Finite-Sample Effects on the Standardized Returns of the Tokyo Stock Exchange

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1 Available online a Pocedia - Social and Behavioal Sciences 65 ( 01 ) Inenaional Congess on Inedisciplinay Business and Social Science 01 (ICIBSoS 01) Finie-Sample Effecs on he Sandadized Reuns of he Tokyo Sock Exchange Tesuya Takaishi a* a Hioshima Univesiy of Economics, Hioshima , Japan Absac We sudy he finie-sample effecs on euns sandadized by he ealized volailiy consuced as a sum of squaed inaday euns and analyze he nomaliy of he sandadized euns which is expeced o appea fom he mixueof-disibuions hypohesis. Using high-fequency daa on he Tokyo Sock Exchange we exac he kuosis of he sandadized eun by aking ino accoun of he finie-sample effecs and also deemine he vaiance a he micosucue noise fee limi. Ou esuls sugges ha he nomaliy of he sandadized euns on he Tokyo Sock Exchange is well ecoveed and hus he sock pice dynamics is consisen wih he mixue-of-disibuion hypohesis. 01 The Published Auhos. by Published Elsevie by Ld. Elsevie Selecion Ld. Open and/o access pee-eview unde CC BY-NC-ND unde esponsibiliy license. of JIBES Univesiy, Jakaa Selecion and pee-eview unde esponsibiliy of JIBES Univesiy, Jakaa Keywods: Realized volailiy; High-fequency daa; Micosucue noise; Mixue-of-disibuions hypohesis 1. Inoducion Saisical popeies of asse euns have been exensively sudied and i is well-known ha he eun disibuions exhibi fa-ailed disibuions which canno oiginae fom he andom walk model wih a consan volailiy and also indicae ha he usual cenal limi heoem does no apply o his case. A possible explanaion fo his non-nomaliy of he eun disibuions is poposed by Clak (1973), which * Coesponding auho. Tel.: addess: -aka@hue.ac.jp The Auhos. Published by Elsevie Ld. Open access unde CC BY-NC-ND license. Selecion and pee-eview unde esponsibiliy of JIBES Univesiy, Jakaa doi: /j.sbspo

2 Tesuya Takaishi / Pocedia - Social and Behavioal Sciences 65 ( 01 ) ofen called he Mixue-of-Disibuions Hypohesis (MDH). In he MDH he asse pice dynamics can be descibed by he Gaussian pocess wih ime-vaying volailiy. The uncondiional eun disibuion fom such pice dynamics can deviae fom he Gaussian disibuion. Moeove Clak elaes he oigin of he volailiy change wih he ading volume which measues he speed of pice change. The MDH can be checked by he euns sandadized by he volailiy which should show he nomaliy unde he MDH. The dawback of his check is ha he volailiy is laen in he makes and one canno obain i diecly fom make obsevables. Howeve ecen availabiliy of high-fequency inaday eun daa ovecomes his difficuly. Namely he so-called ealized volailiy (Andesen e al. 1997, 001a, 001b) can be consuced as a sum of squaed inaday euns and i is poved o be a consisen esimae of he inegaed volailiy unde ideal cicumsances. Using he ealized volailiy he sandadized euns have been invesigaed fo boh sock and exchange ae euns (Andesen e al. 000a, 001a, 001b) and i is confimed ha he sandadized euns show nomaliy, a leas appoximaely. Fuhe elaboae sudies also confim his (Andesen e al. 007, 010). Recenly he nomaliy of he sandadized euns on he Tokyo Sock Exchange was also invesigaed sepaaely in he moning session (MS) and afenoon session (AS) because of he exisence of lunch beak in ading and i is agued ha he nomaliy of he sandadized euns is ecoveed (Takaishi e al. 01). Howeve some kuosis esuls ae consideably smalle han 3, which is inconsisen wih he nomal disibuion. This smalle kuosis migh be explained by aking ino accoun of he finie-sample effecs claimed by Pee & De Vilde (006). The finie-sample effecs dominae when he numbe of inaday obsevaions used fo he ealized volailiy calculaions is small. Fo such case he sandadized eun disibuions deviae fom he sandad nomal disibuion consideably. Alhough vaiance eceives no finie-sample effec, kuosis becomes smalle depending on he numbe of obsevaions. Some empiical sudies also documen such finie-sample effecs (Andesen e al. 007, 010; Fleming & Paye, 011). In his sudy we calculae he sandadized euns a vaious sampling fequencies and invesigae he finie-sample effecs on he sandadized euns of he Tokyo Sock Exchange. Then aking ino accoun of he finie-sample effecs we y o exac he kuosis a he infinie-sample limi by fiing he esuls o he appopiae fiing fomula. Alhough he vaiance is no affeced by he finie-sample effec he micosucue noise enes in he disoion of he vaiance. We also fi he vaiance daa a vaious sampling fequencies o he fiing fomula and obain he vaiance a he micosucue noise bias fee limi.. Realized Volailiy The ealized volailiy is a model-fee esimae of volailiy consuced as a sum of squaed inaday euns (Andesen e al. 1997, 001a, 001b). Le us assume ha he logaihmic pice pocess lnp(s) follows he sochasic diffeenial equaion. d ln p( s) ( s) dw ( s), (1) whee W(s) sands fo a sandad Bownian moion and (s) is he spo volailiy. Using (s) he inegaed volailiy is defined by h h IV ( s) ds. () Using n inaday eun obsevaions he ealized volailiy is given by n RV ( ) i, (3) i 1

3 970 Tesuya Takaishi / Pocedia - Social and Behavioal Sciences 65 ( 01 ) whee is a sampling ime ineval and n is he numbe of obsevaions given by n=h/. Hee noe ha high-fequency sampling coesponds o small sampling ime ineval. Wihou any bias i is poved ha RV goes o IV h fo n. Howeve i is well-known ha sampled inaday euns ae conaminaed by he micosucue noise and he ealized volailiy consuced fom such inaday euns is lagely affeced a high-fequency sampling ime. Unde he assumpion ha he micosucue noise is independen he inaday euns we obseve in he makes ae given by *, (4) whee sands fo noise wih mean 0 and vaiance he ealized volailiy is eplaced by RV * ( ) RV ( ) n i i i 1 i 1 n i (Zhou, 1996). Wih he micosucue noise. (5) Since i i 1 * n. Thus RV diveges fo n. Such divegen behaviou is acually obseved in he makes and is assumed o be independen he bias in he ealized volailiy is given by n can be illusaed in he volailiy signaue plos (Andesen e al., 000b). 3. Sandadized Reun which gives In he MDH, euns ae descibed by =, whee is he volailiy a ime and is independen sandad nomal vaiables. Using he ealized volailiy as a poxy of, he sandadized eun ~ is ~ 1/ given by / RV. If 1/ RV sicly holds he disibuion of ~ should be N(0,1). Howeve his agumen is only ue if he numbe of inaday eun obsevaions used fo he consucion of he ealized volailiy is infiniy. Fo a finie numbe of obsevaions he sandadized eun disibuion is poved o be given by (Pee & De Vilde, 006) ~ ( n 3) / ( / ) ( ~ n f ) 1. (6) n (( n 1) / ) n This disibuion conveges o he sandad nomal disibuion only fo n. The even momens of his disibuion is also calculaed o be k k n (k 1)(k 3) 1 m. (7) ( n k )( n k 4) n Fom Eq.(7) we find ha vaiance is given by m =1 and kuosis, m 4 /(m ) =3n/(n+). Thus fo he finie-sample sandadized eun i is clea ha he kuosis deviae fom 3. If he numbe of obsevaions is big enough his deviaion migh be small. Due o he lunch beak in ading, howeve, he ading hous on he Tokyo Sock Exchange ae divided ino wo sessions, (MS and AS) and hus he numbe of obsevaions fo he ealized volailiy consucion in each session could be small. Fo insance fo 5min sampling fequency, he numbe of obsevaions is n=4(30) fo MS (AS) and he kuosis is calculaed o be.77(.81). The small kuosis obseved in Takaishi e al, (01) whee 5min sampling ime is used

4 Tesuya Takaishi / Pocedia - Social and Behavioal Sciences 65 ( 01 ) migh be explained by his finie-sample effec. As seen lae, acually we obseve such finie-sample effec on he kuosis of he ealized volailiy on he Tokyo Sock Exchange. 4. Resuls on he Tokyo Sock Exchange 4.1. Daa Ou analysis is based on he high-fequency pice daa of 5 socks (1:Misubishi Co., :Nomua Holdings Inc., 3:Nippon Seel, 4:Panasonic Co. and 5:Sony Co.) aded on he Tokyo Sock Exchange. These socks ae lised in he Topix coe 30 index which includes he 30 mos liquid and highly make capialized socks. Ou daa se begins June 3, 006 and ends Decembe 30, 009. These daa ae he same as he daa analyzed in Takaishi e al. (01). 4.. Sandadized Reun in Two Tading Sessions We calculae he ealized volailiy in each ading session a vaious sampling fequencies fom 1min o 40min. Fig.1 shows he sandadized eun disibuions a 5min and 30min sampling ime inevals in he MS ogehe wih he heoeical cuves of Eq.(6). Fo 5(30) min he numbe of inaday eun obsevaions is n=4(4). The heoeical cuve fo 30 min is vey diffeen fom he Gaussian disibuion. On he ohe hand fo 5min i comes close o he Gaussian one. I is also found ha he empiical sandadized eun disibuions ae consisen wih he heoeical esuls. Fig. shows he vaiance (lef) and kuosis (igh) of he sandadized euns fo Misubishi Co. in he MS and AS as a funcion of sampling ime ineval. I is found ha a small he vaiance is significanly small due o he micosucue noise and i eaches a plaeau aound 1 a low fequency. We find ha he kuosis deceases as inceases. The decease a low fequency is consisen wih he finiesample effecs. On he ohe we also find ha a vey high feuqncy he kuosis inceases apidly as he sampling ime ineval deceases. By Fleming and Paye (011) i is claimed ha his incease is elaed o he micosucue noise. Fig. 1. Sandadized eun disibuions: Theoy and empiical esuls. The ed (blue) line shows he heoeical esuls fo n=4 (4), whee n is he numbe of obsevaions. The geen line shows he nomal disibuion. The squae (cicle) symbols show he empiical sandadized eun disibuions obained fom 5 socks in he MS fo n=4 (4).

5 97 Tesuya Takaishi / Pocedia - Social and Behavioal Sciences 65 ( 01 ) Fig.. Vaiance (lef) and kuosis (igh) as a funcion of sampling ime ineval. The esuls of Misubishi Co. ae shown as a epesenaive one. The black solid lines show he fiing esuls. Table 1. Vaiance and kuosis fom unnomalized and nomalized euns. Unnomalized euns Nomalized euns 1/ / RV Vaiance( Misubishi Co. Nomua Co. Nippon S. Panasonic Co. Sony Co ) MS AS Kuosis MS AS Vaiance MS AS Kuosis MS AS In ode o exac he vaiance and kuosis we fi he esuls a vaious finie sampling fequencies o he coesponding fiing fomula. Fo he vaiance we use /(1 A / ) as a fiing fomula whee and A ae fiing paamees. The vaiance is given by he paamee. The paamee A accouns fo he micosucue noise bias. Fo he kuosis we use whee k and B ae fiing paamees. k The kuosis is given by k. Since a vey high-fequency we find ha he kuosis inflaes and deviaes fom he expeced finie-sample fomula we only use he daa fom 5min o 40min sampling ime inevals fo fiing. The esuls of vaiance and kuosis obained by fiing ae lised in Table 1 wih he esuls of unnomalized euns. Boh vaiance and kuosis of he unnomalized euns ae vey fa fom he nomaliy. On he ohe hand i is found ha he vaiances a he noise fee limi ae vey close o 1 as expeced fom he nomaliy. I is also uned ou ha he kuoses a he high-fequency limi come close o 3 alhough hose in he MS (AS) give slighly smalle (bigge) values han Conclusion We have sudied he finie-sample effecs on he sandadized euns on he Tokyo Sock exchange and invesigaed he vaiance and kuosis of sandadized euns. We find ha he vaiances obained a he noise fee limi come close o 1 and he kuoses a he infinie-sample limi ae nea 3. Ou esuls 1 B /

6 Tesuya Takaishi / Pocedia - Social and Behavioal Sciences 65 ( 01 ) sugges ha he nomaliy of he sandadized euns is ecoveed and he MDH is appopiae fo he sock pice dynamics. We also find he sligh diffeence of he kuosis beween MS and AS which migh indicae ha hee exiss a diffeen ading pocess in he wo sessions. Acknowledgemens Numeical calculaions in his wok wee caied ou a he Yukawa Insiue Compue Faciliy and he faciliies of he Insiue of Saisical Mahemaics. This wok was suppoed by Gan-in-Aid fo Scienific Reseach (C) (No.50067). Refeences Andesen, T. G., & Bolleslev, T. (1998). Answeing he Skepics: Yes, Sandad Volailiy Models Povide Accuae Foecass. Inenaional Economic Review, 39, Andesen, T. G., Bolleslev. T., Diebold. F. X., & Labys. P. (000a). Exchange Rae Reuns Sandadized by Realized Volailiy ae (Nealy) Gaussian. Mulinaional Finance Jounal, 4, Andesen, T. G., Bolleslev. T., Diebold. F. X., & Labys. P. (000b). Gea Realizaions, Risk, Mach, Andesen, T. G., Bolleslev. T., Diebold. F. X., & Labys. P. (001a). The disibuion of ealized exchange ae volailiy. Jounal of he Ameican Saisical Associaion, 96, Andesen, T. G., Bolleslev. T., Diebold. F. X., & Ebens. H. (001b). The disibuion of ealized sock eun volailiy. Jounal of Financial Economics, 61, Andesen, T. G., Bolleslev, T., & Dobev, D. (007). No-abiage semi-maingale esicions fo coninuous-ime volailiy models subjec o leveage effecs, jumps and i.i.d. noise: Theoy and esable disibuional implicaions. Jounal of Economeics, 138, Andesen, T. G., Bolleslev, T., Fedeiksen, P., & Nielsen, M. Ø. (010). Coninuous ime models, ealized volailiies, and esable disibuional implicaions fo daily sock euns. Jounal of Applied Economeics, 5, Clak, P. K. (1973). A subodinaed sochasic pocess model wih finie vaiance fo speculaive pices. Economeica, 41, Fleming, J., & Paye, B.S. (011). High-fequency euns, jumps and he mixue of nomals hypohesis. Jounal of Economeics, 160, Pee, R. G., & De Vilde, R. G. (006). Tesing he Coninuous Semimaingale Hypohesis fo he S&P 500, Jounal of Business & Economic Saisics, 4, Takaishi. T., Chen, T. T. & Zheng, Z. (01). Analysis of Realized Volailiy in Two Tading Sessions of he Japanese Sock Make. Pog. Theo. Phys. Supplemen, 194, Zhou, B. (1996). High-fequency daa and volailiy in foeign-exchange aes. Jounal of Business & Economics Saisics, 14, 45-5.