Variance Dynamics in Term Structure Models Job Market Paper

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1 Variance Dynamics in Term Srucure Models Job Marke Paper Cisil Sarisoy a a Deparmen of Finance, Kellogg School of Managemen, Norhwesern Universiy Absrac I design a novel specificaion es for diagnosing he adequacy of affine erm srucure models o describe he observed yield variance dynamics, and derive he associaed limi heory necessary for carrying ou he es. The es saisic uilizes model free esimaors of insananeous variances based on inraday daa as well as model-free prices of variance swaps. Hence, i enables a direc esing of variance dynamics, independen of any specific modeling assumpions. I implemen he es saisic in Eurodollar fuures and opions markes and find ha affine erm srucure models canno accommodae he yield variance dynamics observed in he daa, especially during he crisis period of However, a logarihmic affine specificaion of variances provides a remarkably improved fi. Keywords: Affine Term Srucure Models, Ineres raes, Sochasic volailiy JEL: C52, C58, E43, G12, G13 I would like o hank Torben G. Andersen, and Vikor Todorov for heir invaluable commens and many discussions. I would also like o hank Dobrislav Dobrev, Liudas Giraiis, Ferenc Horvah, Bryan Kelly, Anh Le, Yingying Li, Elena Pikulina, Eric Renaul, Bernd Schwaab, Neil Shephard, Emil N. Siriwardane, Karin Sürmer, Ruey Tsay and seminar and conference paricipans a he Kellogg School of Managemen, he 2017 Sociey of Financial Economerics Fixed Income Workshop (Brussels), he 2017 NBER NSF Time Series conference for helpful commens. E mail address: cisil.sarisoy@kellogg.norhwesern.edu November 28, 2017

2 1. Inroducion Low dimensional affine erm srucure models sand as a cornersone in he erm srucure lieraure. These models provide a very successful fi o he cross secional properies of bond yields, see, e.g., Dai and Singleon (2002). However, heir implicaions for yield variances are sill no clear. Alhough hese models can generae he paerns of uncondiional yield volailiy, see, e.g., Dai and Singleon (2003), he condiional volailies implied by sandard affine mulifacor models can be uncorrelaed or even negaively correlaed wih he ime series of condiional volailiies esimaed via a GARCH approach, see, e.g., Collin Dufresne, Goldsein, and Jones (2009), and Jacobs and Karoui (2009). These findings may raise concerns abou he presence of some sor of misspecificaion in affine erm srucure models. This is an imporan problem because sysemaic biases caused by incorrec specificaion of underlying models may mislead inference abou yield volailiy dynamics, yield risks, and heir respecive pricing implied by he models. Therefore, guidance for diagnosing he presence of such misspecificaion may be useful paricularly for applicaions ha demand correc model implicaions for yield volailiy dynamics. In his paper, I design a novel specificaion es for diagnosing he adequacy of affine erm srucure models o describe he observed yield variance dynamics, and derive he associaed limi heory necessary for carrying ou he ess. I implemen he es saisic in Eurodollar fuures and opions markes and explore wheher he affine erm srucure model class provides a saisfacory characerizaion of he observed yield variance dynamics. The proposed es saisic relies on wo measures ha are highly sensiive o variance saes: model free prices of variance swaps and model free esimaors of insananeous volailiy based on inraday daa. Hence, i enables a direc esing for variance dynamics, being independen from any specific modelling assumpions. This novel es saisic is useful in several dimensions. Firs, i enables a direc esing of variance implicaions of he affine model class wihou making any reference o 2

3 unspanned sochasic volailiy, i.e. a form of bond marke incompleeness ha variance risk can no be hedged wih bonds, see, e.g., Collin Dufresne and Goldsein (2002) and Joslin (2017). Second, one may es he variance specificaion a any given poin in ime, which can be informaive abou he poenial sources of model failures. Third, he developed es saisic serves as a diagnosic ool for he affine specificaion under he pricing measure and does no hinge upon any parameric assumpions regarding he dynamics under he physical measure beyond wha is implied by he no arbirage condiion. In ha aspec, i serves as a valid diagnosic ool for a large se of models including he so called essenially affine class of Duffee (2002), exended affine classificaion of Cheridio, Filipović, and Kimmel (2007) and he models wih a non affine drif under he physical measure, see, e.g., Duare (2003) 1. Fourh, no specific ime series model for he condiional variances is required under his approach. GARCH ype models are ofen employed in he lieraure o judge he feasibiliy of he model implied variances, see, e.g., Collin Dufresne, Goldsein, and Jones (2009), Jacobs and Karoui (2009). Insead, he proposed es saisic relies on non parameric measures of spo volailiy. One approach o miigae he harmful effecs of misspecificaion of variances is o incorporae daases ha provide valuable informaion abou variances in he esimaion. Along his line, several recen sudies explore poenial benefis of incorporaing he ineres rae derivaives daa in model esimaions. For insance, Almeida, Graveline, and Joslin (2011) documen ha relying on derivaives daa ogeher wih underlying ineres raes in esimaion improves he precision of he risk neural parameer esimaes of he underlying affine models. However, hey find ha even when esimaed wih derivaives daa, sandard hree facor affine models fail o mach he cross secion of condiional yield volailiies. Cieslak and Povala (2016) employ realized covariances of zero coupon raes as well as he opions daa in heir proposed affine model s esimaion. They documen an improved fi of he affine model o he condiional yield variances and he cross secion of yields wih mauriies above wo years bu wih he cos of exra 1 See Piazzesi (2010) for a review on nonlineariies in erm srucure lieraure. 3

4 sae variables. Anoher srand of he lieraure along he lines of sochasic volailiy in ineres rae markes invesigaes wheher he erm srucure of yields is relaed o he yield variances. An implicaion of affine erm srucure models is ha he saes driving he yield variances are a linear combinaion of yields. Collin Dufresne and Goldsein (2002) documen low explanaory power of swap raes in explaining reurns on a he money sraddles, which are paricularly sensiive o he volailiy risk. Accordingly, hey propose a sub family of he affine erm srucure models and erm i as unspanned sochasic volailiy models. However, here is mixed evidence on he empirical performance of hese models for capuring he cross secional and ime series properies of yields, see, e.g., Bikbov and Chernov (2009), Collin Dufresne, Goldsein, and Jones (2009) and Joslin (2017). I highligh one fundamenal variance implicaion of he arbirage free affine erm srucure models ha earlier lieraure has no focused on: he insanenous variances of he discoun raes are linearly spanned by he conamporaneous erm srucure of variance swap raes. This relaion is he focus of he esing mechanisms devised in his paper. Inuiively, any naural esing mechanism of his propery requires relaing he measures of insanenous variances o he cross secion of variance swap raes. However, boh he insanenous variances and he variance swap raes are unobserved for discoun raes. Tesing mechanism designed in his paper requires wo crucial seps. The firs one is he model free recovery of he volailiy realizaions. This paper relies on he localized versions of he realized variance esimaors based on inraday daa for inferring insananeous variances. In paricular, for each day in he sample, I form model free esimaors of he spo volailiy by using 10 minue observaions on implied discoun raes. The second sep is he consrucion of he model free measure of variance swap raes on implied discoun raes. Building on previous work of Neuberger (1994), Demeerfi, Derman, Kamal, and Zou (1999), Carr and Wu (2009), and Mele and Obayashi (2013), I show 4

5 ha variance swap raes on implied discoun raes can be synhesized using a porfolio of pu and call opions in a model free way. Specifically, I provide he conrac designs of such variance swaps and he derivaion of model free prices of hese variance swaps wih a porfolio of pu and call opions. In ha conex, model free refers o he circumsance ha he heoreical value of he variance swap raes can be replicaed by a porfolio of European syle opions, wihou hinging upon any parameric assumpions. Based on hese resuls, I approximae he variance swap raes on implied discoun raes each day by using a panel of daily opion prices. I perform diagnosic ess for he popular hree facor affine erm srucure models for Eurodollar fuures and opions daa. More specifically, I examine he variance implicaions of hree facor models wih one and wo sochasic volailiy facors, denoed by A 1 (3) and A 2 (3) respecively. 2 The resuls from esing he variance dynamics of he model wih one sochasic volailiy facor (A 1 (3)) are sriking. For he 6 monh mauriy variance, he es saisic implies large violaions of he affine specificaion, especially during he crisis period of 2008 o Moreover, he affine model sruggles occasionally wih capuring he variance dynamics before Alhough he afffine model wih wo sochasic volailiy facors (A 2 (3)) performs beer especially before 2007, i is srongly rejeced during he crisis period. Given he failure of he affine specificaion wih one and wo volailiy facors, I explore wheher an affine specificaion of logarihmic variances is suppored by he daa. There is subsanial empirical evidence in he equiy and foreign exchange lieraure ha he disribuions of he logarihms of daily realized variances are approximaely Gaussian, see for example Andersen, Bollerslev, Diebold, and Labys (2000, 2001, 2003). Accordingly, Andersen, Bollerslev, and Diebold (2007) sudy volailiy forecasing via modelling he logarihms of realized variances and find subsanial improvemens in he forecasing perfomance (see, also, Andersen, Bollerslev, and Meddahi (2005) for volailiy forecas evaluaions). These resuls regarding equiy and foreign exchange markes pro- 2 The A 1 (3) and A 2 (3) models refer o he affine model classificaions where he covariance of he enire sae space sysem are driven solely by one sae and wo saes respecively. 5

6 vide a foundaion for exploring he affine specificaion on logarihmic variances. I design a es saisic o judge he logarimic affine specificaion of variances and provide he associaed limi heory. I find ha he logarihmic affine specificaion wih wo facors provides a remarkably improved fi for he variance dynamics of implied discoun raes a several mauriies including he shor end of he yield curve. In sark conras o he case of affine variance specificaion, he es saisic exhibis much more moderae values consisenly hroughou he sample, including he crisis period. The findings of his paper indicae he necessiy of furher exensions o he volailiy modelling in affine erm srucure models. Such exensions may be especially imporan for applicaions ha demand correc model implicaions for yield volailiy dynamics, such as ineres rae volailiy hedging, and yield volailiy forecasing. Broadly, presence of misspecificaion in he underlying models may lead o biases in esimaion and may disor he model-implied inference abou volailiy risks, yield risks, and heir respecive pricing. Therefore, prevening poenial biases caused by misspecificaion may offer a fresh perspecive on bond risk premia as well as variance risk premia dynamics. The remainder of his paper is organized as follows. Secion 2 inroduces he class of affine erm srucure models explored in his paper. Moreover, his secion clarifies he links beween Eurodollar fuures variances and variance swap raes in he conex of affine erm srucure models. Secion 3 provides he mehodologies underlying he model free consrucion of spo variances and he variance swap raes ogeher wih he associaed heory. In Secion 4, I develop he diagnosic ess for boh he affine and logarihmic affine variance specificaions and derive he limiing heory. Secion 5 describes he Eurodollar fuures and opions daa se, and oulines he calculaions of he insananous variances and he variance swap raes based on he inraday and he opions daa. Secion 6 presens he main empirical resuls. Secion 7 concludes. 6

7 2. The Model 2.1. The Affine Model Specificaion I consider he class of erm srucure models where he spo ineres rae, r, of a hypoheical AA qualiy bond is an affine funcion of he N dimensional Markov sae variable, X, ha follows an affine diffusion process under he risk neural measure. Following Duffie and Kan (1996) and Dai and Singleon (2000), he sae variable, X, solves he sochasic differenial equaion (SDE), dx = K (Θ X ) d + Σ S dw Q, (2.1) where W Q is an N dimensional vecor of independen Brownian Moions under he risk neural measure Q, K and Σ are N N marices, Θ is an N 1 vecor and S is a diagonal N N marix wih diagonal componens given by [S ii = α i + βi X, wih α i being a scalar and β i an N 1 vecor. The shor rae is an affine funcion of X such ha N r = δ 0 + δ i X i = δ 0 + δx X, (2.2) i=1 where δ x is an N dimensional vecor. Given he risk neural dynamics of X and he specificaion of he shor rae, he ime price of a zero coupon bond wih mauriy a + τ, wih τ being measured in years, is given by P (τ) = E Q [e +τ r sds. (2.3) Following Dai and Singleon (2000), I impose he following parameer consrains o guaranee admissibiliy for he model specificaion in Equaion (2.1) under Q: 1. M j=1 K i,jθ j > 0, 1 i M, 2. K i,j = 0, 1 i M, M + 1 j N, 3. K i,j 0, 1 i j M, 7

8 4. S ii = X i, 1 i M, 5. S jj = α j + Σ M k=1 [β j k X k, M + 1 j N, 6. Σ i,j = 0, 1 i M, M + 1 j N, where α j 0, [β j k 0. The firs hree consrains guaranee ha he drif of he facors driving he volailiy is posiive and ha he non volailiy facors do no feed ino he drif of he volailiy facors. Moreover, hey guaranee posiive feedbacks among volailiy facors. The las hree condiions ensure he posiive semi definieness of he covariance marix of he process. In paricular, in his specificaion, he covariance marix of N facors would be deermined solely by he M volailiy facors, which are auonomous of he N M condiionally Gaussian saes. Hence, under he admissibiliy condiions (1 o 6), he covariance marix of he N facors is affine in he M variance saes and is given by: ΣS Σ = ΣDiag(α)Σ + G 0 + M ΣDiag(Γ row,i )Σ X i i=1 M G i 1X. i (2.4) i=1 In he res of he paper, even hough I provide he heoreical resuls for a general A M (N) model where M saes drive he volailiy, I will use A 1 (3) as an illusraive model where M = 1 facor drives he volailiy. Wihin his seing, one can obain he zero coupon bond prices in closed form. Specifically, Duffie and Kan (1996) show ha he ime price of a zero coupon bond wih mauriy τ is given by P (τ) = e A(τ)+B(τ) X, (2.5) where he loadings A(τ) R and B(τ) R N saisfy he Riccai ODEs, A (τ) = δ 0 + Θ K B(τ) B (τ) = δ x K β(τ) N i=1 N i=1 [ B(τ) Σ 2 i α i, (2.6) [ B(τ) Σ 2 i β i,

9 wih iniial condiions A(0) = 0 and B(0) = 0. Nex, I will provide he pricing of he Eurodollar fuures conracs Eurodollar fuures conracs Eurodollar fuures conracs a ime wih mauriy τ f are quoed as F ED (τ f ) = 100 ( 1 f ED (τ f ) ), (2.7) where f ED (τ f ) is he fuures rae for hree monh LIBOR rae. Eurodollar fuures are cash seled conracs wih final selemen prices being ied o he hree monh London Inerbank Offer Rae (LIBOR). In paricular, he erminal price, i.e., he price a mauriy, of a hree monh Eurodollar fuures conrac wih ime o mauriy τ f is given by F ED +τ f (0) = 100 ( 1 L +τf (τ L ) ), (2.8) where τ L = 90/360 and L +τf (τ L ) is he hree monh LIBOR rae a ime + τ f. 3 The LIBOR is a simple rae and is linked o he zero coupon bond price, 1 + τ L L (τ L ) = 1 P (τ L ). (2.9) Consequenly, he hree monh LIBOR rae a ime + τ f can be wrien as L +τf (τ L ) = 1 ( ) 1 τ L P +τf (τ L ) 1. (2.10) Under he risk neural measure, Q, he ime price of a Eurodollar fuures conrac wih ime o mauriy τ f follows a maringale process and can be obained by aking he condiional expecaion of is delivery prices under Q, F ED (τ f ) = E Q [ 100 ( 1 L+τf (τ L ) ). (2.11) 3 The LIBOR is quoed in annualized erms and he day coun convenion is 360 days for he USD. 9

10 By using he link beween he LIBOR and zero coupon bond prices in Equaion (2.10) and he exponenial affine form of zero coupon bond prices in Equaion (2.5), f ED (τ f ) is given by f ED (τ f ) = 1 τ L e A(τ L) E Q [e B(τ L) X +τf 1 τ L (2.12) The risk neural expecaion in Eq. (2.12) can be obained by using he condiional momen generaing funcion. Duffie, Pan, and Singleon (2000), henceforh DPS, derive closed form expressions for a class of ransforms of affine jump diffusion processes. One implicaion of heir analysis is ha he condiional momen generaing funcion of X +τ f given X can be obained in closed form wihin he affine seing of he shor rae, Equaion (2.2), and he affine diffusion dynamics of X under Q, Equaion (2.1). In paricular, by seing δ 0 = 0, δ x = 0 N 1, a closed form expression for he condiional momen generaing funcion of X + τ f given X can be obained by Equaions ( ) in DPS. Accordingly, he fuure LIBOR rae akes he form f ED (τ f ) = 1 e A(τ L)+A f (τ f )+B f (τ f ) X 1, (2.13) τ L τ L where A f (τ f ) and B f (τ f ) solve he same Riccai ODEs as in Equaion (2.6) wih δ 0 = 0, δ x = 0 N 1, and iniial condiions A f (0) = 0 and B f (0) = B(τ L ). Tranforming he fuures rae, f ED (τ f ) in Equaion (2.13), ino is equivalen hree monh discoun rae leads o an exponenial linear form in sae variables. Specifically, he ime implied hree monh gross rae is given by Ψ τ f = e A(τ L)+A f (τ f )+B f (τ f ) X (2.14) = 1 + τ L f ED (τ f ). An applicaion of Iô s lemma reveals he dynamics of Ψ τ f,τ L, dψ τ f Ψ τ f = µ Ψ (X, τ f ) d + σ Ψ (X, τ f ) dw Q, (2.15) 10

11 wih he insananeous variance := σ Ψ (X, τ f ) σ Ψ (X, τ f ) ) M = B f (τ f ) (G 0 + G i 1X i B f (τ f ) i=1 = B f (τ f ) G 0 B f (τ f ) + M B f (τ f ) G i 1B f (τ f )X i i=1 Φ τ f 0 + Φτ f X G, (2.16) where Φ τ f 0 Bf (τ f ) G 0 B f (τ f ) and Φ τ f = ( Φ τ f 1,..., Φτ f M ) wih Φi B f (τ f ) G i 1B f (τ f ). Since S is affine in he variance sae vecor, X G (X 1, X 2,..., X M ), he insananenous variance,, is also affine in X G. This resul esablishes he fundamenal link beween he M variance saes X G and he cross secion of insanenous variances of implied discoun raes. Virually in all low dimensional affine erm srucure models of he form A M (N) (see he canonical forms in Collin Dufresne, Goldsein, and Jones (2009) and Joslin (2017)), he ime variaion in he insanenous volailiy of he implied discoun raes is driven solely by he M variance saes X G. In he res of he paper, implied hree monh gross rae, Ψ τ f, will be he main quaniy of ineres, raher han he Eurodollar fuures prices, F ED (τ f ). The reason for his is ha given he affine model specificaion under he risk neural measure in Equaions ( ), Ψ τ f Equaion (2.14). Hence, is an exponenial affine funcion of he underlying saes, see dψ τ f Ψ τ f volailiy saes X G = (X 1, X 2,..., X M ) The Condiional Mean of +τ v has spo variance dynamics ha are affine only in he u du I now analyze he implicaions of he underlying A M (N) model for he firs momen of inegraed variance o enable esing based only on volailiy sensiive quaniies. In order o derive he condiional expecaion of inegraed variance, i is useful o firs sar wih he condiional expecaion of X G u. The condiional expecaion of X G u is given by E Q [ X G u F = Θ G ( I e KG (u ) ) + e KG (u ) X G. (2.17) 11

12 Now, using he resul above, he risk neural condiional mean of inegraed variance over [, + τ v for a general A M (N) model is given by E Q [ +τv u du F = Λ τ f,τ v 0 + Λ τ f,τ v 1 X G, (2.18) wih ( ) ( ) Λ τ f,τ v 0 = Φ τ f 0 + Φτ f Θ G τ v Φ τ f K G 1 K G 1 e KG τ v Θ G, Λ τ f,τ v 1 = Φ τ f ( K G 1 K G 1 e KG τ v ). A deailed proof is provided in Appendix A. Hence, he condiional expecaion of he inegraed variance is ied o he conemporaneous variance saes via an affine mapping. In all A M (N) syle models, he ime variaion in he risk neural expecaion of he inegraed variance is solely deermined by he ime variaion of he M variance saes. This feaure is in common wih he spo variance, see Equaion (2.16). Noe ha he condiional expecaion of he inegraed variance given in Equaion (2.18) is essenially equal o he variance swap rae on he implied discoun rae, Ψ τ f, denoed henceforh by SW τv. A variance swap is a forward conrac on he fuure inegraed variance, wih a payoff a expiraion given by he inegraed variance over he conrac horizon minus he variance swap rae. Variance swaps do no require a paymen o ener, consequenly hey represen he risk neural expeced value of fuure inegraed volailiy. In Secion 3.2, I show ha he variance swap rae on he implied discoun rae Ψ τ f can be obained in a model free fashion by using ou of he money pu and call opions on Ψ τ f. Several sudies in he erm srucure lieraure focus on low dimensional affine erm srucure models where a single facor drives he yield variaion, such as A 1 (3) and A 1 (4). I has been documened ha hese models, when esimaed wih bond yields as well as wih bond derivaies, capure he ime variaion in yield volailiy reasonably well and he price dynamics in boh fixed income and fixed income derivaives, see, e.g., Almeida, 12

13 Graveline, and Joslin (2011). 4 Hence, I close his secion by illusraing he wo key quaniies for he A 1 (3) model, specificaion. and SW τv, ha enable he esing of variance A 1 (3) Model: In his canonical form, one of he sae variables, here X 1, deermines he spo volailiy of all hree sae variables. Under he admissibiliy condiions, he A 1 (3) specificaion leads o Equaion (2.16) being an affine funcion of X 1 only. In paricular, The insananeous variance, is given by = Φ τ f 0 + Φτ f X 1. The inegraed variance is given by SW τv = E Q [ +τv u du F = Λ τ f,τ v 0 + Λ τ f,τ v 1 X 1, (2.19) where and Λ τ f,τ v 0 = Φ τ f 0 τ v + Φ τ f 1 Λ τ f,τ v 1 = Φτ f 1 ( θ 1 τ v θ ) 1 ( 1 e κ 11τ v ), κ 11 κ 11 ( 1 e κ 11τ v ). Hence, under he A 1 (3) specificaion, boh he insanenous variance variance swap rae SW τv driving he covariance of all hree facors Affine Variance Spanning Condiion and he are deermined by he facor X 1, which is he only facor The analyical soluions for he variance swap rae in Equaion (2.18) and for he insanenous variance in Equaion (2.16) se he baseline for a variance spanning condiion implied by he A M (N) model. In his secion, I derive his variance spanning condiion. A variance swap a ime wih mauriy + τ v is a conrac wih payoff a expiraion given by he inegraed variance over he horizon of he conrac minus he variance swap 4 See also Collin Dufresne, Goldsein, and Jones (2009), Bikbov and Chernov (2009) among ohers. 13

14 rae. In principle, a variance swap conrac can be designed for any mauriy τ v, which yields a erm srucure of variance swap raes. The resul in Equaion (2.18) esablishes he fundamenal link beween he erm srucure of he variance swap raes and he variance saes X G. The M 1 vecor of variance swap raes is denoed by V S = (SW τ1, SW τ2,..., SW τ M ). Defining he M 1 vecor Ξ 0 = (Λ τ1 0, Λτ2 0,..., Λτ M 0 ) and he M M marix Ξ 1 = ( τ Λ f,τ 1 1, Λ τ f,τ 2 1,..., Λ τ f,τ M ), 1 Equaion (2.18) can be wrien as a sysem of equaions: V S = Ξ 0 + Ξ 1 X G. (2.20) We can inver his sysem in order o express he variance saes X G as an affine funcion of he swap raes V S. This yields, X G = Ξ 1 1 Ξ 0 + Ξ 1 1 V S = ζ + Ξ 1 1 V S. (2.21) Now, since he insananeous variaion of he implied discoun rae is an affine funcion of he variance saes X G (shown in Equaion (2.16)), for any τ f, one can find a se of consans α τ f j, j = 0,..., M such ha M = α τ f 0 + j=1 α τ f j SW τj. (2.22) Hence, he insanenous variaion is ied o he erm srucure of he conemporaneous variance swap raes via an affine mapping in Equaion (2.22). Noe ha his is a sric realizaion by realizaion ideniy which holds a all imes. The class of low dimensional A M (N) syle affine erm srucure srucure models, where he covariance of all saes is driven solely by he M variance facors, all imply an affine mapping of he spo variance in M swap raes. In paricular, for he model of A 1 (3), he variance spanning condiion boils down o = α τ f 0 + ατ f 1 SW τ1. (2.23) 14

15 The variance spanning condiion in Equaion (2.22) is valid for he unspanned s- ochasic volailiy (USV) models, which impose parameric resricions on he A M (N) canonical form such ha he variance facors do no affec he cross secion of yields (see Joslin (2017)). In principle, such USV models are nesed under he A M (N) syle affine erm srucure models. The spo variance on he lef hand side of Equaion (2.22) is a laen process and is no direcly observable. Mos of he exising sudies in low-dimensional affine erm srucure models focus on he condiional volailiy yield dynamics backed ou from he esimaed underlying model. Such parameric condiional variance esimaes are sensiive o he misspecificaion of he underlying model and can in fac even be unrelaed o ime series of GARCH ype esimaes of he condiional variances, see, e.g., Collin Dufresne, Goldsein, and Jones (2009). While he parameric model in Equaion (4.31) specifies he dynamics under he risk neural measure, Q, here are implicaions for he daa generaing measure, P under he assumpion of no arbirage. In paricular, he no arbirage condiion implies ha he insananeous diffusive variance is invarian o changes of he probabiliy measure. Hence, he spo variance,, says he same under boh he daa generaing measure, P, and he (parameerized) equivalen maringale measure, Q. Thanks o he availabiliy of he high frequency daa, he insananeous variance can be esimaed in a model free way. Specifically, hese esimaors enable non parameric inference, i.e., hey are no based on a parameric specificaion of he daa generaing law, X, and accordingly, do no rely on he daa generaing law of Ψ τ f,τ L. In his paper, I employ hese esimaors o measure spo variance. The variance swap raes on he righ hand side of Equaion (2.22) can also be measured in a model free way by using a porfolio of pu and call opions. Building on he previous work of Neuberger (1994) and Demeerfi, Derman, Kamal, and Zou (1999), Carr and Wu (2009), and Mele and Obayashi (2013), I show ha his quaniy can be synhesized using a porfolio of pu and call opions on Ψ τ f,τ L in a model free way. I 15

16 provide he conrac designs of such variance swaps and he derivaion of model free prices of hese variance swaps wih a porfolio of pu and call opions in Secion 3.2. I develop a esing procedure in Secion 3 for he variance spanning condiion in Equaion (2.22) implied by he class of parameric A M (N) models. Such ess serve as a diagnosic ool for he affine specificaion of he model wih M variance facors under he risk neural measure. The es saisic employs model free measures of spo variance and variance swap raes SW τj, which I inroduce in he nex secion. 3. Tesing he Affine Variance Spanning Condiion This secion inroduces a specificaion es for evaluaing he affine variance spanning condiion in Equaion (2.22) and saes he asympoic resuls for he es saisic. The srucure of his secion is as follows: I sar wih defining he model free measures (non parameric esimaors) of he spo variance and variance swap raes SW τj. Then, I develop he limi heory necessary o devise he formal specificaion ess for he variance spanning condiion. Las, I provide he es saisic and he associaed limi heory. The developed es saisic is a diagnosic ool for he variance dynamics implied by he underlying model under he pricing measure, and does no resric P dynamics of he saes over wha is implied by he no arbiage condiion. In ha sense, i relies on he hypohesis ha he model is correcly specified under he pricing measure, free from he parameric specificaions abou he dynamics of he saes under he physical measure Non Parameric Inference for Spo Volailiy I sar by defining he non parameric esimaors of spo variance ha are used in his paper. Thanks o he availabiliy of high frequency daa, realized variance and realized power variance esimaors have been exensively sudied boh empirically and heoreically, see Andersen and Bollerslev (1998), Andersen, Bollerslev, Diebold and Labys (2001, 2003), Barndorf Nielsen and Shephard (2001, 2002a,b, 2003, 2004) among many ohers. The advanage of hese esimaors is ha he pahwise realizaions of 16

17 volailiy a specific imes can be recovered non paramerically. They employ in fill asympoics; hey are consisen as he observaions are sampled more frequenly and feaure an asympoic variance ha can be esimaed by using he observed prices of he underlying asse. The deails abou he non parameric esimaor o recover he volailiy realizaions used in his paper are as follows. The ime uni is normalized o a year. Suppose we have equidisan high frequency observaions y τ f s = log(ψ τ f s ) over h o wih grid size δ = h n (e.g. 5 min.). Then, he annualized spo volailiy esimaor V Ψ,k n,n is given by V Ψ,K n,n = 1 K n δ ( j y τ f ) 2, (3.24) wih I = K n + 1,..., 0 inraday incremens j Ψ = y τ f +jδ yτ f +(j 1)δ. The block size K n < n is a deerminisic sequence of inegers. This esimaor is he localized counerpar of he realized variance esimaor. 5 V Ψ,K n,n is consisen for V Ψ τ f j I as K n and K n δ 0. For he consrucion of he es saisic devised in he laer secions, he spo volailiy needs o be esimaed only a finie sample of poins. In he empirical implemenaion, 10 minues observaions of he Eurodollar fuures prices jus prior o he close a 2pm (CDT) are used o calculae he spo variance esimaor V Ψ,K n,n for each day. Noe ha Eurodollar fuures prices are ransformed o Ψ for he corresponding mauriies via Equaion (2.14). To be precise, in he empirical implemenaion, he uni of ime is a year, where h = refers o a day. The rading par of he day is divided ino n inervals and K n observaions prior o he close of he day are employed for he consrucion of he spo variance esimaor V Ψ,Kn,n. 5 In seings wih jumps, we can consruc spo counerpars of he inegraed runcaed variaion esimaors of Mancini (2001), see also Jacod and Proer (2012). 17

18 3.2. Model free Consrucion of Variance Swap Raes on Ψ τ f There is an exensive lieraure focusing on he pricing of equiy volailiy, see Demeerfi, Derman, Kamal, and Zou (1999), Bakshi and Madan (2000), Carr and Madan (2009) among many ohers. In fac, he heoreical resuls provided for he model free replicaion of he heoreical price of he variance swaps are used by he CBOE o calculae he popular VIX index. The VIX index has become a benchmark over years for measuring and rading US equiy marke volailiy. Alhough pricing of fuure volailiy is well undersood in equiy markes, i is sill in is infancy for fixed income markes. Design and pricing of variance swaps in fixed income markes is a delicae issue because of he sochasic ineres raes. In his secion, I provide conrac designs for variance swaps on he implied discoun rae Ψ τ f and provide he heory regarding he model free consrucion of hese variance swap raes. In ha conex, model free refers o he circumsance ha he heoreical value of he variance swap raes can be replicaed by a porfolio of European syle opions, wihou hinging upon any parameric assumpions. 6 A variance swap is a forward conrac such ha a mauriy one pary pays he quadraic variaion over he conrac horizon and he oher pary pays a fixed rae in exchange, which is ermed as he variance swap rae. Variance swaps require no iniial paymen. In paricular, I consider a variance swap based on Ψ τ f wih he conrac horizon from o + τ v. The payoff a mauriy on he long side of he swap is equal o where SW τv rae SW τv 1 τ v +τv u du SW τv, (3.25) is he fixed variance swap rae deermined a ime. The value of he swap is deermined a ime. A variance swap coss zero o ener. Consequenly, under he assumpion ha he shor rae is uncorrelaed wih inegraed volailiy, 7 he 6 Excluding mild assumpions such as absence of arbirage and he fricionless markes 7 See he lieraure on unspanned sochasic volailiy 18

19 variance swap rae is given by SW τv = 1 τ v E Q [ +τv u du. (3.26) The variance swap rae SW τv can be replicaed by he coninuum of European ou of he money pu and call opions. A proof for he following resul is provided in Appendix C. SW τv = 2 τ Ψ f τ v 0 P u (τ v, K Ψ ) K Ψ 2 P (τ v ) dk Ψ + 2 τ v Ψ τ f Call (τ v, K Ψ ) K Ψ 2 P (τ v ) dk Ψ + ɛ τv,(3.27) where P u (τ v, K Ψ ) and Call (τ v, K Ψ ) are ime prices of he ou of he money European pu and call opions wih srike K Ψ and wih expiraion a ime + τ v, wrien on he simple implied hree monh rae Ψ τ f. I documen, wih simulaions based on empirically relevan parameer values for he Eurodollar fuures marke, ha he effec of he las par is no significan for pracical purposes. Hence, I exclude his erm in he implemenaion and obain he variance swap rae in a model free fashion by a porfolio of ou of he money pu and call opions. The simulaion resuls are presened in Appendix C.2. Equaion (3.27) means ha on each day one can synhesize he variance swap rae wih a horizon τ v by using he opions (wih corresponding mauriy) available on ha day. Since Equaion (3.27) holds for any τ v, variance swap raes for various horizons, i.e. erm srucure of swap raes, can be consruced depending on he availabiliy of he opion daa. Consequenly, variance swap raes are approximaed by a discreizaion of he porfolio of he coninuum of opions such ha ŜW τ v = 2 τ v i ( Õ (τ v, KΨ i ) K Ψ P (τ v ) KΨ2 1 F Ψ,τv i τ v KΨ 0 1) 2, (3.28) where O (τ v, K i Ψ ) is he ime price of an ou of he money opion wih srike price K Ψ and mauriy + τ v. P (τ v ) is he ime price of a zero coupon bond wih mauriy τ v. F Ψ,τv is he forward level approximaed from he opion prices (via he srike where he 19

20 absolue difference beween he call and pu is smalles), and KΨ 0 is he firs available srike below he forward level F Ψ,τv. The las par in Equaion (3.28) is a correcion for he error inroduced by he subsiion of K 0 Ψ insead of he forward price. K i Ψ se as Ki 1 Ψ +Ki+1 Ψ 2 for all srikes, excluding he smalles and he larges srikes. For he smalles and he larges srikes, K i Ψ is se as he disance o he nex higher srike and nex lower srike respecively. Eurodollar fuures conracs do no have consan mauriies over days, insead heir mauriies follow a seesaw paern over days. Then, he swap rae ŜW τ for a fixed ime o mauriy τ can be calculaed from he available opions wih he wo neares mauriies τ v1 and τ v2 via he following linear inerpolaion is ( τ 1 τ v1 ŜW = τ v1 ŜW τ τ v2 τ τ v2 τ v1 τ v2 + τ v2 ŜW τ τ v1 τ v2 τ v1 ), (3.29) τ where SW v1 and τ SW v2 are variance swap raes wih ime o mauriy τv1 and τ v2, respecively. I consruc he variance swap raes by using he selemen prices of opions a each day for which he Eurodollar opion daa is available. The available opion daa covers he period from January 1, 2004 o July 13, A each day in he sample, I consruc a erm srucure of variance swap raes on he implied discoun rae Ψ for fixed mauriies of hree, six, nine monhs, one year and one and a half years by using Equaion (3.29). As he daa is no direcly available on he implied discoun raes and opions on hem bu on Eurodollar fuures and opions, we need o ransform he prices of Eurodollar fuures and opions o he corresponding measures on he implied discoun raes. I provide he deails regarding he consrucion variance swap raes on he implied discoun raes laer in Secion 5.1. This paper is no he firs one o analyze he variance conracs in fixed income markes. Mele and Obayashi (2013) provide he heory underlying he pricing of variance conracs on Treasury fuures. Choi, Mueller, and Vedolin (2017) sudy variance risk 20

21 premiums in he Treasury marke. Grishchenko, Song, and Zhou (2015) examine he role of variance risk premiums based on ineres rae swaps and swapions in predicing Treasury excess reurns. The heoreical and empirical resuls regarding he variance swaps documened in his paper complemen he work above as he focus here is on he pricing of he variance in Eurodollar markes (on implied discoun raes). 4. Tes Saisics and he Limiing Theory This secion formalizes he se up underlying he economeric analysis of he paper. ( ( ) ) Our ineres is on he process, Ψ τ f defined on a probabiliy space Ω (0), F (0) (0), F, P (0) 0 follows he general dynamics under P (0) : dψ τ f Ψ τ f = µ Ψτ f d + σ Ψτ f dw, (4.30) where W is an N dimensional Wiener process, µ Ψτ f processes. The spo variance process is V τ f = σ Ψτ f and σ Ψτ f are càdlàg and F adaped σ Ψτ f akes is values in he se D + (0, ). See Assumpion 1 in Appendix D for precise assumpions regarding regulariy condiions on Ψ τ f. Under he assumpion of no arbirage, here exiss a risk neural probabiliy measure Q (see, e.g., Duffie (2001)) which is locally equivalen o P (0). To be specific, Ψ τ f under Q follows dψ τ f Ψ τ f = µ Ψ,Q d + σ Ψτ f dw Q, (4.31) where W Q µ Ψ,Q is an N dimensional Q Wiener process. Similarly, µ Ψ,Q is càdlàg and adaped. Noe ha I sress ha I do no assume any parameric funcional forms on µ Ψτ f, and σ Ψτ f, i.e. recall ha µ Ψτ f, µ Ψ,Q and σ Ψτ f carry an affine parameric form in he underlying (laen) saes under low dimensional affine erm srucure models. Moreover, he seing in his secion allows he insanenous variance process an Iô semi maringale wih general forms of vol of vol and jumps. o be We have observaions from opions wrien on Ψ τ f a ineger imes (i.e. days) = 21

22 1,..., T, wihin a ime span [0, T. In paricular, a a given ime, we have a cross secion of ou of he money opion prices {O (τ j, K Ψj ); j = 1,..., N K } for some ineger N K. For a enor of τ, here are N τ K number of opions, and hence τ N τ K = N K. Suppose for each enor τ, K m (τ) and K M (τ) represen he minimum and he maximum srikes. To his end, I assume ha he srikes beween K m (τ) and K M (τ) are equidisan, wih grid size K = K M (τ) K m(τ) N. K τ Opion prices are assumed o be observed wih error: Õ (τ, K Ψ ) = O (τ, K Ψ ) + ε τ,kψ, (4.32) where he observaion errors, ε τ,kψ, are defined on space Ω (1). We endow he space Ω (1) wih he produc Borel σ field F (1) and he filraion F (1) = σ(ε τ,kψ s : s ) and he probabiliy P (1) (w (0), dw (1) ). Then, he exended probabiliy space (Ω, F, (F ) 0, P) is given by Ω = Ω (0) Ω (1), F = F (0) F (1), F = s> (F (0) s F (1) s ), ( P dw (0), dw (1)) ( = P (0) dw (0)) ( ) P (1) w (0), dw (1) ). Observaion errors are assumed o be condiionally cenered and o display sochasic volailiy which can also depend on he enor: E [ε τ,kψ F (0) [ = 0, and E ε τ,kψ 2 F (0) = η,kψ,τ. (4.33) See Appendix 2 for addiional assumpions regarding he observaion errors. Now, we are a a sage o esablish he asympoic disribuion of he esimaors. I firs sar wih he limiing disribuion of he high frequency based variance esimaor V Ψ,K n,n given in Equaion (3.24). Theorem 4.1. Le Ψ be a sochasic process soluion o (4.31). Le Assumpion 1 holds. If K n, δ 0 and K n δ 0, hen for all in (0, T, we have V Ψ,K n,n V Ψ τ f ( 2 Ψ,K ) n,n 2 K V n 22 L s U, (4.34)

23 where condiionally on F, U is an N (0, 1) variable. See Alvarez, Panloup, Ponier, and Savy (2012) for a proof. 8 Theorem 4.2. Le Assumpion 2 holds. N τv K is he number of opions on a given day wih enor τ v and wih equally spaced srikes of grid size K beween [K m (τ v ), K M (τ v ). τv Le ŜW is given as in Equaion (3.28) wih F Ψ,τv = KΨ 0. Moreover, assume here exiss P η,kψ,τ v η,kψ,τ v, (4.35) uniformly on [K m (τ v ), K M (τ v ). Then, as K 0, we have where and ŜW τv See Appendix D for a proof. SW τv,m,m K Ŵ N K SW τv,m,m = 2 τ v KM (τ v) K m(τ v) N Ŵ N 4 K τv K = P (τ v ) 2 j=1 L s N (0, 1), (4.36) O (τ v, K Ψ ) K Ψ 2 dk Ψ, (4.37) η,kj,τ v K j Ψ 4 K. (4.38) The limiing disribuion of a es saisic for he affine variance spanning condiion is provided nex. Specifically, he underlying null hypohesis is he affine variance spanning condiion. M = α τ f 0 + j=1 α τ f j τj SW. Theorem 4.3. Le Assumpion 1 and 2 hold. Suppose T, N K wih N K T 0, K n, K n δ 0, wih K n N K ρ and Kn T 0. Ẑ = 2 V Ψ,K n,n ˆα τ f 0 M j=1 ˆατ f j ( Ψ,K ) n,n 2 M K V n + K j=1 (ˆατ f j ŜW τ j )2ŴN K,j L s N (0, 1), (4.39) 8 In a seing wih price jumps, similar resuls have been esablished for he localized versions of he runcaed variaion esimaors (see, e.g., Jacod and Proer (2012), Andersen, Fusari, and Todorov (2015)). 23

24 where (ˆα 0, ˆα 1,..., ˆα M ) ( T Ψ,K n,n τ = argmin {α τ =1 V α f 0 M f j }M j=0 is given as in Theorem 4.2. as See Appendix D for a proof. j=1 ατ f j ŜW τ j ) 2 and Ŵ N K I adjus he variance spanning condiion in Equaion (2.22) for logarihmic variances log M = α τ f 0,log + α τ f τj j,log log SW, where α τ f j,log, j = 0,..., M is a se of consans. I erm his relaion as he logarihmic variance spanning condiion. Nex, I provide he limiing disribuion of a es saisic j=1 under he null of he logarimic affine spanning condiion. Theorem 4.4. Le Assumpion 1 and 2 hold. Suppose T, N K wih N K T 0, K n, K n δ 0, wih K n N K ρ and Kn T 0. Then, we have where Ẑ log = log V Ψ,K n,n τ ˆα f 0,log M j=1 ˆατ f j,log log ŜW M + K j=1 (ˆατ f j )2 Π N K,j 2 K n (ˆα 0,log, ˆα 1,log,..., ˆα M,log ) = and Π N K,j = ŴN K,j. (ŜW τ j ) 2 See Appendix D for a proof. argmin {α τ f j,log }M j=0 τ j M log V Ψ,K n,n τ α f 0,log L s N (0, 1), (4.40) j=1 α τ f j,log log ŜW 2 τ j, 5. Daa and Preliminary Analyses The analysis in his paper relies on Eurodollar fuures and opions daa raded a he CME from January 1, 2004 o July 13, Eurodollar fuures conracs are cash seled conracs wih he delivery based on he 3 monh LIBOR. These conracs are issued quarerly wih mauriies ranging from hree monhs o en years. Consequenly, a each dae here are up o foury conracs available wih conrac monhs March, June, Sepember and December. Noe ha Eurodollar fuures do no have consan 24

25 mauriies. Accordingly in order o ge one ime series, I roll he fuures daa a he end of he monh preceeding he conrac monh. Therefore, he ime o mauriies follow a seesaw paern over days for each mauriy. The CME offers boh quarerly and serial opion conracs. The conrac monhs for quarerly opions are March, June, Sepember and December and hey exercise ino he Eurodollar fuures conracs wih he corresponding mauriies. Alhough he CME recenly inroduced quarerly opions wih mauriies up o four years, he analysis here is based on he opions wih mauriies up o wo years because hese conracs are he mos liquid ones. The serials conrac monhs are he wo non quarerly fron monhs and hey exercise ino he corresponding quarerly Eurodollar fuures conrac immediaely following he serial. Noe ha hese conracs are much less liquid han he quarerly opions. Consequenly, I discard all he serial opions from he sample. Consisen wih he rolling procedure implemened for he Eurodollar fuures, opions are rolled a he end of he monh preceeding he conrac monh Consrucion of he Swap Raes In order o consruc he erm srucure of he variance swap raes in a model free way, I use he daily selemen prices of he CME sandard quarerly opions from January 1, 2004 o July 13, Daily daa on Eurodollar fuures and opions is obained from he CME. 9 I apply he following filers o he opion daa before consrucing he variance swap raes: I eliminae 1) he opions wih zero selemen prices 2) he opions wih zero srikes 3) he opions wih zero ime o mauriy 4) he opions wih zero open ineres. Since he model free consrucion of variance swap raes employs only ou of he money opions, I discard all in he money opions. In order o consruc he variance swap raes on he implied discoun rae Ψ τ f, we need o ransform he Eurodollar fuures and opions prices o he corresponding values for he implied discoun rae. Appendix B oulines he procedures regarding hese 9 I use selemen prices of opions, which prevens issues relaed o sale rading or microsrucure noise. CME calculaes hese prices based on he Globex rades beween and

26 ranformaions. I choose horizons of 6 monh and 1 year and 6 monhs for he synheic variance swap raes. The remaining horizons are excluded because of liquidiy concerns. On a given dae, I choose he wo closes mauriies o a given horizon (i.e. 6 monhs) and consruc he variance swap raes in a model free way by implemening he discreizaion (3.28) for hese wo mauriies and linearly inerpolae he horizon o obain fixed horizons of ineres via Equaion (3.29). Figure 1 plos he ime series of he square roo of he variance swap raes for he mauriies of 6 monhs and 1.5 years. The figure depics all values in annualized basis poin unis. The variance swap raes for 1.5 year mauriy is usually larger han he variance swap raes for 6 monhs mauriy. However, in imes of disress such as he crisis period during 2007 and 2008, he swap raes wih he shor mauriy occasionally go above he swap raes wih he longer mauriy. These findings could be explained by he markes pricing he long erm conracs wih he expecaion ha volailiy would fall back from he crisis levels. Such behaviors regarding he erm srucure of variance swap raes are consisen wih he findings in he lieraure wih regards o he variance swap raes in equiy markes, see, for example Dew-Becker, Giglio, Le, and Rodriguez (2017) Consrucion of Spo Variance In order o consruc a non parameeric esimaor of he spo variance, I obain he inraday series of Eurodollar Fuures prices from TickDaa for he period from January 1, 2004 o July 13, Noe ha Eurodollar fuures are raded boh open oucry (pi rading) and elecronically (Globex); I use inraday rades daa from boh he elecronic and he pi rading sessions. In line wih he pi rading hours, I sar he inraday record a 7:20 am (CDT) and end i a 2:00 pm (CDT). I sample Eurodollar fuures rades a a 10 minues frequency over a 6 hours 40 minues rading period and conver hem ino he corresponding implied discoun raes based on he procedure oulined in Appendix B. Afer implemening he necessary ransformaions, he reurns on log Ψ τ f 26

27 Figure 1: Variance Swap Raes The figure plos he ime series of he square roo of he variance swap raes per annum in basis poins. The red line represens he daily variance swap rae for 6 monhs mauriy. The blue line represens he daily swap rae for 1.5 years mauriy. The sample covers he period from 06/29/2004 o 07/13/2010. are compued. Noe ha, via he rolling procedure, i is possible o obain single series for up o foury mauriies. The mauriy srucure of each of hese series has a seesaw paern over days. Before consrucing he non parameric variance esimaor, I apply he following filers o clean he inraday daa: I eliminae he days wih no rading aciviy, half rading days, and he days wih early marke closures which ypically happens before holidays. Moreover, I exclude he days where here is no price change during he las wo hirds of he day. 10 Consequenly, we have 40 inraday changes on each rading day over approximaely 1600 rading days for he series included in he sample. I consruc he non parameric variance esimaor for each series (wih mauriies up o 4 years) by implemening Equaion (3.24), where n = 40 and K n is se as 30. Noe ha hese series are in annualized erms since h = To obain a spo variance series wih a fixed 10 These evens resul in low rading aciviies for hese days. 27

28 mauriy of S days, I choose he wo spo variance series wih he closes mauriies o S and linearly inerpolae he wo spo variance esimaors on each day. Figure 2 depics he daily values for he square roo of he non parameric variance esimaor for 6 monhs log(ψ) consruced from high frequency daa. The figure shows he series in annualized basis poin unis. Casual inspecion shows ha volailiy follows higher levels during he crisis period from mid 2007 o he end of Afer 2009, i goes back, on average, o he levels observed before he crisis period. Moreover, he 6 monhs volailiy in Eurodollar markes shows occasional spikes, which is in line wih corresponding findings in Treasury markes, see for example, Andersen and Benzoni (2010). Figure 2: Non parameric Spo Volailiy Esimae wih 6 monh mauriy M This figure plos he ime series he non parameric esimae of he insanenous variance of 6 monh implied discoun rae (per annum in basis poins). The sample covers he period from 06/29/2004 o 07/13/

29 6. Evidence on Tesing he Affine Variance Spanning Condiion I now presen my main empirical findings. This secion sudies wheher he affine variance spanning condiion in Equaion (2.22), implied by A M (N) syle affine erm srucure models, is saisfied in he Eurodollar daa. In paricular, I es he affine variance spanning condiion by implemening he es saisic Ẑ (see Theorem 4.3). Recall ha he es saisic relies on he high frequency based non parameric spo variance esimaor as well as on he model free measures of variance swap raes. I focus on he resuls for he 6 monh mauriy spo variance, ha is where τ f = 1 2. The findings for he 3 monhs, 1 year, 1.5 years and 3 years mauriy spo variances are provided in he Appendix. I rely on he model free measures of he variance swap raes for 6 monhs and 1.5 years mauriies o consruc he es saisic. I es he variance spanning implicaion for he wo mos ofen used models, A 1 (3) and A 2 (3), in he low dimensional erm srucure lieraure. I iniially focus on he diagnosic analysis of he variance spanning condiion under he A 1 (3) model, where he covariance of all saes is solely driven by a single facor. Recall ha he variance spanning condiion in ha case boils down o a specificaion wih one facor only (see Equaion (2.23)). Consequenly, he diagnosic analysis of he variance spanning condiion under A 1 (3) model relies on he non parameric measure of he variance swap rae wih 6 monhs mauriy only. Figure 3 depics he daily imes series of he es saisic Ẑ from Theorem 4.3 for he variance spanning condiion implied by he A 1 (3) model. The es saisic is employed dynamically a each poin in ime via a rolling window of lengh of 120 days. Remarkably, he es saisic exhibis very large negaive values (considerably below he 1s percenile value) over he crisis period of 2008 o 2010, which indicaes he failure of he affine variance spanning condiion under he one volailiy facor model. Moreover, he A 1 (3) model sruggles occasionally during he calm period of From an economeric perspecive, he A 1 (3) model is rejeced. Casual inspecion shows persisen behavior of he es saisic over periods where 29