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2 B-S Granger SARV SV GMM QMLE MCMC Gibbs sampler(single-move sampler ) {h t } T t=1 single-move sampler sampling Gibbs sampler(multi-move sampler ) {h t } T t=1 multi-move sampler sampling

3 Gibbs sampler A-R M-H/A-R SV SV h t

4 1 stochastic volatility;sv SV Clark(1973) information arrival( ( Tauchen and Pitts(1983) information arrival ( ) ( ) Hull and White(1987) information arrival SV Taylor(1986) ARCH(Autoregressive Conditional Heteroskedasticity) SV Taylor SV SV SV ARCH SV ARCH, 1

5 ( Bollerslev, Engle and Nelson(1994) ) Black-Scholes(B-S) ARCH SV 3 SV (Markov-chain Monte Carlo;MCMC) 4 B-S,.1 B-S

6 .1 B-S, Bachelier.Black and Scholes(1973) Merton(1990)..1.1., , ( t) S t. t I t, [t, t + h] S t+h S t I t. I t,, ( ) St+h S t E t = 1 E t S t+h 1 < + S t S t.,e t ( ) E( I t )., I t,, ( ) St+h S t V t = 1 V S t St t S t+h < ( ) h E St+h S t t h +0 µ S (I t ). S t 3 S t

7 E t S t+h S t hµ S (I t )S t, d dτ E t(s τ ) = µ S (I t )S t, a.s. (1) τ=t.(1) E t (ds t )= µ S (I t )S t dt.. 1 ( ) h V St+h S t t h +0 σs (I t). V t S t+h V t S t hσs (I t)st ( V t(s t )=0),, d dτ V t(s τ ) = σs (I t)st, a.s. () τ=t,v t (ds t )=σs (I t)st dt. σ S (I t ) S t [ σ S (I t ) lim h +0 1 h V t ( St+h S t S t )] 1 (3).1., ds t = µ S (I t )S t dt + σ S (I t )S t dw t (4).,W t.(4), 4

8 .µ S (I t ) σ S (I t ) t,. Black and Scholes(1973),., σ S (I t ),. 1,.1.. Merton(1976), ds t S t =(µ λk)dt + σdw t + dq t (5), Y ( ), { Y 1 ( λdt) dq t = 0 ( 1 λdt).λ,dw t dq t,k = E Y [Y 1](E Y Y ). (5).1..,..1. SV. 5., 5

9 .,,,,. B-S., ds t = µ S S t dt + σ S S t dw t (6),µ S σ S. K t + h, { S t+h K, S t+h K [S t+h K] + = (7) 0, S t+h <K Black and Scholes(1973),.,. 0, Black- Scholes.,,,. (7),. B-S 6

10 ..,,, Q.,,. ds t = r t dt + σ S dw Q t (8) S t C t = C(S t,k,h,t)=v(t, t + h)e Q t [S t+h K] + (9). W Q t Q, E Q t Q, v(t, t + h) t + h t, r t. Y (t, t + h) 3 1 r t = lim Y (t, t + h) = lim log v(t, t + h) h 0 h 0 h 1 = lim (log v(t, t + h) log v(t, t)) h 0 h = d dτ v(t, τ) τ=t (10), ( W Q t ) [ t+h ] v(t, t + h) = exp r τ dτ (11) t,r t,(9), C t [S t+h K] +. S t S t+h 7

11 ,(9) t. C t = S t [Φ(d t ) e xt Φ(d t σ S h)] (1) x t d t = + σ S h (13) σ S h S t x t = log (14) Kv(t, t + h) Φ( ), (1) Black- Scholes, C t S t K v(t, t + h).x t moneyness 3 x t =0,. at the money. x t > 0, in the money. x t < 0, out of the money. B-S,,,... Hull and White(1987). W t. ds t = r t dt + σ St dw t S t (15) {σ St } t [0,T ], {W t } t [0,T ], 8

12 ,, r t.,(s t,σ St ) (S, σ S ), (1).(9). C t = v(t, t + h)e t [S t+h K] + = v(t, t + h)e t {E t [(S t+h K) + {σ Sτ } t τ t+h ]} (16), I t {σ Sτ } t τ t+h S t+h., W t,(1) v(t, t + h)e t [(S t+h K) + {σ Sτ } t τ t+h ] = S t E t [Φ(d 1t ) e xt Φ(d t )] (17), d 1t = x t γ(t, t + h) h d t = d 1t γ(t, t + h) h γ (t, t + h) = 1 h t+h t + γ(t, t + h) h σsτ dτ (18), Hull and White C t = S t E t [Φ(d 1t ) e xt Φ(d t )] (19) γ(t, t + h) ( ). 9

13 Hull and White(H-W) (19).,.(1) ()., P θ,θ Θ.,H-W. C t = S t F [σ St,x t,θ 0 ] (0) θ 0. (19)., (x t,θ), F [,x t,θ] 1 1,F [,x t,θ] (19) σ imp t = G[S t,c t,x t,θ] (1). Bajeux and Rochet 1 1., θ 0,,, (0)., SV,,., 1, Black-Scholes,.B-S 10

14 ω imp (t, t + h) C t = S t [Φ(d 1t ) e xt Φ(d t )] x t d 1t = ω imp (t, t + h) h + ωimp (t, t + h) h d t = d 1t ω imp (t, t + h) h x t = log S t Kv(t, t + h) () C t Hull and White B-S (19) () B-S ω imp (t, t+h) γ(t, t+h) ω imp (t, t+h) ( H-W ) at the money at the money x t =0 d t = d 1t Φ(d 1t ) e xt Φ(d t ) = Φ(d 1t ) 1 (19) () at the money B-S ω imp (t, t + h) ( ω imp (t, t + h) ) ( h γ(t, t + h) ) h Φ = E t Φ 0 h ω imp (t, t + h) E t γ(t, t + h) =E t [ 1 h t+h t ] 1 σ Sτ dτ B-S ω imp (t, t + h) 11

15 .,.,. (a) Thick tails 1960, Mandelbrot Fama. (b) Volatility clustering,.volatility clustering thick tails,..arch ( ) ( ).ARCH, Engle(198), SV volatility clustering. (c) Leverage effects Black(1976) leverage effect. leverage 1

16 ,.,Schwert(1989) leverage. (d) Long memory and persistence persistent..3 SV ARCH Granger I t t {x t }, {z t } I t {z s } t s=0 I t {z s } t s=0.1 ( Granger ) 4 E(x n t I t 1) =E(x n t I t 1 {z s } s=0 t 1 ), n =1, Granger z t x t 13

17 .3.1 ds t = µ t dt + σ t dw t S t du t = γ t dt + δ t dw U t Cov(dW t,dw U t )=ρ tdt (3).3 µ t,σ t,γ t,δ t,ρ t I U t = σ[u s,s t] 5 I t = σ[u τ,s τ ; τ t] (4).3. Granger R t = S t S t 1. S t 1 (3) R t+1 = µ(u t )+σ(u t )z t+1 (5) z t, leverage effects..4 z t i.i.d. Granger U t z t 14

18 .5 Granger z t U t.5.3.,.4 E[z t+1 (U τ,z τ ; τ t)] = E[z t+1 z τ ; τ t] =0, E[R t+1 (R τ ; τ t)] = E[E[R t+1 (U τ,r τ ; τ t)] (R τ ; τ t)] = E[E[R t+1 (U τ,z τ ; τ t)] (R τ ; τ t)] = E[µ(U t ) (R τ ; τ t)] (6),.4 E[z t+1 (U τ,z τ ; τ t)] = E[z t+1 z τ ; τ t] =1, Var[R t+1 (R τ ; τ t)] = E[Rt+1 (R τ; τ t)] (E[R t+1 (R τ ; τ t)]) = E[σ (U t )zt+1 (R τ ; τ t)] = E[σ (U t )E[zt+1 (U τ,r τ ; τ t)] (R τ ; τ t)] = E[σ (U t )E[zt+1 (U τ,z τ ; τ t)] (R τ ; τ t)] = E[σ (U t ) (R τ ; τ t)] (7), I R t (R τ ; τ t) (8) µ(u t ) I R t -. 15

19 (6),(7). E[R t+1 I R t ] = µ(u t) (9) Var[R t+1 I R t ] = E[σ (U t ) I R t ] (30) lagged autoregressive random variance models contemporaneous autoregressive random variance models.4. stochastic autoregressive volatility(sarv), µ(u t )=0 lagged autoregressive random variance models. R t+1 = σ t z t+1 (31) Cov(σ t+1,z t+1 I R t ) 0 (3) (3) leverage effect., contemporaneous autoregressive random variance models. R t = σ t z t (33) Cov(σ t+1,z t I R t ) 0 (34) 16

20 σ t,(3) (3) (33) (34). (3) (33) leverage effect.4. SARV ɛ t = σ t z t contemporaneous autoregressive random variance models R t = µ t + ɛ t (35).6 µ t I R t 1-,t 1,,ɛ t t 1., E(R t I R t 1 ) = µ t (36) Var(R t I R t 1 ) = E(σ t IR t 1 ) (37). R t (37). (1) (37). (). (a) z t. (b)z t, σt,ɛ t 3 ( 5.1 ). 1 17

21 σ t AR(1) Andersen(1994) σ q t = g(k t ), q {1, } K t = w + βk t 1 +[γ + αk t 1 ]u t (38) ũ t = u t GARCH(1,1),(38) K t = σ t,γ =0,u t = z t 1, σ t = w + βσ t 1 + ασ t 1z t 1.(35) (38) u t = zt 1,GARCH. ARCH.,SV,(38) K t = log σt,α=0,γũ t+1 = η t+1,w+γ = ω, β = φ, log σ t+1 = ω + φ log σ t + η t+1 (39)., η t.cov(η t+1,ɛ t ) 0 leverage effects. 18

22 .4.3 SARV. ɛ t = σ t z t σ q t = g(k t ), q {1, } (40) K t = w + βk t 1 +[γ + αk t 1 ]u t, 0 ũ t = u t 1, K t =(w + γ)+(α + β)k t 1 +[γ + αk t 1 ]ũ t (41).,w γ. α β..,andersen(1994) (,α = 0). K = w + γ 1 α β ρ = α + β (4) δ = γ α (41). K t = K + ρ(k t 1 K)+(δ + K t 1 )Ūt,Ūt = αũ t., K t 3 E(K t ),E(K t ),E(K tk t 1 ),3 K, ρ, δ 19

23 .,. (1) Y t σ t. () σ t K t. (1) z t.,z t, E( y t )= π E(σ t) E( y t y t j )= π E(σ tσ t j ) (43) E( yt y t j ) = π E(σ t σ t j).() g ũ t.,(39) SV,σ t = e Kt, ũ t, [ ] n E(σt n ) = exp E(K t)+ n Var(k t ) 8 E(σ m t σn t j )=E(σm t )E(σn t j ) exp [ mncov(kt,k t j ) 4 Cov(K t,k t j )=φ j Var(K t ) (44) ]. 0

24 3 SV.4.3 SV ɛ t = σ t z t, z t i.i.d.n(0, 1) (45) log σ t = ω + φ log σ t 1 + η t, η t i.i.d.n(0,σ η ) (46), z t η t,,leverage effects. 6 SV ARCH 3.1 (generalized method of moments;gmm) (quasi-maximum likelihood estimation;qmle) 3. MCMC.3.3 MCMC. 3.1 GMM QMLE Jacquier,Polson and Rossi(1994) GMM.(45),(46) SV θ =(ω, φ, ση ). T,m E ɛ c t ɛd t p m 1 g T (θ), W T, ˆθ T = arg max g T (θ) W T g T (θ) θ 1

25 ., GMM.Anderson,Chung and Sorensen(1999),. SV QMLE,Nelson(1988),.(45),, log ɛ t = log σ t + log z t (47),(47) (46),,(47) log zt, 3. MCMC MCMC h t log σt ω 1 φ ( ) ω σ r exp (1 φ)

26 (45),(46) ɛ t = ( ) ht σ r exp z t, z t i.i.d.n(0, 1) (48) h t = φh t 1 + η t, η t i.i.d.n(0,ση ) (49) (φ, σ η,σ r ) noninformative 7 f(φ) I[ 1, 1], f(σr) 1, f(σ σ η) 1 r ση (50) I[ 1, 1] [ 1, 1] Gibbs sampler(single-move sampler ) (φ, σ r,σ η ) f(φ, σ r,σ η {ɛ t} T t=1 ) Gibbs sampler( 5. ) f(φ σ r,σ η, {ɛ t} T t=1 ) f(σ r φ, σ η, {ɛ t } T t=1) f(σ η φ, σ r, {ɛ t} T t=1 ) 3

27 {h t } T t=1 8 f(φ σr,σ η, {h t} T t=1, {ɛ t} T t=1 ) (51) f(σr φ, ση, {h t } T t=1, {ɛ t } T t=1) (5) f(ση φ, σ r, {h t} T t=1, {ɛ t} T t=1 ) (53) f(h t φ, σr,σ η, {h s } s t, {ɛ t } T t=1), (t =1,,T) (54) (51),(5),(53) ( 5.5 ) ( T ) t= φ N h th t 1 ση T, T I[ 1, 1] (55) t= h t 1 t= h t 1 σr Gamma σ η Gamma ( ( ) T, T t=1 ɛ t exp( h t ) T 1, ) T t=1 (h t φh t 1 ) (54) h t Gibbs sampler (56) (57) σ (0) r,σ (0) η, {h (0) t } T t=1 4

28 1 f(φ σr (0),ση (0), {h (0) t } T t=1, {ɛ t} T t=1 ) sampling φ(1) f(σr φ (1),ση (0), {h (0) t } T t=1, {ɛ t } T t=1) sampling σr (1) f(ση φ(1),σr (1), {h (0) t } T t=1, {ɛ t} T t=1 ) sampling σ(1) η f(h 1 φ (1),σr (1),ση (1), {h (0) t } T t=, {ɛ t } T t=1) sampling h (1) 1 f(h φ (1),σr (1),ση (1),h (1) 1, {h (0) t. f(h T φ (1),σr (1),ση (1), {h (1) t } T 1 } T t=3, {ɛ t} T t=1 t=1, {ɛ t} T t=1 ) sampling h(1) ) sampling h(1) T. n f(φ σ (n 1) r,ση (n 1) f(σ r φ(n),σ (n 1) η f(σ η φ (n),σ (n) r f(h 1 φ (n),σ (n) r f(h φ (n),σ (n) r. f(h T φ (n),σ (n) r, {h (n 1) } T t=1, {ɛ t } T t=1) sampling φ (n) t, {h (n 1) t } T t=1, {ɛ t} T t=1 ) sampling σ(n) r, {h (n 1) } T t=1, {ɛ t } T t=1) sampling σ (n) t,ση (n),ση (n),h (n),ση (n) n, (φ (n),σ (n) r, {h (n 1) t } T t=, {ɛ t} T t=1 ) sampling h(n) 1 1, {h (n 1) t } T t=3, {ɛ t } T t=1) sampling h (n), {h (n) t,ση (n) } T 1 t=1, {ɛ t } T t=1) sampling h (n), {h (n) } T t=1) f(φ, σ r,σ η, {h t} T t=1 {ɛ t} T t=1 ) 5 t η T

29 n (φ (n),σ (n) r,ση (n), {h (n) t } T t=1) f(φ, σ r,σ η, {h t} T t=1 {ɛ t} T t=1 ) (M ) N (φ (M+1),σ (M+1) r (φ (M+N),σ (M+N) r,σ (M+1) η.,σ (M+N) η, {h (M+1) t } T t=1), {h (M+N) t } T t=1) f(φ, σ r,σ η, {h t } T t=1 {ɛ t } T t=1) ˆφ = 1 N ˆσ r = 1 N ˆσ η = 1 N M+N i=m+1 M+N i=m+1 M+N i=m+1 φ (i) σ (i) r σ (i) η 3.. {h t } T t=1 single-move sampler sampling (54) ( 5.6 ) f(h t θ, {h s } s t, {ɛ t } T t=1) ( exp h ) ( t exp ɛ t σr ) exp( h t ) exp ( (h ) t µ t ) σ t (58) 6

30 µ t = h φ, t =1 φ(h t+1 h t 1 ), t T 1 1+φ φh T 1, t = T (59) σ t = σ η φ, t =1 σ η 1+φ, t T 1 ση, t = T (60) c h t f(h t ) cg(h t ) g( ) A-R ( 5.3 ) f( ) C, C exp( h t ) h t log f(h t ) = log C h t (h t µ t ) ɛ t exp( h σ t σr t ) log C h t (h t µ t ) ɛ t exp( h t σ )(1 (h t h t )) t = log C + log C + log σ r 1 π σt (h t µ t ) σ t (61) 7

31 ., ( µ ɛ t = µ t + σ t t ( f (h t ) = exp h t ( g (h t ) = exp h t = g(h t ) = c = CC σ r ) exp( h t ) 1 ) ( exp ɛ t σr (h t µ t ) σ ( t C exp (h t µ t ) π σt σ t ( 1 exp (h t µ t ) π σt σ t ) exp( h t ) exp ( (h ) t µ t ) σ t ) ɛ t exp( h σr t )(1 (h t h t )) ) ).A-R f(h t ) cg(h t ) = f (h t ) g (h t )., µ t, σ t f (h t ) g (h t ), 1 f (h t ) g (h t ), f(h t ) h t., h t,,g(h t ) f (h t ) g (h t )., f (h t ) h t, g(h t ) h t., ĥ t arg max h t f (h t ) (6) 8

32 ĥt = µ t. h t = log σ r ɛ t (ĥt µ t σ t + 1 ) (63) 3..3 Gibbs sampler(multi-move sampler ) Shephard and Pitt(1997) {h t } T t=1 singlemove sampler φ 1 Gibbs sampler persistent φ 1 Shephard and Pitt(1997) Gibbs sampler {h t } T t=1 L Gibbs sampler 1 f(φ σr (0),ση (0), {h (0) s }T s=1, {ɛ s} T s=1 ) sampling φ(1) f(σr φ(1),ση (0), {h (0) s }T s=1, {ɛ s} T s=1 ) sampling σ(1) r f(ση φ(1),σr (1), {h (0) s }T s=1, {ɛ s} T s=1 ) sampling σ(1) η f({h s } k 1 s=1 θ(1), {h (0) s }T s=k 1 +1, {ɛ s} T s=1 ) sampling {h(1) s }k 1 s=1 f({h s } k s=k 1 +1 θ(1), {h (1) s }k 1. s=1, {h (0) s }T s=k +1, {ɛ s} T s=1 ) sampling {h (1) s }k s=k 1 +1 f({h s } T s=k L +1 θ(1), {h (1) s }k L s=1, {ɛ s } T s=1 ) sampling {h(1) s }T s=k L +1. 9

33 n f(φ σ (n 1) r,ση (n 1) f(σ r φ (n),σ (n 1) η f(σ η φ(n),σ (n) r, {h s (n 1) } T s=1, {ɛ s} T s=1 ) sampling φ(n), {h (n 1) } T s=1, {ɛ s } T s=1) sampling σ (n) s, {h s (n 1) } T s=1, {ɛ s} T s=1 ) sampling σ(n) η f({h s } k 1 s=1 θ(n), {h s (n 1) } T s=k 1 +1, {ɛ s} T s=1 ) sampling {h(n) s } k 1 s=1 f({h s } k s=k 1 +1 θ(n), {h (n) s } k 1. s=1, {h s (n 1) } T s=k +1, {ɛ s} T s=1 ) sampling r {h (n) s } k s=k 1 +1 f({h s } T s=k L +1 θ(n), {h (n) s } k L s=1, {ɛ s } T s=1 ) sampling {h(n) s } T s=k L +1. θ (i) =(φ (i),σ (i) r,3..1.,ση (i) ) {h t } T t=1 multi-move sampler sampling {h s } t+k s=t 1 f({h s } t+k s=t {h s } t 1 s=1, {h s } T s=t+k+1, {ɛ s } T s=1,θ) sampling {h s } t+k s=t Shephard and Pitt(1997) {h s } t+k s=t {η s } t+k s=t f({η s } t+k s=t {h s } t 1 s=1, {h s } T s=t+k+1, {ɛ s } T s=1,θ) (64) h t 1 {η s } t+k s=t (49) {h s } t+k s=t 30

34 (64) ( [ ]) hs,l(h s ) + ɛ s exp( h σr s ) log f({η s } t+k s=t {h s } t 1 s=1, {h s } T s=t+k+1, {ɛ s } T s=1,θ) = + log f({η s } s=t t+k ) + log f({ɛ s} s=t t+k h s t+k t+k t+k = + log f(η s )+ log f(ɛ s h s,σr) s=t = 1 σ η = 1 σ η s=t t+k t+k ηs s=t s=t [ hs + ɛ s σ r s=t,θ) ] exp( h s ) t+k t+k ηs + l(h s ) (65) s=t (65) {ĥs} s=t t+k ({ĥs} t+k s=t ) g 1 σ η = 1 σ η log g t+k t+k ηs + s=t t+k s=t s=t t+k ηs + s=t s=t [ l(ĥs)+(h s ĥs)l (ĥs)+ 1 ] (h s ĥs) l (ĥs) [ ] 1 l (ĥs) ĥ s l (ĥs) l (ĥs) h s l (ĥs) = 1 ( ɛ s σ r ) exp( ĥs) 1 l (ĥs) = ɛ s σ r exp( ĥs) y s = ĥs l (ĥs), s = t,,t+ k (66) l (ĥs) 31

35 y s h s ( ) y s = h s + ξ s, ξ s N 0, 1 (67) l (ĥs) h s = φh s 1 + η s, η s N(0,ση ) (68) de Jong and Shephard(1995) simulation smoother {η s } t+k s=t g a t = φh t 1,P t = σ η i = 0,,k e t+i,d t+i,k t+i, (i =0,,k) e t+i = y t+i a t+i (69) D t+i = P t+i 1 l (ĥt+i) (70) K t+i = φp t+i (71) D t+i L t+i = φ K t+i (7) a t+i+1 = φa t+i + K t+i e t+i (73) P t+i+1 = φp t+i + σ η (74) U t+k =0,r t+k =0 e t+i,d t+i,k t+i, (i =0,,k) 3

36 i = k,, 0 {η s } t+k s=t C t+i = 1 U t+i (75) ξ t+i N(0,C t+i ) (76) η t+i = r t+i + ξ t+i (77) V t+i = U t+i L t+i (78) r t+i 1 = e t+i D t+i + L t+i r t+i V t+iξ t+i C t+i (79) U t+i 1 = 1 + L D t+iu t+i + V t+i (80) t+i C t+i (76) ξ t+i 0, C t+i g {η s } t+k s=t M-R/A-R f (1) {ηs 0}t+k s=t [ f({η ()g {η s } t+k 0 s=t min s } s=t) t+k ] g({ηs 0}t+k s=t), 1 [ f({η 0 1 min s } s=t) t+k ] g({ηs} 0 s=t), 1 t+k (3) (3) α({ηs 0 }t+k s=t, {η s} s=t t+k )= 1, { f({ηs 0}t+k s=t) g({ηs 0}t+k s=t) g({ηs 0}t+k s=t) f({ηs 0}t+k s=t), f({ηs} 0 t+k s=t) >g({ηs} 0 s=t) t+k f({η s } t+k s=t) g({η s } s=t) t+k [ f({ηs } t+k ] { min s=t)g({ηs 0}t+k s=t) f({ηs} 0 s=t)g({η t+k s } s=t), 1 f({ηs 0, }t+k s=t) >g({ηs 0}t+k s=t) t+k f({η s } t+k s=t) <g({η s } s=t) t+k 33

37 (4)() {η s } s=t t+k α({ηs 0}t+k s=t, {η s } s=t) t+k 1 α({η 0 s} t+k s=t, {η s } t+k s=t) {ĥs} t+k s=t {ĥs} t+k s=t (76) {y s } t+k s=t (67) (68) {h s } t+k s=t h t t 1 = φh t 1,P t t 1 = ση i =0,,k h t+i t+i 1,P t+i t+i 1,h t+i t+i,p t+i t+i, (i = 0,,k) h t+i t+i 1 = φh t+i 1 t+i 1 (81) P t+i t+i 1 = φ P t+i 1 t+i 1 + σ η (8) h t+i t+i = h t+i t+i 1 + P t+i t+i 1 F t+i ν t+i (83) P t+i t+i = P t+i t+i 1 P t+i t+i 1 F t+i (84) ν t+i = y t+i h t+i t+i 1 (85) F t+i = Pt+i t+i 1 1 l (ĥt+i) (86) h t+i t+i 1,P t+i t+i 1,h t+i t+i,p t+i t+i, (i = 0,,k) i = k,, 0 h t+i t+k, (i =0,,k) 34

38 h t+i t+k = h t+i t+i + P t+i(h t+i+1 t+k h t+i+1 t+i ) (87) P t+i t+k = P t+i t+i + P t+i (P t+i+1 t+k P t+i+1 t+i ) (88) P t+i = φ P t+i t+i P t+i+1 t+i (89) h t+i t+k, (i =0,,k) (76) {y s } t+k s=t (67) (68) {h s } t+k s=t {h s } t+k s=t {ĥs} t+k s=t Shephard and Pitt(1997) U i [0, 1] [ k i = int T i + U ] i, i =1,,L L + int[x] x 3.3 MCMC 9,,(48),(49) 35

39 {ɛ s } T s=1.{h s} T s=1 single-move sampler Gibbs sampler Gibbs sampler CD(convergence diagnostic) : 90 CD φ [0.8717, 0.97] ση [0.0059, ] σr [0.8799, ] SV,ARCH. SV ARCH,.,SV., GMM,QMLE MCMC 36

40 ,, SV. 37

41 5 5.1 σ t z t., E(ɛ t σ t )=σ t E(z t σ t )=σ t E(z t )=σ t, E(σ t )=E(E(ɛ t σ t )) = E(ɛ t ). Jensen E(ɛ 4 t ) = E(σ4 t )E(z4 t ) = 3E((σ t ) ) 3(E(σ t )) = 3(E(ɛ t )),. E(ɛ 4 t ) (E(ɛ t )) 3 5. Gibbs sampler (θ 1,θ,θ 3 ) 3 f(θ 1,θ,θ 3 ) f(θ 1,θ,θ 3 ) f(θ 1 θ,θ 3 ) f(θ θ 3,θ 1 ) f(θ 3 θ 1,θ ) 38

42 (θ 1,θ,θ 3 ) f(θ 1,θ,θ 3 ) Gibbs sampler (θ (0),θ(0) 3 ) f(θ 1 θ (0),θ(0) 3 ) θ (1) 1 θ (1) 1 f(θ θ (0) 3,θ(1) 1 ) θ(1) θ (1) 1 θ(1) f(θ 3 θ (1) 1,θ (1) ) θ (1) 3 1 (θ (1),θ (1) 3 ) (θ () 1,θ (),θ () 3 ) n (θ (n) 1,θ(n),θ(n) 3 ) n (θ (n) 1,θ (n),θ (n) 3 ) f(θ 1,θ,θ 3 ) 5.3 A-R f(x) ( g(x) ) f(x) A-R(acceptance-rejection; ) g(x) x f(x) cg(x)( c ) f(x) (1)g(x) x [0, 1] u 39

43 () f(x ) cg(x ) x f(x) ( ) 1 f(x ) cg(x ) (1) u f(x ) cg(x ) x u > f(x ) (1) cg(x ) ( Pr x u f(x) ) cg(x) = = ( Pr u f(x) cg(x) ( Pr u f(x) cg(x) f(x) cg(x) g(x) f(x) = cg(x) g(x)dx = f(x) ) x g(x) ) x g(x)dx f(x) f(x)dx g(x) f(x) g(x) cg(x) c 5.4 M-H/A-R 5.3 A-R x f(x) cg(x) g(x) f(x) M-H/A-R(Metropolis-Hasting/acceptancerejection) 5.3 g(x) M-H/A-R f(x) N (X 1,,X N ) 40

44 (1)n =1 x n >1 x = X n 1 [ ] f(x) ()A-R min cg(x), 1 y X n (3) α(x, y) = 1, { f(x) cg(x) cg(x) f(x), f(x) >cg(x) f(y) cg(y) [ ] { f(y)g(x) min f(x)g(y), 1 f(x) >cg(x), f(y) <cg(y) (4)() y α(x, y) 1 α(x, y) X n = y X n = x (5)n <N n = n +1 () n = N f(x) cg(x) α(x, y) =1 A-R 41

45 5.5 SV,h 1,f(h 1 θ) c( ). f(φ ) f(θ, {h s } T s=1, {ɛ s} T s=1 ) = f(θ)f({h s } T s=1 θ)f({ɛ s} T s=1 {h s} T s=1,θ) = f(φ)f(ση)f(σ r)f(h T h T 1,θ) f(h 1 θ)f({ɛ s } T s=1 {h s } T s=1,σr) T f(φ) exp ( (h ) t φh t 1 ) t= ση ) ( P T t= φ hth t 1 P T t= exp h t ση P T t= h t I[ 1, 1] f(σ η ) =f(σ η ) dσ η dση f(θ, {h s } T s=1, {ɛ s} T s=1 )σ4 η = f(θ)f({h s } T s=1 θ)f({ɛ s } T s=1 {h s } T s=1,θ)σ 4 η = f(φ)f(ση )f(σ r )f(h T h T 1,θ) f(h 1 θ)f({ɛ s } T s=1 {h s} T s=1,σ r )σ4 η T f(ση 1 )σ4 η exp ( (h ) t φh t 1 ) σ t= η ση ( ) =(ση ) T 1 1 exp ση P T t= (ht φh t 1) 4

46 f(σ r ) =f(σ r ) dσ r dσr f(θ, {h s } T s=1, {ɛ s} T s=1 )σ4 r = f(θ)f({h s } T s=1 θ)f({ɛ s} T s=1 {h s} T s=1,θ)σ4 r = f(φ)f(ση )f(σ r )f(h T h T 1,φ,ση ) f(h 1 θ)f({ɛ s } T s=1 {h s} T s=1,σ r )σ4 r T ( ) f(σr)σ r 4 1 σ t=1 r exp(h t /) exp ɛ t σr exp(h t ) ( ) (σr ) T 1 exp σr P T t=1 ɛ t exp( ht) 5.6 SV h t 5.5,h 1,f(h 1 θ) c( ). f(h t θ, {h s } s t, {ɛ s } T s=1 ) = f({ɛ s } T s=1 θ, {h s } T f(θ, {h s } T s=1 s=1) ) f(θ, {h s } s t, {ɛ s } T s=1) = f({ɛ s } T s=1 θ, {h s } T f(θ, {h s } s t ) s=1)f(h t θ, {h s } s t ) f(θ, {h s } s t, {ɛ s } T s=1 ) f(ɛ 1 θ, h 1 ) f(ɛ T θ, h T ) f({h s} T s=1 θ) f({h s } s t θ) f(ɛ t θ, h t )f({h s } T s=1 θ) = f(ɛ t θ, h t )f(h T h T 1,θ) f(h 1 θ) f(ɛ 1 θ, h 1 )f(h h 1,θ), t =1 f(ɛ t θ, h t )f(h t h t 1,θ)f(h t+1 h t,θ), t T 1 f(ɛ T θ, h T )f(h T h T 1,θ), t = T 43

47 t =1 f(h t θ, {h s } s t, {ɛ s } T s=1) ( ) 1 σ r exp(h 1 /) exp ɛ 1 1 exp ( (h ) φh 1 ) σr exp(h 1 ) σ η ση ( exp h ) ( ) 1 exp ɛ 1 exp( h σr 1 ) exp (h 1 h φ ) t T 1 σ η φ f(h t θ, {h s } s t, {ɛ s } T s=1) ( exp h ) ( ) t ɛ t exp σr exp(h exp t) ( exp h ) ( 1 exp ɛ t σr t = T ) exp( h t ) ( (h t φh t 1 ) exp σ η ) exp ( h t φ(h t 1+h t+1 ) σ η 1+φ 1+φ ) ( (h ) t+1 φh t ) σ η f(h t θ, {h s } s t, {ɛ s } T s=1) ( ) 1 σ r exp(h T /) exp ɛ T 1 exp ( (h ) T φh T 1 ) σr exp(h T ) σ η ση ( exp h ) ( ) T exp ɛ T exp( h σr T ) exp ( (h ) T φh T 1 ) ση 44

48 6 1 ( ) ds E WqY = E qy [(α λk)dt + dq] S V WqY ( ds S ) = E Y [(α λk)dt +(Y 1)λdt] =αdt = E WqY [ λkdt + dw + dq] E qy [σ dt +(dq) λkdtdq] (σ + E Y (Y 1) )dt. E WqY I t dw t,dq t,y. 3 [t, t + h], v(t, t + h) = exp( hy (t, t + h)) Y (t, t + h) = 1 log v(t, t + h) h. 4 (1988) Granger (1988) Granger. x t (predicton mean squared error:pmse) σ (x t I t 1 ) σ (x t I t 1 {z s } s=0 t 1 ) σ (x t I t 1 ) t 1 PMSE σ (x t I t 1 {z s } t 1 s=0 ) t 1 {z s} t 1 s=0 ) PMSE 6.1 (Granger ) σ (x t I t 1 )=σ (x t I t 1 {z s } t 1 s=0 ) Granger z t x t 45

49 Granger x t z t PMSE 5 t µ t,σ t,γ t,δ t,ρ t I U t - 6 z t η t,(46) z t 1 ɛ t = σ t z t log σ t = ω + φ log σ t 1 + λz t 1 + η t leverage effects. 7 φ [ 1, 1] ση σ r log ση log σ r [, ] σ η σr f(ση)=f(log σr) d log σ η dση 1 ση f(σ r )=f(log σ r ) d log σ r dσ r 1 σr 8 f(φ, σr,σ η, {h t} T t=1 {ɛ t} T t=1 ) {h t} T t=1 (φ, σr,σ η, {h t} T t=1 ) Gibbs sampler 9 GAUSS. 10 (6),f (h t ) log f (h t )., (log f (h t )). 11 CD {θ (i) } i= {θ (i) } 3000 i= {θ(i) } i=7001 {θ (i) } 3000 i=1 θ 1 ˆσ 1 {θ (i) } i=7001 θ ˆσ CD θ 1 CD = θ ˆσ ˆθ 3000, 3000 {θ (i) } 3000 i= {θ(i) } i=7001

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51 [9] Clark, P. A Subordinated Stochastic Process Model with Finite Variance for Speculative Process Econometrica, Vol.41, (1973), pp [10] de Jong, P. and Shephard, G. The Simulation Smoother for Time Series Biometrika, Vol.8, (1995), pp [11] Engle, R.F. Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation Econometrica, Vol.55, (198), pp [1] Ghysels, Harvey and Renault Stochastic Volatility in G.S.Maddala and D.L.Rao(eds), The Handbook of Econometrics, North-Holland, (1996), pp [13] Hull, J. and White, A. The pricing of options on assets with stochastic volatilities J.Finance, Vol.4, (1987), pp [14] Jacquier, E., Polson, N.G., and Rossi, P.E. Bayesian analysis of stochastic volatility models (with discussion) J.Business Econom.Statist., Vol.1, (1994), pp [15] Merton, R.C. Option pricing when underlying stock returns are discontinuous J.Financ.Econom., Vol.3, (1976), pp [16] Merton, R.C. Continuous Time Finance, Basil Blackwell, Oxford, (1990) 48

52 [17] Nelson, D.B. The Time Series Behavior of Stock Market Volatility and Returns unpublished doctoral dissertation, Department of Economics, M.I.T. [18] Schwert, G.W. Business cycles, financial cycles and stock volatility Carnegie-Rochester Conference Series on Public Policy, Vol.39, (1989), pp83-16 [19] Shephard, G. and Pitt, M.K. Likelihood Analysis of Non- Gaussian Measurement Time Series Biometrika, Vol.84, (1997), pp [0] Tauchen, G. and Pitts, M. The Price Variability-Volume Relationship on Speculative Markets Econometrica, Vol.51, (1983), pp [1] Taylor, S.J. Modeling Financial Time Series, John Wiley Sons, (1986) 49

03.Œk’ì

03.Œk’ì HRS KG NG-HRS NG-KG AIC Fama 1965 Mandelbrot Blattberg Gonedes t t Kariya, et. al. Nagahara ARCH EngleGARCH Bollerslev EGARCH Nelson GARCH Heynen, et. al. r n r n =σ n w n logσ n =α +βlogσ n 1 + v n w