Trading Collar, Intraday Periodicity, and Stock Market Volatility. Satheesh V. Aradhyula University of Arizona. A. Tolga Ergun University of Arizona

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1 Trding Collr, Inrdy Periodiciy, nd Sock Mrke Voliliy Sheesh V. Ardhyul Universiy of Arizon A. Tolg Ergun Universiy of Arizon My, 00 Absrc: Using 5 minue d, we exmine mrke voliliy in he Dow Jones Indusril Averge in he presence of rding collrs. We use polynomil specificion for cpuring inrdy sesonliy. Resuls indice h mrke voliliy is 3.4 percen higher in declining mrkes when rding collrs re in effec. Resuls lso suppor U-shped inrdy periodiciy in voliliy. Key Words: Trding Collr, DJIA, GARCH, Voliliy, Inrdy, Sesonliy JEL Clssificion Codes: C, G8 Pper cceped for presenion he AAEA nnul meeings, Long Bech, CA. July 00. Corresponding Auhor: Professor Sheesh Ardhyul, Economics Building # 3, PO Box 003, Universiy of Arizon, Tucson, AZ USA. E-mil: sheesh@g.rizon.edu. Phone: Fx:

2 Trding Collr, Inrdy Periodiciy, nd Sock Mrke Voliliy In response o he sock mrke crsh of 987, he New York Sock Exchnge (NYSE) esblished se of regulions o conin excessive mrke voliliy nd o regin invesor confidence. The mos fmous of hese regulions re Rule 80A (rding collr), insiued on Augus, 990 nd Rule 80B (circui breker), insiued on Ocober 9, 988. The Rule 80A resrics index rbirge form of progrm rding in componen socks of he S&P 500 sock price index. When implemened originlly in 990, Rule 80A imposed resricions on index rbirge rding in componen socks of he S&P 500 index whenever he Dow Jones Indusril Averge (DJIA) moved bove or below he previous dy s closing level by 50 poins or more. Once imposed, hese rding resricions will be removed when he DJIA reurns o wihin 5 poins of previous dy s closing level. In Februry 999, percen bsed riggering levels (s opposed o bsolue vlues of 50 nd 5) were implemened. These riggering levels re djused qurerly nd re nnounced by he NYSE he beginning of ech qurer. Tble liss hisoricl collr levels h hve been se by he NYSE. Numerous reserchers hve sudied he voliliy of he sock mrkes nd GARCH models hve become ubiquious for modeling mrke voliliy. See Aydemir (998) nd Diebold nd Lopez (995) for recen surveys of voliliy modeling in finnce. A gre mjoriy of he GARCH models for mrke voliliy hve no ken he rding collrs ino modeling considerion. Forecss from models h do no incorpore he insiuionl deils migh no be s ccure.

3 The presence of rding collrs could conceivbly ler voliliy dynmics nd voliliy models should ccoun for his. To successfully model rding collrs nd heir effec on voliliy one needs o use high frequency inrdy d becuse rding collrs re imposed during rding session nd ypiclly do no ls for long ime periods. For exmple, one cnno discern he presence of rding collrs from dily closing vlues of he DJIA. In his sudy, we esime GARCH models h explicily ccoun for he NYSE s rding collr rules using high frequency d. Finncil mrkes exhibi srong periodic dependencies cross he rding dy ypiclly voliliy is highes he open nd owrd he close of he dy nd filure o ccoun for his my seriously disor he inferences mde from he models (Bolleslev 00). Two pproches hve been used in he lierure o cpure inrdy sesonl perns in voliliy in he conex of ARCH models: use of dummy vribles in he condiionl vrince equion (e.g., Billie nd Bollerslev 990, nd Ederingon nd Lee, 00) nd use of Flexible Fourier forms (e.g., Andersen nd Bollerslev 997, 998 nd Mrens 00). In his pper, we use polynomil funcion o cpure sysemic inrdy periodiciies in voliliy nd esime he sesonl componens simulneously wih he res of he model. The polynomil funcionl form, like he Fourier form inroduced by Andersen nd Bollerslev (997, 998), cn be viewed s flexible form for pproximing he rue, unknown sesonl pern. By incresing he number of polynomil erms, he funcion could be mde rbirry close o rue sesonl componen. Also, using simple prmeric resricions, he funcion could be mde coninuous nd smooh s i cycles from one dy o he nex. Andersen nd Bollerslev (997, 998) nd Mrens (00) esime he sesonl componen firs nd use he esimed sesonliy o desesonlize he reurns d. Then,

4 3 hey fi ARCH models o he desesonlized d. In his pper, we esime he sesonl componens nd he ARCH prmeers simulneously. By esiming he sesonliy (inrdy effecs) simulneously wih he GARCH prmeers, his pproch provides efficien esimes nd voids he shor-flls of using esimed d (Pgn). The polynomil form dvoced in his pper is prsimonious in prmeers nd is esy o esime. For high frequency d, he dummy vrible pproch requires lo of prmeers o compleely specify he inrdy sesonliy. For exmple, for he 5 minue inervl d we use, i would ke s mny s 78 prmeers using he dummy vribles pproch o cpure he ime of he dy effecs on condiionl voliliy. Wih polynomil specificion, sufficienly flexible sesonl pern cn be esimed ofen imes using jus 4 or 5 prmeers. This is he firs sudy o exmine he mrke voliliy in he DJIA in he conex of he rding collrs while simulneously ccouning for inrdy sesonliy. D To model mrke voliliy nd how i is ffeced by he presence of rding collr, one should use high frequency d for observing inrdy flucuions in reurns h migh rigger he collr. We use five-minue inervl d for he DJIA. We obined he d from he Tick D Incorpored. The smple period is from Februry 6, 999 o Augus 3, 00, giving ol of 49,00 observions on reurns. The smple period corresponds o he curren regime of collr levels which re se bsed on percen of he DJIA. Overnigh reurns re excluded so he smple conins exclusively 5 minue reurns. Three dummy vribles, D b, D, nd D, hve been creed o represen he periods when he collr is in effec. Dummy vrible D b kes vlue of one when he collr is in

5 4 effec nd if he collr hs been riggered from below. Thus, D b represens he periods when he Rule 80A is in effec due o decreses in he DJIA. Similrly, D kes vlue of whenever he collr is in effec due o increses in he DJIA. Finlly, dummy vrible D is creed which kes he vlue of one whenever he collr is in effec nd zero oherwise. Obviously, D + D b = D. The smple mens for D b, D, nd D, respecively, re , nd indicing h he Rule 80A hs been in effec for bou 6.0% of smple observions. During he smple period, which is bou wo nd yers long, he collr ws riggered ol of 99 imes. Once riggered, he collr syed on for bou 30 observions on verge. Given he 5 minues d used in his sudy, his corresponds o bou wo hours nd 30 minues. Thus, he verge lengh of rding collr is bou wo nd hlf hours. The smple men nd sndrd deviion for 5 minue reurns for he DJIA re 8.540x0-6 nd.8x0-5. Assuming reurns re uncorreled, he sndrd error for he 5 men equls , 00 = 5.340x0-8 mking he men indisinguishble from zero sndrd significnce levels. The smple skewness nd kurosis re nd.0 indicing h he disribuion for reurns hve hick ils. A Jrque-Ber es sic of 745,644.8 provides srong evidence of deprure from normliy. To ccommode hick ils, we use Suden disribuion for modeling reurns in our nlysis. The Model We specify he following model for log-reurns: r ~ (0, h, v)

6 5 where, r ln( DJIA / DJIA ), is Suden disribuion wih /v degrees of freedom, zero men nd vrince of h, nd nd v re prmeers o be esimed. To cpure voliliy dynmics during he periods of rding collrs, we esime he following four specificions for he condiionl vrince: Model A: h h ( D ) Model B: h h D b b ( D ) Model C: h ( D ) ( D ) ( D ) 0 h Model D: h ( D D ) ( D D ) ( D D ) b b b b b b 0 0 h Where is inrdy sesonliy erm, D, D, nd D b re dummy vribles defined erlier, nd b b b b,,,,,,,,,,,, nd re prmeers o be esimed. The, inrdy sesonliy componen is defined s: 3 k exp 0 S S 3S... k S where, S is sesonliy ime index vrible h kes vlues beween 0 nd nd 0,,..., k re prmeers o be esimed. D on S re obined s follows. Le n represen he number of periods in dy (which is he lengh of he sesonliy cycle). Then, for given rding dy wih n five minue reurns, he ih observion of vrible S for h dy is given by i/n. The sesonliy erm cn be mde coninuous by resricing, S0 S i.e., k i 0. Similrly, he sesonliy erm cn be mde smooh s i cycles from one dy i o he nex by imposing d ds k d, i.e., i i 0. All four models reduce o ds S0 S i

7 6 convenionl GARCH models wih no sesonliy when he prmeers,,, k re resriced o zero. Given previous dy s closing vlue for he DJIA (which is will be in he condiioning informion se), he models cn endogenously deermine wheher collr would be in effec for he nex period. Thus, condiionl voliliy forecss from hese models ppropriely ccoun for he presence of collrs, if ny. Models A nd B llow he condiionl vrince dynmics o be differen during he rding collr periods wihou llowing he GARCH prmeers o be differen when he rding collr is in effec. Models C nd D llow condiionl vrince dynmics o be differen during he rding collrs by llowing he GARCH prmeers o vry when he rding collr is in effec. Models A nd C re up nd down mrkes he sme while models B nd D llow he voliliy dynmics o behve differenly during up nd down mrkes. Model D is he mos generl model nd ness he oher hree models in i while model A is he mos resricive nd ness in he oher hree models. Resuls Mximum likelihood esimes of he prmeers for he four models re given in ble. One over he degrees of freedom prmeer (/v) is significnly differen from zero for ll four models supporing he choice of disribuion over he norml. Esimed degrees of freedom for ll four models is remrkbly close 7.. Esimed GARCH coefficiens, nd, re highly significn in ll four models. Esimed mgniudes of he coefficiens re similr cross he four models. In ll models, coefficien is esimed o be much higher hn nd he sum of esimed GARCH

8 7 coefficiens is close o indicing persisence of memory. This is expeced in high frequency d. In model A, is posiive nd significn. An esimed vlue of 0.0 indices h mrke voliliy is.0% higher during he periods of rding collr. However, model A does no disinguish rising nd declining mrkes. To disinguish he effecs of rding collr in rising nd declining mrkes, we look resuls from model B. The coefficien is insignificn indicing h in rising mrkes, he presence of collr hs no bering on mrke voliliy. In conrs, b is significn indicing h mrke voliliy is pprecibly higher in declining mrkes. A vlue of indices h mrke voliliy is 3.4% higher when he collr is in effec in declining mrkes. All hree rding collr coefficiens (,, ) re significn in Model C implying h voliliy dynmics re differen during rding collr regimes. However, resuls from Model D indice h voliliy dynmics re no ffeced by he presence of rding collr during rising mrkes. However, voliliy dynmics re significnly ffeced by he presence of rding collr in declining mrkes. 0 In he esimion, hird order polynomil ws found o dequely model he inrdy sesonl perns. Resuls in ble indice h he esimed sesonliy prmeers,,, nd, re highly significn for ll four models. These sesonliy prmeers re mrkedly similr for ll he four models indicing h he esimed inrdy sesonliy polynomil is robus cross he four model specificions. The null hypohesis on no inrdy sesonliy ( ) is soundly rejeced for ll four models. Empiricl resuls indice he 0 presence of significn inrdy sesonl perns in he DJIA. Figure is grphicl depicion of he esimed inrdy sesonl pern in voliliy for model D. Sesonl perns for oher hree models re similr. Figure indices h he 0

9 8 DJIA is bou hree imes more volile he open nd he end of he dy hn middy. Esimed sesonliy polynomil suppors U shped voliliy pern repored by erlier sudies for oher equiy mrkes (e.g., Andersen nd Bollerslev, 997). Likelihood rio ess fvor Model D over he oher hree models. Tesing one ime, he null hypohesis of resricive model (A,B, nd C) is rejeced in fvor of model D sndrd levels of significnce. Akike Informion Crieri given in ble lso indice h Model D is he preferred model. Thus, empiricl resuls sugges h voliliy dynmics re significnly differen when rding collr is imposed nd h hese dynmics re no idenicl during up nd down mrkes. Conclusions In his pper, we esime voliliy models for he DJIA h ccoun for he presence of rding collrs h hve been insiued by he NYSE. Using polynomil specificion, we simulneously esime inrdy sesonliy in voliliy. Resuls suppor U-shped inrdy periodiciy in mrke voliliy. We esime h mrke voliliy is % higher when rding collrs re in effec nd 3.4% higher when rding collrs re in effec in declining mrkes.

10 9 References: Andersen, T. G., nd T. Bollerslev, 997, Inrdy periodiciy nd voliliy persisence in finncil mrkes, Journl of empiricl finnce, 4, Andersen, T. G., nd T. Bollerslev, 998, Deusche Mrk-Dollr voliliy: Inrdy civiy perns, mcroeconomic nnouncemens, nd longer run dependencies, Journl of finnce, 53, Aydemir, A. B., 998, Voliliy modeling in finnce in: J. Knigh nd S. Schell, eds., Forecsing voliliy in he finncil mrkes (Buerworh Heinemnn) -46. Bollerslev, T., 00, Finncil economerics: Ps developmens nd fuure chllenges, Journl of Economerics 00, 4-5. Billie, R. T., nd Bollerslev, T., 990, Inr dy nd iner mrke voliliy in foreign exchnge res, Review of Economic Sudies 58, Diebold F. nd J.A. Lopez, 995, Modeling voliliy dynmics in: K. Hoover ed., Mcroeconomerics: Developmens, ensions nd prospecs(kluwer Acdemic), Ederingon, L., nd J. H. Lee, 00, Inrdy voliliy in ineres-re nd foreign-exchnge mrkes: ARCH, nnouncemen, nd sesonliy effecs, Journl of fuures mrkes,, Mrens, M., 00, Forecsing dily exchnge re voliliy using inrdy reurns, Journl of Inernionl Money nd Finnce 0, -3.

11 0 Tble. New York Sock Exchnge s Hisoricl Trding Collr Levels Period Collr Level Augus, Februry 5, Februry 6, Mrch 3, April, June 30, July, December 3, Jnury 3, Mrch 3, April 3, June 30, July 3, Mrch 30, April, 00 - June 9, July, 00 - Sepember 8,

12 Tble. Mximum likelihood esimes of mrke voliliy models for he DJIA. Prmeer Model A Model B Model C Model D (.7) (-78.65) -8.6 (-3.49) 38.5 (0.94) (3.05) (87.94) 0.00 (5.4) (.67) (-9.6) -8. (-9.57) 38. (5.87) (4.98) (3.74) (-0.7) b (7.63) (.65) -6.8 (-8.3) (-3.43) (.00) (.03) (99.93).8 0 (.76) (3.6) (-3.4) (.60) (-99.75) -8.0 (-.05) 37.6 (7.99) (4.93) 0.89 (5.98) (3.0) b.56 0 (.46) (-0.06) b (.94) (-.35) b (-.6) / (34.) (34.6) (34.) (34.6) Log-likelihood Akike Informion Crierion Noe: Numbers in prenheses re -rios. Esimion period is /6/999 8/3/00 wih smple size of 49,00.

13 Figure. Esimed inrdy sesonl perns in voliliy of he Dow Jones Indusril Averge Volliy of reurns :30 0:00 0:30 :00 :30 :00 :30 3:00 3:30 4:00 4:30 5:00 5:30 6:00 Time of he dy