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1 ( ) 19 1
2 ,.,.,,. [15],.,.,,., , 1., , 1., 1,., 1,,., 1. i
3 t t ii
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5 ,000-3, ,000-3, , (2,000, ) (2,000, ) (2,000, ) (2,000, ) (2,000, ) (2,000, ) (2,000-3,000, ) (2,000-3,000, ) (3,000-30,000, ) (3,000-30,000, ) ,000 ( 1) ,000 ( 2) ,000 ( 3) ,000-3, ,000-30, ,000 ( 1) ,000 ( 2) ,000 ( 3) ,000-3, ,000-30, Up Down t t t t iv
6 1 1.1,.,,.,.,,., ( ),.,,,,.,., [14], ( ), ( ), ,.,,,. [15],,.,,, (Up Down). Up Down 1. Up Down 1,., 1, , , : 2 :, ; 3 :,, ; 1
7 4 :., 1, , 1., 1, 2 ; 5 : 1,, ; 6 :. 2
8 2 2.1, 2.,,., ( ). (CDA: continuous double auction ), Cohen et al. [2], Luckock [7], Tang and Tian [13], Maslov [8], Slanina [11], Daniels et al. [3], Farmer et al. [4], Smith et al. [12].,,.,,,., [15], ( ) ( ),. 2.2,.,, /.,.,., ( ) ( ) ( ),,.,,.,,., 2.1.,,. 3
9 2.1:,, 2.,.,,.,. (depth). (Ask) (Bid) ( 2.2). 2.2:,.,., ( ) 4
10 ( ) ( ), ( ) ( ) ( )., :,, 2 ( ) ( ) 4.,. M/M/1.,., : (µa ) ra r A (2.1) f A (t) = t e (µ A+λ A )t I ra (2 µ A λ A t). (2.2) f B (t) =, λ A (µb λ B f A (t), f B (t) : ( ) ; ) rb r B t e (µ B+λ B )t I rb (2 µ B λ B t). r A, r B : ; µ A, λ A, µ B, λ B : 4 ; I r (z) 1, I r (z) = I r (z) = ( ( z )r z 2n 2) 2 n=0 n!(r + n)!., { 1, ρ A 1, (2.3) P A = P (T A ) = r ρ A A, ρ A > 1. { 1, ρ B 1, (2.4) P B = P (T B ) = r ρ B B, ρ B > 1. 5
11 , T A, T B P A, P B : ( ) ; : ( ) ; ρ A, ρ B : (ρ A = λ A /µ A, ρ B = λ B /µ B ).,, : ( (2.5) f U (t) = f A (t) 1 ( (2.6) f D (t) = f B (t) 1 t 0 t 0 ) f B (τ)dτ, ) f A (τ)dτ., ( ) P U (P D ), (2.7) P U = P (T U ) = (2.8) P D = P (T D ) = f U (t)dt = P A f A (t) t f D (t)dt = P B f B (t) t f B (τ)dτ. f A (τ)dτ.,,,, 1,.,,,,.., ( ), ( ),., ( ), ( ), ( ) 2 ( ),.,,., 2, r U A,r U B,r D A,r D B, 1. : 6
12 (2.9) P = ( ) P UU P UD P DU P DD (2.10) P UU = P U (r A U, r B U ) = f U (t)dt = P A f A (t)dt t f B (τ)dτ. (2.11) P DU = P U (r A D, r B D ) = f U (t)dt = P A f A (t)dt t f B (τ)dτ. (2.12) P UD = P D (r A U, r B U ) = f D (t)dt = P B f B (t)dt t f A (τ)dτ. (2.13) P DD = P D (r A D, r B D ) = f D t)dt = P B f B (t)dt t f A (τ)dτ. r A U, r B U, r A D, r B D :, ; P UU :, ; P UD :, ; P DU :, ; P DD :, 7
13 3 3.1,.,,., Niederhoffer and Osborne [10], Fielitz and Bhargava [6], Fielitz [5], McQuenn and Thorley [9].,,,. Fielitz and Bhargava [6], ,, 3 (Up, Down, No movement). 2., 200 (Order) ;,.,. McQuenn and Thorley [9], NYSE,, 2 (Up and Down) 2. 2., Niederhoffer and Osborne [10],., 6, /8dollar, Anderson and Goodman [1],.,,.,,, 1, , Nonparametric,,.,,,.,. 8
14 , 2 ( ), 2 1. : 1. S = U, D : ; 2. X t, t = 0, 1, 2,... :. X t = U X t = D; 3. n ijk : i, j, k ( n U,U,D 2 ); 4. n ijkl : i, j, k, l ; 5. P ij = P {X t = j X t 1 = i}; 6. P ijk = P {X t = k X t 1 = j, X t 2 = i}; 7. P ijkl = P {X t = l X t 1 = k, X t 2 = j, X t 3 = i};, Anderson and Goodman [1] 1 2.,,.,,,., (null hypothesis) u (uth-order), (alternative hypothesis) u + 1.,, P ijk = P jk, for i = 1, 2,..., m (i, k S; j S u ). (likelihood ratio criterion) : (3.1) λ = i,j,k( ˆP jk / ˆP ijk ) n ijk. P jk ˆP jk. (3.2) ˆPjk = i S n ijk / i S n ijk. k S 2 log λ, m u (m 1) 2 χ 2. m 2 (U D), u. 5%.,., m m m (3.3) 2 log λ = 2 n ijk (log P ijk log P jk ) i=1 j=1 k=1 9
15 m(m 1) 2 = 2, 2 log λ > 5.991,., 2. (3.4) 2 log λ = 2 m m m i=1 j=1 k=1 l=1 m n ijkl (log P ijkl log P jkl ) m 2 (m 1) 2 = 4, 2 log λ > 9.488,.,.,,., 5%, K. (3.5) K k=0 ( ) n 0.95 n k 0.05 k = k, n. K,. K, (, 1 ), (, 2 ) (, 3 ). 1,., : 1. ; 2. ;
16 , 1 1,497, 2 1,527, 3 1,609.,,,,., 1,, 1.,. : 1. :.. (a) :. (b) :. 2. :,.,, 3.1., 1.,,. 30,000,. 30,000, 2,000, 2,000 3,000 3,000 30, ,000 3,000 3,000 30,000,.,,. 3.1: 2, ,000,000 1,000 3, ,000,000 10,000 30, ,000,000 50,000 50, ,000, , ,
17 3.3.2,,, 1,.,.,,., 2.. 9,.,,,. 5%,.,.,,,,.., ( ) 20,,.,
18 4 4.1,,. 4.2,,,,. Group. Total. G1 G5,, 300 1, 5. G1 300, G5 4. Top100 Top : 1 ( ) 2 ( ) ( ) Group Total 1 76,808 1, ,633 1, ,459 2,190 Top50 9,116 76,808 18,248 8, ,633 19,591 12, ,459 31,305 Top100 5,503 76,808 12,671 5, ,633 13,348 7, ,459 20,592 G1 1,461 76,808 6,125 1, ,633 6,552 2, ,459 9,682 G , ,752 1, ,305 1,361 G G G ,., 3 2,.,, 2,000 Group Top50.,, (,, ).,,.,,.,,., 2,000-3,000 3,000-30,000 G1(Top300). 13
19 4.3: 1-1.5tick 1.5-2tick 2-3tick 3-4tick 4-5tick >=5tick K-tick. K-tick, K., K-tick,,., 3. 6, K-tick, 6,., %. 30%.,, : 3,000-3,
20 4.5: 2,000-3, : 2,
21 4.2, 1.,,,, 1,, , ( ) 2 (Up Down),, 1., 5% ,. : ( (4.1), P UU = P DU = n UU n UU + n UD ; P UD = n DU n DU + n DD ; P DD = P = n UD n UU + n UD ; P UU P DU n DD n DU + n DD n ij i, j. P UD P DD,, 1 ( 4511, ). ( ) , ,. 1,. ) 16
22 4.4: - 4.5: - 17
23 . 4.7: - 1(2,000, ) P uu P ud P du P dd P uu P ud P du P dd Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD
24 4.8: - 1(2,000, ) P uu P ud P du P dd P uu P ud P du P dd Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD
25 4.9: - 2(2,000, ) P uu P ud P du P dd P uu P ud P du P dd Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD
26 4.10: - 2(2,000, ) P uu P ud P du P dd P uu P ud P du P dd Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD
27 4.11: - 3(2,000, ) P uu P ud P du P dd P uu P ud P du P dd Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD
28 4.12: - 3(2,000, ) P uu P ud P du P dd P uu P ud P du P dd Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD
29 4.13: (2,000-3,000, ) P uu P ud P du P dd P uu P ud P du P dd Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD : (2,000-3,000, ) P uu P ud P du P dd P uu P ud P du P dd Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD
30 4.15: (3,000-30,000, ) P uu P ud P du P dd P uu P ud P du P dd Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD : (3,000-30,000, ) P uu P ud P du P dd P uu P ud P du P dd Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD Mean SD
31 , ( ) ( ), , ( ) ( ), , 2 ( ).,, ,,,.,.. T, 1, F, : 1 2,000 ( 1) T T T T T T T T T T F T T T T T T T T T T T T T T T 26
32 4.18: 2,000 ( 2) F F T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T 27
33 4.19: 1 2,000 ( 3) F F T T T T F F F F F F T T F F T T F F T T F F T T T T F F T T F F F T F F F F 28
34 4.20: 1 2,000-3, T T F T T T T T F F T T T F F F F F 4.21: 1 3,000-30, T T T T T T F F F F T T T T T T T F, , 3, , 1, 2,., 1 2.,, 1 1., 1, 2. 2, 1 29
35 . 1, 1, 3 (3.3), 2, 3 (3.4).,. 4.22: 2 2,000 ( 1) T T T T T T T T T T T T T T T T T T T T T T T T T T 30
36 4.23: 2 2,000 ( 2) T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T 4.24: 2 2,000 ( 3) T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T T F 31
37 4.25: 2 2,000-3, T T T T T T T T T T T T T T F T T T 4.26: 2 3,000-30, T T T T T T F F T T T T T T T T T T, 1, 2,. 2, 1.,,. 32
38 5,, 1.,,, , 3., 3 11,.,.,, 2, ,. 5.28,., u : ; d : ; u i d i : i ; : i ; R u ui = u i u ; R d d i = d i d.,. 4, 6. R u ui R d di,., 1, ( ). 33
39 5.27: Up u2 R u u2 u3 R u u3 u4 R u u4 u5 R u u5 u6 R u u6 u7 R u u u2 R u u2 u3 R u u3 u4 R u u4 u5 R u u5 u6 R u u6 u7 R u u
40 5.28: Down d2 R d d2 d3 R d d3 d4 R d d4 d5 R d d5 d6 R d d6 d7 R d d d2 R d d2 d3 R d d3 d4 R d d4 d5 R d d5 d6 R d d6 d7 R d d
41 5.2, 1,.,.,,,,.,, 1.,, 1., 1,,., ( ),, ( )., ( ), ( ),.,, 1, 1., 1, t t,,, 1., 1. ( ) ( ), 5% t , 5%. F,, T, 36
42 5.29: t t , , F , , T , , F , ,153, F , , F , , F ,690, ,380, T , , F , , F , , F , , F , , F , , T , , F , , F , , F , , F , , T , , F , , F , , F , , F 37
43 5.30: t t , , T , , T , , F , ,007, F , , F , , F ,411, ,344, T 127 2,214, ,779, F , , F , , F , , F , , F , , T , , F , , F , , F , , F , , F , , F , , F , , F , , F 38
44 ,.,, 1.,, t, 1 1., 1. ( ) 1, 5% t. 5.31: 1 t t ,406 21,506 21,101 16, F 3,153 23,239 19,259 16, F ,314 52,658 63,261 62, T 10,185 49,008 66,734 57, T ,614 12,251 25,201 14, T 1,975 9,155 23,244 12, T ,364 97,176 10,725 81, F 3,010 71,060 10,542 95, T ,010 71,060 10,542 95, T 4,182 12,339 23,851 14, T ,252 10,421 46,141 11, T 6,592 11,597 43,218 11, T ,781 13,952 60,832 12, F 3,964 10,626 59,209 12, T ,900 13,133 21,847 12, F 4,466 11,284 21,422 12, T ,287 23,237 36,739 19, F 6,580 23,927 40,447 19, F ,112 19,222 35,113 21, T 4,004 13,167 35,809 20, T ,162 7,130 96,878 6, F 16,156 6, ,484 6, T 39
45 5.32: 1 t t ,195 20,823 20,203 14, F 3,725 18,099 19,480 14, F ,138 50,597 64,602 47, F 7,682 44,458 67,734 46, T ,898 12,556 24,767 12, T 2,766 10,816 22,691 10, T , ,408 9,680 80, F 2,741 92,273 9,356 73, F ,297 13,423 24,393 14, T 5,888 12,674 20,669 13, T ,040 9,217 47,488 9, T 4,993 10,505 45,256 9, F ,478 11,882 59,013 11, F 5,463 12,987 55,716 11, F ,316 9,556 20,564 10, T 4,183 10,025 19,779 10, T ,661 18,885 35,919 16, F 6,166 20,367 38,427 16, F ,721 15,028 33,647 18, T 3,424 12,293 31,756 18, T ,769 5,740 93,533 6, T 15,955 6, ,641 5, F,. 1 1., 1. 1.,. 40
46 6 [2006] 1.,. 1, 1, ,,.,, , 1., , 1., 1., 1,,., 1.,,. 41
47 [1] Anderson, T.W., Goodman, L.A., Statistical inference about Markov chains, Annals of Mathematical Statistics 28(1957) [2] Cohen, K., Conroy, R. and Maier, S., Market Making and the Changing Structure of the Securities Industry, Rowman and Littlefield, Lanham (1985) [3] Daniels, M., Farmer, J., Iori, G. and Smith, E., Quantitative Model of Price Diffusion and Market Friction Based on Trading as a Mechanistic Random Process, Physical Review Letters, 90(2003) [4] Farmer, J., Patelli, P. and Zovko, I., The Predictive Power of Zero Intelligence in Financial Markets, Technical report, Santa Fe Institute Working Paper (2003). [5] Fielitz, B.D., On the stationarity of transition matrices of common stocks, Journal of Financial and Quantitative Analysis, 10(1975) [6] Fielitz, B.D., Bhargava, T.N., The behavior of stock-price relatives: A Markovian analysis, Operations Research, 21(1973) [7] Luckock, H., A Steady-State Model of the Continuous Double Auction, Quantitative Finance, 3(2003) [8] Maslov, S., Simple Model of a Limit Order-Driven Market, Physica A: Statistical Mechanics and Its Applications, 278(2000) [9] McQuenn, G., Thorley, S., Are stock returns predictable? A test using Markov chains, Journal of Finance, 46(1991) [10] Niederhoffer, V., Osborne, M., Market making and reversal on the stock exchange, Journal of the American Statistical Association, 61(1966) [11] Slanina, F., Mean-Field Approximation for a Limit Order Driven Market Model, Physical Review E, 64(2001) [12] Smith, E., Farmer, J., Gillemot, L. and Krishnamurthy, S., Statistical Theory of the Continuous Double Auction, Quantitative Finance, 3(2003) [13] Tang, L. and Tian, G., Reaction-Diffusion-Branching Models of Stock Price Fluctuations, Physica A: Statistical Mechanics and Its Applications, 264(1999)
48 [14],, [15],,, 16(2006)
49 ,,,.,,.,,.,. 44
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