THE TERM STRUCTURE OF INTEREST RATES IN A MARKOV SETTING

THE TERM STRUCTURE OF INTEREST RATES IN A MARKOV SETTING Rober J. Ellio Haskayne School of Business Universiy of Calgary Calgary, Albera, Canada rellio@ucalgary.ca Craig A. Wilson College of Commerce Universiy of Saskachewan Saskaoon, Saskachewan, Canada cwilson@commerce.usask.ca May 8, 2003 Absrac An ineres rae model is described in which randomness in he shor-erm ineres rae is due enirely o a Markov chain. We model randomness hrough he mean-revering level raher han hrough he ineres rae direcly. The shor-erm ineres rae is modeled in a risk-neural seing as a coninuous process in coninuous ime. This allows he valuaion of ineres rae derivaives using he maringale approach. In paricular, a closed-form soluion is found for he value of a zero-coupon bond. We also show how o incorporae he iniial erm srucure o calibrae he model for arbirage-free bond prices. The model is shown o encapsulae several empirically observed feaures associaed wih ineres rae ime series. Key Words: Ineres Rae Modeling, Term Srucure, Derivaives, Markov Chain 1

1 Inroducion Curren models of he shor-erm ineres rae ofen involve reaing he shor rae as a diffusion or jump diffusion process in which he drif erm involves exponenial decay oward some value. The basic models of his ype are Vasiček [1] and Cox, Ingersoll and Ross [2], where he disincion beween hese wo ineres rae models ress wih he diffusion erm. The drif erm, (of boh models), ends o cause he shor rae process o decay exponenially owards a consan level. This feaure is responsible for he mean-revering propery exhibied by hese processes. An exension o hese models has come in he form of allowing he drif o incorporae exponenial decay oward a manifold, raher han a consan. This is known as he Hull and Whie [3] model, and i allows he shor rae process he endency o follow he iniial erm srucure of ineres raes. This is an imporan exension, because wih a judicious choice of he manifold, he iniial erm srucure prediced by he model can exacly mach he exising erm srucure. In general, his canno be done wih a consan mean-revering level. Alhough here are many oher exensions o he basic models incorporaing sochasic volailiy, non-linear drif (so decay is no longer exponenial), and jumps, for example he Hull-Whie exension is he mos applicable o our sudy. The Hull-Whie model is described under he risk-neural probabiliy by he sochasic differenial equaion dr = (){ r() r } d + σ()r ρ dw, (1) where r represens he shor-erm, coninuously compounded ineres rae, and {W } is a Brownian moion under he risk-neural probabiliy. The parameer ρ akes one of he wo values 0 or 1/2, depending on wheher i exends he Vasiček or Cox-Ingersoll-Ross model. The parameer funcions (), r(), and σ() exend he basic models, in which hese parameers are jus consans. The randomness in his model comes from he Brownian moion, and for he exended Vasiček model when ρ = 0, i can be inerpreed as adding whie noise o he shor rae. For he exended Cox-Ingersoll-Ross model he noise is muliplicaive, bu i is sill applied direcly o he shor rae process. 2

The main problem wih his model is in he way i handles he cyclical naure of ineres raes. A ime series of ineres raes ends o appear cyclical because he supply and demand for money is closely relaed o income growh, which flucuaes wih he business cycle. This has implicaions for real (adjused for inflaion) ineres raes. For example, a a business cycle peak shor-erm raes should be rising and a a rough raes should be falling. This also has implicaions for he slope of he erm srucure i should be seeper a a peak and flaer a a rough. Roma and Torous [4] find ha his propery of real ineres raes canno be explained by a simple addiive noise ype model, such as Vasiček. The Hull-Whie exension can provide a correcion for his problem o a degree, bu since he parameer funcions are deerminisic, i implies ha he business cycle effecs are known wih cerainy, which does no allow for he possible variaion in lengh and inensiy. In addiion, when he cenral bank arges a consan rae of inflaion, his flucuaion is ransferred o nominal ineres raes, so he same characerisics could apply o hem. We approach his problem by modeling he mean-revering level direcly as a random process, and have he shor rae chase he mean-revering level in a linear drif ype model. The mean-revering level is assumed o follow a finie sae, coninuous ime Markov chain. The swiching of he Markov chain o differen levels produces a cyclical paern in he shor rae ha is consisen wih he above effec, and he randomness inheren in he Markov chain prevens he business cycle lenghs and inensiies from being compleely predicable. The remainder of his aricle is planned ou as follows. Secion 2 discusses our model for he shor rae. We look a deails of he Markov chain and how i relaes o he shor rae. In Secion 3 we use our shor rae model o value a zero-coupon bond. A closed-form soluion is provided. This also provides a closed-form soluion for longer-erm ineres raes by invering he bond price. We also show ha he model can be calibraed using he iniial erm srucure of zero-coupon bond prices. Finally Secion 4 concludes, and some mahemaical deails are included in he appendix. 3

2 The Model We model uncerainy in he economy wih a probabiliy space denoed (Ω, F, P ), where Ω is a se of possible saes of he economy ha is assumed o be large enough o suppor he Markov chain defined below, F is a sigma field over subses of Ω ha is assumed o be large enough o suppor he filraion generaed by he Markov chain, and P is a risk-neural probabiliy measure. Several properies of he Markov chain are described in he following subsecion, and afer ha we describe how he shor-erm ineres rae is o be modeled. 2.1 The Markov Chain We denoe by {X } a homogeneous, righ coninuous Markov chain aking values in he sae space of uni vecors S = {e 1,..., e N }, where e i is a vecor in R N wih 1 in he i h coordinae and 0 elsewhere. We denoe he righ coninuous compleion of he filraion generaed by he Markov chain as {F X }. The filraion models he informaion abou he realised sae of he economy, ω Ω, ha is revealed by observing he pah of he Markov chain. The Markov chain, X, is assumed o change saes according o a ransiion inensiy marix A, in which he columns sum o zero and he offdiagonal enries are non-negaive. Therefore he risk-neural probabiliy of going from sae e i o e j beween ime and + s is he enry from he j h row and he i h column of he marix exponenial e As. We also assume ha he ransiion inensiy marix, A, is symmeric. This is a resricive assumpion because i implies ha he risk-neural probabiliy of swiching from one sae o anoher is equal o he probabiliy of swiching back over any equivalen ime inerval. We mainain his assumpion because i allows for convenien closed-form soluions of he zero-coupon bond price and oher quaniies of ineres. Our specificaion of he sae space allows he sochasic process X o have he following semi-maringale represenaion X = 0 AX s ds + M, (2) 4

where he vecor process {M } is a righ coninuous, square inegrable, zeromean maringale wih respec o he filraion {F X } and he measure P, (see Ellio [5]). From Equaion 2 we can wrie he dynamics of X as dx = AX d + dm, (3) and by applying Fubini s heorem he condiional expecaion of X +s given is F X E[X +s F X ] = X + s AE[X u X ] du = e As X. (4) This condiional expecaion is he condiional probabiliy disribuion vecor ha he Markov chain will be in each sae. 2.2 The Shor-Term Ineres Rae The shor-erm ineres rae is modeled by he sochasic process denoed by {r }, which is assumed o have dynamics given by he following ordinary differenial equaion dr = (){ r(, X ) r } d. (5) The quaniy r(, X ) is he level ha he process ends oward, and (), which is assumed o be a posiive valued funcion, is he rae a which he mean-revering level is approached. In general, hese dynamics do no describe a saionary process, because he parameers may vary wih ime, (his is also rue of he Hull-Whie model); however, he process can be made saionary by requiring ha he parameers depend on ime only hrough he Markov chain. Jus as in he Hull-Whie model, he ime dependence is allowed so ha he iniial erm srucure can be mached exacly. In paricular we can choose how r(, e i ) follows ime in each sae in order o mach he iniial erm srucure of ineres raes and () so ha he erm srucure of volailiies is mached. The main difference beween he sochasic process described in Equaion 5 and he Hull-Whie dynamics of Equaion 1 is ha insead of incorporaing noise ino he shor rae using a diffusion erm, randomness eners he shor rae hrough he mean-revering level. This is a subsanial change because i means ha ineres rae sample pah is differeniable. We use a 5

Markov chain o generae randomness raher han a Brownian moion because i seems inuiively beer suied o deal wih he business cycle effec. The soluion o Equaion 5 saring a ime s is obained by variaion of consans, ( ){ ( u ) } r = exp (u) du r s + exp (v) dv (u) r(u, X u ) du. (6) s s s We wrie he filraion generaed by his sochasic process as {F r }, and by noicing ha r only depends on X u for u, we can see ha {r } is adaped o {F X }. Noice ha because he mean-revering rae, (), does no depend on he Markov chain, he exponenial erm in Equaion 6 is a deerminisic quaniy. We will make use of his observaion for valuing zero-coupon bonds under he model. The expeced fuure shor rae can be easily deermined from Equaions 4 and 6. Applying Tonelli s heorem o ake he expecaion operaor hrough he inegral sign and wriing r(u, X u ) = r(u) T X u, where r(u) = ( r(u, e 1 ),..., r(u, e n )) T, gives ( E[r Fs r ] = exp s { r s + ) (u) du (7) ( u ) } exp (v) dv (u) r(u) T e A(u s) E[X s Fs r ] du. s s We analyse his expression in he conex of a long-erm mean-revering propery in Subsecion 3.2 in he case of consan parameer funcions. 3 Zero-Coupon Bonds This secion provides he deails of deermining he value of a defaul free zero-coupon bond ha pays $1 wih cerainy a a fixed mauriy ime, T, wih no oher cash flows. We denoe he value a ime of a zero-coupon bond mauring a ime T by B(, T ), and under he risk-neural measure, is value is deermined by solving [ ( T B(, T ) = E exp r u du ) F r ]. (8) 6

We firs proceed o solve for his quaniy in he general case, and hen we specialise he general resuls o he paricular case when he parameer funcions do no vary direcly wih ime. This special case will be imporan when we calibrae he model o he iniial erm srucure of zero-coupon bond prices. 3.1 The General Case To solve for he quaniy in Equaion 8, we mus firs deermine he inegral of he shor rae process. This can be done by inegraing Equaion 6; however i will also be convenien o have he erm r(, X ) in he ouer inegral he reason for his is made clear in he appendix where Resul 1 is proven. We can accomplish his by inerchanging he order of inegraion according o Tonelli s heorem. This gives T r u du = r T where { T β(, u, T, X u ) = u ( u ) T exp (v) dv du + β(, u, T, X u ) du, (9) ( v ) } ( u ) exp (w) dw dv exp (w) dw (u) r(u, X u ). (10) The firs erm in Equaion 9 is F r -measurable and i can be deermined a leas by numerical inegraion i can be deermined analyically for some special cases, in paricular when () is consan. This means our ask in solving Equaion 8 is reduced o finding E[exp( T β(, u, T, X u) du) F r ]. We find his value in hree seps. Firs we consruc a vecor sochasic process by pos muliplying he exponenial erm by he Markov chain, X, hen we find he expeced value of ha vecor process by making use of he semi-maringale represenaion in Equaions 2 and 3, and from ha we can deermine he expeced value of he exponenial erm as desired. The soluion we find is in erms of a linear sysem of ordinary differenial equaions, bu he approach is somewha echnical, so we summarise his in he following resul and leave he deails o he appendix. Resul 1 If P is a risk-neural probabiliy and he risk free shor-erm ineres rae is characerised by Equaion 5, hen he value a ime of a 7

zero-coupon bond paying $1 a ime T is T ( u ) } B(, T ) = exp { r exp (v) dv du 1 T Φ(, T )E[X F r ], (11) where 1 is an N-dimensional column vecor wih 1 in each enry, and Φ(, v) = diag [ e v β(,u,t,e 1)du,..., e v β(,u,t,e N )du ] Qdiag[e λ 1(v ),..., e λ N (v ) ]Q 1 (12) is he fundamenal marix saring a ime for he N-dimensional linear sysem of ordinary differenial equaions, y (u) = {A diag[β(, u, T, e 1 ),..., β(, u, T, e N )]}y(u) (13) evaluaed a ime u = T, and β(, u, T, e i ) is given in Equaion 10. Here Q is an N N marix whose columns are eigenvecors of he ransiion inensiy marix A, and he λ i s are he corresponding eigenvalues. From Resul 1 we can see ha our model of he shor-rae leads o an affine erm srucure. To deermine longer-erm ineres raes we jus solve for he yield of he zero-coupon bonds, R(, T ) = ln{b(, T )}/(T ). 3.2 Consan Parameers An imporan special case occurs when he parameer funcions do no vary direcly wih ime. As menioned earlier his resuls in a saionary shor rae process, which is an imporan feaure for analysing ineres rae ime series, paricularly for real ineres raes ha have been adjused for inflaion. This secion replaces he main resuls from he general case wih inegraion performed when possible. We rewrie he shor rae dynamics as we rewrie he soluion o Equaion 14 as dr = { r(x ) r } d, (14) r = e {r ( s) s + e (u s) r(x } u ) du, (15) s 8

and he funcion β defined in Equaion 10 akes he form β(, u, T, X u ) = β(t u, X u ) = {1 e (T u) } r(x u ). (16) Taking he condiional expecaion of Equaion 15 gives { } E[r Fs r ] = e ( s) r s + r T e (u s) e A(u s) due[x s Fs r ]. (17) Noe ha he above inegral exiss in any case; however if he marix A+I is inverible, (which i will be for all choices of alpha excep N paricular values ha depend on he ransiion rae marix A), we can wrie he soluion as follows E[r F r s ] = e ( s) {r s + r T (A + I) 1 {e ( s) e A( s) I}E[X s F r s ]}. (18) Furhermore, if we assume ha he Markov chain is recurren and ergodic, hen aking he limi gives s lim E[r F r s ] = r T (A + I) 1 π, (19) where π is he limiing probabiliy disribuion vecor for he Markov chain. Therefore in he consan parameer case, he expeced ineres rae converges o a consan ha is independen of boh he iniial ineres rae and he iniial sae of he Markov chain. where Applying Resul 1 gives a bond price of 1 e B(, T ) = exp { r (T ) } 1 T Φ(, T )E[X F r ], (20) [ Φ(, v) = diag exp {( v + e (T v) (T ) }] e ) r i Qdiag[e λ1(v ),..., e λ N (v ) ]Q 1. (21) Here again Q is an N N marix whose columns are eigenvecors of A, he λ i s are he corresponding eigenvalues and r i = r(e i ). 9

3.3 Maching he Iniial Term Srucure Since he erm srucure of zero-coupon bond prices can be exacly mached by a one dimensional sysem, such as he Hull-Whie model, i seems clear ha he general form of our model will possess oo much freedom o have he funcions r(, e i ) be uniquely deermined by he iniial erm srucure. In fac, whenever he Markov chain and he associaed funcion r(, X ) have a non-rivial sae space, (so he Markov chain has more han one sae and r(, e i ) r(, e j ) for some saes e i and e j and for all in some subse of ime wih posiive Lebesgue measure), here is very lile srucure imposed on he mean-revering manifold. Inuiively, his is because whaever one r(, e i ) funcion is, we can choose anoher o cancel i ou on average when calculaing zero-coupon bond prices. Because of his freedom, in order o make use of he iniial erm srucure as a useful inpu o he model, we have o specify some relaionship beween mean-revering funcions associaed wih differen saes. One simple possibiliy for accomplishing his is o have he various mean-revering funcions shifed from each oher by consan amouns. This is he approach we presen here. Suppose ha he mean-revering rae () is a consan as in he previous subsecion, and ha he mean-revering level is he sum of wo quaniies, r(, X ) = r() + r T X. (22) Here he r wihou an argumen is a column vecor of consans, and he r() is a real-valued funcion. We assume ha he consans and r i are known, bu he funcion r() is no. The goal is o find r() as a funcion of he oher parameers and he curren erm srucure. We can wrie he funcion β from Equaion 10 as β(, u, T, X u ) = {1 e (T u) } r(u) + {1 e (T u) } r T X u. (23) This means ha he erms from he fundamenal marix Φ(, v) in Equaion 12, e v β(,u,t,e i) = e v {1 e (T u) } r(u) du e v {1 e (T u) } r i du, (24) ake he form of he produc of wo exponenial funcions, he firs of which is independen of e i, and he second involves only he consan parameers. 10

This means ha we can rea he firs erm as a scalar muliple and pull i ou of he diagonal marix so ha he fundamenal marix akes he form Φ(, v) = e v {1 e (T u) } r(u) du Φ cons (, v), (25) where Φ cons (, v) is he fundamenal marix for he consan case described in Equaion 21. Therefore he bond price in his siuaion can be wrien as where B(, T ) = A(, T )B cons (, T ), (26) { T } A(, T ) = exp (1 e (T u) ) r(u) du, (27) and B cons (, T ) is he bond price calculaed from he consan parameer case given in Equaion 20. Since he consan parameers are presumed o be known, he value B cons (, T ) is known, so o find he bond price, we only have o deermine A(, T ). We do his by examining wo quaniies associaed wih he iniial erm srucure he raio of wo zero-coupon bonds and he slope of he erm srucure. We leave he derivaion o he appendix, and jus quoe he resul. Resul 2 If P is a risk-neural probabiliy and he risk free shor-erm ineres rae is characerised by Equaion 5, where () = is a consan and r(, X ) is as given in Equaion 22, hen he value a ime of a zero-coupon bond paying $1 a ime T is given by Equaion 26, where ln{a(, T )} = ln { B(0, T ) B(0, ) } { B cons (0, T )} ln B cons (0, ) ) 1 e (T { ln{b(0, )} B cons (, T ) is given in Equaion 20, and ln{b cons (0, )} ln{b cons(0, )} }, { = r 0 e 1 + B cons (0, ) exp 1 e } r 0 1 T {A diag[(1 e ) r i ]}Φ cons (0, )E[X 0 F r 0 ], where Φ cons (0, ) is given in Equaion 21 wih T replaced by. 11 (28) (29)

From Equaion 27 he funcion r() can be deermined quie easily by differeniaion r(t ) = 1 A(, T ) A(, T ) T 1 T { 1 A(, T ) A(, T )}, (30) T for any [0, T ], and where A(, T ) is obained from Resul 2. As a cavea o his procedure, we should menion ha he well-known problem of inerpolaing erm srucure observaions, (see Jordan [6] for a brief overview of he problem), can cause considerable error in esimaing r() because of he insabiliy of he differeniaion operaion. However, a discussion of his issue is bes lef o an empirical es of he model. 4 Conclusion We modeled he shor-erm ineres rae using a mean-revering equaion wih a random mean-revering level. A Markov chain is used o model he randomness in he mean-revering level. We argue ha his model can explain a business cycle effec in he shor rae ha encompasses boh regulariy in he business cycle and randomness in boh he duraion and severiy of business cycles. From his shor-erm ineres rae model we deermine he value of zero-coupon bonds and we show how o calibrae he model using he iniial erm srucure o obain arbirage-free bond prices. A Appendix A.1 Derivaion of he Zero-Coupon Bond Price According o he discussion following Equaions 9 and 10 we need only deermine E[Z,T F r ], where ( T ) Z,T = exp β(, u, T, X u ) du, (31) and he funcion β is defined in Equaion 10. Now for T fixed, we define a new sochasic process ( v ) Z,T (v) = exp β(, u, T, X u ) du (32) 12

for v T. Thus Z,T = Z,T (T ), and since T eners he above expression in a deerminisic way, Z,T (v) is F X v -measurable. The dynamics of Z,T (v) are given by dz,t (v) = β(, v, T, X v )Z,T (v) dv. (33) Applying Iô s inegraion by pars for general semi-maringales and subsiuing he dynamics from Equaions 3 and 33 gives Z,T (v)x v Z,T ()X = = v v X u dz,t (u) + β(, u, T, X u )Z,T (u)x u du + + [Z,T ( ), X] v, v v Z,T (u ) dx u (34) Z,T (u ){AX u du + dm u } where f(u ) is he lef limi of f as u, and he square bracke erm is he general quadraic covariaion. Equaion 34 can be simplified in a number of ways. Firs, because he Markov chain akes values as uni vecors, he funcion β can be wrien in he form β(, u, T ) T X u, where β(, u, T ) = (β(, u, T, e 1 ),..., β(, u, T, e N )) T. Also, Z,T (v) is differeniable in v, so i is of finie variaion. This means ha he square bracke process jus adds up he producs of he jumps of Z and X. Bu Z,T (v) is coninuous, so i has no jumps. Thus he square bracke erm is idenically zero and he lef limi of Z akes is value, Z,T (u ) = Z,T (u) almos surely. Since X is righ coninuous wih lef limis exising, i has a mos a counable number of disconinuiies, all of he jump ype, so X u = X u for Lebesgue-almos every u [, T ], and he lef limi can be replaced by is righ limi inside he inegral wihou affecing he value. Finally, we can wrie ( β(, u, T ) T X u )X u = diag[ β(, u, T )]X u, again because of our choice of he Markov chain s sae space. (This is why i was imporan for us o apply Tonelli s heorem in Equaion 9 o ge β as a funcion of X u raher han a funcion of he inegral of X u.) This means ha Equaion 34 akes he form Z,T (v)x v Z,T ()X = v (A diag[ β(, u, T )])Z,T (u)x u du+ v Z,T (u) dm u. (35) Because Z,T (v) is lef coninuous and adaped, hence a predicable process, and also bounded on [, T ], and M is a square inegrable maringale, he 13

sochasic inegral of Z wih respec o dm is a zero-mean maringale wih respec o P and he filraion {F X }. Thus he condiional expecaion of he hird erm of Equaion 35 given he informaion F X is zero. Taking condiional expecaion and applying Fubini s heorem o ake he expecaion hrough he inegral sign gives E[Z,T (v)x v F X ] Z,T ()X = v (A diag[ β(, u, T )])E[Z,T (u)x u F X ] du. (36) This is he inegral version of he linear sysem of ordinary differenial equaions described in Equaion 13. Since he coefficien marix, A diag[ β(, u, T )], saisfies he Lappo- Danilevskiĭ condiion of being muliplicaively commuaive wih is own inegral, (a consequence of he ransiion inensiy marix being consan and symmeric), he fundamenal marix of his sysem is jus he marix exponenial of he inegral of he coefficien marix. (For more deails on his resul see Adrianova [7] 4.2 for example.) Therefore he fundamenal marix, call i Φ(, v), akes he form described in Equaion 12, which is clearly deerminisic. The soluion o Equaion 36 is E[Z,T (v)x v F X ] = Φ(, v)z,t ()X. (37) Now because he Markov chain sae space consiss of uni vecors, Z,T (v) = 1 T Z,T (v)x v ; also F r F X and Z,T () = 1, so E[Z,T (T ) F r ] = 1 T Φ(, T )E[X F r ], (38) and he zero-coupon bond value is as given in Equaion 11. This concludes he proof of Resul 1. A.2 Maching he Iniial Term Srucure In his subsecion we prove ha A(, T ) has he form given in Equaion 28. Firs, from Equaions 26 and 27 ln { B(0, T ) B(0, ) } T = (1 e (T u) ) r(u) du + 0 0 (1 e ( u) ) r(u) du + ln { B cons (0, T ) B cons (0, ) 14 }

and Thus ln{b(0, )} T = (1 e (T u) ) r(u) du (39) {1 e (T ) } ln{a(, T )} = ln { B(0, T ) B(0, ) 0 e ( u) r(u) du + ln { B cons (0, T ) B cons (0, ) = 0 (1 e ( u) ) r(u) du + ln{b cons(0, )} = e ( u) r(u) du + ln{b cons(0, )}. (40) 0 } { B cons (0, T )} ln B cons (0, ) ) 1 e (T { ln{b(0, )} ln{b cons(0, )} }. }, (41) Using Equaion 20 we can see ha he derivaive of he consan parameer bond price wih respec o he mauriy is B cons (0, ) From Equaion 21 we find ha Φ cons (0, ) = r 0 e 1 e exp { r 0 1 e + exp { r } 0 [ = diag exp [ +diag exp {( + 1 e {( + 1 e } 1 T Φ cons (0, )E[X 0 F r 0 ] 1 T Φ cons(0, ) E[X 0 F r 0 ]. (42) ) r i }{ 1 + e } r i ] Qdiag[e λ i ]Q 1 ) r i }] Qdiag[e λ i λ i ]Q 1. (43) Since a diagonal marix wih produc enries is equal o he produc of he diagonal marices, he marix Qdiag[λ i ] = AQ, and A is symmeric, he above expression akes he following form Φ cons (0, ) = {A diag[(1 e ) r i ]}Φ cons (0, ). (44) 15

Puing his quaniy ino he consan parameer bond price derivaive gives B cons (0, ) = r 0 e 1 e B cons (0, ) + exp { r } 0 (45) 1 T {A diag[(1 e ) r i ]}Φ cons (0, )E[X 0 F0 r ] so ln{b cons (0, )} { = r 0 e 1 + B cons (0, ) exp 1 e } r 0 1 T {A diag[(1 e ) r i ]}Φ cons (0, )E[X 0 F r 0 ] (46) and rearranging hese equaions shows ha A(, T ) is given as in Equaion 28. This ends he proof of Resul 2. References [1] O. Vasiček, An equilibrium characerizaion of he erm srucure, Journal of Financial Economics, 5, 1977, 177 188. [2] J. C. Cox, J. E. Ingersoll and S. A. Ross, A heory of he erm srucure of ineres raes, Economerica, 53(2), 1985, 385 408. [3] J. Hull and A. Whie, Pricing ineres-rae derivaive securiies, Review of Financial Sudies, 3(4), 1990, 573 592. [4] A. Roma and W. Torous, The cyclical behavior of ineres raes, Journal of Finance, 52(4), 1997, 1519 1542. [5] R. J. Ellio, New finie dimensional filers and smoohers for noisily observed Markov chains, IEEE Transacions on Informaion Theory, 39(01), 1993, 265 271. [6] J. V. Jordan, Term srucure modeling using exponenial splines: Discussion, Journal of Finance, 37(2), 1982, 354 356. [7] L. Ya. Adrianova, Inroducion o Linear Sysems of Differenial Equaions (Rhode Island, American Mahemaical Sociey, 1995). 16