures conrac expires. Addiionally, here are also some unusual cases where he VIN and he VIF do no sraddle he VIX (in erms of expiraion daes) due o he iming of he rollover of he S&P 500 opion conracs occurring he week before he expiraion of he VIX fuures conrac, creaing even larger iming differences beween he expiraions of he fuures conrac, he VIN, and he VIF. 11

12 IC Var ( F ) E ( VIX ) F 2 2 T T (11) where IC is he implied convexiy, Var (F T ) is he expeced variance of fuure VIX fuures prices beween ime and ime T, E VIX is he forward-saring variance swap rae compued in (10), 2 ( T) and 2 F is he square of he curren price of he VIX fuures conrac expiring a ime T. 5. Daa We obain daily daa from TradeSaion for Augus 25, 2008 hrough he end of 2011 for he VIN and VIF series described in he previous secion. 6 When necessary, he series are adjused according o equaions (8) and (9), respecively. The VIN and VIF roll over exacly one week prior o he hird Friday of each monh, wih he hird Friday being he expiraion of he S&P500 opions. Thus, any unusual behavior in he opions in he las week (as par of he VIN) does no influence he indices. The VIN horizon (n 1 ) is calculaed as he number of minues remaining beween ime and one week prior o he hird Friday of each monh. The VIF horizon (n 2 ) is calculaed as he number of minues remaining beween ime and one week prior o he hird Friday of he following monh. The adjused VIN and VIF are used o esimae he forward-saring variance swap rae as presened in equaion (10). VIX fuures daa are obained from he CQG Daa Facory for all conracs expiring beween Augus 2008 and December The VIX fuures series employed here is composed of only he nearby conracs, which are rolled over o he nex nearby conrac on he Monday before he hird Friday of every monh. The las bid and ask of he VIX fuures prices comprise he daily closing values for he analysis. Finally, he daily closing level of implied convexiy is deermined from he 6 Daa are no available for he nearby and far-erm VIX series prior o Augus 25,

13 daily closing VIX fuures mid-price and he conemporaneous forward-saring variance swap rae using equaion (11). 6. Implied convexiy properies and resuls 6.1. Implied convexiy: he normal range Given he lack of prior sudies on he behavior of implied convexiy, a reasonable saring poin is o examine he ime series hisory of his measure. Figure 1 displays implied convexiy levels in conjuncion wih VIX index levels from Augus 2008 hrough December This figure shows ha he highes levels of implied convexiy occur during periods when he VIX also peaks. Visually, he implied convexiy exhibis large posiive spikes; moreover, he speed of mean reversals in he implied convexiy series is faser han for he VIX. Furhermore, he implied convexiy is fairly sable as he VIX reverses o lower levels. The imporan characerisics observed in Figure 1 are logical for wo reasons. Firs, implied convexiy is he marke s forecas of he fuure realized variance of VIX fuures prices beween now and he fuures conrac s expiraion. Thus, he implied convexiy represens he variance of fuures prices, no fuures reurns. This is imporan because for a given level of reurn variance, a higher price will display a higher level of variance, simply due o he scaling effec. Addiionally, since a high spo VIX level produces high VIX fuures prices, his will generae high implied convexiy values, since he implied convexiy is an esimae of he fuure realized variance of VIX fuures prices. Conversely, lower VIX levels will generae lower VIX fuures prices and herefore lower implied convexiy values. In conclusion, he size of he convexiy adjusmen is srongly posiively correlaed o he level of he VIX, as refleced by Figure 1. The second reason why he resuls in Figure 1 are sensible is he fac ha mean-reversion 13

14 occurs faser for he convexiy adjusmen han i does for he VIX iself. If he marke anicipaes he VIX o gradually coninue on a downward rend afer an iniial decline, hen he convexiy adjusmen should be affeced in wo ways: Firs, he VIX will rend owards a lower value and herefore he fuure realized fuures variance associaed wih ha lower VIX level should decrease (as explained in he previous paragraph). Second, o he exen ha his downward rend is no expeced o be unusually erraic, he variance of he VIX fuures price reurns should also decrease. Since he implied convexiy is an esimae of he fuures price variance, and since he variance of a price is a funcion of boh he variance of is reurns and of he level of ha price, 7 he combinaion of hese wo effecs will lower he convexiy adjusmen, hus acceleraing he mean reversion of he implied convexiy compared o ha of he underlying VIX. Figure 1 also shows ha he implied convexiy dips below zero in many insances (41.21% of he ime), a counerinuiive phenomenon for a no-arbirage siuaion. This argumen is based on he fac ha he variance of he VIX fuures prices can never be negaive. Consequenly, by definiion, implied convexiy should always be posiive, since i is a forecas of he variance of VIX fuures prices from ime unil conrac expiraion, According o equaion (11), a negaive implied convexiy occurs when he square of he VIX fuures prices F 2 is larger han he forward-saring variance swap rae esimaed in equaion (10). 8 The fac ha negaive implied convexiy values are a common occurrence in realiy shows ha VIX fuures ofen rade above heir heoreical upper bounds, an observaion consisen wih possible mispricing of VIX fuures prices. The resuls from Figure 1 raise he issue of wha ypical implied convexiy values should be, 7 Given a cerain reurn volailiy, a higher price will imply flucuaion levels of larger magniude, ranslaing ino a higher price variance. 8 The upper bound of he VIX fuures price is equal o he square roo of he forward-saring variance swap rae, as shown in Carr and Wu (2006). This is because he price of he VIX fuures is he expeced value of he square roo of fuure expeced variance, which is always smaller han or equal o he square roo of he expeced value of fuure expeced variance. As such, implied convexiy is negaive if and only if he VIX fuures price is above is heoreical upper bound, and posiive if and only if he VIX fuures price is below is heoreical upper bound. 14

15 a leas in erms of how he marke currenly values VIX fuures ha include negaive implied convexiies. Table 1 displays he summary saisics for he implied convexiies by year in Panel A and by implied convexiy quinile in Panel B. Panel A shows ha implied convexiy exhibied is highes value of and is lowes value of during he las four monhs of Since 2008 he volailiy in implied convexiy has seadily declined. Overall, implied convexiy exhibied posiive skewness and a relaively large kurosis. However, during he las quarer of 2008 implied convexiy exhibied negaive skewness. The resuls in Panel B also show ha high levels of posiive implied convexiy (quinile 5) possess a very large sandard deviaion of values. Figure 2 addiionally repors he implied convexiy s frequency disribuion over he sample period. In paricular, Figure 2 shows a large aggregaion of daa poins around zero ha appear almos normally disribued, as well as he presence of a large number of posiive ouliers and a negaively skewed disribuion of ouliers. Table 2 describes he hisorical level perceniles of implied convexiy overall, as well as by years. Over he sample ime period, he median daily closing implied convexiy level is The 50 h, 75 h, and 95 h inerquarile ranges show ha implied convexiy oscillaes wihin a narrow range he majoriy of he ime. However, as marke volailiy periodically changes, so does implied convexiy. The resuls in Table 2 clearly show ha subsanial year-o-year variaions in implied convexiy levels exis. Such resuls are o be expeced, since he VIX iself flucuaes considerably during he sample period, causing an increase in he volailiy of he VIX and he relaed VIX fuures prices, which in urn direcly affecs implied convexiy levels. Alhough negaive implied convexiy values are found in all of he sampled years, since 2008 he median implied convexiy levels have remained slighly above zero. In fac, he 5% and 95% cuoff values for implied convexiy in Table 2 have seadily rended owards zero during his period. Moreover, generally he enire implied 15

16 convexiy range declined wih ime since 2008, which is consisen wih an improvemen in pricing efficiency. 9 Overall, hese resuls show ha he disribuion of convexiy values has evolved over ime Implied convexiy in relaion o sock marke volailiy In his secion we invesigae he relaion beween implied convexiy and marke volailiy. Theoreically, he implied convexiy componen of he VIX fuures price is he curren expecaion of he fuure realized variance of he VIX fuures prices from he presen o fuures expiraion. Therefore, if marke volailiy is high, he VIX is high, VIX fuures are high, and consequenly he variance of he VIX fuures also should be high. The opposie relaion is rue when marke volailiy is low. Thus, a direc link should exis beween implied convexiy and various marke volailiy measures. Table 3 displays he regression resuls beween implied convexiy and he prior 30-day hisorical variance of he S&P 500 index. 10 The purpose of his regression is o measure he exen o which prior volailiy in he marke is associaed wih curren implied convexiy. The resuls show ha recen realized marke volailiy and curren implied convexiy levels possess a posiive and significan relaion in general, wih he hisorical variance of he S&P 500 index explaining approximaely 18 percen of he variaion in implied convexiy for he enire ime period. Grouping he daa ino implied convexiy quiniles shows ha only he very high (quiniles 4 and 5) and very low (quinile 1) levels of curren implied convexiy are significanly relaed o recen realized hisorical marke volailiy, wih 8%, 24%, and 12% R-squares, respecively. As such, he resuls 9 In paricular, during 2009 he implied convexiy levels were beween and (a range of ) a he 90 h percenile, compared o a range of in 2010 and in The regressions beween implied convexiy and differen marke measures use a consan 1-year horizon implied convexiy. To obain he 1-year consan mauriy implied convexiy, we divide he implied convexiy backed ou of equaion (11) by he number of calendar days lef o he VIX fuures expiraion and muliply i by

17 show ha he relaion beween hisorical marke volailiy and curren implied convexiy becomes eviden for exreme values of implied convexiy; oherwise, any relaion is undeecable. A similar relaion is found beween implied convexiy and implied marke volailiy. The resuls in Table 4 show ha implied marke volailiy (he VIX) has a posiive and significan posiive relaion wih he implied convexiy for quiniles 4 and 5, explaining approximaely 11% and 36% of he oal variaion in implied convexiy. However, for quinile 1 he relaion is negaive and significan, explaining approximaely 22% of he oal variaion. The relaion beween VIX and implied convexiy can be visualized in Figure 3, where he posiive relaion is eviden for he larger values of implied convexiy (or for he VIX). In oher words, large implied convexiy values are almos always posiive. Overall, he resuls show ha large implied convexiy levels conain informaion associaed wih hisorical and expeced marke volailiy Implied convexiy as an esimae of he realized VIX fuures variance The implied convexiy componen should reflec he marke s expecaion of he fuure realized variance of VIX fuures prices beween he presen ime and conrac expiraion. Figure 4 displays he mean value of he fuure VIX fuures price variance and he implied convexiy by calendar days lef unil VIX fuures conrac expiraion. The figure shows ha boh he VIX fuures variance and he implied convexiy seadily decline as he VIX fuures conrac becomes shorer in duraion, which is expeced wih less ime remaining unil he fuures conrac s expiraion. As such, Figure 4 shows ha implied convexiy is posiively relaed o VIX fuures ime o expiraion. This relaion brings ino quesion wheher implied convexiy is relaed o oher characerisics of he VIX fuures conrac or is underlying asse, such as he volume (liquidiy) of he VIX fuures and he 17

18 presence of jumps in he VIX series. 11 Table 5 shows he resuls of he muliple regression analysis of all hree explanaory variables on he level of he implied convexiy. The resuls show ha he level of he implied convexiy is relaed o he VIX fuures ime o expiraion in he enire sample. However, he sign of he implied convexiy (i.e. wheher negaive or posiive) is no relaed o eiher VIX fuures ime o expiraion, VIX fuures liquidiy, or he dummy variable deermining he presence of a jump in he VIX series (a one for a jump and zero oherwise). Figure 4 also shows ha implied convexiy exhibis a pronounced jagged paern as well as a dipping below zero wih 13 days o expiraion. 12 Alernaively, he fuure realized VIX fuures variance exhibis a smooher paern, and by definiion remains above zero when he VIX fuures conrac is close o expiraion. Addiionally, he VIX fuures variance says relaively consan hroughou he 38 calendar days before expiraion. The larges difference beween he implied convexiy and he realized VIX fuures variance (0.0042) occurs 13 days from VIX fuures expiraion, whereas he smalles difference (0.0001) occurs 9 days before expiraion. Overall, he implied convexiy forecass he VIX fuures variance relaively closely Violaions of he VIX fuures upper bound Previously we found ha implied convexiy values are negaive nearly 50% of he ime, meaning ha he VIX fuures conrac ofen rades above heir heoreical upper bound. To examine his phenomenon more closely, we deermine he VIX fuures price ha would eliminae his violaion (i.e. he value ha would make he negaive implied convexiy values equal o zero) for each day in he sample. We find ha his adjusmen value is almos always ouside of he VIX 11 Jumps are examined in deail in he following secion. 12 Thus, in Figure 4 he average values for he implied convexiies for each day before expiraion are posiive (excep for day 13), whereas Figures 2 and 3 show ha almos 50% of he observaions in he sample possess negaive convexiy values. 18

19 fuures bid-ask spread (94.93% of all violaions), excep for a few cases when he violaion is very small. Figure 5 shows he frequency disribuion of he adjusmen relaive o he size of he bid-ask spread, showing ha he majoriy of he ime he adjusmen is in excess of 100% of he bid-ask spread, and ofen i is much larger. Given he significan size of he adjusmen needed o correc for he VIX fuures upper bound violaion, we explore is relaion o implied convexiy. Table 6 shows he resuls of he regression where he adjusmen is he dependen variable and he implied convexiy is he independen variable Overall, he resuls show a significanly large regression R- squared (93.14%), meaning ha implied convexiy is srongly relaed o he upper bound adjusmen. However, when he daa is separaed ino quiniles based on he size of he implied convexiy, he resuls show ha only he lowes (quinile 1) and highes (quinile 5) implied convexiy levels explain a large amoun of he variaion in he upper bound violaions, whereas he middle quiniles explain much less. Saisically, he resuls show ha implied convexiy and he size of he VIX fuures upper bound violaion are srongly relaed. However, as implied convexiy values end owards zero, i becomes a less imporan explanaory variable for he size of he VIX fuures upper bound violaions. 7. Poenial violaions of he model All models are based on assumpions designed o allow for racable closed-form soluions. The Carr and Wu (2006) approach of presening a lower and upper bound for he price of he VIX fuures is based on several such assumpions. In conjuncion wih Jensen s inequaliy, he lower bound of he VIX fuures is he forward-saring volailiy swap rae, and he upper bound is he 19

20 square roo of he forward-saring variance swap rae. The convexiy adjusmen is subraced from he price of he forward variance o deermine he fair value of he VIX fuures. In he following secions we discuss he Carr and Wu (2006) simplifying assumpions. Alhough each assumpion can be empirically analyzed in deail, we argue how possible assumpion violaions should be minimal; herefore, rare violaions should no maerially affec he resuls. Moreover, an exensive examinaion of each assumpion is beyond he scope of his paper, is consrained by aricle lengh, and is lef for fuure research if needed Jumps in he volailiy process A consideraion wih our implied convexiy analysis sems from he fac ha he proof in Carr and Wu (2006) assumes ha he volailiy price process is coninuous. If insead one allows for jumps, he fuure realized variance is hen a combinaion of he variances arising from he quadraic variaion of he diffusion componen of he process plus he quadraic variaion of he jump componen of he process. In fac, Broadie and Jain (2008) noe ha he effec of ignoring jumps in compuing he fair variance swap rae from he porfolio of S&P 500 opions used in he VIX definiion could be significan. Therefore, we proceed o es for he presence of jumps in he (spo) VIX as a means o verify wheher he coninuous volailiy assumpion in Carr and Wu (2006) could be violaed. Various non-parameric mehods designed o idenify he occurrence of jumps in sochasic processes have emerged in he las few years (for example, see Ai-Sahalia (2002), Carr and Wu (2003), Barndorff-Nielsen and Shephard (2006), and Jiang and Oomen (2008), o name a few). We choose o implemen he sraighforward es found in Lee and Mykland (2008), boh for is inuiive 20

21 appeal as well as for is abiliy o ouperform he nonparameric jump ess of Barndorff-Nielsen and Shephard (2006) and Jiang and Oomen (2008). The saisic L(i) ess a ime i wheher here is a jump in he process S() from i-1 o i and is defined as: Li () ln[s( ) / S ( )] ( ) i i 1 (12) i where i 1 1 ( ) ln[s( ) / S( )] ln[s( ) / S( )] (13) i j j 1 j 1 j 2 K 2 j 1 ( K 2) Using daily frequencies, we adop he Lee and Mykland (2008) recommended opimal window size of K se o 16 days. For n, he number of observaions, and c 2/, le C n 1/2 [2ln( n)] ln( ) ln[ln( n)] 1 and S 1/2 n 1/2 c 2 c[2ln( n)] c[2ln( n)] (14) The acual es is hen wheher he raio L() i Cn S n is above he criical value corresponding o he chosen significance level. Since under he null hypohesis of no jumps he raio has a cumulaive x disribuion funcion of he form exp( e ), he criical value corresponding o he 1% upper ail will herefore be * ln( ln(0.99)). We es for he presence of jumps in he VIX over our enire sample period a he 1% significance level. We deermine ha only five likely insances of jumps occur. Since he sample period conains 847 rading days, his is he equivalen of less han one-and-a-half jumps per year. For comparison, Lee (2012) invesigaes he predicabiliy of jump arrivals in U.S. sock markes, idenifying he number of jumps experienced by individual equiies from he DJIA and he S&P 500 index from January 4, 1993 o December 21, The associaed average number of jumps per securiy is per year. We herefore conclude ha he assumpion of a coninuous process by 21

22 Carr and Wu (2006) is a reasonable one for he VIX, a leas in he conex of our sudy. Accordingly, our compued variance swap rae is consisen wih he conclusion ha i is essenially free of any bias due o jumps Approximaion error The Carr and Wu (2006) lower and upper bound proof is based on a coninuous range of opion srikes, whereas he acual VIX (and VIN and VIF) employ discree srikes for ou-of-hemoney opions wih non-zero bids unil wo adjacen non-zero bids occur. Jiang and Tian (2007) examine biases associaed wih he discree and runcaion calculaion procedure of he CBOE, leading o a misspecificaion of rue volailiy, especially in regards o he erm srucure of volailiies. This bias ranslaes o he VIN being relaively more downward biased han he VIF a high volailiy levels, causing he upper bound o be downward biased. However, he imporance of his bias is suspec, since he simulaion sudy employed by Jiang and Tian o achieve he resuls employs an unrealisic large range of srike prices relaive o real-life srikes. Moreover, only he relaive difference beween VIN and VIF would affec he erm srucure volailiy. Consequenly, using he acual srike series employed by he markes would have a minimal bias on he oucomes found in his sudy Selemen procedure As examined in Pavlova and Daigler (2008), a selemen bias exiss due o he procedure employed o deermine he individual opion prices used o calculae he VIX fuures selemen price. In paricular, any opion employed o calculae he VIX a he open on he Wednesday selemen day uses he rade price a 8:31 a.m. Cenral Time if a rade exiss, raher han he average 22

23 of he bid-ask price uilized o calculae he VIX, VIN, and VIF indices. Pavlova and Daigler (2008) do deermine ha large differences exised hrough mid-may 2007 for cerain expiraions beween he cash VIX and he VIX fuures selemen value. However, only six of 33 monhs had biases where he VIX fuures was larger han he cash VIX a fuures expiraion, i.e. siuaions consisen wih a negaive implied convexiy. Thus, any biases due o selemen in pas expiraions did no promoe he negaive implied convexiies consisen wih arbirage opporuniies. Moreover, marke paricipans repor ha he selemen bias in recen years is near zero. Consequenly, previous lieraure does no suppor he selemen bias causing he negaive implied convexiies. 8. Conclusions The VIX fuures price heoreically includes a convexiy adjusmen componen o compensae for he difference beween he forward sar swap rae and he squared VIX fuures price, wih ha difference equal o he fuure variance of he VIX fuures price. One approach o dealing wih his issue is o model he fuure VIX fuures variance based on some assumed sochasic behavior of he VIX. We approach he problem differenly by solving for he convexiy implied by deermining he difference beween he forward-saring swap rae (deermined by he adjused value of he nex wo expiraions of he S&P 500 opions series) and he square of he curren VIX fuures price, and hen proceed o sudy he properies of his implied convexiy. We examine he characerisics of he implied convexiy adjusmen using daily daa. The resuls show ha he implied convexiy daily values are ofen negaive and herefore ouside he noarbirage range. Moreover, he implied convexiy is relaed o he S&P 500 volailiy. Finally, he implied convexiy underesimaes realized VIX fuures variance approximaely 71% of he ime, which is he opposie of he relaion beween implied volailiy and realized volailiy. 23

24 Furher research is needed o undersand why implied convexiy possesses he ime series properies and relaional characerisics discovered in he paper. In paricular, hese resuls sugges ha arbirage profis exis for a very liquid fuures conrac. Such furher research should provide insighs o VIX fuures pricing in general, and herefore o a superior VIX fuures pricing model. 24

25 References Ai-Sahalia, Y. (2002). Telling from discree daa wheher he underlying coninuous-ime model is a diffusion. Journal of Finance, 57, Becker, R., Clemens, A., and Whie, S. (2007). Does implied volailiy provide any informaion beyond ha capured in model-based volailiy forecass? Journal of Banking and Finance, 31, Barndorff-Nielsen, O. E., and Shephard, N. (2006). Economerics of esing for jumps in financial economics using bipower variaion. Journal of Financial Economerics, 4, Broadie, M., and Jain, A. (2008). The effec of jumps and discree sampling on volailiy and variance swaps. Inernaional Journal of Theoreical and Applied Finance, 11(8), Carr, P., and Wu, L. (2003). Wha ype of process underlies opions? A simple robus es. Journal of Finance, 58, Carr, P., and Wu, L. (2006). A ale of wo indices. Journal of Derivaives, 13, Exchange, C. B. O. (2003). The VIX whie paper. URL: hp://www. cboe. com/micro/vix/vixwhie. pdf. Dupoye, B., Daigler, R., and Chen, Z. (2011). A simplified pricing model for volailiy fuures. The Journal of Fuures Markes, 31, Grunbichler, A., and Longsaff, F. A. (1996). Valuing fuures and opions on volailiy. Journal of Banking and Finance, 20, Heson, S. (1993). A closed-form soluion for opions wih sochasic volailiy wih applicaions o bond and currency opions. The Review of Financial Sudies, 6, Huskaj, B., and Nossman, M. (2013). A erm srucure model for VIX fuures. The Journal of Fuures Markes, 33, Jiang, G., and Tian, Y. (2007). Exracing model-free volailiy from opion prices: An examinaion of he VIX index. Journal of Derivaives, 14, Jiang, G. J., and Oomen, R. C. (2008). Tesing for jumps when asse prices are observed wih noise - a swap variance approach. Journal of Economerics, 144, Lee, S. (2012). Jumps and informaion flow in financial markes. Review of Financial Sudies, 25,

26 Lee, S., and Mykland, P. (2008). Jumps in financial markes: a new nonparameric es and jump dynamics. Review of Financial Sudies, 21, Lin, Y. N. (2007). Pricing VIX fuures: Evidence from inegraed physical and risk-neural probabiliy measures. The Journal of Fuures Markes, 27, Lu,. J., and Zhu, Y. Z. (2010). Volailiy componens: The erm srucure of VIX fuures. The Journal of Fuures Markes, 30, Mencia, J., and Senana, E. (2013). Valuaion of VIX derivaives. Journal of Financial Economics, 108, Pavlova, I., and Daigler, R. T. (2008). The non-convergence of he VIX fuures a expiraion. Review of Fuures Markes, 17, Poon, S. H., and Clive W. J. G. (2003). Forecasing volailiy in financial markes: A review. Journal of Economic Lieraure, 41, Wang, Z., and Daigler, R. (2013). The opion SKEW index and he volailiy of volailiy. Financial Managemen Meeings, Chicago. Zhang, J. E., and Huang, Y. (2010). The CBOE S&P 500 hree-monh variance fuures. Journal of Fuures Markes, 30, Zhang, J. E., Shu, J. and Brenner M. (2010). The new marke for volailiy rading. The Journal of Fuures Markes, 30, Zhang, J. E., and Zhu, Y. (2006). VIX fuures. The Journal of Fuures Markes, 26, Zhu, S. P., and Lian, G. H. (2012). An analyical formula for VIX fuures and is applicaions. The Journal of Fuures Markes, 32,

27 Figure 1 Daily closing levels of he VIX and implied convexiy (IC) from Augus 2008 o December VIX IC Zero V I X I C Aug 08 Dec 08 Apr 09 Aug 09 Dec 09 Apr 10 Aug 10 Dec 10 Apr 11 Aug Noe: This figure shows he hisorical level of he VIX (lef verical axis) and of he implied convexiy, IC, (righ verical axis) from Augus 2008 o December The Zero gradien line corresponds o he lower bound of implied convexiy. 27

28 Figure 2 Frequency disribuion of implied convexiy 140 F r e q u e n c y Implied convexiy level Noe: This figure shows he disribuion of he implied convexiy daily closing values from Augus 2008 o December

29 Figure 3 Relaion beween VIX and implied convexiy I m p l i e d c o n v e x i y VIX level Noe: This figure plos he implied convexiy s daily closing values (verical axis) agains he VIX daily closing values (horizonal axis). The sample period is Augus 2008 o December

30 Figure 4 Average realized VIX fuures variance (Var VIX) and implied convexiy (IC) values by days lef unil VIX fuures expiraion Average of VarVIX Fuures Average of IC L e v e l Days Remaining o VIX Fuures Mauriy Noe: This figure shows he average implied convexiy daily closing levels (IC) and he average fuure realized VIX fuures variance by days remaining o VIX fuures mauriy. A any poin in ime, he VIX fuures conrac is he nearby conrac. 30

31 Figure 5 Frequency disribuion of he downward VIX fuures price adjusmen needed o avoid a VIX fuures upper bound violaion (as a percenage of he VIX fuures bid-ask spread) F r e q u e n c y Adjusmen as a percenage of bid-ask spread Noe: This figure shows he frequency disribuion for he VIX fuures downward adjusmen needed in order o preven an upper bound violaion of he VIX fuures, as defined by Carr and Wu (2006). The adjusmen is expressed as a percenage of he VIX fuures bid-ask spread. Only violaions ha fall ouside of he bid-ask spread are shown (94.93% of all violaions), all of which fall below he VIX fuures bid price. 31

32 Table 1 Summary saisics for daily closing levels of implied convexiy No. obs. Mean Median Min. Max. Sd. Dev. Skewness Kurosis Panel A: by year All Panel B: by Implied Convexiy quinile Noe: Panel A presens he summary saisics for he daily closing levels of implied convexiy for he enire sample and by year. Panel B shows he implied convexiy saisics by quiniles, where quinile 1 is he lowes quinile and quinile 5 is he highes quinile. Means ha are saisically differen from zero are in ialics. Table 2 Normal ranges for daily closing levels of implied convexiy over he sample period Year No. obs. 5% 10% 25% 50% 75% 90% 95% All Noe: This able shows he disribuion of daily closing levels of implied convexiy by perceniles for he enire daa se and for each year of he daa. Table 3 Regression resuls for implied convexiy regressed on he prior monh for he S&P 500 reurn variance Quinile 1 Quinile 2 Quinile 3 Quinile 4 Quinile 5 All Consan *** (0.0168) *** (0.0010) *** (0.0006) *** (0.0012) *** (0.0244) *** (0.0071) SP500 variance *** (0.1038) (0.0169) (0.0102) *** (0.0159) *** (0.0776) *** (0.0428) Regression saisics R-squared No. obs Noe: This able provides he regression where he daily closing level of implied convexiy is he dependen variable and he S&P 500 reurn variance over he prior 30 days is he independen variable. The daa is annualized for consisency. Quinile 1 represens he smalles implied convexiy values. Sandard errors are repored in parenheses below he regression coefficiens.*, **, *** shows significance a he 10%, 5%, and 1% levels, respecively. 32