Long-Term Factorization of Affine Pricing Kernels

Transcription

1 arxiv: v2 [q-fin.mf] 27 Jul 2017 Long-Term Facorizaion of Affine Pricing Kernels Likuan Qin and Vadim Linesky Deparmen of Indusrial Engineering and Managemen Sciences McCormick School of Engineering and Applied Sciences Norhwesern Universiy Absrac This paper consrucs and sudies he long-erm facorizaion of affine pricing kernels ino discouning a he rae of reurn on he long bond and he maringale componen ha accomplishes he change of probabiliy measure o he long forward measure. The principal eigenfuncion of he affine pricing kernel germane o he long-erm facorizaion is an exponenial-affine funcion of he sae vecor wih he coefficien vecor idenified wih he fixed poin of he Riccai ODE. The long bond volailiy and he volailiy of he maringale componen are explicily idenified in erms of his fixed poin. A range of examples from he asse pricing lieraure is provided o illusrae he heory. This paper is based on research suppored by he gran CMMI from he Naional Science Foundaion. likuanqin2012@u.norhwesern.edu linesky@iems.norhwesern.edu 1

2 1 Inroducion The sochasic discoun facor (SDF) is a fundamenal objec in arbirage-free asse pricing models. I assigns oday s prices o risky fuure payoffs a alernaive invesmen horizons. I accomplishes his by simulaneously discouning he fuure and adjusing for risk. A familiar represenaion of he SDF is a facorizaion ino discouning a he risk-free ineres rae and a maringale componen adjusing for risk. This maringale accomplishes he change of probabiliies o he risk-neural probabiliy measure. More recenly Alvarez and Jermann (2005), Hansen e al. (2008), Hansen and Scheinkman (2009) and Hansen (2012) inroduce and sudy an alernaive long-erm facorizaion of he SDF. The ransiory componen in he long-erm facorizaion discouns a he rae of reurn on he pure discoun bond of asympoically long mauriy (he long bond). The permanen componen is a maringale ha accomplishes a change of probabiliies o he long forward measure. Qin and Linesky (2017) sudy he long-erm facorizaion and he long forward measure in he general semimaringale seing. The long-erm facorizaion of he SDF is paricularly convenien in applicaions o he pricing of long-lived asses and o heoreical and empirical invesigaions of he erm srucure of he risk-reurn rade-off. In addiion o he references above, he growing lieraure on he long-erm facorizaion and is applicaions includes Hansen and Scheinkman (2012), Hansen and Scheinkman (2017), Borovička e al.(2016), Borovička e al.(2011), Borovička and Hansen(2016), Bakshi and Chabi-Yo (2012), Bakshi e al.(2015), Chrisensen(2017), Chrisensen (2016), Qin and Linesky (2016), Qin e al. (2016), Backus e al. (2015), Filipović e al. (2017), Filipović e al. (2016). Empirical invesigaions in his lieraure show ha he maringale componen in he long-erm facorizaion is highly volaile and economically significan (see, in paricular, Bakshi and Chabi-Yo (2012) for resuls based on pricing kernel bounds, Chrisensen (2017) for resuls based on srucural asse pricing models connecing o he macro-economic fundamenals, and Qin e al. (2016) for resuls based on explici parameerizaions of he pricing kernel, where, in paricular, he relaionship among he measures P, Q and L is empirically invesigaed). Thefocusofhepresen paperisonheanalysisoflong-ermfacorizaioninaffine diffusion models, boh from he perspecive of providing a user s guide o consrucing long-erm facorizaion in affine asse pricing models, as well as employing affine models as a convenien laboraory o illusrae he heory of he long-erm facorizaion. Affine diffusions are work-horse models in coninuous-ime finance due o heir analyical and compuaional racabiliy(vasicek(1977), Cox e al.(1985), Duffie and Kan (1996), Duffie e al. (2000), Dai and Singleon (2000), Duffie e al. (2003)). In his paper we show ha he principal eigenfuncion of Hansen and Scheinkman (2009) ha deermines he long-erm facorizaion, if i exiss, is necessarily in he exponenialaffine form in affine models, wih he coefficien vecor in he exponenial idenified wih he fixed poin of he corresponding Riccai ODE. This allows us o give a fully 2

3 explici reamen and illusrae dynamics of he long bond, he maringale componen and he long-forward measure in affine models. In paricular, we explicily verify ha when he Riccai ODE associaed wih he affine pricing kernel possesses a fixed poin, he affine model saisfies he sufficien condiion in Theorem 3.1 of Qin and Linesky (2017) so ha he long-erm limi exiss. In Secion 2 we review and summarize he long-erm facorizaion in Brownian moion-based models. In Secion 3 we presen general resuls on he long-erm facorizaion of affine pricing kernels. The main resuls are given in Theorem 3.2, where he marke price of Brownian risk is explicily decomposed ino he marke price of risk under he long forward measure idenified wih he volailiy of he long bond and he remaining marke price of risk deermining he maringale componen accomplishing he change of probabiliies from he daa-generaing o he long forward measure. The laer componen is deermined by he fixed poin of he Riccai ODE. In Secion 4 we sudy a range of examples of affine pricing kernels from he asse pricing lieraure. 2 Long-Term Facorizaion in Brownian Environmens We work on a complee filered probabiliy space (Ω, F,(F ) 0,P). We assume ha alluncerainy inheeconomy isgeneraedbyan n-dimensional BrownianmoionW P and ha (F ) 0 is he (compleed) filraion generaed by W P. We assume absence of arbirage and marke fricions, so ha here exiss a sricly posiive pricing kernel process in he form of an Iô semimaringale. More precisely, we assume ha he pricing kernel follows an Iô process ( denoes vecor do produc) ds = r S d S λ dw P wih r 0 s ds < and he marke price of Brownian risk vecor λ such ha he process M 0 = e 0 λs dwp s λs 2 ds is a maringale (Novikov s condiion E P [e λs 2ds ] < for each > 0 suffices). Under hese assumpions he pricing kernel has he risk-neural facorizaion S = 1 A M 0 = e 0 rsds M 0 ino discouning a he risk-free shor rae r deermining he risk-free asse (money marke accoun) A = e 0 rsds and he exponenial maringale M 0 wih he marke price of Brownian risk λ deermining is volailiy. We also assume ha E P [S T /S ] < for all T > 0. The inegrabiliy of he SDF S T /S for any wo daes T > 3

4 ensures ha ha zero-coupon bond price processes P T := E P [S T/S ], [0,T] are well defined for all mauriy daes T > 0 (E [ ] = E[ F ]). Since foreach T het-mauriy zero couponbond price process P T can be wrien as P T = M TPT 0 /S, where M T = S P T/PT 0 = EP [S T]/E P 0 [S T] is a posiive maringale on [0,T], we can apply he Maringale Represenaion Theorem o claim ha dm T = M T λt dw P wih some λ T, and furher claim ha he bond price process has he represenaion i=0 dp T = (r +σ T λ )P T d+p T σ T dw P wih he volailiy process σ T = λ λ T. Following Qin and Linesky (2017), for each fixed T > 0 we define a self-financing rading sraegy ha rolls over invesmens in T-mauriy zero-coupon bonds as follows. Fix T and consider a self-financing roll-over sraegy ha sars a ime zero by invesing one uni of accoun in 1/P0 T unis of he T-mauriy zero-coupon bond. A ime T he bond maures, and he value of he sraegy is 1/P0 T unis of accoun. We roll he proceeds over by re-invesing ino 1/(P0 T PT 2T ) unis of he zero-coupon bond wih mauriy 2T. We coninue wih he roll-over sraegy, a each ime kt re-invesing he proceeds ino he bond P (k+1)t kt. We denoe he valuaion process of his self-financing sraegy B T: ( k ) 1 B T = P (k+1)t, [kt,(k +1)T), k = 0,1,... P (i+1)t it For each T > 0, he process B T is defined for all 0. The process S B T exends he maringale M T o all 0. I hus defines he T-forward measure Q T F = M TP F on F for each 0, where T now has he meaning of he lengh of he compounding inerval. Under he T-forward measure Q T exended o all F, he roll-over sraegy (B T ) 0 wih he compounding inerval T serves as he numeraire asse. Following Qin and Linesky (2017), we coninue o call he measure exended o all F for 0 he T-forward measure and use he same noaion, as i reduces o he sandard definiion of he forward measure on F T. Since he roll-over sraegy (B T) 0 and he posiive maringale M T = S B T are defined for all 0, we can wrie he T-forward facorizaion of he pricing kernel for all 0: S = 1 M T B T. We now recall he definiions of he long bond and he long forward measure from Qin and Linesky (2017). 4

5 Definiion 2.1. (Long Bond) If he wealh processes (B T ) 0 of he roll-over sraegies in T-mauriy bonds converge o a sricly posiive semimaringale (B ) 0 uniformly on compacs in probabiliy as T, i.e. for all > 0 and K > 0 we call he limi he long bond. lim P(sup Bs T T B s > K) = 0, s Definiion 2.2. (Long Forward Measure) If here exiss a measure Q equivalen o P on each F such ha he T-forward measures converge srongly o Q on each F, i.e. lim T QT (A) = Q (A) for each A F and each 0, we call he limi he long forward measure and denoe i L. The following heorem, proved in Qin and Linesky (2017), gives a sufficien condiion ha ensures convergence o he long bond in he semimaringale opology which is sronger han he ucp convergence in Definiion 1 and convergence of T- forward measures o he long forward measure in oal variaion, which is sronger han he srong convergence in Definiion 2 (we refer o Qin and Linesky (2017) and he on-line appendix for proofs and deails). Theorem 2.1. (Long Term Facorizaion and he Long Forward Measure) Suppose ha for each > 0 he raio of he F -condiional expecaion of he pricing kernel S T o is uncondiionalexpecaionconvergeso a posiive limiin L 1 as T (under P), i.e. for each > 0 here exiss an almos surely posiive F -measurable random variable which we denoe M such ha E P [S T] E P [S T ] L 1 M as T. (2.1) Then he following resuls hold: (i) The collecion of random variables (M ) 0 is a posiive P-maringale, and he family of maringales (M T ) 0 converges o he maringale (M ) 0 in he semimaringale opology. (ii)the longbondvaluaion process(b ) 0 exiss, and he roll-oversraegies (B T ) 0 converge o he long bond (B ) 0 in he semimaringale opology. (iii) The pricing kernel possesses he long-erm facorizaion S = 1 B M. (2.2) (iv) T-forward measures Q T converge o he long forward measure L in oal variaion on each F, and L is equivalen o P on F wih he Radon-Nikodym derivaive M. The process B has he inerpreaion of he gross reurn earned saring from 5

6 ime zero up o ime on holding he zero-coupon bond of asympoically long mauriy. The long bond is he numeraire asse under he long forward measure L since he pricing kernel becomes 1/B under L. The long-erm facorizaion of he pricing kernel (2.2) decomposes i ino discouning a he rae of reurn on he long bond and a maringale componen encoding a furher risk adjusmen. Suppose he condiion (2.1) in Theorem 2.1 holds in he Brownian seing of his paper. Then he long bond valuaion process is an Iô semimaringale wih he represenaion db = (r +σ λ )B d+b σ dw P wih some volailiy process σ such ha he process M = S B saisfying dm = M λ dw P wih λ = λ σ is a maringale (he permanen componen in he long-erm facorizaion). Thus, he long-erm facorizaion Eq.(2.2) in he Brownian seing yields a decomposiion of he marke price of Brownian risk λ = σ +λ ino he volailiy of he long bond σ and he volailiy λ of he maringale M. The change of probabiliy measure from he daa-generaing measure P o he long forward measure L is accomplished via Girsanov s heorem wih he L-Brownian moion W L = W P + 0 λ s ds. 3 Long Term Facorizaion of Affine Pricing Kernels We assume ha he underlying economy is described by a Markov process X. We furher assume X is anaffine diffusion andhe pricing kernel S is exponenial affine in X and he ime inegral of X. Affine diffusion models are widely used in coninuousime finance due o heir analyical racabiliy (Vasicek (1977), Cox e al. (1985), Duffie and Kan (1996), Duffie e al. (2000), Dai and Singleon (2000), Duffie e al. (2003)). We sar wih a brief summary of some of he key facs abou affine diffusions. We refer he reader o Filipović and Mayerhofer (2009) for deails, proofs and references o he lieraure on affine diffusion. TheprocessweworkwihsolveshefollowingSDEonhesaespaceE = R m + Rn for some m,n 0 wih m+n = d, where R m + = { x R m : x i 0 for i = 1,...,m } : dx = b(x )d+σ(x )dw P, X 0 = x, (3.1) where W P is a d-dimensional sandard Brownian moion and he diffusion marix α(x) = σ(x)σ(x) (here denoes marix ranspose o differeniae i from superscrip 6

7 T ) and he drif vecor b(x) are boh affine in x: α(x) = a+ d x i α i, b(x) = b+ i=1 d x i β i = b+bx for some d d-marices a and α i and d-dimensional vecors b and β i, where we denoe by B = (β 1,...,β d ) he d d-marix wih i-h column vecor β i, 1 i d. The firs m coordinaes of X are CIR-ype and are non-negaive, while he las n coordinaes are OU-ype. Define he index ses I = {1,...,m} and J = {m + 1,...,m + n}. For any vecor µ and marix ν, and index ses M,N {I,J}, we denoe by µ M = (µ i ) i M, ν MN = (ν ij ) i M,j N he respecive sub-vecor and sub-marix. To ensure he process says in he domain E = R m + Rn, we need he following assumpion (cf. Filipović and Mayerhofer (2009)) Assumpion 3.1. (Admissibiliy) (1) a JJ and α i,jj are symmeric posiive semi-definie for all i = 1,2,...,m, (2) a II = 0, a IJ = a JI = 0, (3) α j = 0 for j J, (4) α i,kl = α i,lk = 0 for k I\{i} for all 1 k,l d, (5) b I 0, B IJ = 0, and B II has non-negaive off-diagonal elemens. Thecondiionb I 0onheconsanerminhedrifofheCIR-ypecomponens ensures ha he process says in he sae space E. Making a sronger assumpion b I > 0ensures ha he process insananeously reflecs fromheboundary E andreeners heinerior ofhesaespace ine = R m ++ Rn,where R m ++ = { x R m : x i > 0 for i = 1,...,m }. For any parameers saisfying Assumpion 3.1, here exiss a unique srong soluion of he SDE(3.1)(cf. Theorem 8.1 of Filipović and Mayerhofer(2009)). Denoe by P x he law of he soluion X x of he SDE (3.1) for x E, P x (X A) := P(X x A). Then P (x,a) = P x (X A) defined for all 0, Borel subses A of E, and x E defines a Markov ransiion semigroup (P ) 0 on he Banach space of Borel measurable bounded funcions on E by P f(x) := E f(y)p (x,dy). As shown in Duffie e al. (2003), his semigroup is Feller, i.e., i leaves he space of coninuous funcions vanishing a infiniy invarian. Thus, he Markov process ((X ) 0,(P x ) x E ) is a Feller process on E. I has coninuous pahs in E and has he srong Markov propery (cf. Yamada and Waanabe (1971), Corollary 2, p.162). Thus, i is a Borel righ process (in fac, a Hun process). We make he following assumpion abou he pricing kernel. Assumpion 3.2. (Affine Pricing Kernel) We assume ha he pricing kernel is exponenial-affine in X and is ime inegral: i=1 S = e γ u (X X 0 ) 0 δ X sds, (3.2) where γ is a scalar and u and δ are d-vecors and denoes marix ranspose. The pricing kernel in his form is a posiive muliplicaive funcional of he Markov 7

8 process X. The associaed pricing operaor P is defined by P f(x) = E P x [S f(x )] for a payoff f of he Markov sae. We refer he reader o Qin and Linesky (2016a) for a deailed reamen of Markovian pricing operaors. The pricing kernel in he form (3.2) is called affine due o he following key resul ha shows ha he erm srucure of pure discoun bond yields is affine in he sae vecor X (cf. Filipović and Mayerhofer (2009) Theorem 4.1). Proposiion 3.1. Le T 0 > 0. The following saemens are equivalen: (i) E P [S T0 ] < for all fixed iniial saes X 0 = x R m + R n. (ii) There exiss a unique soluion (Φ( ),Ψ( )) : [0,T 0 ] R R d of he following Riccai sysem of equaions up o ime T 0 : Φ () = 1 2 Ψ J() a JJ Ψ J ()+b Ψ()+γ, Φ(0) = 0, Ψ i () = 1 2 Ψ() α i Ψ()+β i Ψ()+δ i, i I, (3.3) Ψ J () = B JJ Ψ J()+δ J, Ψ(0) = u. In eiher case, he pure discoun bond valuaion processes (wih uni payoffs) are exponenial-affine in X: P T = E P [S T/S ] = (P T 1)(x) = P(T,X ) = e Φ(T ) (Ψ(T ) u) X x (3.4) for all 0 T +T 0 and he SDE iniial condiion x R m + R n. Since in his paper our sanding assumpion is ha E P [S ] < for all, in his case he Riccai ODE sysem has soluions Ψ() and Φ() for all, and he bond pricing funcion enering he expression (3.4) for he zero-coupon bond process P(,x) = (P 1)(x) = e Φ() (Ψ() u) x (3.5) is defined for all 0 and x E. We nex show ha an affine pricing kernel always possesses he risk-neural facorizaion wih he affine shor rae funcion. Theorem 3.1. (Risk-Neural Facorizaion of Affine Pricing Kernels) Suppose X saisfies Assumpion 3.1 and he pricing kernel saisfies Assumpion 3.2 ogeher wih he assumpion ha E P x[s ] < for all 0 and every fixed iniial sae X 0 = x R m + Rn. (i) Then he pricing kernel admis he risk-neural facorizaion S = e 0 r(xs)ds M 0 wih he affine shor rae r(x) = g +h x, (3.6) 8

9 wih g = γ 1 2 u J a JJu J +b u, h i = δ i 1 2 u α i u+β i u, i I, h J = δ J +B JJ u J (3.7) and he maringale M 0 = e 0 λ sdws P λs 2 ds wih he marke price of Brownian risk (column d-vecor) λ = σ(x ) u, (3.8) where σ(x) is he volailiy marix of he sae variable X in he SDE (3.1) and λ 2 = λ λ = u α(x )u. (ii) Under he risk-neural measure Q defined by he maringale M, he dynamics of X reads dx = (b(x ) α(x )u)d+σ(x )dw Q, (3.9) where W Q = W P + 0 λ sds is he sandard Brownian moion under Q. Proof. (i) Define a process M 0 := S e 0 r(xs)ds. I is also in he form of Eq.(3.2) wih γ replaced by γ g and δ replaced by δ h. Thus, Proposiion 3.1 also holds if we replace S wih M 0, replace γ wih γ g and replace δ wih δ h, i.e. E P [M T /M ] = e Φ(T ) (Ψ(T ) u) X x, where Φ () = 1 2 Ψ J() a JJ Ψ J ()+b Ψ()+γ g, Φ(0) = 0, Ψ i() = 1 2 Ψ() α i Ψ()+β i Ψ()+δ i h i, i I, Ψ J() = B JJ Ψ J()+δ J h J, Ψ(0) = u. Wih he choice of g and h in Eq.(3.7), he soluion o he above ODE is Φ() = 0 and Ψ(0) = u, which implies E P [M T /M ] = 1. This shows ha M 0 is a maringale. Furhermore, using he SDE for he affine sae X, we can cas M 0 in he exponenial maringale form e 0 λ sdws P λs 2ds. wih λ given in (3.8). (ii) The SDE for X under Q follows from Girsanov s Theorem. We nex urn o he long erm facorizaion of he affine pricing kernel. Theorem 3.2. (Long Term Facorizaion of Affine Pricing Kernels) Suppose he soluion Ψ() of he Riccai ODE (3.3) converges o a fixed poin v R d : lim Ψ() = v. (3.10) Then he following resuls hold. (i) Condiion Eq.(2.1) is saisfied and, hence, all resuls in Theorem 2.1 hold. 9

10 (ii) The long bond is given by B = e λπ(x ) π(x 0 ), (3.11) where π(x) = e (u v) x (3.12) is he posiive exponenial-affine eigenfuncion of he pricing operaor P wih he eigenvalue e λ wih P π(x) = e λ π(x) inerpreed as he limiing long-erm zero-coupon yield: λ = γ 1 2 v J a JJv J +b v (3.13) lnp(,x) lim for all x. (iii) The long bond has he P-measure dynamics: = λ (3.14) db = (r(x )+(σ ) λ )B d+b (σ ) dw P, where he (column vecor) volailiy of he long bond is given by: σ = σ(x ) (u v). (3.15) (iv) The maringale componen in he long-erm facorizaion of he PK M = S B can be wrien in he form M = e 0 (λ s ) dws P λ s 2ds, (3.16) where λ = λ σ = σ(x ) v. (3.17) (v) The long-erm decomposiion of he marke price of Brownian risk is given by: λ = σ +λ, where σ is he volailiy of he long bond (3.15) and λ given in (3.17) defines he maringale (3.16). (vi) Under he long forward measure L he sae vecor X solves he following SDE where W L = W P + 0 λ s dx = (b(x ) α(x )v)d+σ(x )dw L, (3.18) ds is he d-dimensional Brownian moion under L, and he 10

11 long bond has he L-measure dynamics: db = (r(x )+ σ s 2 )B d+b (σ ) dw L. Proof. Since he soluion of he Riccai ODE Ψ() converges o a consan as, he righ hand side of Eq.(3.3) also converges o a consan. This implies ha Ψ () also converges o a consan. This consan mus vanish, oherwise Ψ() canno converge o a consan. Thus, he righ hand side of Eq.(3.3) also converges o zero. All hese imply ha Ψ() = v is a saionary soluion of he Riccai equaion Eq.(3.3). Applying Proposiion 3.1 o he affine kernel of he form 1/B, where B is he process defined in (3.11), i hen follows ha π(x) defined in Eq.(3.12) is an eigenfuncion of he pricing operaor wih he eigenvalue (3.13). We can hen verify ha M := S e λπ(x ) π(x 0 ) is a maringale (wih M 0 = 1). We can use i o define a new probabiliy measure Q π F := M P F associaed wih he eigenfuncion π(x). The dynamics of X under Q π follows from Girsanov s Theorem. We sress ha π(x) is he eigenfuncion of he pricing semigroup operaor, raher han merely an eigenfuncion of he generaor. I is generally possible for an eigenfuncion of he generaor o fail o be an eigenfuncion of he semigroup. Tha case will lead o a mere local maringale. In our case, π(x) is an eigenfuncion of he semigroup by consrucion, and he process M is a maringale, raher han a mere local maringale. We now show ha he condiion (2.1) holds under our assumpions in Theorem 3.2. We firs re-wrie i under he probabiliy measure Q π : lim T EQπ [ P T P T 0 B ] 1 = 0. (3.19) We will now verify ha his indeed holds under our assumpions. Firs observe ha by Eq.(3.4): P T P T 0 B = e λ (Φ(T ) φ(t)) (Ψ(T ) v) (X X 0 ). Since lim T Ψ(T) = v and lim T Φ (T) = λ, we have ha lim T P T P T 0 B almos surely. Nex, we show L 1 convergence. Firs, we observe ha for any ǫ > 0 here exiss T 0 such ha for all T > T 0 = 1 Ψ i (T ) v i ǫ 11

12 for all i I and Thus, P T P0 T B e λ (Φ(T ) φ(t))+(ψ(t) v) X 0 1+ǫ P T P0 T B 1+(1+ǫ) e k X. k i =±ǫ Since X remains affine under Q π, by Theorem 4.1 of Filipović and Mayerhofer (2009) here exiss ǫ > 0 such ha e k X is inegrable under Q π for all vecors k such ha k i = ±ǫ. Thus, by he Dominaed Convergence Theorem, Eq.(3.19) holds. This proves (i) and (ii) (Eq.(3.14) follows from Eq.(3.5) and he fac Φ () λ as ). (iii) follows from Eq.(3.11) and Io s formula. To prove (iv), we noe ha by Theorem 2.1 M is a maringale. By Iô s formula, is volailiy is λ. This proves (iv). Par (v) follows from Eq.(3.17). To prove (vi), firs noe ha Eq.(3.16) and Girsanov s heorem implies ha W L = W P + 0 λ s ds is an L-Brownian moion. The dynamics of X and B under L hen follows. The economic meaning of Theorem 3 is ha he exisence of a fixed poin v of he soluion o he Riccai equaion is sufficien for exisence of he long erm limi. The fixed poin v iself idenifies he volailiy of he long bond in Eq.(17) and he long-erm zero-coupon yield in Eq.(16) via he principal eigenvalue (15). We noe ha he condiion in Theorem 3.2 of Qin and Linesky (2017) is auomaically saisfied in affine models. Indeed, from Eq.(3.5) when he Riccai equaion has a fixed poin v, from Theorem 3.2 in his paper we have P(T,x) lim T P(T,x) = e λ, and we can wrie P(,x) = e λ L x (), where L x () = e λ P(,x) is a slowly varying funcion of ime for each x. By Eq.(3.14), he eigenvalue λ is idenified wih he asympoic long-erm zero-coupon yield. We noe ha since Ψ() = v is a saionary soluion of he Riccai ODE (3.3), he vecor v saisfies he following quadraic vecor equaion: 1 2 v α i v +β i v δ i = 0, i I, B JJ v J δ J = 0. However, in general his quadraic vecor equaion may have muliple soluions leading o muliple exponenial-affine eigenfuncions. In order o deermine he soluion ha defines he long-erm facorizaion, if i exiss, i is essenial o verify ha v is he limiing soluion of he Riccai ODE, i.e. ha Eq.(3.10) holds. In his regard, we recall ha Qin and Linesky (2016) idenified he unique recurren eigenfuncion π R of an affine pricing kernel wih he minimal soluion of he quadraic vecor equaion (see Appendix F in he on-line e-companion o Qin and Linesky (2016)). We recall ha, for a Markovian pricing kernel S (see Hansen and Scheinkman (2009) and 12

13 Qin and Linesky (2016)), we can associae a maringale M π = S e λπ(x ) π(x 0 ) wih any posiive eigenfuncion π(x). In general, posiive eigenfuncions are no unique. Qin and Linesky (2016) proved uniqueness of a recurren eigenfuncion π R defined as such a posiive eigenfuncion of he pricing kernel S, i.e. E P x[s π(x )] = e λ π(x) for some λ, ha, under he locally equivalen probabiliy measure (eigen-measure) Q π R defined by using he associaed maringale M π R as he Radon-Nikodym derivaive, he Markov sae process X is recurren. However, in general, wihou addiional assumpions, he recurren eigenfuncion π R associaed wih he minimal soluion o he quadraic vecor equaion may or may no coincide wih he eigenfuncion π L germane o he long-erm limi and, hus, he long forward measure may or may no coincide wih he recurren eigenmeasure (he fixed poin v of he Riccai ODE may or may no be he minimal soluion of he quadraic vecor equaion). Under addiional exponenial ergodiciy assumpions he fixed poin of he Riccai ODE is necessarily he minimal soluion of he quadraic vecor equaion and π R = π L. If he exponenial ergodiciy assumpion is no saisfied, hey may differ, or one may exis, while he oher does no exis. We refer he reader o Qin and Linesky (2016) and Qin and Linesky (2017) for he exponenial ergodiciy assumpion. Analyical racabiliy of affine models allows us o provide fully explici examples o illusrae hese heoreical possibiliies. In he nex secion we give a range of examples. 4 Examples 4.1 Cox-Ingersoll-Ross Model Suppose he sae follows a CIR diffusion (Cox e al. (1985)): dx = (a κ P X )d+σ X dw P, (4.1) where a > 0, σ > 0, κ P R, and W P is a one-dimensional sandard Brownian moion (in his case m = d = 1 and n = 0). Consider he CIR pricing kernel in he form (3.2). The shor rae is given by (3.6) wih g = γ + au and h = δ uκ P u 2 σ 2 /2. For simpliciy we choose γ = au and δ = 1+uκ P +u 2 σ 2 /2, so ha he shor rae can be idenified wih he sae variable, r = X. The marke price of Brownian risk is λ = σu X. Under Q he shor rae follows he process (3.9), which is again a CIR diffusion, bu wih a differen rae of mean reversion: κ Q = κ P +σ 2 u. 13

14 The fixed poin v of he Riccai ODE Ψ () = 1 2 σ2 Ψ 2 () κ P Ψ()+δ wih he iniial condiion Ψ(0) = u can be readily deermined. Since 1 2 u2 σ 2 uκ P + δ = 1 > 0, we know ha Ψ(0) = u is beween he wo roos of he quadraic equaion 1 2 σ2 x 2 κ P x +δ = 0. This immediaely implies ha Ψ() converges o he larger roo, i.e. lim Ψ() = κ 2 P +2σ 2 δ κ P σ 2 = κ 2 Q +2σ2 κ P where we inroduce he following noaion: κ L = κ 2 Q +2σ2. Thus, he long bond in he CIR model is given by σ 2 B = e λ κ L κ Q σ 2 (X X 0 ) = κ L κ P σ 2 =: v, wih and he long bond volailiy λ = a(κ L κ Q ) σ 2 (4.2) σ = κ L κ Q X. σ Under he long forward measure he sae follows he process (3.18), which is again a CIR diffusion, bu wih he differen rae of mean reversion κ L > κ Q. The fixed poin v is proporional o he difference beween he rae of mean reversion under he long forward measure L and he daa generaing measure P. I defines he marke price of risk under L via λ = vσ X. We noe ha if one selecs u = ( κ P ± κ 2 P 2σ2 )/σ 2 in he specificaion of he pricing kernel, hen v = 0 and λ = 0, so he margingale componen in he long erm facorizaion is degenerae, and he pricing kernel is in he ransiion independen form. In his case, κ P = κ L so ha he daa-generaing measure coincides wih he long-forward measure. This is he condiion of Ross recovery heorem (see Qin and Linesky (2016) for more deails). Since he closed form soluion for he CIR zero-coupon bond pricing funcion is available(cox e al.(1985)), hese resuls can also be recovered by direcly calculaing he limi P(T,y) lim = e λπ(y) T P(T,x) π(x) 14

15 wih he eigenvalue λ given by Eq.(4.2) and he eigenfuncion π(x) = e κ L κ Q σ 2 x. Remark 4.1. Borovička e al. (2016) in heir Example 4 on p.2513 also consider an exponenial-affine pricing kernel driven by a single CIR facor. However, heir specificaion of he PK is in a special form such ha h = 0 in Eq.(4) for he shor rae (which corresponds ohe choice δ = uκ P +u 2 σ 2 /2inour parameerizaion). Thus, all dependence on he CIR facor is conained in he maringale componen in he riskneural facorizaion of heir PK, wih he shor rae being consan. In his special case he long bond is deerminisic and he long forward measure is simply equal o he risk-neural measure since he shor rae is independen of he sae variable. In his special case he pricing operaor has wo disinc posiive eigenfuncions. One of he eigenfuncions is consan. This eigenfuncion defines he risk-neural measure, which coincides wih he long forward measure in his case due o independence of he shor rae and he eigenfuncion of he sae variable. The second eigenfuncion (Eq.(19) in Borovička e al. (2016)) defines a probabiliy measure, which is disinc from he risk-neural measure and, hence, disinc from he long forward measure as well. Depending on he specific parameer values of he CIR process, eiher one of he wo eigenfuncions may serve as he recurren eigenfuncion. The eigenmeasure associaed wih he oher eigenfuncion will no be recurren, as he CIR process will have a non-mean revering drif under ha measure. 4.2 CIR Model wih Absorpion a Zero: L Exiss, Q π R Does No Exis We nex consider a degenerae CIR model (4.1) wih a = 0, σ > 0, and κ R. When a vanishes, he diffusion has an absorbing boundary a zero, i.e. here is a posiive probabiliy o reach zero in finie ime and, once reached, he process says a zero wih probabiliy one for all subsequen imes. Consider a pricing kernel in he form of Eq.(3.2). The shor rae is given by (3.6) wih g = γ and h = δ uκ P 1 2 u2 σ 2. We assume γ = 0 and δ = 1+uκ P u2 σ 2 > 0, so ha shor rae r akes values in R +. The marke price of Brownian risk is λ = σu X, and under Q he shor rae follows he process (3.9), which is again a CIR diffusion wih an absorbing boundary a zero, bu wih a differen rae of mean reversion κ Q = κ P +σ 2 u. I is clear ha under any locally equivalen measure, zero remains absorbing and hus no recurren eigenfuncion exiss. Neverheless, we can proceed in he same way as in our analysis of he CIR model o show ha B = e κ L κ Q σ 2 (X X 0 ) wih κ L = κ 2 Q +2σ2 is he long bond and X solves he CIR SDE (4.1) wih a = 0 and mean-revering rae κ L under L. In fac, he reamen of he long bond and he long forward measure is exacly he same as in he non-degenerae example wih a > 0, even hough his case is ransien wih absorpion a zero. The eigenvalue 15

16 degeneraes in his case, λ = 0, and he asympoic long-erm zero-coupon yield vanishes, corresponding o he evenual absorpion of he shor rae a zero. 4.3 Vasicek Model Our nex example is he Vasicek (1977) model wih he sae variable following he OU diffusion: dx = κ(θ P X )d+σdw P wih κ > 0, σ > 0 (in his case m = 0, n = d = 1). Consider he pricing kernel in he form (3.2). The shor rae is given by (3.6) wih g = γ+uκθ P 1 2 u2 σ 2 and h = δ uκ. For simpliciy we choose γ = uκθ P u2 σ 2 and δ = 1+uκ, so ha he shor rae is idenified wih he sae variable, r = X. The marke price of Brownian risk is consan in his case, λ = σu. Under Q he shor rae follows he process (3.9), which in his case is again he OU diffusion, bu wih a differen long run mean θ Q = θ P σ2 u κ (he rae of mean reversion κ remains he same). The explici soluion o he ODE Ψ () = κψ()+δ wih he iniial condiion Ψ(0) = u is Ψ() = ( δ κ +u)e κ + δ κ, and he limi yields he fixed poin lim Ψ() = δ =: v. Thus, he long bond in he κ Vasicek model is given by wih he long-erm yield and he long bond volailiy B = e λ 1 κ (X X 0) λ = θ Q σ2 2κ 2 σ = σ κ. Under he long forward measure he shor rae follows he process (3.18), which is again he OU diffusion, bu wih a differen long run mean θ L = θ Q σ2 κ 2 (he rae of mean reversion remains he same). 16

17 4.4 Non-mean-revering Gaussian Model: Q π R Exiss, L Does no Exis Suppose X is a Gaussian diffusion wih affine drif and consan volailiy dx = κ(θ X )d+σdw P, bu nowwihκ<0, sohaheprocess isnomean-revering. Consider arisk-neural pricing kernel ha discouns a he rae r = X, i.e. S = e 0 Xsds. Then he pure discoun bond price is given by P T = P(X,T ) wih P(x,) = A()e xb(), } B() = 1 e κ, A() = exp {(θ σ2 σ2 κ 2κ2)(B() ) 4κ B2 (). I is easy o see ha he raio P(y,T )/P(x,T) does no have a finie limi as T and, hence, P T/PT 0 does no converge as T. Thus, he long bond and he long forward measure L do no exis in his case. However, he recurren eigenfuncion π R and he recurren eigen-measure Q π R do exis in his case and are explicily given in Secion of Qin and Linesky (2016). Under Q π R, X is he OU process wih mean reversion (since κ < 0): 4.5 Breeden Model dx = (σ 2 /κ κθ+κx )d+σdw Qπ R. Our nex example is a special case of Breeden (1979) consumpion CAPM considered in Example 3.8 of Hansen and Scheinkman(2009). There are wo independen facors, a sochasic volailiy facor X v evolving according o he CIR process dx v = κ v(θ v X v )d+σ v X v dw v,p and a mean-revering growh rae facor X g evolving according o he OU process dx g = κ g (θ g X g )d+σ g dw g,p. Here i is assumed ha κ v,κ g > 0, θ v,θ g > 0, σ g > 0, σ v < 0 (so ha a posiive incremen o W v reduces volailiy), and 2κ v θ v σv 2 (so ha volailiy says sricly posiive). Suppose ha equilibrium consumpion evolves according o dc = X g d+ X v dw v,p +σ c dw g,p, where c is he logarihm of consumpion C. Thus, X g models predicabiliy in he growh rae and X v models predicabiliy in volailiy. Suppose also ha he 17

18 represenaive consumer s preferences are given by [ ] E e bc1 a 1 1 a d for a,b > 0. Then he implied pricing kernel S is ( S = e b C a = exp a Xs g ds b a X v s dw v,p s Using he SDEs for X g and X v i can be cas in he affine form (3.2): ( S = exp γ a σ v (X v Xv aσc 0 ) σ g (X g X g 0 ) aκv σ v 0 Xv sds (a+ aσcκg σ g ) ) 0 Xg sds, where γ = b aκvθv σ v aσcκgθg σ g. ) a σ c dw g,p. 0 Proposiion 4.1. If κ g > 0 (mean-revering growh rae) and κ v + κ 2 v +2aκ v σ v + aσ v > 0, Eq.(3.10) holds and, hus, Theorem 3.2 applies. The long bond is given by ( B = exp λ+( a ) v 1 )(X v σ Xv 0 )+(aσ c v 2 )(X g X g 0 v σ ), g where λ = γ 1 2 σ2 gv2+κ 2 v θ v v 1 +κ g θ g v 2, v 1 = ( κ 2 v +2aκ v σ v κ v )/σv, 2 v 2 = a(1/κ g + σ c /σ g ), and he sae variables have he following dynamics under L: dx (κ v = v θ v ) κ 2 v +2aκ vσ v X v d+σ v X v dw v,l, dx g = κ g ( θ g aσ2 g κ 2 g Proof. In his model Eq.(3.3) reduces o aσ cσ g κ g X g ) d+σ g dw g,l. Φ () = 1 2 σ2 g Ψ 2() 2 +κ v θ v Ψ 1 ()+κ g θ g Ψ 2 ()+γ, Φ(0) = 0, Ψ 1 () = 1 2 σ2 v Ψ 1() 2 κ v Ψ 1 ()+ aκ v σ v, Ψ 1 (0) = a σ v, Ψ 2 () = κ gψ 2 ()+a+ aσ cκ g σ g, Ψ 2 (0) = aσ c σ g. In his special case Ψ 1 () and Ψ 2 () are separaed and hus can be analyzed independenly. I is easy o see ha if κ g > 0 hen Ψ 2 () converges o v 2. When κ v + κ 2 v +2aκ a vσ v +aσ v > 0, σ v is greaer han he smaller roo of he second order equaion 1 2 σ2 v Ψ 1() 2 κ v Ψ 1 ()+ aκv σ v, which implies ha Ψ 1 () converges o he larger roo of he second-order equaion for v 1. The eigenvalue and he dynamics of he sae variable can be compued accordingly.. 18

19 The proof essenially combines he proofs in Examples 4.1 and 4.3. Similar o hese examples, we observe ha he rae of mean reversion of he volailiy facor is higher under he long forward measure, κ 2 v +2aκ vσ v > κ v, while he rae of mean reversion of he growh rae remains he same, bu is long run level is lower under L. 4.6 Borovička e al. (2016) Coninuous-Time Long-Run Risks Model Our nex example is a coninuous-ime version of he long-run risks model of Bansal and Yaron (2004) sudied by Borovička e al. (2016). I feaures growh rae predicabiliy and sochasic volailiy in he aggregae consumpion and recursive preferences. The model is calibraed o he consumpion dynamics in Bansal and Yaron (2004). The wo-dimensional sae modeling growh rae predicabiliy and sochasic volailiy follows he affine dynamics: ] X 1 d[ X 2 = ([ ] [ ][ ]) X 1 [ ] X 2 d+ X 1 d [ ] W 1,P W 2,P where W i,p, i = 1,2, are wo independen Brownian moions. Here X 1 is he sochasic volailiy facor following a CIR process and X 2 is an OU-ype mean-revering growh rae facor wih sochasic volailiy. The aggregae consumpion process C in his model evolves according o dlogc = d+X 2 d+ X dW 3,P, where W 3,P is a hird independen Brownian moion modeling direc shocks o consumpion. Numerical parameers are from Borovička e al. (2016) and are calibraed o monhly frequency (here ime is measured in monhs). The represenaive agen in his model is endowed wih recursive homoheic preferences and a uniary elasiciy of subsiuion. Borovička e al. (2016) solve for he pricing kernel: [ ] dlogs = d X 1 d X2 d X dw P where he hree-dimensional Brownian moion W P = (W i,p ) i=1,2,3 is viewed as a column vecor. We now cas his model specificaion in he hree-dimensional affine form of Assumpion 3.2. To his end, we inroduce a hird facor X 3 = logs. We can hen wriehepricing kernel inheexponenial affineforms = e X3, where hesaevecor (X 1,X2,X3 )followsahree-dimensionalaffinediffusiondrivenbyahree-dimensional Brownian moion: dx = (b+bx )d+ X 1 ρdw P, where he numerical values for enries of he hree-dimensional vecor b and ,,

20 marices B and ρ are given above. We can now direcly apply our general resuls for affine pricing kernels. Firs, by Theorem 3.1, he shor rae is r(x ) = X 1+X2 and depends only on he facors X 1 and X 2 and is independen of X 3. The risk-neural (Q-measure) dynamics is given by: X X d X 2 1 = X 2 X 3 d+ XρdW 1 Q X 3, where ρ = The vecor Ψ() = (Ψ 1 (),Ψ 2 (),Ψ 3 ()) solves he ODE (here α := ρρ ): Ψ 1 () = 1 2 Ψ() αψ()+b 11 Ψ 1 ()+B 21 Ψ 2 ()+B 31 Ψ 3 (), Ψ 2 () = B 22Ψ 2 ()+B 32 Ψ 3 (), Ψ 3 () = 0 wih Φ(0) = Ψ 1 (0) = Ψ 2 (0) = 0,Ψ 3 (0) = 1. I is immediae ha and, since B 22 < 0, Ψ 3 () 1 and Ψ 2 () = B 32 B 22 (1 e B 22 ) lim Ψ 2() = B 32 /B 22 = := v 2. To see Ψ 1 () convergence, noice ha we can wrie 1 2 Ψ() αψ() + B 11 Ψ 1 () + B 21 Ψ 2 ()+B 31 Ψ 3 () = c 1 (Ψ 1 ()) 2 +c 2 Ψ 1 ()+c 3 (Ψ 2 ()) 2 +c 4 Ψ 2 ()+c 5,wherec 1,c 2,c 3,c 4,c 5 < 0. Since Ψ 1 (0) = Ψ 2 (0) = 0, we have Ψ 1(0) < 0. Since Ψ 2 () > 0 and i is easy o see ha Ψ 1 () < 0. Since Ψ 2 () < v 2, we have c 1 (Ψ 1 ()) 2 + c 2 Ψ 1 () + c 3 (Ψ 2 ()) 2 + c 4 Ψ 2 () + c 5 > c 1 (Ψ 1 ()) 2 + c 2 Ψ 1 () + c 3 v c 4 v 2 + c 5. We can check ha c 1 (Ψ 1 ()) 2 +c 2 Ψ 1 ()+c 3 v 2 2 +c 4v 2 +c 5 = 0 has wo negaive roos. Denoe he larger roo v 1, we see ha Ψ 1 () > v 1. Combining hese facs, we see ha Ψ 1 () converges o v 1. The exac value of v 1 has o be deermined numerically. The numerical soluion yields v 1 = lim Ψ 1 () = In Figure 1, we plo he funcions Ψ 1 () and Ψ 2 (), as well as he gross reurn B +T on he T-bond over he period [0,] as a funcion of T. In his numerical example we ake = 12 monhs, so we are looking a he one-year holding period reurn, and assume ha he iniial sae X 0 and he sae X are boh equal o he saionary 20

21 mean under P. We observe ha in his model specificaion Ψ() and B +T very close o he fixed poin for around 30 years (360 monhs). are already Ψ 1 Ψ B +T T Figure 1: Plo of Ψ 1 (), Ψ 2 () and B +T. Time is measured in monhs. ByTheorem3.2,heeigenfunciondeermininghelongbondisπ(x) = e v 1x 1 v 2 x 2, corresponding o he eigenvalue (noe his is no annualized yield since ime uni is in monh) he long bond is given by he maringale componen is given by λ = b 1 v 1 +b 2 v 2 b 3 = , B = e λ v 1(X 1 X1 0 ) v 2(X 2 X2 0 ), M = e λ v 1(X 1 X1 0 ) v 2(X 2 X2 0 )+X3, and he sae vecor (X 1,X2,X3 ) has he following dynamics under he long forward measure L: X X d X 2 1 = X 2 X 3 d+ XρdW 1 L X 3. 21

22 As already observed by Borovička e al. (2016), in his model he sae dynamics under he long forward measure L is close o he sae dynamics under he risk-neural measure Q and is subsanially disinc from he dynamics under he daa-generaing measure P due o he volaile maringale componen M. However, our approach o he analysis of his model is differen from he analysis of Borovička e al. (2016). We cas i as a hree-facor affine model and direcly apply our Theorem 3.2 for affine models ha is, in urn, a consequence of our Theorem 2.1 for semimaringale models. We only need o deermine he fixed poin (3.10) of he Riccai equaion. Exisence of he long bond, he long erm facorizaion of he pricing kernel, and he long forward measure hen immediaely follow from Theorem 3.2, wihou any need o verify ergodiciy. In fac, he hree-facor affine process (X 1,X2,X3 ) is no ergodic, and no even recurren, as is immediaely seen from he dynamics of X 3. In conras, he approach in Borovička e al. (2016) relies on he wo-dimensional mean-revering affine diffusion(x,x 1 ). 2 Namely, since heperron-frobeniusheoryofhansen and Scheinkman (2009) requires ergodiciy o single ou he principal eigenfuncion and ascerain is relevance o he long-erm facorizaion, Borovička e al. (2016) implicily spli he pricing kernel ino he produc of wo sub-kernels, a muliplicaive funcional of he wo-dimensional Markov process (X 1,X2 ) and he addiional facor in he form e Xs 1dW3,P s. The Perron-Frobenius heory of Hansen and Scheinkman (2009) is hen applied o he muliplicaive funcional of he wo-dimensional Markov process (X 1,X2 ). In conras, in our approach we do no require ergodiciy and work direcly wih he non-ergodic hree-dimensional process and verify ha he Riccai ODE possesses a fixed poin, which is already sufficien for exisence of he long-erm facorizaion in affine models by Theorem Conclusion This paper consrucs and sudies he long-erm facorizaion of affine pricing kernels ino discouning a he rae of reurn on he long bond and he maringale componen ha accomplishes he change of probabiliy measure o he long forward measure. I is shown ha he principal eigenfuncion of he affine pricing kernel germane o he long-erm facorizaion is an exponenial-affine funcion of he sae vecor wih he coefficien vecor idenified wih he fixed poin of he Riccai ODE. The long bond volailiy and he volailiy of he maringale componen are explicily idenified in erms of his fixed poin. When analyzing a given affine model, a research needs o esablish wheher he Riccai ODE possesses a fixed poin. If he fixed poin is deermined, he long-erm facorizaion hen follows. I is shown how he long-erm facorizaion plays ou in a variey of asse pricing models, including single facor CIR and Vasicek models, a wo-facor version of Breeden s CCAPM, and he hree-facor long-run risks model sudied in Borovička e al. (2016). 22

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