independenly fllwing square-r prcesses, he inuiive inerpreain f he sae variables is n clear, and smeimes i seems dicul nd admissible parameers fr whic

Transcription

1 A MONTE CARLO FILTERING APPROACH FOR ESTIMATING THE TERM STRUCTURE OF INTEREST RATES Akihik Takahashi 1 and Seish Sa 2 1 The Universiy f Tky, Kmaba, Megur-ku, Tky Japan 2 The Insiue f Saisical Mahemaics, Minami-Azabu, Mina-ku, Tky Japan (Received il 14, 2000; Revised e 23, 2000) Absrac. We develp new mehdlgy fr esimain f general class f erm srucure mdels based n a Mne Carl lering apprach. We uilize he generalized sae space mdel which can be naurally applied he esimain f he erm srucure mdels based n he Markv sae prcesses. I is als pssible inrduce measuremen errrs in he general way wihu any bias. Mrever, he Mne Carl ler can be applied even he mdels in which he zer-cupn bnds' prices can n be analyically bained. As an example, we apply he mehd LIBORs (Lndn Iner Bank Oered Raes) and ineres raes swaps in he Japanese marke and shw he usefulness f ur apprach. Key wrds and phrases: Generalized Sae Sapce Mdel, Mne Carl Inegrain, Ineres Rae Mdel, Self-Organizing Mehd 1. Inrducin We prpse a new framewrk f he esimain f he erm srucure f ineres raes based n he sae space mdel. In paricular, we develp a Mne Carl lering apprach fr esimaing he erm srucure mdels based n muli-dimensinal Markv sae variables. Fr example, ur mehd can be applied he erm srucure mdels based n he dynamic general equilibrium hery f Cx, Ingersll and Rss (15a,b) which includes muli-facr CIR (Cx, Ingersll and Rss) mdels used by Chen and Sc (13), Pearsn and Sun (14), Singh (15) and Due and Singlen (19), and he schasic vlailiy mdel develped by Lngsa and Schwarz (12). I is well-knwn ha a leas w sae variables are necessary explain he dynamics f he erm srucure in he real wrld. Chen and Sc (13), Pearsn and Sun (14), Singh (15), and Due and Singlen (19) cncluded ha ne-facr mdels are n enugh describe he variain f he erm srucure by analyzing reasury r swap markes in he Unied Saes. Hwever, muli-facr mdels fen esed in he analysis are n necessarily he bes amng he candidaes. Fr insance, in he muli-facr CIR mdel where he sp ineres rae is described by he sum f several sae variables 1

2 independenly fllwing square-r prcesses, he inuiive inerpreain f he sae variables is n clear, and smeimes i seems dicul nd admissible parameers fr which he sae variables are nn-negaive ver he enire sample perid. (See Due and Singlen (19).) One f he reasns why he mdels which d n explain he daa very well are fen emplyed in empirical analyses is ha hey allw analyic sluins f zer cupn bnds' prices. I is mainly due he limiain f he mehds applied he esimain. In paricular, Chen and Sc (13), Pearsn and Sun (14) and Due and Singlen (19) apply he maximum likelihd mehd while Singh (15), and Lngsa and Schwarz (12) emply he hree-sage leas square mehd wih he principal cmpnen analysis and he generalized mmen mehd(gmm) wih GARCH, respecively. Hwever, i is subsanially dicul apply hse mehds wihu analyic sluins f he zer-cupn bnds' prices. Mrever, exising researches end replace unbservable sae variables such as he sp rae and he vlailiy by sme bservable variables (Chan e al. (12), Lngsa and Schwarz (12)), bu hey may subsanially suer frm he measuremen errrs. While sme f hem ake he measuremen errrs in accun explicily, he ways f he cnsiderain are n naural and smewha ad hc. (Chen and Sc (13), Due and Singlen (19)) We prpse a Mne Carl lering apprach based n he generalized sae space mdel vercme he prblems f exising researches. The sae space mdel cnsiss f he sysem mdel describing he prcesses f sae variables and he bservain mdel represening he funcinal relain beween he sae variables and he bservainal daa in he real wrld, which implies ha he mehd can be naurally applied he esimain f he erm srucure mdels based n he Markv sae prcesses. I is als pssible inrduce measuremen errrs in he general way wihu any bias. Mrever, he Mne Carl ler can be applied much brader class f he erm srucure mdels, especially even he mdels in which he zer-cupn bnds' prices can n be analyically bained. The paper is rganized as fllws. In secin w, we will rs summarize he sae space mdel and he erm srucure mdels based n Markv sae prcesses. Then, we will clarify he relain beween ineres rae mdels as well as bservainal daa and he sae space mdel. Nex, we will give a cncree algrihm f he Mne Carl ler applied he empirical analysis. In secin hree, we will shw he usefulness f he Mne Carl ler by he analysis f LIBORs (Lndn Iner Bank Oered Raes) and ineres rae swaps in he Japanese marke. In secin fur, we will give he cnclusin. 2

3 2. The Esimain f he Term Srucure Based n he Sae Space Mdeling 2.1 Sae Space Mdel fr Term Srucure We explain in his secin he esimain mehd fr he cmmn facrs and he parameers f erm srucure mdels based n he sae space mdeling. Firs, we briey explain erm srucure mdels based n Markv sae prcesses. (See Bjrk (16), Due (16) r Hull (19) fr he deail) Given he lered prbabiliy space (; F; ff g;p) wih he ime hrizn [0;T ] fr sme T < 1, we suppse ha a N-dimensinal vecr f sae variables dened by Y fllws a N-dimensinal Markv prcess: (2.1) dy = (Y ;)d + S(Y ;)db ; where B is he d-dimensinal sandard Brwnian min under he lered prbabiliy space, and (Y ;) and S(Y ;) dene real-valued funcins f R N [0;T ] 7! R N and R N [0;T ] 7! R N d, respecively. Suppse als ha he insananeus shr-erm ineres rae a ime dened by r(y ;) and a zer cupn bnd's price a wih he mauriy T dened by P (Y ;; T ) are sme funcins f Y where 2 [0;T ] and T 2 [; T ]. Here, a zer cupn bnd wih he mauriy T means a bnd wih n cupns and wih he face value, 1 which is redeemed a ime T. We ne ha he se f he zer cupn bnds' prices fp (Y ;; T )g T 2[;T ] represens he erm srucure f ineres raes a ime : The assumpin reecs he idea ha he whle erm srucure can be explained by relaively small number f facrs, Y while wha he facrs represen depends n he specicain f a mdel. Then, based n he arbirage-free argumen f nancial ecnmics P (Y ;; T ) saises a parial dierenial equain(pde) 1 2 race(ss0 P YY )+[, ] 0 P Y + P, rp =0; wih he erminal bundary cndiin, P (Y ;T; T ) = 1, where P P 0 : Here, R N -valued vecr, denes s called he risk premium which is a funcin f Y and, (Y ;) and fr insance, i can be deermined by he general equilibrium asse pricing hery f ecnmics such as CIR(15a). Mrever, i is well knwn ha he sluin f his PDE is represened by he cndiinal expecain given infrmain a ime, (2.2) R P (Y ;; T )=E Q [e, T r(y u;u)du jf ]; where E Q [jf ] denes he cndiinal expecain perar given infrmain a ime under he risk-neural prbabiliy measure Q. (See Bjrk (16).) I is als knwn ha under he measure Q, he vecr f sae variables Y fllws a schasic dierenial equain, (2.3) dy = f(y ;), (Y ;)gd + S(Y ;)db 3

4 where B denes he d-dimensinal sandard Brwnian min under he measure Q. (See Bjrk(16) and Chaper 7 and Appendix E f Due (16).) Fr applicain f hese erm srucure mdels bserved daa, we inrduce he general frm f sae space mdels. (See Kiagawa and Gersh(16) fr he deail.) A sae space mdel cnsiss f he fllwing sysem mdel and he bservain mdel. Tha is, 8 >< Y = F (Y, ;v ) sysem mdel (2.4) >: Z = H(Y ;u ) bservain mdel where Y, Z and dene a N-dimensinal sae vecr, a M-dimensinal bservain vecr a ime and he ime inerval f bservainal daa respecively while v and u dene he N-dimensinal sysem nise and he M-dimensinal bservainal nise whse densiy funcins are given respecively by q(v) and (u). F (; ) and H(; ) are generally nn-linear funcins, and he iniial sae vecr Y 0 is assumed be a randm variable whse densiy funcin is given by p 0 (Y ). Nex, we clarify hw he sae space mdels can be applied he esimain f erm srucure mdels. When is sucienly small, he Euler apprximain he equain (2:1) can be used fr he sysem mdel, Y = F (Y, ;v ). Tha is, (2.5) Y = Y, + (Y, ;, ) + S(Y, ;, )v p where he sysem nise v fllws he N-dimensinal sandard nrmal disribuin. Of curse, he her apprximain schemes culd be applied he discreizain f (2:1). (Fr insance, see secin D and nes f chaper 11 in Due(16).) Mrever, when Y is explicily slved given Y, as in he case f a linear schasic dierenial equain, i is beer use ha represenain: Namely, suppse ha in (2:1), Y is represened by a linear schasic dierenial equain. (2.6) dy =(AY + ())d + SdB where () and A dene R N -valued funcins f he ime parameer and N N marix wih cnsan elemens respecively, and S denes an N d marix wih cnsan elemens s ha = SS 0 is psiive denie. Then, given Y,, Y can be expressed as (2.7) Y = e A Y, + Z, = FY, + ()+v () e (,s)a (s)ds + v () : where F e A is an NN marix wih cnsan elemens and () R, e(,s)a (s)ds is an N 1 vecr funcin f ime. Here, v () fllws he nrmal disribuin wih he mean zer and he variance cvariance marix dened by (2.8) Z 0 e sa e sa0 ds: 4

5 In his case, he sysem mdel is given by (2:7) and he densiy funcin q(v) f he sysem nise v is he nrmal disribuin wih he variance cvariance marix specied by (2.8). In he bservain mdel, he vecr f he bservain a ime dened by Z, can be expressed as a funcin f k( 1) unis f zer-cupn bnds' prices and he bservain nise vecr u. (2.9) Z = h(p (Y ;; + T 1 ); ;P(Y ;; + T k )) + u : Tha is, each elemen fz is an bserved bnd price r ineres rae which is represened by a funcin f zer cupn bnds wih pssibly dieren mauriies (T i ;i = 1; ;k) and a measuremen errr. Mrever, Z can be als expressed as a funcin f Y, because each P (Y ;; T i ) is a funcin f Y. (2.10) Z = H(Y )+u In h(), P (Y;; + T i ) can be evaluaed by he cmpuain f he equain (2:2) under he prcess (2:3). In addiin, we assume hereafer ha he densiy funcin (u) f he nise vecr u is given by ha f he muli-dimensinal nrmal disribuin wih he mean 0, and he variance-cvariance marix u. Here, u is assumed be a M M diagnal marix wih psiive diagnal elemens where M is dened be he dimensin f he bservainal daa a each ime. LIBORs and ineres rae swap raes a ime are he ypical examples f he bservain vecr Z. In his case, h() can be specied by he hereical relain beween LIBORs/ineres rae swap raes and he zer-cupn bnds' prices. Tha is, LIBOR wih he erm n dened by L (Y ; n ) and swap raes wih he erm n dened by S (Y ; n ) are expressed by he zer-cupn bnds' prices as fllws. (See fr insance Bjrk(16), Due(16) and Due and Singlen(19).) (2.11) (2.12)! 1 1 L (Y ; n ) = P (Y ;; + n ), 1 n S (Y ; n ) = 1, P (Y ;; + n ) P n= i=1 P (Y ;; + i) ; where denes he inerval f cash ws. Fr example, =0:5 is sandard in he swap marke f Japanese yen. We will explain acual daa f LIBORs and swap raes used fr he empirical analysis in he nex secin. Frm he discussin abve, we shw ha a erm srucure mdel represened by (2.1) { (2.3) can be re-inerpreed wihin he framewrk f he generalized sae space frm (2.4). Hence we can bain he esimaes f Y in a erm srucure mdel by esimaing saes in he crrespnding sae space mdel. 5

6 2.2 Esimain f Term Srucure by he Mne Carl Filer We discuss abu ur esimain mehd in deail. We ne ha he sandard Kalman ler cann be applied he esimain because bh he sysem mdel and he bservain mdel described abve are generally nn-linear. Thus we uilize he Mne Carl ler. While several appraches are prpsed fr he Mne Carl ler (see Duce, Bara, and Duvau(15), Durbin, and Kpman(19), Grdn, Salmnd, and Smih(13), Tanizaki(13), fr insance), we adp he apprach develped by Kiagawa(16). In he fllwing, we describe he uline f he algrihm f he Mne Carl ler applied he empirical analysis in he nex secin. Firs, we summarize he nain fllwing Kiagawa(16). p(y jz, ), called \ne sep ahead predicr" denes he cndiinal densiy funcin f Y given Z, where is he inerval f ime series. p(y jz ), called \ler" denes he cndiinal densiy funcin f Y given Z. fp (1) ; ;p (m) g and ff (1) ; ;f (m) g represen he vecrs f he realizain f m rials f Mne Carl frm p(y jz, ) and p(y jz ), respecively. Then, if we se ff (1) 0 ; ;f (m) 0 g as he realizain f randm draws frm p 0 (Y ), he densiy funcin f he iniial sae vecr Y 0, he algrihm f he Mne Carl ler is as fllws. [The algrihm f he Mne Carl ler] (i) Generae he iniial sae vecr ff (1) 0 ; ;f (m) 0 g. (ii) Apply he fllwing seps (a)(d) each ime = 0; ; 2; ; (T, );T where T denes he nal ime pin fhedaa. (a) Generae he sysem nise v (j), j =1; ;m accrding he densiy funcin q(v). (b) Cmpue fr each j =1; ;m, p (j) = F (f (j),;v (j) ). (c) Evaluae (j) = n[x ;0; u ], where n[x ;0; u ] he densiy funcin f N(0; u ) a x = Z, H(p (j) ), fr j = 1; ;m. Here, H() represen fr insance, he equain (2:11) and (2:12) which are regarded as funcins f he sae vecr Y. The prices f zer-cupn bnds in hse equains are cmpued hrugh he equain (2:2) by using he prcess (2:3), and if i is n evaluaed analyically, sme numerical mehd such as Mne Carl simulain is implemened. ; ;p (m) g. Mre pre- (d) Implemen resampling f ff (1) ; ;f (m) cisely, bain f (i) g frm fp (1), i =1; ;mby he sampling wih replacemen frm fp (1) wih he prbabiliy Prb.(f (i) = p (j) jz )= (j) P m ; j =1; ;m; i =1; ;m: k=1 (k) ; ;p (m) g 6

7 The esimain f unknwn parameers is based n he maximum likelihd mehd. If denes he vecr represening all unknwn parameers, he lg-likelihd l() is given by l() = lg p(z ; ;Z T j) = T X k=1 lg p(z k jz ; ;Z (k,1) ;) where p(z jz 0 ) = p 0 (Z ). Since each erm in he lg-likelihd can be apprximaed as p(z k jz ; ;Z (k,1) ;) ' 1 m mx j=1 (j) k ; he lg-likelihd l() is cmpued wihin he framewrk f he Mne Carl ler by l() ' T X k=1 lg m X j=1 (j) k 1 A T, lg m: Then, The maximum likelihd esimar ^ is bained by maximizing l() wih respec. Fr maximizain, grid search r a self-rganizing mehd is applied. (See Kiagawa(19) fr deails f a self-rganizing sae-space mdel.) Finally, we uilize AIC (Akaike Infrmain Crierin) as a crierin selec he erm srucure mdel if here are several candidaes. Tha is, he mdel wih he smaller AIC can be regarded as he beer mdel (Akaike (13) ). 3. Analysis f he Term Srucures in he Japanese Marke In his secin, we examine he validiy f ur mehd using he ime series f ineres raes in he Japanese marke. The daa used fr he analysis is summarized as fllws. The perid and he frequency f he daa: daily daa f 19/1/1-19/7/22 (662 bservains). Japanese yen LIBORs: six-mnh, welve-mnh. Japanese yen swap raes: w-year, hree-year, fur-year, ve-year, seven-year, enyear. Figures 1(1)(3) shw he bservainal daa fr he perid f he analysis; (1) and (2) shw he series f LIBORs and hse f swap raes respecively while (3) shws he spread beween en-year and w-year. Fr an ineres rae mdel, we use Hull and Whie(14) in which he dynamics f a sae vecr Y can be represened by a linear schasic dierenial equain (2:6), where Y = (Y i );i = 1; 2 is a w dimensinal sae vecr and () is a R 2 -valued funcin f he ime parameer. They als assume ha he sp rae r is expressed as 7

8 a funcin f Y 1, r = g(y 1 ) where g() is sme real-valued funcin. Fr a funcinal frm f r = g(y 1 ), if we ake g(y 1 )=Y 1,we allw negaive ineres raes because f he nrmaliy fhe sp rae while we can bain an analyic sluin f P (Y ;; T ), which subsanially reduce he cmpuainal burden. In paricular, he mdel implies relaively high prbabiliy f he negaive ineres raes in such lw ineres raes envirnmen f recen Japan and his seems inapprpriae. Hence, we specify g(y 1 ) based n Hull and Whie(19) (chaper 21, pp.588) such ha g(y 1 )=Y 1 fr Y 1 ", and g(y 1 )="e (Y 1,") " fr Y 1 < ", where " is sme predeermined psiive cnsan. Clearly, g() is psiive, mnnically increasing, and lim y!,1 g(y) =0. We ne ha frm abve specicain, Y 1 can be cnsidered be a facr which has a large impac n he shr-erm secr f he erm srucure. On he her hand, Y 2 will be characerized afer he ing f he mdel he daa. We nex deermine he bservain mdel as he equain (2:9) wih he equains (2:11) and (2:12). Mrever, he zer-cupn bnds' prices in h() f(2:9) are cmpued by he equain, (3.1) P (Y ;; T )=E Q [e,r T g(y 1u )du jy ]: We ne ha (3.1) shuld be cmpued by sme numerical mehd such as Mne Carl simulains because i can n be evaluaed analyically. We apply he algrihm f he Mne Carl ler described in he previus secin he esimain. The sae space mdel applied his empirical analysis is described by (3.2) (The sysem mdel) wih Y =(Y 1 ;Y 2 ) 0 and (The bservainal mdel) Y = FY, + ()+v () ; (3.3) wih (3.4) Z n; = 8 >< >: Z =(Z 1; ; ;Z 8; ) 0 L (Y ; n )+u n; (fr n =1; 2, wih n =0:5; 1) S (Y ; n )+u n; (fr n =3; ; 8, wih n =2; 3; 4; 5; 7; 10) ; where he dimensin f he bservainal daa is M =8. In he cmpuain f P (Y ;;T) in L (Y ;T n ) and S (Y ;T n ), we apply a numerical inegrain fr he inegral, R T g(y 1u )du in (3.1) and a Mne Carl inegrain fr he cndiinal expecain: Namely, (3.5) P (Y (i) ;: T ) 1 ' J JX j=1 exp 0 X g(y (i;j) l=0 1 1;+lA ; 8

9 where Y (i) denes he value f Y 1 fr he i-h paricle f he sae vecr in he Mne Carl ler, and Y (i;j) 1;+l denes he value f Y 1;+l in he j-h pah f he sae vecr saring frm Y (i), which are generaed frm Y (i;j) +l = FY (i;j) +(l,1) + ( + l)+v(i;j) ( + l); Y (i;j) = Y (i) : In he Mne Carl inegrain, J = 300, he randm numbers are generaed by ran2 in Numerical Recipes (Secnd Ediin) by Press e. al.(12) and he mehd f aniheic variaes (Kals and Whilck (16) ) is uilized, in which we se v (i;j) +l =,v (i;j,1) +l ;l = 1; 2; :::; (T=) fr even j (i.e. j =2; 4; ::::; 300). Mrever, he number f paricles in he Mne Carl ler is m = 5000 and we use 80 parallel cmpuer fr his cmpuain (SGI 2800 sysem). In esimain, i is hard implemen sandard numerical pimizain mehds such as quasi-newn mehd because cmpuainal diculy arises due he nnlineariy and large number f parameers in he mdel. Hence, fr " in g() we se " = 0:0005 which is smaller han he lwes level f six-mnh LIBOR bserved during he sample perid s ha he chice f " des n have large impac n he shape f he yield curve, and fr he her parameers we adp he fllwing esimain mehd. (Sep1) Apply Kalman-Filer as if he ineres rae funcin were g(r) = r. (Sep2) Apply he self-rganizing mehd using he esimaes bained in (Sep1) as iniial values. (Sep2') Implemen he self-rganizing mehd again seing he average f each parameer ver he sample perid as he iniial value. The prcedure is ieraed unil he lg-likelihd is n signicanly imprved because sligh imprvemen is n reliable due he randm naure f he esimaed lg-likelihd funcin. (Sep3) Apply a grid search arund he esimaes bained in (Sep2') unil he lg-likelihd is n signicanly imprved. Here, he self-rganizing mehd was prpsed by Kiagawa (19); in his mehd, he riginal sae vecr, Y is augmened wih he unknwn parameer vecr, as Y =(Y ;) 0, and we assume is ime-varying such ha =, + v () where v () is nrmally disribued. (Fr example, see Higuchi and Kiagawa (2000) r Kiagawa and Sa (2000).) Finally, we briey explain he resul f his analysis. Firs we invesigae he ime series f he esimaed facrs. Figures 2(1)(2) shw he relain beween he esimaed facrs Y i, i = 1; 2 bained by averaging all he samples f lers and he frward raes cmpued frm bserved LIBORs and swaps. The sample crrelain beween he level f each facr and he crrespnding frward rae is als lised belw 9

10 f each gure. Frm he graphical bservain and he sample crrelain, here seems be srng relain beween Y 2 and en-year frward rae which is dened be he raes wih he erm 0.5 year saring frm 9.5 years frward, and he variain f Y 1 is similar ha f six-mnh LIBOR. Nex, we examine he ing f he mdel he daa. The ime series f each bservain and crrespnding esimae are shwn in Figures 3(1)(8), and he explanain pwer f each esimae which is dened by max 1 he variance f residuals, he variance f bservains ; 0 is lised abve crrespnding gure. We ne ha he esimaes f LIBORs and swap raes are based n esimaed parameers and esimaed facrs. They shw ha he mdel s very well he swap raes while i des n LIBORs: The explanain pwers are mre han.5% fr all he swap raes while hey are 39.4% and 77.3% fr six-mnh and welve-mnh LIBORs respecively. In he w-facr mdel, esimaed erm srucures are generally ed well he real nes excep LIBORs. In Figures 4(1)(4), we shw he examples in which he ing LIBORs are gd (9/30/), average (6/24/), and bad (12/4/, 5/14/). Hence, we can cnclude ha he mdel explains he variains f he swap raes very well while i des n explain hse f LIBORs. The resul f he w-facr case implies ha anher facr may be necessary imprve he ing LIBORs. Hence, we nex implemen he case in which he sae vecr Y =(Y i ), i =1; 2; 3 is hree dimensinal, and has he same frm f (3:2). We repr ha he explanain pwers are mre han.5% fr all he raes and mre han % excep fr welve-mnh LIBOR and w-year swap rae. Especially, we ne ha he mdel remarkably imprves he ing LIBORs. (See Figure 5) Figures 6(1)(4) shw he bservains and esimaes f he erm srucures a fur daes(/6/24, /12/4, /9/30, /5/14, which are same as he w-facr case.), which implies ha he mdel can replicaes he real erm srucures including LIBORs very clsely. Mrever we ne ha AIC in his case(aic= ) is subsanially imprved by mre han Hwever, when we implemen mre suble mehd f mdel diagnsics described in Kim, Shephard and Chib (19), i urns u ha here are sme prblems fr he innvain f he mdel. Especially, he independence f he innvains and he nrmaliy f he ransfrmed innvains are rejeced by using Bx- Ljung saisic and Bwman and Shenn (15) nrmaliy saisic. We shw he resuls fr six-mnh LIBOR and ve-year swap rae in Figure 7. Thus i seems necessary invesigae he brader class f ineres rae mdels. 4. Cnclusin We develp a new framewrk fr he empirical analysis f he erm srucure f ineres raes based n he generalized sae space mdel. Our apprach is useful fr he esimain and he mdel diagnsics f varius ypes f ineres rae mdels, which are 10

11 usually cnsidered ugh ask. As an example, we apply he Mne Carl ler he ime series f LIBORs and f ineres rae swaps in he Japanese marke, and cnrm he validiy f ur mehd. Furhermre, we will uilize his apprach analyze mre cmplicaed mdels. Acknwledgemen We hank edirs and annymus referees fr useful cmmens n he previus versin. References Akaike, H. (13). Infrmain hery and an exensin f he maximum likelihd principle, Secnd Inernainal Sympsium n Infrmain Thery (eds. B. N. Perv and F. Csaki), 267{281, Akademiai Kiad, Budapes. Bjrk, T. (16) Ineres rae hery, Financial Mahemaics Bressanne ( eds. W. J. Runggaldier),Springer- Verlag, Belrin. Chan, K.C., Karlyi, G.A., Lngsa, F.A. and Sanders, A.B. (12). An empirical cmparisn f alernaive mdels f he shr-erm ineres rae, Jurnal f Finance, 47, 1209{1227. Chen, R. and Sc, L. (13). Maximum likelihd esimain fr a mulifacr equilibrium mdel f he erm srucure f ineres raes, Jurnal f Fixed Incme, ember, 14{31. Cx, J.C., Ingersll, J.E. and Rss, S.A. (15a). An inerempral general equilibrium mdel asse prices, Ecnmerica, 53, 363{384. Cx, J.C., Ingersll, J.E. and Rss, S.A. (15b). Ecnmerica, 53, 385{408. A hery f he erm srucure f ineres raes, Duce, A., Bara, E. and Duvau, P. (15). A Mne Carl apprach recursive Bayesian sae esimain, Prceedings IEEE Signal Prcessing/Ahs Wrkshp n Higher Order Saisics, Girna, Spain. Due, D. (16). Dynamic Asse Pricing Thery(secnd ediin), Princen Universiy Press, Princen, NJ. Due, D. and Singlen, K.J.(19). An ecnmeric mdel f he erm srucure f ineres-rae swap yields, Jurnal f Finance, 52, 1287{1321. Durbin, J. and Kpman, S.J. (19). Mne Carl maximum likelihd esimain fr nn-gaussian sae space mdels, Bimerika, 84, 669{684. Grdn, N., Salmnd, D.J. and Smih, A.F.M. (13). Nvel apprach nnlinear/nn-gaussian Bayesian sae esimain," IEE Prceedings-F, 140, 2, 107{113. Higuchi, T. and Kiagawa, G. (2000). Knwledge discvery and self-rganizing sae space mdel, IEICE Transacins n Infrmain and Sysems, E83-D, 1, 36{43. Hull, J. (19). Opins, Fuures, and Oher Derivaives (furh ediin), Prenice-Hall, Upper Saddle River, NJ. Hull, J. and Whie, A. (19). Taking raes he limis, Risk,., 19, 168{169. Hull, J. and Whie, A. (14). Numerical prcedures fr implemening erm srucure mdels II:Twfacr mdels, Jurnal f Derivaives, 2, 37{48. Kals, M. H. and Whilck, P. A. (16). Mne Carl Mehds Vlume I, Wiley-Inerscience Publicain, New Yrk. 11

12 Kim, S., Shephard, N. and Chib, S. (16). Schasic vlailiy: Likelihd inference and cmparisn wih ARCH mdel, Review f Ecnmic Sudies, 65, 361{393. Kiagawa, G.(16). Mne Carl Filer and Smher fr Nn-Gaussian Nnlinear Sae Space Mdels, Jurnal f Cmpuainal and Graphical Saisics, 5, 1, 1{25. Kiagawa, G.(19). A Self-Organizing Sae-Space Mdel, Jurnal f he American Saisical Assciain, 93, 443, 1203{1215. Kiagawa, G. and Gersch, W. (16). Smhness Prir Analysis f Time Series (Lecure Nes in Saisics N. 116), Springer-Verlag, Belrin. Kiagawa, G. and Sa, S. (2000). Nnlinear sae space mdel apprach nancial ime series wih ime-varying variance, Prceedings f he Hng Kng Inernainal Wrkshp n Saisics and Finance: An Inerface (eds. W.S. Chan, W.K. Li and H. Tng), Imperial Cllege Press, Lndn. Lngsa, F. and Schwarz, E.S. (12). Ineres rae vlailiy and he erm srucure:a w-facr general equilibrium mdel, Jurnal f Finance, 47, 1259{1282. Pearsn, N.D. and Sun, T.S.(14). Expliing he cndiinal densiy in esimaing he erm srucure: An applicain he Cx, Ingersll, and Rss mdel, Jurnal f Finance, 49, 1279{1304 Press, W.H., Flannery, B.P., Teuklsky, S.A. and Veerling, W.T. (12). Numerical Recipes in C (2nd Ediin), Cambridge Universiy Press, Cambridge. Singh, M.K. (15). Esimain f mulifacr Cx, Ingersll, and Rss erm srucure mdel, Jurnal f Fixed Incme, ember, 8{28. Tanizaki, H. (13). Nnlinear Filers (Lecure Nes in Ecnmics and Mahemaical Sysems, 400), Springer-Verlag, Belrin. 12

13 (1) LIBORs (3) Spread (Swap10Y - Swap2Y) LIBOR-1Y LIBOR-6M (BP) Aug Nv Jul Mar Aug Nv Jul (2) SWAP raes SWAP-10Y SWAP-7Y SWAP-5Y SWAP-4Y SWAP-3Y SWAP-2Y Aug Nv Fig. 1. Jul Mar ime series f bserved LIBORs and swap raes (1) Libr6M(slid), Y1 (ds) (Scale:lef) (Scale:righ) (2) Frward10Y(slid), Y2 (ds) (Scale:lef) (Scale:righ) Crr= 0.90 Crr= 0. {Parameer 0 esimaes: 1 0 1,0:08 0:08 A A, 0:32 10,4,0:16 10,3 A, 0,0:13,0:16 10,3 0:17 10,2 u = diag(0:11 10,7 ; 0:56 10,8 ; 0:78 10,9 ; 0:63 10,9 ; 0:63 10,9 ; 0:53 10,9 ; 0:58 10,9 ; 0:16 10,8). Fig. 2. esimaed facrs (w-facr case) 13

14 (1) Libr6M (2) Libr1Y R2= 39.4 % R2= 77.3 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) (3) Swap2Y (4) Swap3Y R2=.7 % R2=.5 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) (5) Swap4Y (6) Swap5Y R2=.6 % R2=.7 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) (7) Swap7Y (8) Swap10Y R2=.8 % R2=.6 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) Fig. h3. bservains and esimaes (w-facr i case); he explanain pwer (R2) is dened by max 1 he variance f residuals, ; he variance f bservains 0. 14

15 (1) 24,19 * * * * * * * * (2). 4, 19 ** * * * * * * (3). 30, 19 * ** * * * * * (4) 14, 19 ** * * * * * * (Yr) (Yr) (Yr) (Yr) Observain(*), Esimaes() Fig. 4. Observain(*), Esimaes() Observain(*), Esimaes() he erm srucures(w-facr case) Observain(*), Esimaes() (1) Libr6M (2) Libr1Y R2=.6 % R2=.6 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) Fig. 5. bservains and esimaes f LIBORs (hree-facr case) (1) 24,19 (2). 4, 19 (3). 30, 19 (4) 14, * * * * * * * * ** * * * * * * ** * * * * * * ** * * * * * * (Yr) (Yr) (Yr) (Yr) Observain(*), Esimaes() Fig. 6. Observain(*), Esimaes() Observain(*), Esimaes() he erm srucures (hree-facr case) Observain(*), Esimaes() 15

16 M-Libr Nrmalized Innvains BL(30) = nrm Q-Q pl f nrmalized innvains B-S = Crrelgram f Nrmalized Innvains Quaniles f Sandard Nrmal Y-Swap Nrmalized Innvains BL(30) = nrm Q-Q pl f nrmalized innvains B-S= Crrelgram f Nrmalized Innvains Quaniles f Sandard Nrmal Fig. 7. The Resul f mdel diagnsics fr hree-facr case (We shw he resuls fr nly six mnh LIBOR and ve year Swap), where innvains are cmpued by u M +1 = Pr(Zk;+1 Z k;+1 jz ) ' 1 P M M i=1 Pr(Z k;+1 Z (i) k;+1jy +1 ), nrmalized innvains are n M =,1 (u M ) (he inverse f he nrmal disribuin), Z k;+1 means bserved value, and BL(30) and B-S is dened as Bx-Ljung saisic wih 30 lags and Bwman and Shenn (15) nrmaliy saisic ( 2 (2)), respecively. (See Kim, Shephard and Chib (19).) 16