Effects of Finite-Sample and Realized Kernels on Standardized Returns on the Tokyo Stock Exchange
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1 Effecs of Finie-Sample and Realized Kernels on Sandardized Reurns on he Toyo Soc Exchange The Third Inernaional Conference High-Frequency Daa Analysis in Financial Mares /7/0 Tesuya Taaishi Hiroshima Universiy of Economics
2 Ouline of presenaion Inroducion Realized Volailiy Empirical Resuls Higher Momens Auocorrelaion of Sandardized Reurns Realized Kernels Bipower Variaion Realized Volailiy Disribuion Conclusion
3 3 Sepember 0 The OPERA experimen announced ha neurinos could be faser han ligh. Faser-han-ligh neurinos experimen was wrong! The anomalous resul appears o come from a bad connecion beween a fiber opic cable ha connecs o he GPS receiver used o correc he iming of he neurinos' fligh and an elecronic card in a compuer.
4 Inroducion Sylized properies of asse reurns Absence of auocorrelaions Slow decay of auocorrelaion in absolue reurns Fa-ailed (heavy ail disribuions Volailiy clusering Leverage effec Volume/volailiy correlaion..
5 Gopirishnan e. al(999
6 Gopirishnan e. al(999
7 Tsallis and Aneneodo(003 Suden- disribuion
8 How can we explain he sylized facs? Some of he sylized facs are explained by he mixure of disribuions hypohesis (MDH Clar(976 Reurns are described by Gaussian wih ime-varying volailiy R ~N(0, Consisen Absence of auocorrelaions Slow decay of auocorrelaion in absolue reurns Fa-ailed (heavy ail disribuions ( # of informaion Volume, # of ransacions
9 We assume r E 0 r Absence of auocorrelaions E E rr E E E 0 Non-vanishing auocorrelaion in absolue reurns ( r c( r c E E c 0 c E No independen r Slow decay (long auocorrelaion ime E
10 exp( ( ( / r r P 0 ( ( ( d r P P r P /( (ln ( h e h h P h e h h P / ( Uncondiional reurn disribuion Lognormal disribuion Inverse gamma disribuion h Clar(973 Parez(97 Volailiy disribuion Suden- disribuion We do no now he form of volailiy disribuions Fa-ailed disribuions
11 Consisency chec of he MDH R Sandardized reurns will be Gaussian variables wih mean 0 and variance R ~N(0, Volailiy is unobserved in he mares. Volailiy is esimaed by using high-frequency daa. R RV / Variance = Kurosis=3 Normaliy of sandardized reurns Andersen e al. (000 :Exchange Rae Reurns Andersen e al. (00,(007, (00 :Soc Reurns Fleming & Paye (0:Soc Reurns ec Here we analyze soc reurns on he TSE
12 Realized Volailiy Le us assume ha he logarihmic price process follows a sochasic diffusion as d ln p( ( dw( Andersen, Bollerslev (998 (:spo volailiy W(: Sandard Brownian moion ( T T ( s ds Inegraed volailiy (IV for T period Realized volailiy is defined by a sum of squared finely sampled reurns. N RV r i T i* N IV T ( 0 : sampling period r i ln p( i ln p( i reurn calculaed using high-frequency daa
13 Non-rading hours issue Le us consider daily volailiy Usually soc exchange mares are no open for a whole day. Domesic soc rade a he Toyo soc exchange brea sar end 09:00 :00 :30 brea 5:00 brea morning session afernoon session How o deal wih he inraday reurns during he breas?
14 RV wihou including reurns in he breas underesimaed Hansen and Lunde (005 inroduced an adjusmen facor Correc RV so ha he average of RV maches he variance of he daily reurns c: adjusmen facor T: rading days c T T ( R R RV 0 Variance of daily reurns Average of original realized volailiies 0 RV crv Original realized volailiy For sandardized reurns his changes he value of variance bu no urosis
15 In order o avoid non-rading hours issue we calculae RV in he wo rading sessions separaely. sar 09:00 :00 :30 brea Open P MS, Close Open P MS, P AS, end Close P AS, RV, RV AS, MS : morning session RV :afernoon session RV.. Morning reurn Open Close R MS, ln PMS, ln PMS, Afernoon reurn Open Close R AS, ln PAS, ln PAS, R RV MS, / MS, R RV AS, / AS,
16 sar 09:00 :00 :30 brea Open P MS, end Close P AS, 3. Morning session + Afernoon session Open Close R Inra, ln PMS, ln PAS, ( RV MS, R Inra, RV AS, / Larger variance is expeced This could be underesimaed
17 Microsrucure noise ( ( ( r r (0, : ( WN rue noise Observed reurns are also conaminaed by noise Price discreeness, bid-as spreads, ec. N i N i i i i N i i i i N i r r r r RV ( Noise erms ( ( ln ( ln P P Observed prices are conaminaed by microsrucure noise ( ( ( In he presence of noise RV is calculaed as follows N Zhou(996
18 RV N RV T RV RV N RV RV Sampling frequency (period T N RV When N is large, he conribuion of he noise erms becomes large. 5-min frequency is ofen used for RV consrucion
19 Empirical Resuls Our analysis is based on 5 socs on he Toyo Soc Exchange :Misubishi Co. :Nomura Holdings Inc. 3:Nippon Seel Co. 4.Panasonic Co. 5.Sony Co. Our daa se begins June 3, 006 and ends December 30, 009.
20 Volailiy signaure plo for Misubishi Co. RV ( RV A 4% bias a 5min 3% bias a 5min Sampling frequency min
21 Variance of sandardized reurns Noise conribuion a / d Morning session Afernoon session Sampling frequency
22 Misubishi Co. Kurosis of sandardized reurns Afernoon session Rapid increase Morning session Linear decrease Sampling frequency min
23 3/ ( / (( / ( ( y y P Peers and De Vilder (006 Sandardized reurn disribuion m m y E m m / ( Momens 3( 3 ( 4 y E ( y E Variance Kurosis Gaussian = # of samples
24 Sandardized reurn disribuion(heory # of samples
25 Sandardized reurn disribuion(soc daa MS 5min (4 30min (4
26 Fiing Resuls Misubishi Co. Afernoon session A 0 A / d Morning session
27 Nomura Afernoon session A 0 A / d Morning session
28 a / d Misubishi Co.
29 Variance Fiing resuls Misubishi Nomura Nippon S. Panasonic Sony MS AS MS+AS Kurosis Misubishi Nomura Nippon S. Panasonic Sony MS AS MS+AS
30 Higher momens ( 6 y E ( 8 y E ( 0 y E
31 6 h momen Misubishi MS f ( a 4 N a 4.7
32 8 h momen f ( a N a 0
33 0 h momen f ( a N a 895
34 Auocorrelaion of sandardized reurns We assume r E E Auocorrelaion of reurns is insignifican rr E E E 0 ( r c( r c E E c 0 c E Auocorrelaion of absolue reurns is no necessarily zero r For sandardized reurns r Auocorrelaion is always zero no only for reurns bu also for absolue reurns
35 Misubishi Co. Morning session Afernoon session
36 Realized Kernels h q h q q h RV RV (0 ( h i i h n i h r r ( x x x x x x Barndorff-Nielsen, Hansen, Lunde, and Shephard [008] Hansen and Lunde [008] h q h q q h RV RV (0 ( x x ( 0 / ( / 0 ( 3 x x x x x x x Parzen
37 Volailiy signaure plo Misubishi RV ( q RV (0 q h q h h ( x x
38 Volailiy signaure plo Misubishi
39 Variance of sandardized reurns by RK Misubishi
40 Variance of sandardized reurns by RK Misubishi
41 Kurosis of sandardized reurns by RK
42 Kurosis of sandardized reurns by RK
43 Variance of sandardized reurns by RK MS Parzen Kernel q=0 q=0
44 Kurosis of sandardized reurns by RK MS Misubishi Parzen Kernel q=0 q=0
45 Mone Carlo Simulaions Consan volailiy case r N(0,0.04 Mae T=0000 ime series T=0000 Calculae RV a various sampling frequencies Repea he process 5000 imes We also inroduce arificial microsrucure noise N(0,0.0
46 Mone Carlo Simulaion Variance of sandardized reurns by RK RV ( q RV (0 q h q h h ( x x
47 Variance of sandardized reurns by RK Wih noise
48 Kurosis of sandardized reurns by RK
49 (q / RV R y (/ ( ( ( ( n(n ] [ 4 (0 ( A O h n h H h H q h A r r A E q h RV r r r E RV r E y E q h h q h j i j i h q h n i j i j i i q n i i Variance h q h q q h RV RV (0 ( n i r i RV A (0 ( q h h H Kurosis 4 4 ] [ / (/ ( ( 8 ( (3 n(n 3 ] [ ] [ q h y E A O h n h H h H n y E y E Assume A is independen Approximae expressions for variance and urosis?
50 Variance (Theory q=0 q=0
51 Kurosis (Theory
52 Comparison of Theory and MC Variance MC Theory MC MC Theory Theory
53 Bipower variaion n i i i r r BPV n i RV r i m j j ds s ( Jump componen Bipower variaion eliminaes jump componen ds s ( n i i i r r n n BPVn Finie sample correcion
54 n i i i r r BPV Misubishi MS n i i i r r n n BPVn Variance of sandardized reurns by BPV
55 Kurosis of sandardized reurns by BPV Misubishi
56 T=000 Mone Carlo simulaions N(0,0.0 (0,0.04 N noise n i i i r r BPV n i i i r r n n BPVn
57 N(0,0.04 Mone Carlo simulaions T=000 noise N(0,0.0 BPV or BPVn RV BPV also has finie-sample effecs
58 Realized volailiy disribuion Inverse gamma disribuion loos good, someimes. Remember he inverse gamma disribuion gives Suden s -disribuion for uncondiional reurn disribuion
59 Sandard sochasic volailiy log log If he volailiy disribuion is log-normal log Normal disribuion If he volailiy disribuion is inverse gamma, wha ind of ransformaion gives normal? P / ( ( e ( Normal disribuion ( is beer ransformaion?
60 Box-Cox ransformaion 0 ( ( f 0 log( f f ( ( Non-linear sochasic volailiy model Yu and Yang Volailiy disribuion may be a hin for more suiable volailiy models?
61 Conclusion We analyze he normaliy of sandardized reurns by using realized volailiies in he wo rading sessions of he Toyo Soc Exchange. Variance of sandardized reurns is largely affeced by microsrucure noise. Kurosis of sandardized reurns receives he finie-sample effecs. Higher momens also show expeced finie-sample effecs Auocorrelaion of absolue sandardized reurns is consisen wih he MHD RK and BPV also receive finie sample effecs. Need appropriae finie-sample formula
62 Reurn ime series in he differen ime zones for Misubishi Co. Morning Session Lunch brea Afernoon session Overnigh brea
63 Mone Carlo Simulaions Consan volailiy case r N(0,0.04 Mae T=0000 ime series T=0000 Calculae RV a various sampling frequencies Repea he process 5000 imes We also inroduce arificial microsrucure noise N(0,0.04
64 Volailiy signaure plo
65 Variance of sandardized reurns Wihou noise Wih noise Fiing formula B0 /( Bd
66 0000 lengh A d Sampling frequency
67 40000 lengh A d Slope depends on lengh
68 No difference is seen in urosis
69 HL adjusmen facor Noise conribuions Daily reurn Open_o_Close reurn Morning session + Afernoon session HL adjusmen facor also adjuss microsrucure noise effecs. Sampling frequency min
70 Misubishi Co. Morning session + Afernoon session HL Variance Variance of sandardized reurn wihou HL adjusmen R E crv HL 0 R E RV 0 c d sampling period min
71 Nomura
72 Mone Carlo simulaions N(0,0.04 T=000 noise N(0,0.0 BPV RV
73 Variance of sandardized reurns by BPV Misubishi BPV n i r i ri
74 Fuure Wor Wha is he rapid increase of he urosis a he high frequencies? Deference beween cloc ime and ic ime? Oher momens? E y Same analysis for exchange rae More clear resuls?
75 daily reurns for 49 larges soc of he Naional Soc Exchange (NSE in India over he period Nov 994 June 00. Maia, Pal, Sanley, Saluna(003
76 T.G. Andersen, T. Bollerslev, F.X. Diebold and P. Labys, 000, Exchange Rae Reurns Sandardized by Realized Volailiy are (Nearly Gaussian, Mulinaional Finance Journal 4 (000,
77 T.G. Andersen, T. Bollerslev, F.X. Diebold and H. Ebens, 00, The disribuion of realized soc reurn volailiy, Journal of Financial Economics 6, 43 76
78 Spin model S.Bornhold, In.J.Mod.Phys.C( S i Agens live a sies on an n-dimensional laice (In his sudy we use -dimensional laice. Each sie has a spin. S j S i aes + or - Buy Sell We may assign + sae o Buy order and - sae o Sell order
79 M ( S j ( n i j n h ( J S ( S j ij j Difference beween buy and sell orders i ( M ( 0 Local ineracion Global ineracion Spins are updaed by he following probabiliy S ( p /( exp( h ( i S i ( - p i Local ineracion: Majoriy effec Global ineracion: Minoriy effec
80 M ( S j ( n j L=00 bea= alpha=0
81 Reurn ime series M ( M ( / r(
82 Reurn disribuions
83 Cumulaive reurn disribuions
84 Realized volailiy in Spin model L=5 5, β=.0, α=0 We define = corresponds o one spin updae. T=5x5=565
85 Realized volailiy d=
86 Reurn disribuion Sampling frequency d= r r Kurosis:43. Sd. dev.: Kurosis:.9 Sd. dev.:0.996
87 Variance of Sandardized reurns d
88 Kurosis of sandardized reurns A0 (5*5 / d
89 Spin model r r
90 Misubishi Co. Morning session + Afernoon session
91 Nomura Morning session + Afernoon session
92 Nomura
93 Morning session
94 Realized volailiy, ransacions and volume Price change p ( p( p( p( 3 ln p( ln p( ln p( Price change in Δ # of ransacions in Δ ln p( N ( i ln p ran. ( i Price change beween i-h and i+ h ransacions.
95 ln p( N ( N i w ( ( ln p Diffusion process ran. ( i Plerou e. al.(000 Variance of price change a one ransacion ran. w( ( ln p ( i Realized volailiy and # of ransacions p( ( ln RV RV RV ( N ( w ( RV N ( ( w Are here any correlaions beween N( and ( w?
96
97 Volume and ransacions V N
98 V N
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