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1 c Takahashi and Sao 2001 he insananeous shor rae ATSM 2 Takahashi and Sao 2001 Hull and Whie 1994, ; QZD12245@nifyserve.or.jp ; sao@ism.ac.jp

2 Cox, Ingersoll and Ross 1985a, 1985b CIR CIR Chen and Sco 1993 Pearson and Sun 1994 Singh 1995 Duffie and Singleon 1997 h : R N R h(y) =α + β y y R N α R β R N ATSM Duffie 1996 Chaper 7 Dai and Singleon 2000 ATSM US Takahashi and Sao 2001 Dai and Singleon 2000 Dai and Singleon 2000 p ATSM Duffie and Kan 1996 Vasicek 1977 Langeieg 1980 Hull and Whie 1990, 1994 CIR Cox e al. 1985b Chen and Sco 1993 Pearson and Sun 1994 Singh 1995 Duffie and Singleon 1997 Longsaff and Schwarz Chen 1996 alduzzi e al rennan and Schwarz 1982 lack and Karasinski 1991 Hull and Whie 1994, 1997 GMM Chan e al Longsaff and Schwarz 1992 Simulaed Momen Mehod SMM Dai and Singleon 2000 hree-sage leas square mehod wih he principal componen analysis Singh 1995 Chen and Sco 1993 Pearson and Sun 1994 Duffie and Singleon 1997 Chan e al Longsaff and Schwarz 1992 measuremen error Chen and Sco 1993 Duffie and Singleon 1997

3 jörk 1996 Duffie 1996 T < [0,T ] (Ω, F, {F },P) N Y N (3.1) dy = µ(y,)d + S(Y,)d, d µ(y,) S(Y,) R N [0,T ] R N R N [0,T ] R N d r(y,) T P (Y,; T ) Y [0,T ] T [, T ] 1 {P (Y,; T )} T [,T ] Y P (Y,; T ) 1 2 race(ss P YY)+[µ φ] P Y + P rp =0, P (Y,T; T )=1, P YY 2 P,P Y Y Y P Y R N φ Y φ(y,) Cox e al. 1985a (3.2) P (Y,; T )=E Q [e R T r(y u,u)du F ], E Q [ F ] Q jörk 1996 Q Y (3.3) dy = {µ(y,) φ(y,)}d + S(Y,)d Q d jörk 1996 Duffie 1996 Chaper 7 Appendix E

4 Kiagawa and Gersh 1996 ( Y = F (Y,v ) (3.4) Z = H(Y,u ) Y Z N M v u N M q(v) ψ(u) F (, ) H(, ) Y 0 p 0(Y ) 3.1 Y = F (Y,v ) (3.5) Y = Y + µ(y, ) + S(Y, )v v N 3.1 Shoji and Ozaki 1998 Shoji 1998 Y Y 3.1 Y (3.6) dy =(AY + β ())d + Sd β () A R N N N S Σ = SS N d Y Y (3.7) Y = e A Y + Z = FY + β()+v ( ) e ( s)a β (s)ds + v ( ) R F e A N N β() ea( s) β (s)ds N 1 v ( ) 0 Σ (3.8) Σ Z 0 e sa Σe sa ds. 3.7 v q(v) 3.8 Z k( 1) u (3.9) Z = h(p (Y,; + T 1),...,P(Y,; + T k )) + u Z (T i,i=1,...,k)

5 137 P (Y,; T i) Y Z Y (3.10) Z = H(Y )+u. h( ) P (Y,; + T i) u ψ(u) 0 Σ u Σ u M M M LIOR London Iner ank Offered Raes Z LIOR LIOR h( ) LIOR τ n (R (Y,τ n)) LIOR (L (Y,τ n)) (S (Y,τ n)) jörk 1996 Duffie 1996 Duffie and Singleon 1997 (3.11) (3.12) (3.13) R 1 (Y,τ n)= log P (Y,; + τ n) τ n L (Y,τ n)= S (Y,τ n)= P (Y,; + τ n) τ n P 1 P (Y,; + τn) τ δ n/δ, i=1 P (Y,; + iδ) δ δ = Y 4. Douce e al Durbin and Koopman 1997 Gordon e al Tanizaki 1993 Kiagawa 1996 Kiagawa 1996 Kiagawa 1996 p(y Z ) Z Y p(y Z ) Z Y {p (1),...,p (m) }, {f (1),...,f (m) } p(y Z ) p(y Z ) m {f (1) 0,...,f(m) 0 } Y 0 p 0(Y )

6 [ ] 1. {f (1) 0,...,f(m) 0 } N(µ f0,σ f0 ) Σ f0 2. a d =0,, 2,..., (T ),T T a v (j), j =1,...,m N(0,Σ ) ( 3.7 N(0,I) 3.5 b j =1,...,m p (j) = F (f (j),v(j) ) F c N(0,Σ u) n[x;0,σ u] x = Z H(p (j) ), j =1,...,m α (j), j =1,...,m H( ) Y d {f (1),...,f (m) } {p (1),...,p (m) } f (i), i =1,...,m {p (1),...,p (m) } Prob.(f (i) = p (j) α (j) Z )= P m j=1 α(j) i =1,...,m θ L(θ) p(z Z 0)=p 0(Z ) L(θ) =p(z,...,z T θ) = T Y k=1 p(z k Z,...,Z (k 1),θ) l(θ) l(θ) = T ψ X k=1 log mx j=1 α (j) k! T log m θ ˆθ Kiagawa 1998 AIC Akaike s Informaion Crierion Akaike 1973 AIC = 2l(ˆθ) + 2(he number of parameers)

7 Hull and Whie 1994, 1997 LIOR Takahashi and Sao ATSM 2 1. Hull and Whie 1994, (5.1) Y =(Y 1,Y 2) Y = FY + β()+v ( ), (5.2) (5.3) Z n, = ( Z =(Z 1,,...,Z 8,) L (Y,τ n)+u n, (n =1, 2 S (Y,τ n)+u n, (n =3,...,8 ), LIOR M =8,τ 1 =0.5,τ 2 =1,τ 3 =2,τ 4 =3,τ 5 = 4,τ 6 =5,τ 7 =7,τ 8 =10 L (Y,T n) S (Y,T n) P (Y,,T) (5.4) P (Y,; T )=E Q [e R T g(y u)du Y ] g( ) g( ) (5.5) Y (i) Y (i) P (Y (i),; T ) 1 J JX j=1 0 X T/ exp@ l=0 1 g(y (i,j) +l ) A, Y i Y (i,j) Y (i,j) (i,j) +l = FY + +(l 1) β ( + l )+v (i,j) ( + l ), +l Y (i,j) = Y (i) Y +l j 3.3

8 Dai and Singleon 2000 (5.6) dy = K(Θ Y )d + Σ S d, Σ N N S() (S()) ii = α i + γ iy (5.7) r = δ 0 + Q NX i=1 δ iy i. (5.8) dy = K (Θ Y )d + Σ S d K = K + ΣΦ Θ = K 1 (KΘ Σψ) Φ =(λ 1γ 1,λ 2γ 2,...,λ N γ N) ψ =(λ 1α 1,λ 2α 2,...,λ Nα N ) marke prices of risk Λ() Λ() = p S()λ λ =(λ 1,λ 2,...,λ N ). (5.9) P (Y,; + τ) =e b0(τ) (τ) Y b0(τ),(τ ) b0(0) = 0,(0) = 0 (5.10) (5.11) db0(τ) dτ d(τ ) dτ = θ K (τ )+ 1 2 = K (τ ) 1 2 NX i=1 NX i=1 Λ Σ 2 (τ ) αi δ0, i Λ Σ 2 (τ ) γi + δy. i δ y =(δ 1,δ 2,...,δ N )

9 141 2 ψ! Y =(Y 1,Y 2) k 11 0 (5.12), K =, Θ =(θ 1, 0), 0 k 22 (5.13) Σ = ψ 1 0 σ 21 1!, ψ β1y! 1 0 S = 0 α2 + β 2Y 1 (5.14) r = δ 0 + Y 1 + Y 2. he marke price of risk λ 0 (5.15) R 1 (Y,τ)= log P (Y,; + τ) = b0(τ )+b1(τ)y1 + b2(τ )Y2. τ b0(τ),(τ )=(b1(τ),b2(τ)) (5.16) (5.17) Y 1,n+1 = Y 1,n + k 11(θ 1 Y 1,n) + p β 1Y 1,nv 1,n Y 2,n+1 = Y 2,n k 22Y 2,n + σ 21 p β1y 1,nv 1,n + p α 2 + β 2Y 1,nv 2,n. v 1,n,v 2,n i.i.d. N(0, ) n Z 1,n,...,Z 6,n (5.18) Z 1,n Z 2,n Z 3,n Z 4,n Z 5,n Z 6,n 1 C C A = c(τ 1) c(τ 2) c(τ 3) c(τ 4) c(τ 5) c(τ 6) 1 C C A + H 1(τ 1) H 2(τ 1) H 1(τ 2) H 2(τ 2) H 1(τ 3) H 2(τ 3) H 1(τ 4) H 2(τ 4) H 1(τ 5) H 2(τ 5) H 1(τ 6) H 2(τ 6) 1 Cψ C A! Y 1,n Y 2,n + u 1,n u 2,n u 3,n u 4,n u 5,n u 6,n 1 C C A (5.19) τ 1 =1, τ 2 =2, τ 3 =5, τ 4 =7, τ 5 =10, τ 6 =20 (5.20) c(τ) = b 0(τ,k 11,k 22,θ 1,β 1,β 2,σ 21,α 2,δ 0) H 1(τ )=b 1(τ,k 11,k 22,β 1,β 2,σ 21) H 2(τ )=b 2(τ,k 22).

11 143 2.

12 Y Y 1 + Y

13 145 Akaike, H Informaion heory and an exension of he maximum likelihood principle, Second Inernaional Symposium on Informaion Theory eds.. N. Perov and F. Csáki, , Akademiai Kiádo, udapes. alduzzi, P., Das, S. R., Foresi, S. and Sundaram, R A simple approach o hree facor affine erm srucure models, Journal of Fixed Income, 6, jörk, T Ineres rae heory, Financial Mahemaics ressanone ed. W. J. Runggaldier, Springer, erlin. lack, F. and Karasinski, P ond and opion pricing when shor raes are log-normal, Financial Analyss Journal, July-Augus, rennan, M. J. and Schwarz, E. S An equilibrium model of bond pricing and a es of marke efficiency, Journal of Financial and Quaniaive Analysis, 17 3, Chan, K. C., Karolyi, G. A., Longsaff, F. A. and Sanders, A An empirical comparison of alernaive models of he shor-erm ineres rae, Journal of Finance, 47, Chen, L Sochasic Mean and Sochasic Volailiy A Three-Facor Model of he Term Srucure of Ineres Raes and Is Applicaion o he Pricing of Ineres Rae Derivaives, lackwell, Oxford, U.K. Chen, R. and Sco, L Maximum likelihood esimaion for a mulifacor equilibrium model of he erm srucure of ineres raes, Journal of Fixed Income, December, Cox, J. C., Ingersoll, J. E. and Ross, S. A. 1985a. An ineremporal general equilibrium model asse prices, Economerica, 53, Cox, J. C., Ingersoll, J. E. and Ross, S. A. 1985b. A heory of he erm srucure of ineres raes, Economerica, 53, Dai, Q. and Singleon, K. J Specificaion analysis of affine erm srucure models, Journal of Finance, LV 5, Douce, A., ara, E. and Duvau, P A Mone Carlo approach o recursive ayesian sae esimaion, Proceedings IEEE Signal Processing/Ahos Workshop on Higher Order Saisics, Girona, Spain. Duffie, D Dynamic Asse Pricing Theory, 2nd ed., Princeon Universiy Press, Princeon, New Jersey. Duffie, D. and Kan, R A yield-facor model of ineres raes, Mah. Finance, 6, Duffie, D. and Singleon, K. J An economeric model of he erm srucure of ineres-rae swap yields, Journal of Finance, 52, Durbin, J. and Koopman, S. J Mone Carlo maximum likelihood esimaion for non-gaussian sae space models, iomerika, 84, Gordon, N., Salmond, D. J. and Smih, A.F.M Novel approach o nonlinear/non-gaussian ayesian sae esimaion, IEE Proceedings-F, 140 2, Hull, J. and Whie, A Pricing ineres rae derivaive securiies, The Review of Financial Sudies, 3 4, Hull, J. and Whie, A Numerical procedures for implemening erm srucure models II: Two-facor models, Journal of Derivaives, 2, Hull, J. and Whie, A Taking raes o he limis, Risk, December, Kiagawa, G Mone Carlo filer and smooher for non-gaussian nonlinear sae space models, J. Compu. Graph. Sais., 5 1, Kiagawa, G A self-organizing sae-space model, J. Amer. Sais. Assoc., 93,

14 Kiagawa, G. and Gersch, W Smoohness Prior Analysis of Time Series, Lecure Noes in Sais., No. 116, Springer, erlin. Langeieg, T. C A mulivariae model of he erm srucure, Journal of Finance, 35, Longsaff, F. and Schwarz, E. S Ineres rae volailiy and he erm srucure: A wo-facor general equilibrium model, Journal of Finance, 47, Pearson, N. D. and Sun, T. S Exploiing he condiional densiy in esimaing he erm srucure: An applicaion o he Cox, Ingersoll, and Ross model, Journal of Finance, 49, Shoji, I Approximaion of coninuous ime sochasic processes by a local linearizaion mehod, Mah. Comp., 67, Shoji, I. and Ozaki, T A saisical mehod of esimaion and simulaion for sysems of sochasic differenial equaions, iomerika, 85, Singh, M. K Esimaion of mulifacor Cox, Ingersoll, and Ross erm srucure model, Journal of Fixed Income, Sepember, Takahashi, A. and Sao, S Mone Carlo filering approach for esimaing he erm srucure of ineres raes, Ann. Ins. Sais. Mah., 53 1, Tanizaki, H Nonlinear Filers, Lecure Noes in Econom. and Mah. Sysems, No. 400, Springer, erlin. Vasicek, O. A An equilibrium characerizaion of he erm srucure, Journal of Financial Economics, 5,

15 Proceedings of he Insiue of Saisical Mahemaics Vol. 50, No. 2, (2002) 147 An Applicaion of Mone Carlo Filer for Esimaing he Term Srucure of Ineres Raes Akihiko Takahashi (Deparmen of Mahemaical Science, Universiy of Tokyo) Seisho Sao (The Insiue of Saisical Mahemaics) We have developed a new mehodology for esimaing he general class of erm srucure models based on a Mone Carlo filering approach. We uilize he generalized sae space model, which can be naurally applied o he esimaion of erm srucure models based on Markovian processes. I is also possible o inroduce measuremen errors in a general way wihou any bias. Moreover, we illusrae our mehod using an affine erm srucure model and JG daa. Key words: Generalized sae space model, Mone Carlo filer, ineres rae model, affine erm srucure model.

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