39, 1, , Empirical Analysis on Jump Detection in High-Frequency Data Hiroki Masuda and Takayuki Morimoto, (realized volatility, RV).,, ( )

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1 39, 1, , Empirical Analysis on Jump Detection in High-Frequency Data Hiroki Masuda and Takayuki Morimoto, (realized volatility, RV).,, ( ) realized bipower variation (BPV)., BPV, Lee and Mykland (2008).,,. Estimation and prediction of daily financial volatility have been central issues in financial econometrics. In particular there have been many literatures reporting on a study of realized volatility (RV) based on high-frequency data. In addition, much attention is currently focused on realized bipower variation (BPV). BPV is theoretically defined in the framework of a continuous time price process independent of specific model structure. Moreover it enables us to effectively extract robust information about the diffusive volatility against jumps. In this paper, as an application of BPV, we introduce an empirical analysis of non-parametric jump-detection using the statistics proposed by Lee and Mykland (2008). Given a sufficiently large sample size, the statistics allows us to apply the test to data at any sampling frequency and to extract information of timing and signed size of jumps. Thus it is informative when investigating market microstructure in financial markets. :, Bipower variation,, Lee-Mykland, 1.,,, ( Press (1967), Merton (1976), Beckers (1981), Ball and Torous (1983) )., ( Chernov et al., ( hiroki@math.kyushuu.ac.jp)., ( morimot@kwansei.ac.jp).

2 (2003), Eraker et al. (2003), Aït-Sahalia (2004) ), : ; ; ;, ( ),.,. Andersen et al. (2003, 2007b),,,.,,.,. (realized volatility, RV), RV., RV,., realized bipower variation ( BPV) Barndorff- Nielsen and Shephard (2004, 2006),. BPV RV,,,,., BPV,, 1) Lee and Mykland (2008, Lee-Mykland),,., Barndorff-Nielsen and Shephard (2006), 1) RV BV, Huang and Tauchen (2005) BPV RV, RV., Jiang and Oomen (2008), (swap variance measure) SwV SwV RV. SwV

3 35 10, Lee-Mykland, 3 S&P 500,., Huang (2006), 10 S&P 500 5, Lahaye et al. (2007),,,,,., Beine et al. (2006),, Dungey et al. (2007), (co-jumps)., Lee-Mykland,,,,,. Lee-Mykland,,,, ( ),., Barndorff-Nielsen and Shephard (2006) Aït-Sahalia and Jacod (2009).,,, : RV, BPV, ( 2 ); ; Dungey et al. (2007),,., 2 BPV Lee-Mykland, 3. 4,,. 2. Lee-Mykland,.

4 I := [0, T ], T > 0,, S = (S t ) t I n + 1 (t i ) n i=0. n, 0 t 0 < t 1 < < t n T. X := log S : t t X t = X 0 + µ s ds + σ s dw s + ζ j. (2.1) 0 0 j N t, w, µ σ 2),, (N t ) t I λ t (N, N t := N t lim s t N s = 1 a.s., t I), ζ j I j ζ j 0 a.s. ( j 1). N t = 0 0. J t t, (2.1) : dx t = µ t dt + σ t dw t + J t dn t., σ t > 0 a.s. ( t I), I X a.s. :, λ I a.s.. (2.1), I i := (t i 1, t i ] i X := X ti X ti 1 ti ti = µ s ds + σ s dw s + ζ j t i 1 t i 1 N ti 1 <j N ti ti ti =: µ s ds + σ s dw s + i Z (2.2) t i 1 t i 1 ( Z t := j N t ζ j ): i N = N ti N ti 1 = 0 X I i, i Z = 0., X I i, ((2.2) ) ((2.2) )., ( - ),., I ( i X) n i=1, µ, σ,., max 1 i n (t i t i 1 ) 0 2),, X.

5 37, µ, σ,, BPV., Lee-Mykland. 2.2 : RV BPV I ( i X) n i=1, RV n RV n = ( i X) 2, n ( ) i=1 T RV n p σsds 2 + ( X s ) 2 =: QV c + QV d 0 0<s T. RV n I, I QV c (X ), QV c., X, QV d QV c,, RV n,. Barndorff-Nielsen and Shephard (2004, 2006) QV c BPV,. BPV, 2 : n 1 BP V n = i X i+1 X. i=1, BP V n BP V n p 2 π QV c,, π 2 BP V n Z QV c. RV n BP V n µ σ, Z,., BPV. X QV d 0, RV n QV c, X,, RV n :, Barndorff-Nielsen and Shephard (2006).

6 Lee-Mykland, Lee-Mykland., t i = it/n, h = h n := t i t i 1 ( ) Lee-Mykland,,, n I i = (t i 1, t i ]., I i µ ti 1 h σ ti 1, I., I i i X, (2.2) i X µ ti 1 h σ ti 1 h iw + iz (2.3) h σ ti 1 h. U i := iw h N(0, 1) i.i.d.. I i, i Z 0, (2.3), h 0, 1 0, U i ( ± )., (2.3) ( ), I i. µ σ µ ti 1 σ ti 1. Lee-Mykland, I i, t i 1 K = K n ( N) W i := [t i K, t i 1 ],., 1/2 < γ < 1 K n = O(n γ ) ; γ, W i K n h n = O(n γ 1 ) 0.

7 39 ˆµ i (γ) := 1 K 1 ˆσ i (γ) := ( 1 K 2 i 1 j=i K+1 i 1 j=i K+2 j X = X t i 1 X ti K K 1 ) 1/2 j 1 X j X, I Lee-Mykland {T i (γ)} i n : T i (γ) = ix ˆµ i (γ). (2.4) ˆσ i (γ) Lee-Mykland,, W i BPV. Lee-Mykland Theorem 1.1 2, {T i (γ)} i n n, i n { 1 T i(γ) c 2 ( U i 1 K 1 i 1 j=i K+1, ) } i Z U j + = o p (1) (2.5) c 2 σ ti K h. c 2 := 2/π. W i, (2.3), i Z σ ti 1 σ ti K. K T i (γ)., (2.5), I i : i Z = 0 T i (γ) ; i Z 0 T i (γ) (± )., n, i, T i (γ) I i 1, I i., ( ), Lee-Mykland. (2.5), I, n T i (γ) T 0 i (γ) := 1 c 2 ( U i 1 K 1 i 1 j=i K+1 U j )

8 {Ti 0 (γ)}, Galambos (1978, Theorem 3.8.2) : ( ) x R : lim [b Pr n max T i 0 (γ, 2) a n n i n a n b n : 2 log n a n = c 2 b n = c 2 2 log n. log π + log log n 2c 2 2 log n, J n = {i n : X I i } (I i ) i Jn ] x = exp( e x ). (2.6) (2.6), ( ) α (0, 1), (I i ) i Jn I 1 { T 0 i (γ) } i n, i, T i (γ) α I i. α.. 1. α := 1 exp( e x ) ( ), x β := log{ log(1 α)} : β. 2. ( i X) i n, i (2.4) T i (γ), {T i (γ)} K i n : K,,. 3. (2.6), i n, b n ( T i (γ) a n ) > β I i, b n ( T i (γ) a n ) β I i : I, h I i 1. α = exp( e β ) = 1 α = , β = log( log(0.9999)) = b n ( T i (γ) a n ) > β, I i. K., K, K

9 41 BV. Lee-Mykland,, n, γ (1/2, 1) K :, 1, 1, 1, 30, 15, 5, 7, 16, 78, 110, 156, 270 (Lee-Mykland Section 1.3 ). K. 1 N, K = N 1/2 (2.7). N 1/2 N 1/2. Lee-Mykland. Lee-Mykland, ;,., 3., Lee-Mykland. I ( ) Barndorff- Nielsen and Shephard Aït-Sahalia and Jacod,, Lee-Mykland. Lee-Mykland, BPV {T i (γ)},, BPV m ( 3) multipower variation (MPV), BPV. MPV, Barndorff-Nielsen et al. (2006)., QV c BPV, 3 MPV., Corsi et al. (2008), MPV.,, MPV Lee-Mykland,., Lee-Mykland BPV.

10 (Black-Scholes ) X, Lee-Mykland. X : X t = 0.2t + w t + Z t. (2.8) Z w, N (0, δ 2 ) (δ 2 ). I = [0, 1] i/n (i n) X, 1000 (2.7) K N n Lee-Mykland ;, (0, K]. I 100 :, 0 < u 1 < u 2 < < u 100 < U(0, 1)-. Lee-Mykland α = 0.05, α 0.05, 0.01, 0.001, ,., α β : (α, β) = (0.05, 2.97), (0.01, 4.60), (0.001, 6.91), (0.0001, 9.21). (2.9) α β. α,,., l, Lee-Mykland J l, J l / ( J l ) 1000 l=1.,, :, Lee-Mykland., J l /100 l. α 0. (2.8), δ 2 = 0.5, 0.25., 1000

11 43 1 (α, β) = (0.0001, 9.21) 1000 Lee-Mykland mean(j) s.d.(j), (2.12) mrv. α = δ 2 = 0.5 n K mean(j) s.d.(j) mean(mrv ) s.d.(mrv ) δ 2 = 0.25 n K mean(j) s.d.(j) mean(mrv ) s.d.(mrv ) P mean(j) := q s.d.(j) := 1000 l=1 J l/100 P 1000 l=1 { J l/100 mean(j)} 2 2 (α, β) = (0.001, 6.91) 1000 Lee-Mykland mean(j) s.d.(j), (2.12) mrv. α = δ 2 = 0.5 n K mean(j) s.d.(j) mean(mrv ) s.d.(mrv ) δ 2 = 0.25 n K mean(j) s.d.(j) mean(mrv ) s.d.(mrv ) P mean(j) := q s.d.(j) := 1000 l=1 J l/100 P 1000 l=1 { J l/100 mean(j)} 2 mean(j) := J l /100, (2.10) 1000 l=1 s.d.(j) := { J l /100 mean(j)} , (2.11) l=1,, (2.9) (α, β) 1 4.

12 (α, β) = (0.01, 4.60) 1000 Lee-Mykland mean(j) s.d.(j), (2.12) mrv. α = 0.01 δ 2 = 0.5 n K mean(j) s.d.(j) mean(mrv ) s.d.(mrv ) δ 2 = 0.25 n K mean(j) s.d.(j) mean(mrv ) s.d.(mrv ) P mean(j) := q s.d.(j) := 1000 l=1 J l/100 P 1000 l=1 { J l/100 mean(j)} 2 4 (α, β) = (0.05, 2.97) 1000 Lee-Mykland mean(j) s.d.(j), (2.12) mrv. α = 0.05 δ 2 = 0.5 n K mean(j) s.d.(j) mean(mrv ) s.d.(mrv ) δ 2 = 0.25 n K mean(j) s.d.(j) mean(mrv ) s.d.(mrv ) P mean(j) := q s.d.(j) := 1000 l=1 J l/100 P 1000 l=1 { J l/100 mean(j)} 2. Lee-Mykland, I, 1, 30, 15, 5, n = 6000, 12000, 24000, , n = 5000, 10000, 15000, mean(j) Lee-Mykland, δ 2 = 0.5, 0.25, α n

13 45. δ 2 = 0.25,, δ 2 = ,, α mean(j) s.d.(j)., α.,, α ( α mean(j) ). 3) 1 4, RV ( 1 ). Lee-Mykland I i = ((i 1)/n, i/n] mrv n := n n ˆ J n j n,j / ˆ J n ( j X) 2 (2.12), I. ˆ J n Lee- Mykland I i i, J ˆ n J ˆ n. RV (mrv m modified ). (2.8) 1, mrv 1, n., mrv 1, Lee-Mykland mrv. α,., 4 Lee-Mykland,. 3) α, α = 0.5, 0.1, α = 0.001,

14 , Lee-Mykland,,., 502 : TOPIX (TOPIX) TOPIX 500 (TOPIX500), ; (USD/JPY), , USD/JPY bid ask (mid-quote).,, 5 4)., TOPIX TOPIX500 ( ) (60/5) = 54, USD/JPY (60/5) = 288.,, 2,, Lee-Mykland,., USD/JPY (UTC), TOPIX TOPIX500., Lahaye et al. (2007), α = Lee-Mykland., T, K (2.7)., TOPIX, TOPIX500, USD/JPY 502, Lee-Mykland., : N (days), N (jump days), JF (jump day) = N (jump days)/n (days),, 4) RV, 5,. Bandi and Russell (2006).

15 47 5 N (days) N (jump days) JF (jump day) USD/JPY N (days) N (jump days) JF (jump day) N (days) = ( ), N (jump days) = ( ), JF (jump day) = N (jump days)/n (days)., (Jump-Frequency)., USD/JPY, USD/JPY 24, 1 TOPIX TOPIX500., 6, ,,.,., 7 5,, ˆµ (jumps) , 5 1.

17 N N (jumps) JF (jumps) N N (jumps) JF (jumps) N = ( ), N (jumps) = ( ), JF (jumps) = N (jumps)/n., y, x., 2 (FUNAI ELECTRICS NTT DoCoMo) (Monex Beans KINDEN), USD/JPY, TOPIX, TOPIX500.,,?, ( ), ( ),, ( Seven & I ),.,,, ( ),.,,, ( Sumitomo Forestry ).,

18 N N (jumps) ˆµ (jumps) N N (jumps) ˆµ (jumps) N = ( ), N (jumps) = ( ), ˆµ (jumps) = ( ).,., RV,.,., (New York Stock Exchange, NYSE), x, y RV U. (Tokyo Stock Exchange, TSE),, 4 W.,,? 2 3, : 2, USD/JPY, 5 ; 3,, TOPIX TOPIX500 5.

19 51 2 (USD/JPY) 3 (TSE)

20 , USD/JPY 12:30, 13:30, 24:00. UTC, (Eastern Standard Time, EST), 12:30 5 7:30, 4 8:30., 13:30 8:30 (EST) 9:30 (EST), 24:00 19:30 (EST) 20:30 (EST)., (Japan Standard Time, JST), JST UTC 9, 12:30 21:30 (JST), 13:30 22:30 (JST), 24:00 9:00 (JST).,, NYSE 9:30 (EST) TSE 9:00 (JST).,, USD/JPY., USD/JPY,., 3,.,,,.,. 3.2, TOPIX, TOPIX500, USD/JPY 5,, ( )., ( ),.,, 15 4 USD/JPY. Lahaye et al. (2007),,. (Suprises) :,. ( ),. (Scheduled news) :,,.

21 53.,, Lee-Mykland,,,., ( ) 2,.,,,, (jump term) (diffusion term).,,.,,., 2., 5, USD/JPY 1417,.,,,, USD/JPY,, (FOMC), 3., (suprises) :., EST, UTC 5).. (1) 11 ( ): 12:46:30 (UTC), 8:46:30 (EST) (2) 175 ( ): 13:02:59 (UTC), 9:02:59 (EST) (3) 77 ( ): 13:37:46 (UTC), 9:37:46 (EST) 5)

22 (USD/JPY),. 4, USD/JPY Lee-Mykland., y 9.21 α = , 4.60 α = 0.01., Lee-Mykland., α = 0.01 α = , 5, 11 Lee-Mykland. α = 0.01 α = 0.001, 15 α = 0.01., 3, 3., (1),., (2), (3). 6),, (1), (2) 6), 5,, 5.

23 (USD/JPY),. 7), ( Federal Open Market Committee, FOMC ) ( Foreign Exchange Intervention Operations, FXIO ) 2 8). 2,,., %, 14,., FOMC FXIO,. 6, USD/JPY FOMC FXIO., FOMC 7) (1)-(3), 14:03:11 (UTC) 93,,,,. 8) FOMC FOMC FXIO FXIO, USD/JPY,.

24 FOMC FXIO 8 FOMC FXIO N (news) N (no news) N (jumps news) N (jumps no news) JF (jumps) JF (jumps news) JF (jumps no news) ˆµ (jumps news) ˆµ (jumps no news) ˆσ 2 (jumps news) ˆσ 2 (jumps no news) : N %. FXIO , 2004, 2005.,, 8 :, N ( ), N ( ), JF ( ), JF ( ), ˆµ ( ), ˆσ 2 ( ),,,,,., :

25 57 FOMC, FXIO JF (jumps news) JF (jumps no news) ; FOMC ˆµ (jumps news) ˆµ (jumps no news), FXIO. 2,., JF (jumps news) JF (jumps no news),,. 8, Z 0 := JF (jumps news) JF (jumps no news) JF (jumps) (1 JF (jumps)) (1/N (news) + 1/N (no news)). p P = 2 Pr (Z Z 0 ), P α 0., FOMC Z 0 = , P = , FXIO Z 0 = , P = , α 0 = ,., ˆµ (jumps news) ˆµ (jumps no news),, (Welch )., 8, t 0 = ˆµ (jumps news) ˆµ (jumps no news) ˆσ2 (jumps news)/n (jumps news) + ˆσ 2 (jumps no news)/n (jumps no news) ν t. ν : ν = (A 1 + A 0 ) 2 B 1 + B 0, A 0 = ˆσ 2 (jumps news)/n (jumps news), A 1 = ˆσ 2 (jumps no news)/n (jumps no news), B 0 = {ˆσ 2 (jumps news)/n (jumps news)} 2 /{N (jumps news) 1}, B 1 = {ˆσ 2 (jumps no news)/n (jumps no news)} 2 /{N (jumps no news) 1}. p P, P α 0., FOMC t 0 = , P = , FXIO t 0 = , P = , α 0 = FOMC, FXIO., FOMC

26 ,.,, Lee-Mykland Lahaye et al. (2007),,. 3.3, Lee-Mykland., 10 20,. Lee-Mykland,,,.,, Lee-Mykland., (West Texas Intermediate futures contract deliverable at Cushing, Oklahoma, WTI ) , 1. 7 Lee-Mykland,, N = 6483, α = α = 0.01, α = , 20, , α,. α = , 25 16, WTI., JF (jumps) = , 5 FUNAI ELECTRIC JF (jumps) = , WTI :,., 7,., , 1988, 1991, , 2002, 2002 (

27 59 7 ( ) )., WTI?, , WTI Lee-Mykland. α = 0.01 α = , , 1 17.,,?, 1991, : (a) (1 17 ): ( ); (b) (8 19 ): (8 ). 2, 8 2 (a) (b).,.

28 (1991 ), Lee-Mykland,,,.,,,,,. 4. Lee-Mykland,,, 5,, Lee-Mykland.,,,.,, 1 1,, 10 1.,,.,, FOMC FXIO.

29 61,., FOMC, FXIO.,,, Lee-Mykland.,., I T,. Lee- Mykland,,,, T ( )., I X,., Woerner (2007), ( ).,. Andersen et al. (2007a), RV,,.,.,,,., Corsi et al. (2008),., Fan and Wang (2007).,,,. ( ), Tauchen and Zhou (2007).,.

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31 63 Lee, S. and Mykland, P. A. (2008). Jumps in financial markets: A new nonparametric test and jump dynamics, Rev. Financial Studies, 21, Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous, J. Financial Econ., 3, Press, S. J. (1967). A compound events model for security prices, J. Business, 40, Tauchen, G. and Zhou, H. (2007). Realized jumps on financial markets and predicting credit spreads. Preprint. Woerner, J. H. C. (2007). Inference in Lévy-type stochastic volatility models, Adv. in Appl. Probab., 39,

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kanatani0709wp.dvi Recent Studies on Estimation of Volatility in the Presence of Market Microstructure Noise 2009 9 RV MMN 21730174 522-8522 1-1-1 HP: http://www.biwako.shiga-u.ac.jp/sensei/t-kanatani/ Email: t-kanatani@biwako.shiga-u.ac.jp