Modeling VXX. First Version: June 2014 This Version: 13 September 2014

Transcription

1 Modeling VXX Sebasian A. Gehricke Deparmen of Accounancy and Finance Oago Business School, Universiy of Oago Dunedin 9054, New Zealand Jin E. Zhang Deparmen of Accounancy and Finance Oago Business School, Universiy of Oago Dunedin 9054, New Zealand Firs Version: June 2014 This Version: 13 Sepember 2014 Keywords: VXX; VIX Fuures; Roll Yield; Marke Price of Variance Risk; Variance Risk Premium JEL Classificaion Code: G13

2 Modeling VXX Absrac We sudy he VXX Exchange Traded Noe (ETN), ha has been acively raded in he New York Sock Exchange in recen years. We propose a simple model for he VXX and derive an analyical expression for he VXX roll yield. The roll yield of any fuures posiion is he reurn no due o movemens of he underlying, in commodiy fuures i is ofen called he cos of carry. Using our model we confirm ha he phenomena of he large negaive reurns of he VXX, as firs documened by Whaley (2013), which we call he VXX reurn puzzle, is due o he predominanly negaive roll yield as proposed bu never quanified in he lieraure. We provide a simple and robus esimaion of he marke price of variance risk which uses hisorical VXX reurns. Our VXX price model can be used o sudy he price of opions wrien on he VXX.

3 Modeling VXX 1 1 Inroducion There are hree major risk facors which are raded in financial markes: marke risk which is raded in he sock marke, ineres rae risk which is raded in he bond markes and ineres rae derivaive markes, and volailiy risk which up unil recenly was only raded indirecly in he opions marke. I is well acceped in he lieraure ha boh equiy reurns and variance are random (French, Schwer, and Sambaugh, 1987). I is also well undersood ha he variance risk premium is significan and negaive (Carr and Wu, 2009). Invesors rade volailiy eiher o ake advanage of he opporuniy in he variance risk premium or o hedge agains volailiy risk. One way invesors can rade volailiy would be o buy a-he-money (ATM) opions, bu hese do no necessarily say a-he-money. When he opions are Ou-of-hemoney (OTM) and in-he-money (ITM) hey have smaller volailiy sensiiviy (Vega) and are herefore less effecive for rading volailiy. Opions conracs will no always be able o mee invesors need for volailiy risk managemen as here will no be enough liquidiy in he opions markes when he marke goes down. Also when invesors rade volailiy in he opions marke hey also rade marke risk and possibly ineres rae risk (for longer erm opions), herefore rading volailiy hrough opions is ofen conaminaed by hese oher risk facors making i inefficien for risk managemen. Developing a financial marke o rade volailiy direcly is very imporan for researchers and praciioners. (Zhu and Zhang, 2007) Chen, Chung, and Ho (2011) show ha VIX (if radeable) and VIX opions expand invesor opporuniy se and are useful for diversificaion, in a mean-variance opimizing markowiz framework. In 2003, he mehodology for calculaing he VIX index changed and he index using he old mehodology was renamed o VXO. The VIX is now calculaed using all ou-of-he-

4 Modeling VXX 2 money opions on he S&P 500 which have a bid price. Following his change in 2004, he CBOE launched he much anicipaed VIX fuures and in 2006 VIX opions also sared rading on he CBOE. Boh VIX fuures and opions have consisenly grown in daily dollar rading volume. In 2009, S&P Dow Jones Indices sared reporing several differen VIX fuures indices which represen he reurns of differen VIX fuures posiions. One example of a VIX fuures index is he S&P 500 VIX Shor-Term Fuures Index (SPVXSTR) which racks he performance of a posiion in he neares and second neares mauring VIX fuures. The SPVXSTR is rebalanced daily o creae a consan one monh mauriy VIX fuures posiion. Shorly afer he VIX fuures indices sared were developed Barclays Capial ipah launched he firs ever VIX fuures index Exchange Traded Produc (ETP), he VXX Exchange Traded Noe (ETN). An ETN is unsecured senior deb ha pays no coupons (ineres) and does no have a fixed redempion a mauriy bu raher is redempion value is linked o he performance of some underlying (Bao, Li, and Gong, 2012).The VXX s redempion value, for example, depends on he value of he SPVXSTR a mauriy less an annual managemen fee of 0.89%. There are now many differen VIX fuures ETNs wih differen underlying indices, all of hese combined make he VIX fuures ETN marke. The VIX fuures ETN marke has become vasly popular, all of he ETNs combined have a marke capializaion of nearly 4 billion US dollars and average daily rading volume in excess of 800 million US dollars (Whaley, 2013). One of he main drivers of VIX fuures ETNs growing populariy may be ha Muual funds and Hedge funds are ofen resriced from rading fuures and opions bu hey sill have a need o hedge volailiy risk herefore hey rade in he VIX fuures ETN marke. The VXX is he mos popular of he VIX fuures ETNs and is now he hird mos raded ETP, amongs all ETPs, based on average daily rading volume. The VXX is only

5 Modeling VXX 3 jus behind he ishares MSCI Emerging Markes ETF (EEM), much furher behind he S&P 500 ETF (SPY) and in fron of he ishares Russell 2000 ETF in erms of daily rading volume (IWM) 1. Whaley (2013) is he firs o documen he phenomena of he highly negaive reurns of he VXX, which we will refer o as he VXX reurn puzzle. Eraker and Wu (2013) also show he significan negaive performance of VIX fuures and VIX fuures index ETPs (including he VXX). Deng, McCann, and Wang (2012) show ha ETNs on VIX fuures indices, such as he VXX, are no very effecive hedging/diversificaion ools for equiy and mixed equiy and bond porfolios. Hancock (2013) ess he performance of VIX fuures ETNs and compares hem o hree benchmarks. Hancock (2013) shows ha he VXX and oher VIX fuures ETNs never consisenly ouperform benchmarks even when used o diversify equiy porfolios. These findings hold even when differen holding periods and porfolio weighing mehods are used. Hancock (2013) suggess ha he poor performance is unique o VIX fuures ETNs and is no a propery of volailiy. We documen he VXX reurns in able 1 which shows he summary saisics of he VXX, SPX (S&P 500 index ETP) and VIX reurns from 30 h January 2009 o he 27 h June Noe he abysmal performance of he VXX as can be seen firsly by he -0.32% average daily discree reurn of he VXX and he average daily coninously compounded reurn of -0.39% as opposed o he average daily coninuously compouned and discree reurns of he VIX which were -0.09% and 0.15% respecively. Secondly, he Holding Period Reurn (HPR) shows ha wihin our sample period he VXX has los 99.59% of is value, he VIX has only los 71.66%. The Compound Annual Growh Rae (CAGR) of % of he VXX compared o a CAGR of % of he VIX, furher displays he underperformance of he VXX. We will laer show ha he main reason why he VXX does no follow he 1 ETP daabase websie: as of he 10 h Ocober The average daily rading volume is compued as an average of he daily number of shares of ha ETP raded over he previous 3 monhs.

6 Modeling VXX 4 VIX, as he consan 30-day mauriy VIX fuures does, is due o he roll yield.in figure 2 we plo he VIX index, he VXX price and he consan 30-day-mauriy VIX fuures price, as in Zhang, Shu, and Brenner (2010), so he difference is visually observable. Even wih he well documened and easily observed underperformance he VXX marke has made grea srides in populariy, figure 3 shows us he upward rend in he daily dollar rading volume and he iniial increase in and hen levelling off in marke capializaion of he VXX since incepion. Whaley (2013), Deng, McCann, and Wang (2012), Husson and McCann (2011) and Bao, Li, and Gong (2012) all sugges ha he VXX is subjec o he roll yield of VIX fuures and ha his is he cause for he underperformance of he VXX. None of he aforemenioned aricles quanify he roll yield or aemp o measure i, herefore we will creae a model for he VXX which allows he quanificaion of he roll yield and proves he hypohesis ha he roll yield drives he significan negaive reurns of he VXX. The roll yield of any fuures posiion is he reurn ha a fuures invesor capures when he fuures price converges o he spo price, i is he par of he reurn which is no due o changes in he price of he underlying asse or index. When he marke is in backwardaion (i.e. downward sloping erm-srucure) he price rolls up o he spo price, herefore he roll yield will be posiive. When he marke is in conango (i.e. upward sloping erm-srucure) he price rolls down o he spo price, herefore he roll yield will be negaive. The VIX fuures erm srucure is in conango during normal imes and herefore he roll yield is for example negaive. The VIX fuures erm srucure can be in backwardaion, usually during large economic downurns, and he roll yield will become posiive which can make ETPs on VIX fuures indices profiable (Whaley, 2013). We sudy he VXX price by using he VIX fuures price approximaion from Zhang, Shu, and Brenner (2010), which we review in secion 3.1, o propose he firs sochasic volailiy model of he VXX which accouns for he underlying dynamics of he S&P 500 index (SPX)

7 Modeling VXX 5 and he VIX index. We believe he relaionship beween he VXX, he VIX and he S&P 500 is essenial in building a comprehensive model. We show ha he difference beween he 30-day-mauriy VIX fuures price change and he VXX price change in figure 2 is in fac due o he roll yield. We hen go furher and show ha he roll yields sign is driven, on aggregae, by he negaive marke price of variance risk, λ. Eraker and Wu (2013) use an equilibrium model approach o show ha he Variance Risk Premium (VRP) is he driver of he VXX s negaive reurns. This is consisen wih our finding as he marke price of variance risk, λ, and he VRP are almos proporional as shown by Zhang and Huang (2010). In he nex secion we will explain he mehodology for how he SPVXSTR index is calculaed. Then in Secion 3 we will review he heory behind pricing he VIX and VIX fuures from Zhang and Zhu (2006) and Zhang, Shu, and Brenner (2010) and use his o creae a sochasic model for he VXX price and examine he roll yield of he VXX. In secion 4 we will use he VXX model o develop a simple way of esimaing he marke price of variance risk. In secion 5 we will examine he effec of he rebalancing frequency of he SPVXSTR which will also be a robusness es of our coninuous ime VXX model. Finally in secion 6 we will conclude and discuss on our findings. 2 The SPVXSTR index To model he VXX we mus firs undersand he SPVXSTR. In his secion we will presen he mehodology for calculaing he SPVXSTR index as inerpreed from S&P Dow Jones Indices (2012). The SPVXSTR index seeks o model he oucome of holding a long posiion in shorerm VIX fuures, specifically holding posiions in he neares and second neares mauring VIX fuures. The posiion is rebalanced daily o creae a consan rolling one-monh

8 Modeling VXX 6 mauriy VIX fuures posiion (Barclays, 2013). The index is calculaed by SP V XST R = SP V XST R 1 (1 + CDR + T BR ), (1) where SP V XST R is he index level a ime, SP V XST R 1 is he index level a ime 1, CDR is he Conrac Daily Reurn of he VIX fuures posiion and T BR is he Treasury Bill Reurn earned on he noional value of he posiion. The T BR is given by [ T BR = ] Dela T BAR, (2) where Dela is he number of calendar days beween he curren and previous business days. T BAR 1 is he Treasury Bill Annual Reurn, which is equal o he mos recen weekly high discoun rae for 91-day US Treasury bills effecive on he preceding business day. Usually he raes are announced by he US Treasury on each Monday, bu if he Monday is a holiday hen Fridays raes will apply. The CDR is calculaed by CDR = w 1, 1F T 1 + w 2, 1 F T 2, w 1, 1 F T w 2, 1 F T 2 1 (3) where w i, 1 is he weigh in he i h neares mauring VIX fuures a ime 1, F T i is he marke price of he i h neares mauring VIX fuures conrac a ime and F T i 1 is he marke price of he i h neares mauring VIX fuures conrac a ime 1. 2 The weighs are adjused daily o be and w 1, = dr d, w 2, = 1 dr d, 2 In Equaion (3) we use we use w 1, 1 and w 2, 1 in he numeraor. Deng, McCann, and Wang (2012)CDR use w 1, and w 2, which is inconsisen wih he mehodology from S&P Dow Jones Indices (2012). When calculaing discree reurns of any posiion he weighs should say consan over he period you are calculaing he reurn for and only he prices should change.

9 Modeling VXX 7 where S&P Dow Jones Indices (2012) defines dr =The oal number of business days wihin a Roll Period beginning wih, and including, he following business day and ending wih, bu excluding, he following CBOE VIX Fuures Selemen Dae. The number of business days includes a new holiday inroduced inra-monh up o he business day preceding such a holiday. and d =The oal number of business days in he curren Roll Period beginning wih, and including, he saring CBOE VIX Fuures Selemen Dae and ending wih, bu excluding, he following CBOE VIX Fuures Selemen Dae. The number of business days says consan in cases of a new holiday inroduced inra-monh or an unscheduled marke closure (S&P Dow Jones Indices, 2012, p. 7) Figure 1 shows he deerminaion of dr and d in a diagram for convenience of undersanding. 3 Modeling VXX 3.1 Review of VIX and VIX fuures model To model he VXX we need a model for he VIX index and VIX fuures. Zhang and Zhu (2006) and Zhang, Shu, and Brenner (2010) have developed a model for he VIX and VIX fuures, for compleeness we review and combine he resuls from boh in his secion. The SPX (S&P 500 index) can be modeled by he following diffusion process wih a sochasic process of insananeous volailiy as described by Heson (1993), ds = µs d + V S db P 1,, (4) dv = κ(θ V )d + σ v V db P 2,, (5) where S is he SPX, V is he insananeous variance of he SPX, µ is he expeced reurn from invesing in he SPX, θ is he physical measure for he long run mean level of he insananeous variance, κ is he physical measure for he speed of mean reversion of insananeous variance and σ V measures he he variance of variance. B P 1, and B P 2, are

10 Modeling VXX 8 wo sandard Brownian moions ha describe he random noise in he SPX reurn and variance, respecively, hey are correlaed by a consan correlaion coefficien ρ. The ransformaions beween physical and risk-neural parameers are given by and θ = θ κ κ (6) κ = κ + λ, (7) where κ is he risk-neural speed of mean reversion of volailiy, θ is he risk-neural long run mean level of insananeous variance and λ is he marke price of variance risk. We can hen describe he risk-neural dynamics of he SPX as follows ds = rs d + V S db 1,, (8) dv = κ (θ V )d + σ v V db 2,, (9) where r is he risk free rae, and db 1, and db 2, are wo new sandard Brownian moions which are correlaed by he consan correlaion coefficien, ρ. The VIX is equal o he variance swap rae (Carr and Wu, 2009), which is equivalen o he condiional expecaion in he risk-neural measure where τ 0 = V IX 2 = E [ 1 τ 0 +τ0 ] V s ds = (1 B)θ + BV, (10) 1 e κ τ0 and B = κ τ 0. Then he VIX fuures price formula is given by F T 100 = E (V IX T ) =E ( (1 B)θ + BV T ) + = (1 B)θ + BV T f (V T V )dv T, 0 (11)

11 Modeling VXX 9 where he ransiion probabiliy densiy as given by Cox e al. (1985) is where f (V T V ) = ce u v ( v u) q/2 Iq (2 uv), (12) c = 2κ σ 2 V (1 e κ (T ) ), u = cv e κ (T ), v = cv T, q = 2κ θ where I q (.) is he modified Bessel funcion of he firs kind and of order q. The disribuion funcion is he non-cenral chi-square, χ 2 (2v; 2q + 2, 2u) wih 2q + 2 degrees of freedom and parameer of non-cenraliy 2u proporional o V. Noe ha (T ) is he ime o mauriy of he VIX fuures conrac. (Zhang and Zhu, 2006) Equaion (11) is he accurae formula for he VIX fuures price from Zhang and Zhu (2006) using our own noaion. Zhang, Shu, and Brenner (2010) provide us wih a very good closed form approximaion of equaion (11) given by 3 σ 2 V 1, where F T 100 = F 0 + F 1 + F 2, (13) F 0 = [θ (1 Be κ (T ) ) + V Be κ (T ) ] 1 2, F 1 = σ2 v 8 [θ (1 Be κ (T ) ) + V Be κ (T ) ] 3 2 [ B 2 V e κ (T ) 1 (T ) e κ + θ (1 e κ (T ) ) 2 κ 2κ 3 In Zhang, Shu, and Brenner (2010) θ is assumed o be ime dependan, θ, bu we sick wih he simpler version of he model from Zhang and Zhu (2006) and assume ha θ is consan. ],

12 Modeling VXX 10 F 2 = σ4 v 16 [θ (1 Be κ (T ) ) + V Be κ (T ) ] 5 2 [ 3 B 3 2 V e κ (T ) (1 (T ) e κ ) (1 e κ (T ) ) 3 κ 2 2 θ κ 2 where F 1 +F 2 can be hough of as a convexiy adjusmen from he Taylor series expansion of equaion (11). 3.2 Nearly 30-day VIX fuures Table 2 presens he values of esimaed VIX fuures prices using he full formula from Zhang and Zhu (2006), he closed form approximaion of he full formula from Zhang, Shu, and Brenner (2010), equaion (13), and wo simplificaions of he closed form approximaion, F 0 + F 1 and jus F 0. From able 2 we can see ha for 30-day VIX fuures prices using ], jus F 0 creaes a very small error from he accurae formula, equaion (11). The able shows ha he error from using jus F 0 insead of he accurae formula,, equaion (11), is always wihin 3% when θ = 0.1 and V ranges from 0.04 o 0.2, κ ranges from 4 o 7 and σ v ranges from 0.1 o 0.7. There is one oulier when V = 0.04, κ = 4 and σ V = 0.7, bu he error is only jus ouside 3% a 3.20%. The Roo Mean Squared Error (RMSE) is 1.29% which is very accepable. The resuls of he numerical exercise presened in able 2 lead us o proposiion 1 below. Proposiion 1 The price of nearly 30-day-o-mauriy VIX fuures can be given by F T 100 = [θ (1 Be κ (T ) ) + V Be κ (T ) ] 1 2, (14) wih some small error when compared o he accurae VIX fuures price formula from Zhang and Zhu (2006), as demonsraed in Table 2. For example for he range of parameers σ V = 0.1 o 0.7, V = 0.04 o 0.20, κ = 4 o 7, consan θ = 0.1 and mauriy of 30 days, he RMSE is only 1.29%.

13 Modeling VXX 11 We ake he naural log of equaion (14) o ge an expression for he naural log price of VIX fuures given by ( ) F T ln = ln[θ (1 Be κ (T ) ) + V Be κ (T ) ], (15) where ln( F T ) is he naural log he price of nearly 30-day o mauriy VIX fuures conrac. 100 In Figure 4 we can see he heoreical erm srucure of VIX fuures using equaion (14), he full approximaion of VIX fuures prices, equaion (13) and only he F 0 + F 1 segmen of he full approximaion. We use parameer esimaes of θ = 0.1, κ = 5, σ v = and V = 0.06 o creae an upward sloping VIX fuures erm srucure, as is normal for he VIX fuures marke. The difference beween he poins a + 30 and + 29 is equal o he average one-day roll yield of a 30-day o mauriy VIX fuures conrac. The spo reurn is zero when he underlying insananeous variance is consan which means ha any reurn ha can be seen is due o he roll yield of VIX fuures. I can be seen in he diagram ha as you sep hrough ime from +30 o +29 he reurn will be negaive herefore he one-day roll yield will be negaive when he erm srucure is upward sloping. 3.3 Model of Conrac Daily Reurn We can model he change of nearly 30-day log VIX fuures price by aking he Taylor series expansion of our simple log VIX fuures price formula, equaion (15), his gives us d ln F T = ln F T dv + 1 V 2 2 ln F T V 2 nex we subsiue in he parial derivaives o ge 4 4 Seen he Appendix, secion A. (dv ) 2 + ln F T d, (16)

14 Modeling VXX 12 d ln F T = 1 [ θ 2 Be κ (T ) θ + V 1 [ θ 4 Be κ (T ) θ + V + 1 [ κ (V θ )Be κ (T ) 2 ] 1 dv θ + (V θ )Be κ (T ) ] 2 (dv ) 2 ] d. (17) Proposiion 2 The SPVXSTR index is rebalanced daily o mainain a VIX fuures posiion wih one monh mauriy, herefore we can model he conrac daily reurn (CDR ) of he SPVXSTR as he log reurn of a 30-day o mauriy VIX fuures posiion. From his and equaion (17) we ge where CDR = d ln F T T =+τ0 = d ln F +τ 0 + RY, (18) where τ 0 = 30/365, d ln F +τ 0 VIX fuures conrac and RY d ln F +τ 0 = 1 [ ] θ 1 2 Be κ τ 0 θ + V dv 1 [ ] θ 2 (19) 4 Be κ τ 0 θ + V (dv ) 2 RY = 1 2 [ κ (V θ )Be κ τ 0 θ + (V θ )Be κ τ 0 ] d, (20) is he change in he log price of a consan 30-day o mauriy is he roll yield of he SPVXSTR. The roll yield of he SPVXSTR is he reurn of he underlying VIX fuures posiion due o he mauriy of he posiion changing from 30 days o 29 days, from one rebalancing of he posiion o jus before he nex rebalancing.

15 Modeling VXX VXX model Proposiion 3 We know ha he change in he SPVXSTR index, and herefore he VXX, is composed of he reurn of he fuures posiion, he CDR, and a risk-free reurn on he noional of he fuures posiion, T BR. Therefore we can model he VXX using CDR combined wih a risk free reurn r given by where RY d ln V XX = CDR + rd = d ln F T = d ln F +τ 0 + RY + rd + rd T =+τ0 (21) is he one day roll yield of he VXX going from 30-day mauriy o 29-day mauriy and r is he risk-free reurn on he noional value of he fuures posiion. This model of he log VXX price is o our knowledge he firs aemp in he lieraure o model he VXX using he underlying dynamics of he SPX. The model can be used o derive he marke price of variance risk, λ, from VXX reurns, as is described in Secion 4. We could also use his model o price VXX opions, which are essenially Asian opions on he underlying insananeous variance, V. In he nex secion we use our VXX model o quanify he roll yield and show ha i drives he VXX s reurns. 3.5 VXX roll yield Whaley (2013), Deng, McCann, and Wang (2012) and Husson and McCann (2011) all sugges he roll yield as he reason he VXX s reurns are so negaive. Figure 2 shows us a comparison beween he performance of he VIX, he VXX and a consan 30-day o mauriy VIX fuures conrac. We can see an obvious difference beween he 30-day o mauriy VIX fuures conrac and VXX. From equaion (21) we know ha he difference beween he 30-day VIX fuures price and he VXX can be explained by he roll yield, RY. To examine wha drives he roll yield of he VXX we assume ha he insananeous variance, V, is consan a he physical measure long run mean level of insananeous

16 Modeling VXX 14 variance, θ, o produce he aggregae upward sloping erm srucure of he VXX. If V is consan hen dv = 0, and herefore equaion (18) simplifies o where RY CDR = RY = 1 [ κ (θ θ )Be κ τ 0 ] 2 θ + (θ θ )Be κ τ 0, (22) is he one day roll yield of he aggregae VXX. To examine wha drives he roll yield o be negaive, during normal imes, we can use he ransformaion from he riskneural measure o he physical measure long-run mean level of insananeous variance, equaion (6) and subsiuing his ino equaion (22) we ge RY λκ κ Be κ τ 0 = 1. (23) λ κ Be κ τ 0 As all parameers apar from λ are always posiive and κ = κ + λ > 0 (Zhang, Shu, and Brenner, 2010), from equaion (23) we can see ha λ, he marke price of variance risk, is he driver of sign of he one day roll yield of he VXX, on aggregae. We conclude ha he negaive roll yield of he VXX is driven by he usually negaive, as shown in able 3, marke price of variance risk. 4 The Marke Price of Variance Risk, λ When modelling insananeous volailiy which is no direcly radable i is imporan o incorporae he marke price of risk. Under he Cohen e al. (1972) model you can always perfecly hedge your posiion and herefore you do no need he marke price of variance for your model. When you are modeling somehing ha is no raded his siuaion changes as you will no be able o creae a perfec risk free porfolio and herefore an invesor will require a premium o compensae for he risk, his is he marke price of variance risk. When implemening a sochasic volailiy model, such as in he Heson (1993) framework, esimaing he marke price of variance risk, λ, is essenial. There is no clear consen-

17 Modeling VXX 15 sus on he esimaion of he marke price of variance risk,λ. Table 3 shows some differen auhor s recen esimaes for he marke price of variance risk λ, he risk neural measure of he mean revering speed of variance, κ, and he sample period used. We can see from able 3 ha he esimaion of λ can vasly vary depending on he esimaion mehodology used. Our model les us develop a simple mehod of esimaing λ. If we subsiue V = θ in equaion (21) we ge d ln V XX = 1 2 [ κ (θ θ )Be κ τ 0 θ + (θ θ )Be κ τ 0 where dv = 0, so we are isolaing he aggregae effec of he roll yield. ] d + rd, (24) We can now ake he inegral of equaion (24) and subsiue in he ransformaion from risk-neural o physical measure long run mean level of variance, from equaion (6) o ge ( ) V XXT R = ln = 1 λbe κ τ 0 V XX 0 2 (1 + λ T + rt, (25) Be κ κ τ 0) where R is he coninuously compounded reurn on he VXX over he sample. V XX T is he las VXX price and V XX 0 is he saring VXX price, in he sample period, rt is he risk free reurn and λ is given by λ = λκ κ = λκ κ λ. (26) Proposiion 4 We can use he VXX reurn and a esimae of κ o measure λ, he marke price of variance risk by solving equaion (26) for λ, which gives us and solving equaion (25) for λ we ge λ = λκ, (27) κ + λ

18 Modeling VXX 16 2R E λ =. (1 2R E )Be κ κ τ 0 where R E is he annualized excess reurn of he VXX over he sample period, given by R E = 1 T ln V XX T V XX 0 r (28) We use he parameer esimae of κ = from Luo and Zhang (2012),as heir esimae of κ is he mos recen available one in he lieraure and he closes o our sample period, o demonsrae our new mehodology of calculaing λ. We hen use he VXX prices from incepion V XX 0 = on 30 Jan 2009 and he VXX price a he end of our sample V XX T = 28.86, on 27 Jun T = in years and rt is he cumulaive reasury bill reurn over he same ime period, rt = T BR 0,T = 0.558% as defined in equaion (2) from secion 2 bu cumulaed over he enire sample. The cumulaed TBR is very small bu his is expeced as Treasury bill raes have been almos zero since he financial crisis. We inpu hese parameer esimaes ino equaion (27) and (4) from proposiion 4 o calculae ha λ = wih very lile need for compuing power. This esimae coincides wih oher auhors as i is negaive and of similar magniude, refer o able 3 for comparison. This mehod for esimaing he marke price of variance risk, λ, makes he calibraion of of any Heson (1993) model much simpler, as λ is now a funcion of VXX prices and κ. 4.1 The Marke Price of Variance Risk and he Variance Risk Premium Eraker and Wu (2013) use an economic equilibrium model o show ha he abysmal performance of VIX fuures and VIX fuures index ETPs can be explained by he negaive Variance Risk Premium (V RP ). The V RP and he marke price of variance risk, λ, are 5 VXX price daa from NASDAQ websie:

19 Modeling VXX 17 similar conceps as hey boh measure he amoun of compensaion ha risk adverse invesors require for aking on he variance risk. Variance is negaively relaed wih equiy reurns and herefore he Variance Risk Premium and marke price of variance are boh negaive. Invesors accep he negaive reurns during normal imes, when aking a long posiion in volailiy, in order o hedge agains imes of high volailiy where hey will receive a posiive reurn from his long posiion, such as he 2008 financial crisis. Zhang and Huang (2010) show ha he marke price of variance risk, λ, from he Heson (1993) framework is almos proporional o he Variance Risk Premium, V RP, as defined by Carr and Wu (2009) as long as λτ 0 is small. Their resul is shown by V RP = [( ) ( κτ 0 + O(κ 2 τ 0 ) θ ) ] 3 κτ 0 + O(κ 2 τ0 2 ) V λτ 0 + O(λ 2 τ0 2 ), (29) where O( ) is a funcion of order λ 2 τ 2 0 (Zhang and Huang, 2010). The firs par of he equaion is obviously proporional o λ as i is muliplied by λτ 0. The reason he relaionship beween V RP and λ is almos proporional is because of he O(λ 2 τ 2 0 ) par of equaion (29) which is no proporional o λ bu aslong as λτ 0 is small (relaive o 1) hen λ 2 τ 2 0 will be very small. Our findings are consisen wih hose from Eraker and Wu (2013) as we find a negaive marke price of variance risk drives he reurns of he VXX o be so negaive, hrough he negaive roll yield, and hey find a negaive Variance Risk Premium as he cause of he negaive reurns of VIX fuures posiions and VIX fuures ETNs. 5 Rebalancing Frequency of SPVXSTR In his secion we explore he effec of rebalancing frequency of he SPVXSTR. We sar by replicaing he SPVXSTR index using VIX fuures prices from he 20 h of December

20 Modeling VXX unil he 28 h of March , using he mehodology from S&P Dow Jones Indices (2012). This replicaed SPVXSTR ime series is displayed in figure 5 along wih he acual SPVXSTR ime series, he lines are almos exacly idenical showing ha our replicaion is accurae. Figure 6 shows four ime series of he replicaed SPVXSTR index wih differen rebalancing frequencies of daily, weekly, bi-weekly and monhly rebalancing. The figure shows ha as he rebalancing frequency is decreased from daily o weekly, biweekly and monhly, he SPVXSTR s value decreases. If his effec exiss going from daily o more frequen rebalancing, for example hourly, hen his would be a problem for our coninuous ime model. To examine he effec of he rebalancing frequency on he price of he VXX for smaller ime seps han daily we needed a VIX fuures price ime series ha was inraday, bu real daa for his is only available o us for he las 50 days, herefore we chose o simulae a five year long hourly VIX fuures price ime series. To simulae he hourly ime series of VIX fuures prices we firs need a ime series of insananeous volailiy, which we ge from he physical measure sochasic process of insananeous variance Heson (1993), given by dv = κ(θ V )d + σ v V db. (30) We hen use he simple VIX fuures price approximaion, F 0, from equaion (13) o find a ime series of neares and second neares mauring VIX fuures prices. We use κ = as his is he mos recen esimaion, we λ = as calculaed in secion 2. We propose θ = 0.1 and σ v = 0.4 as reasonable value. The resuls of his secion are no sensiive o wha parameers are used, aslong as hey are reasonable. For simpliciy we assume ha VIX fuures maure every 28 days, ha here are no non-rading days, rading hours are 24 hours of he day and ha he risk free rae is zero. 6 Available a hp://cfe.cboe.com/daa/hisoricaldaa.aspx#vx, accessed on he 20 h of April 2014.

21 Modeling VXX 19 We hen use he mehodology from secion 2 o calculae he SPVXSTR index for five years wih differen rebalancing frequencies and a saring value of one. Figure 7 shows he resuling SPVXSTR hourly ime series for differen rebalancing frequencies from hourly o monhly. We can see in figure 7 ha he simulaed SPVXSTR ime series for hourly and daily rebalancing are almos idenical. The rebalancing effec going from daily o hourly rebalancing is herefore very very small and no a problem for our model. There is however a rebalancing effec if he index is rebalanced less ofen han dail, his is consisen wih our findings using marke VIX fuures prices. To show ha our conclusion on he rebalancing frequency is robus o he erm srucure of VIX fuures we repeaed he above exercise bu holding V consan a differen levels. This allows us o creae a ime series of SPVXSTR wih a upward sloping (in conango) VIX fuures erm srucure, as shown in figure 8, and downward sloping (in backwardaion) VIX fuures erm srucure, as shown in figure 9. From figures 8 and 9 we can see ha he rebalancing frequency does no significanly impac he SPVXSTR for hourly rebalancing. However here is a significcan effec when going o less frequen rebalancing. These resuls are robus o he erm srucure shape of VIX fuures. Therefore we can conclude ha shifing from a daily rebalancing o more frequen rebalancing does no affec he reurns of he SPVXSTR significanly. Also in boh figures he VXX model ime series esimaed using our model is he coninuous limi of he rebalancing ime series. The VXX model line is almos idenical o he daily rebalancing ime series, showing ha our coninuous ime VXX model is adequae o model he discree ime VXX. Figures 8 and 9 also show he imporance of he roll yield as a driver of he SPVXSTR and subsequenly he VXX. The wo figures isolae he effec of he erm-srucure on he reurns of he VXX SPVXSTR, by holding V consan and we know ha he roll yield is a resul of he erm srucure of VIX fuures. When he erm srucure is upward sloping,

22 Modeling VXX 20 causing a negaive roll yield, he simulaed SPVXSTR will end o 0 as in figure 8, and when he erm srucure is downward sloping, causing posiive roll yield, he simulaed SPVXSTR is exponenially increasing as in figure 9. 6 Conclusions and Discussions We sudy he VXX ETP which has been raded very acively on he New York Sock Exchange in recen years. We use he VIX fuures price approximaion from Zhang, Shu, and Brenner (2010) and simplify i for he nearly 30-day VIX fuures conrac. From his simplified formula for VIX fuures prices we develop a model for he VXX. Our model is, o our knowledge, he firs ever model of he VXX which encompasses he dynamics of he SPX index and he VIX index. Our model is he simples way o model he VXX while capuring he relaionship beween he SPX, VIX and he VXX. Our model explains he large negaive reurns of he VXX very well and is in line wih he mehodology from S&P Dow Jones Indices (2012). Our VXX model allows us o show ha he difference in reurns of he consan 30-day mauriy VIX fuures conrac, as in Zhang, Shu, and Brenner (2010), and he VXX is due o he roll yield as suggesed in he lieraure. We hen examine he roll yield and show ha λ, he marke price of variance risk is he main driver of he roll yield. Therefore we have provided an explanaion of he VXX reurn puzzle as he consan 30-day mauriy VIX fuures conrac does no exhibi hese negaive price movemens and is closely relaed o he VIX, herefore he roll yield drives he negaive reurns of he VXX. We have also provided a simple and robus way of measuring he marke price of variance risk, λ using our model and VXX prices. To undersand he economic explanaion for his we sugges examining he economic model for VIX Exchange Traded Noes from Eraker and Wu (2013). Their model finds ha he negaive reurn premium (Variance Risk Premium),

23 Modeling VXX 21 which is almos proporional o λ (Zhang and Zhu, 2006), is an equilibrium oucome because long VIX fuures posiions allow invesors o hedge agains high volailiy and low reurn saes, such as exhibied in a financial crisis. Our coninuously rebalanced VXX model is adequae for modeling he daily rebalanced VXX as he effec of he rebalancing frequency is only significan a less frequen han daily rebalancing. Our model for he VXX is he firs of is kind, as i is he only one ha includes he relaionship beween he SPX, he VIX and he VXX which we believe is fundamenal in undersanding he VXX. Our model could also be used by praciioners o price opions wrien on he VXX, which can be regarded as Asian opions wrien on he underlying insananeous variance. Bao, Li, and Gong (2012) have creaed a model for pricing VXX opions bu hey do no accoun for he dynamics of he S&P 500 or he VIX, which is essenial in modeling he VXX. Our research shows ha he roll yield is he main cause for he negaive performance of he VXX, as suggesed in he lieraure. I would be ineresing o see wheher he roll yield also plays a large par in he reurns of oher VIX fuures ETPs, we expec ha i would. Our model could be expanded by using he full approximaion formula of VIX fuures from Zhang, Shu, and Brenner (2010) and leing θ be ime dependan. One could use a similar approach o ours o explore he effec of he roll yield on oher VIX fuures ETNs bu we advise cauion in using he simplified VIX fuures price formula F 0 as i will be prone o more error a longer mauriies. Furher research is also needed ino he calibraion echnique bes used for our model and is accuracy alhough i is heoreically sound. Exploring similar approaches o he one in his aricle o creae models of oher VIX fuures Exchange Traded Producs could help furher develop he lieraure around hese popular ye myserious invesmen producs.

24 Modeling VXX 22 Appendix A. Solving for CDR model From equaion (15) and Io s lemma we ge d ln F T = ln F T dv + 1 V 2 2 ln F T V 2 (dv ) 2 + ln F T herefore we need o find each of he parial derivaives ln F T V are given by, 2 ln F T V 2 d, (31) and ln F T. These ln F T = 1 V 2 = 1 2 κ(t ) Be [θ (1 Be κ(t ) ) + V Be κ(t ) ] [ ] θ 1, (32) Be κ (T ) θ + V and 2 ln F T V 2 = 1 [ ] θ 2 2 Be κ (T ) θ + V (33) ln F T = 1 2 [ κ (V θ )Be κ (T ) θ + (V θ )Be κ (T ) ]. (34) We hen subsiue all he parial derivaives ino equaion (31) giving us he full funcion of he log fuures reurn as in equaion (17) from secion 3.3.

25 Modeling VXX 23 References Bao, Qunfang, Shenghong Li, and Donggeng Gong, 2012, Pricing VXX opion wih defaul risk and posiive volailiy skew, European Journal of Operaional Research 223, Barclays, 2013, VXX and VXZ Prospecus, hp:// accessed on Carr, Peer, and Liuren Wu, 2009, Variance risk premiums, Review of Financial Sudies 22, Chen, Hsuan-Chi, San-Lin Chung, and Keng-Yu Ho, 2011, The diversificaion effecs of volailiy-relaed asses, Journal of Banking & Finance 35, Cohen, Jerome B, Fischer Black, and Myron Scholes, 1972, The valuaion of opion conracs and a es of marke efficiency, The Journal of Finance 27, Cox, John C, Jonahan E Ingersoll Jr, and Sephen A Ross, 1985, A heory of he erm srucure of ineres raes, Economerica: Journal of he Economeric Sociey 53, Deng, Geng, Craig McCann, and Olivia Wang, 2012, Are VIX fuures ETPs effecive hedges?, The Journal of Index Invesing 3, Duan, Jin-Chuan, and Chung-Ying Yeh, 2010, Jump and Volailiy Risk Premiums implied by VIX, Journal of Economic Dynamics and Conrol 34, Eraker, Bjørn, and Yue Wu, 2013, Explaining he Negaive Reurns o VIX Fuures and ETNs: An Equilibrium Approach, Available a SSRN French, Kenneh R, G William Schwer, and Rober F Sambaugh, 1987, Expeced sock reurns and volailiy, Journal of financial Economics 19, 3 29.

26 Modeling VXX 24 Hancock, GD, 2013, VIX Fuures ETNs: Three Dimensional Losers, Accouning and Finance Research 2, Heson, Seven L, 1993, A closed-form soluion for opions wih sochasic volailiy wih applicaions o bond and currency opions, Review of financial sudies 6, Husson, Tim, and Craig McCann, 2011, The VXX ETN and Volailiy Exposure, PIABA Bar Journal 18, Lin, Yueh-Neng, 2007, Pricing VIX fuures: Evidence from inegraed physical and riskneural probabiliy measures, Journal of Fuures Markes 27, Luo, Xingguo, and Jin E Zhang, 2012, The erm srucure of VIX, Journal of Fuures Markes 32, S&P Dow Jones Indices, 2012, S&P 500 VIX fuures indices Mehodology, hp://us.spindices.com/indices/sraegy/sp-500-vix-shor-erm-index-mcap, accessed on Whaley, Rober E, 2013, Trading Volailiy: A Wha Cos, The Journal of Porfolio Managemen 40, Zhang, Jin E, and Yuqin Huang, 2010, The CBOE S&P 500 hree-monh variance fuures, Journal of Fuures Markes 30, Zhang, Jin E, Jinghong Shu, and Menachem Brenner, 2010, The new marke for volailiy rading, Journal of Fuures Markes 30, Zhang, Jin E, and Yingzi Zhu, 2006, VIX fuures, Journal of Fuures Markes 26, Zhu, Yingzi, and Jin E Zhang, 2007, Variance erm srucure and VIX fuures pricing, Inernaional Journal of Theoreical and Applied Finance 10,

27 Modeling VXX 25 Table 1: Summary saisics of he daily reurns for he SPY, VIX and VXX. This able shows he summary saisics and correlaions of he VXX, SPX (S&P 500 index ETP) and he VIX index reurns from he 2 nd February 2009 o he 13 h Augus R D represens esimaes using discree daily reurns and R C represens esimaes using coninuously compounded daily reurns. The annualised sandard deviaion is calculaed by muliplying he sandard deviaion by 252. The Holding Period Reurn (HPR) is he discree reurn from he firs price o he las price of he sample. The Compound Annual Growh Rae (CAGR) is he consan yearly growh rae ha would lead o he change from he firs price o he las price in he sample, i is calculaed by CAGR = (HP R + 1) 1 T 1, where T is he lengh of he sample in years. SPX VIX VXX R D R C R D R C R D R C Mean 0.08% 0.07% 0.15% 0.09% 0.32% 0.39% significance p-value (0.0121) (0.0217) (0.4323) (0.6226) (0.0020) (0.0001) Sandard Deviaion (σ) 1.13% 1.13% 7.10% 6.86% 3.81% 3.78% Annualised σ 18.00% 18.00% % % 60.55% 60.06% Skew significance p-value (0.0596) (0.0006) (0.0000) (0.0000) (0.0000) (0.0000) Excess Kurosis significance p-value (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) Holding Period Reurn % 97.26% 71.66% % 99.55% % CAGR 19.25% 20.41% 62.43% Correlaions R D R C SPY VIX VXX SPY VIX VXX SPY significance p-value (0.0000) (0.0000) (0.0000) (0.0000) VIX significance p-value (0.0000) (0.0000) VXX 1 1

28 Modeling VXX 26 Table 2: 30-day VIX fuures price esimaion. This able shows he differen VIX fuures price esimaes using four differen formulae and range of parameer esimaes for V, σ V and κ. For his exercise we keep he ime o mauriy consan a 30 days, τ = τ 0 = and θ consan a θ = The firs four columns show he hypoheical θ, V, σ V and κ parameers used in he fuures price esimaes. The firs column of esimaed fuures prices, labelled by F 0, uses he simple approximaion for VIX fuures prices, he F 0 par of equaion (13). The nex column of VIX fuures prices, labelled by F 0 + F 1, uses he simple formula of VIX fuures prices and he firs half of he convexiy adjusmen, F 0 + F 1 from equaion (13). The F 0 + F 1 + F 2 column of VIX fuures prices uses he full approximaion formula, equaion (13), from Zhang, Shu, and Brenner (2010). The final column of VIX fuures prices uses he accurae formula, equaion (11), from Zhang and Zhu (2006). The columns labelled % error, are he percenage difference of he preceding column of prices from he prices esimaed by he accurae formula. Parameers VIX Fuures Price esimaes θ V κ σ V F 0 % error F 0 + F 1 % error F 0 + F 1 + F 2 % error Accurae F % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 37.24

29 Modeling VXX 27 Table 3: λ and κ esimaes by various auhors. This able shows he esimaed value of λ, he marke price of variance risk, and κ, he risk neural speed of mean reversion of variance, from differen auhors using differen sample periods and esimaion mehods. Auhor Daa period κ λ Lin (2007) 21 Apr Apr Duan and Yeh (2010) 2 Jan Dec Zhang and Huang (2010) 18 May Aug Luo and Zhang (2012) 2 Jan Aug Our esimaion 30 Jan Jun Luo and Zhang (2012) do no give he esimae for lambda, bu heir aricle is imporan here as we use heir κ esimae. We use he κ = esimae from Luo and Zhang (2012) and assume ha i is accurae for our sample period.

30 Modeling VXX 28 Figure 1: Undersanding he SPVXSTR Roll Period. This diagram shows how dr and d are deermined for he calculaion of he weighs in each VIX fuures conrac of he SPVXSTR. T i is he selemen dae of he i h neares mauring VIX fuures, which is 30 days before S&P 500 opions mauriy dae (3rd Friday of every monh) and is usually on a Wednesday. T i 1 is he day before i h neares mauring VIX fuures selemen and he las day of he roll period. On he las day of he roll period he neares seling VIX fuures is eliminaed and he second neares seling VIX fuures becomes he neares. The dr and d are he facors used in he calculaion of he weighs of each of he VIX fuures conracs in he SPVXSTR, as shown in secion 2. The roll period represens he ime during which he weigh in he neares seling VIX fuures conrac is gradually replaced by a posiion in he second neares VIX fuures conrac. A he end of he roll period all he weigh will be in he second neares VIX fuures conrac which hen becomes he neares as he old neares maures, hen he nex roll period sars, and he process is repeaed.

31 Modeling VXX 29 Figure 2: Hisorical VIX, 30-day VIX fuures price and VXX price. This figure shows he level of he VIX and he price of 30-day VIX fuures on he primary verical axis and he VXX price on he secondary verical axis. The 30-day VIX fuures conrac is he linearly inerpolaed price of a consan 30-day mauriy VIX fuures conrac, as in Zhang, Shu, and Brenner (2010).

32 Modeling VXX 30 Figure 3: Marke Capializaion and Trading Value of VXX. This figure shows he daily dollar rading volume and marke capializaion of he VXX from he 30 h January 2009 o he 27 h June 2014 in billion US dollars.

33 Modeling VXX 31 Figure 4: VIX Term Srucure. This figure shows he erm srucure of VIX fuures prices from 1 day o 50 day mauriy calculaed using our simple approximaion,f 0, he approximaion wih he firs par of he convexiy adjusmen, F 0 + F 1 and he full approximaion from Zhang, Shu, and Brenner (2010), F 0 + F 1 + F 2. These esimaed VIX fuures prices are calculaed using consan parameer esimaes of θ = 0.1, κ = 5, σ V = and V = 0.06 bu he ime o mauriy varies from 1 day o 50 days.

34 Modeling VXX 32 Figure 5: Replicaed vs. Acual SPVXSTR. This figure shows he acual SPVXSTR ime series and our replicaed SPVXSTR ime series using he mehodology from S&P Dow Jones Indices (2012) from he 20 h December 2005 unil he 28 h March 2014.

35 Modeling VXX 33 Figure 6: Replicaed SPVXSTR, differen Rebalancing frequencies. This figure shows four differen ime series of our replicaion of SPVXSTR. SPVXSTR daily corresponds o daily, SPVXSTR weekly o weekly, SPVXSTR bi-weekly o wo weekly and SPVXSTR monhly o monhly rebalancing. The final values of he indices are for daily, for weekly, for bi-weekly and for monhly rebalancing.

36 Modeling VXX 34 Figure 7: Simulaed index using physical process for V. This figure shows he simulaed SPVXSTR index over our 4 year simulaion period using V 0 = 0.02, σ V = 0.4, he risk-neural parameer esimaes κ = and θ = 0.1, he physical process of dv as described in equaion (30) and he simple VIX fuures price formula, F 0, from equaion eqrefapproxvixfuure from Zhang, Shu, and Brenner (2010). The label of each ime series corresponds o he rebalancing frequency used.

37 Modeling VXX 35 Figure 8: Simulaed index using V = θ < θ. This figure shows he ime series of he simulaed SPVXSTR, when he insananeous variance is se consan a V = θ = < θ = 0.1 forcing a upward sloping VIX fuures erm srucure. To calculae he fuures prices we use he volailiy of volailiy σ V = 0.4 and he risk-neural parameer esimaes κ = and θ = 0.1 are used. The label of each ime series corresponds o he rebalancing frequency used.

38 Modeling VXX 36 Figure 9: Simulaed index using V = 0.14 > θ. This Figure shows he ime series of he simulaed SPVXSTR, when V is se consan a 0.14 which is higher han θ = 0.1 forcing a downward sloping VIX fuures erm srucure. To calculae he fuures prices we use he volailiy of volailiy σ V = 0.4 and he risk-neural parameer esimaes κ = and θ = 0.1 are used. The label of each ime series corresponds o he rebalancing frequency used.