Inrna. J. of Sc. and Eng., Vol. 3(:4-8, Ocobr 0, Maudul Ilam al. ISS: 086-503 Forcang Volaly of Dhaka Sock Exchang: Lnar V on-lnar modl Maudul Ilam #, Lakr Erhad Al *, ahda Afroz #3 # Sac Dcln, Khulna Unvry, Khulna-908, Bangladh Mahmac Dcln, Khulna Unvry, Khulna-908, Bangladh 3 Sac Darmn, Mawlana Bhahan Scnc and chnology Unvry, Sanoh, angal-90, Bangladh colnmaudul@gmal.com lonmah@yahoo.com 3 nahdafrz@yahoo.com Abrac Pror nformaon abou a fnancal mark vry nal for nvor o nv mony on arch har from h ock mark whch can rnghn h conomy. h udy xamn h rlav ably of varou modl o forca daly ock ndx fuur volaly. h forcang modl ha mloyd from ml o rlavly comlx ARCH-cla modl. I found ha among lnar modl of ock ndx volaly, h movng avrag modl rank fr ung roo man uar rror, man abolu rcn rror, hl-u and Lnx lo funcon crra. W alo xamn fv nonlnar modl. h modl ar ARCH, GARCH, EGARCH, GARCH and rrcd GARCH modl. W fnd ha nonlnar modl fald o domna lnar modl ulzng dffrn rror maurmn crra and movng avrag modl aar o b h b. hn w forca h nx wo monh fuur ock ndx rc volaly by h b (movng avrag modl. Kyword Volaly, Sock ndx fuur volaly, EGARCH, GARCH, Rrcd GARCH. IJSE Journal Vol. 3(0 I. IRODUCIO Volaly n ock mark ha bn on of h mo analyzd u n h a dcad. h rm volaly a ky lmn for rcng fnancal nrumn uch a oon, a maur of rad off bwn rurn and rk for allocang a and cloly rlad o orfolo rurn fracl, oon rcng and rk managmn. Fnancal mark volaly alo ha a wdr mac on fnancal rgulaon, monary olcy and macro conomy. h raccal moranc of volaly modlng and forcang n many fnanc alcaon man ha h ucc or falur of volaly modl wll dnd on h characrc of mrcal daa ha hy ry o caur and rdc. A hgh volaly n a ock mark cra a bad mac for h counry conomy. For h raon h volaly an moran u ha concrn govrnmn olcy makng, mark analy, corora and fnancal managr. o mak h mark b ffcn and mak rlabl o h nvor, many bunmn ry o forca h volaly bcau h ock mark on of h ourc for h ndury o ra mony. In h mrcal fnanc lraur, many lnar modl ar ud o dcrb h ock rurn volaly. Porba and Summr (986 cfy a aonary AR ( roc for h volaly of h S&P 500 Indx. Frnch, Schwr and Sambaugh (987 u a non-aonary ARIMA (0,, 3 modl o dcrb h volaly of h S&P 500 Indx. Schwr (990 and Schwr and Sgun (990 u a lnar AR ( a an aroxmaon for monhly ock rurn volaly. h xnv u of uch lnar modl no urrng nc hy rovd good fr ordr aroxmaon o many roc and h acal hory wll dvlod for lnar Gauan modl. Howvr, cran faur of a volaly r canno b dcrbd by lnar m r modl. For xaml, mrcal vdnc how ha ock rurn nd o xhb clur of oulr, mlyng ha larg varanc nd o b followd by anohr larg varanc. Such lmaon of lnar modl hav movad many rarchr o condr nonlnar alrnav. h mo commonly ud nonlnar m-r modl n h fnancal lraur ar h auorgrv condonal hrocdac (ARCH modl of Engl (98, h gnralzd ARCH (GARCH modl of Bollrlv (986, h xonnal GARCH (EGARCH modl of lon(99 and hrhold ARCH(ARCH of Zakoan (990 and Glon, Jaganahan, and Runkl (993. h ARCH-cla modl hav bn found o b uful n caurng cran nonlnar faur of fnancal m-r uch a havy-ald drbuon and clur of oulr. Bra, Bubny and Park (993 nvga h valdy of h convnonal OLS modl o ma omal hdg rao ung fuur conrac. Anohr comlx cla of nonlnar modl calld comonn ARCH modl (rrcd GARCH (, uggd by Bollrlv, Engl, and lon (994. hy conclud ha h comonn ARCH modl a uabl ool for dcrbng hor run movmn and long run lvl of volaly found n fnancal m r. Bangladh a dvlong counry whr h Sock Mark an conomc ndcaor of h counry. Bu h Sock Mark of Bangladh no an ffcn mark. So, makng h mark ffcn and rducng h uncrany ha h nvor nv, h volaly forca ncary for h govrnmn and olcy makr. h uro of h ar o xamn h rlav ably of varou modl o forca daly ock ndx fuur volaly on h ba of rror maurmn and fnd h b forcang modl whch uabl for Bangladh. h ar organzd a follow. Scon conan om mhod ha w ud o analy h daa. Scon 3 dcrb mrcal analy of lcng forcang modl. W conclud n Scon 4. II. DAA AD MEHODOLOGY Daa Daly clong rc daa of DSE-0 ndx bwn January 00 and ovmbr 0 oband from Dhaka Sock Exchang. Snc Mo radng acv ak lac n nar day conrac, only nar-day conrac daa ar xamnd. A 4
Inrna. J. of Sc. and Eng., Vol. 3(:4-8, Ocobr 0, Maudul Ilam al. ISS: 086-503 connuou unc of 739 obrvaon of clong rc daa gahrd ovr h la n-yar rod. h logarhm of rc rlav mulld by 00 ud o calcula rc chang..., r 00 ln[ P / P- ], whr h (uncondonal drbuon of r lokurc and aymmrc(n om ca,corrlaon bwn rurn abn or vry wak and corrlaon bwn h magnud of rurn on narby day ar ov and acally gnfcan. Ang h drbuonal ror of daly ock ndx rc chang, varou dcrv ac ar rord n abl ncludng: man, andard dvaon, kwn, kuro and h Kolmogrov-Smrnov (K-S D ac normaly. h null hyoh of normaly rjcd a h % lvl ung K-S D ac and h dvaon from normaly may b gland by h Krnl Dny grah a wll a h aml kwn and kuro maur. Whl kwn rlavly mall and kuro vry larg for boh DSE-0Indx. Followng Poon, Sr-Huang (005, w ma h volaly of daly rurn by h followng uaon ( r -m whr, r h rurn on day and m h avrag rurn ovr h -day rod. Snc, varanc mly h uar of andard dvaon; mak no dffrnc whchvr maur w u whn w comar h volaly of wo a. Snc volaly lan varabl, many rarchr hav rord o ung daly uar rurn, calculad from daly clong rc, o roxy daly volaly. Loz (00 how ha Î an unbad bu xrmly mrc maor of du o aymmrc drbuon. L, Y m + Î Î z and z : (0,.hn E [ Î j ] E [z j ]. Snc, z : c. ( - - ( Mhodology h focu of h ar on h forcang accuracy of daly ock rc volaly from varou acal modl. h bac mhodology nvolv h maon of varou modl for an nal rod and fnd h b modl on h ba of rror maurmn crra. hn h b modl for h lar daa (n-aml forca and fnally calcula nx wo monh forcad daa, hu for formng ou-of aml forca. h lnar modl mloyd ar: ( a random walk modl, ( Horcal modl, (3 an auorgrv modl, (4 a movng avrag modl, (5 an xonnal moohng modl, (6 a ml rgron modl. h nonlnar modl ulzd hr ar ARCH, GARCH(,,EGARCH(,, GARCH(, and Rrcd GARCH(,Modl. Fg. Sha of ral drbuon of daly rurn of DSE-0 Indx. ABLE I Dcrv ac on ock ndx bwn January, 00 and ovmbr 30, 0. Sac DSE-0 Indx Saml z 739 Man 0.0646 Mdan 0.0500 Sandard. dvaon.6380 Skwn 0.6443 Kuro 78.76 Kolmogrov-Smrnov 0.099 * Indca acal gnfcanc a h 0.0 lvl. Fg. DSE-0 ndx volaly for h rod from January 00 o ovmbr. IJSE Journal Vol. 3(0 5
Inrna. J. of Sc. and Eng., Vol. 3(:4-8, Ocobr 0, Maudul Ilam al. ISS: 086-503 abl II Dcrv ac on ock ndx volaly r of nar day conra bwn January, 00 andovmbr30,0. Sac DSE-0 Indx Saml z 739 Man.646 Mdan 0.309 Sandard. dvaon 6.375 Skwn 8.76 Kuro 56.96 Kolmogrov-Smrnov 0.358 * Indca acal gnfcanc a h 0.0 lvl. Lnar modl Random walk modl Accordng o Random walk modl, h b forca of oday volaly dnd on yrday obrvd volaly. + ( - whr, h daly volaly maur dfnd n uaon (. Horcal avrag modl Undr horcal avrag modl, h condonal xcaon of volaly aumd o b conan and h omal forca of fuur volaly would b h horcal avrag. - + (3 whr, h daly volaly maur dfnd n uaon (. Movng avrag modl For a movng avrag modl of ordr h forca of volaly a m on h avrag of h rcn volal; ha, - + - + +... + - + (4 Exonnal moohng modl Exonnal moohng a ml mhod of adav forcang. Sngl xonnal moohng forca gvn by, ( - a - + a - + (5 whr, 0 < a < h moohng facor. By rad ubuon, h rcuron can b rwrn a a a a - j - ( - - j + ( - + j 0 Sml rgron modl h a on- ahad forca bad on h ml lnar rgron of h volaly a rod on h volaly a rod +. h xron gvn by, + b+ b + (7 + Auo rgrv modl h fr-ordr auorgrv modl dfn a, l - + (8 h gnral form of AR modl of ordr l - + (9 onlnar modl ARCH modl h ARCH modl, fr nroducd by Engl (98, ha bn xndd by many rarchr and xnvly urvyd n (6 Bra and Hggn (993, Chou and Kronr (99, Bollrlv and D-bold and Loz (995. In conra o h horcal volaly modl dcrbd, ARCH modl do no mak u of h a andard dvaon, bu formula condonal varanc, h,of a rurn va maxmum lklhood rocdur. W follow h ARCH lraur hr by wrng h.o llura h, fr wr rurn, r m + r, a h z (0 whr, z : (0, a wh no. h roc z cald by h (h condonal varanc whch n urn a a funcon of a uard rdual rurn. In h ARCH ( roc rood by Engl (98, j j j h w a ( + - wh ω>0 anda j ³ 0 o nur h rcly ov varanc. ycally, of hgh ordr bcau of h hnomnon of volaly rnc n fnancal mark. GARCH (, modl In h andard GARCH (, cfcaon: y x g + w+ a - + b - ( (3 h man uaon gvn n ( wrn a a funcon of xognou varabl wh an rror rm. h (3 n GARCH (, rfr o h rnc of a fr-ordr GARCH rm (h fr rm n arnh and a fr-ordr ARCH rm (h cond rm n arnh. ARCH modl hrhold ARCH wa nroducd ndndnly by Zakoan(990 and Glon, Jaganahan and Runkl (993.h cfcaon for h condonal varanc, w+ a + g d b (4 - - -+ - whr, d f <0, and d 0 ohrw. In h modl, good nw ( <0, and bad nw( >0, hav dffrnal ffc on h condonal varanc good nw ha an mac ofa, whl bad nw ha an mac of a+ g If l > 0,h lvrag ffc x. For hghr ordr cfcaon of h ARCH modl, w + a - + g -d - + j b - (5 EGARCH modl h EGARCH or Exonnal GARCH modl wa rood by lon (99. h cfcaon for h condonal varanc - - log( w + b log( - + a + g - - (6 whr, h lf-hand d h log of h condonal varanc. h ml ha h lvrag ffc xonnal, rahr uadrac and ha forca of h condonal varanc ar IJSE Journal Vol. 3(0 6
Inrna. J. of Sc. and Eng., Vol. 3(:4-8, Ocobr 0, Maudul Ilam al. ISS: 086-503 guarand o b nonngav. h rnc of lvrag ffc can b d by h hyoh ha, g < 0. For hghr ordr cfcaon of EGARCH modl, - - log( w+ b j log( - j + ( a + g j - - (7 Rrcd GARCH (, or Comonn ARCH modl h condonal varanc n h GARCH (, modl, w+ a( - w + b( w (8 - - - how, man rvron o w whch a conan forall m. By conra, h comonn modl allow man rvron o a varyng lvl, modl a, - - -- w+ ( - - w + f( - - - a( - w + b( w (9 r (0 Hr ll h volaly, whl h m varyng long run volaly. h fr uaon dcrb h ranory -, whch convrg o zro wh owr a + b. h cond uaon dcrb h long run comonn, of comonn, whch convrg o w wh owr ofr. Combnng h ranory and rmann uaon afollow, (-a-b ( -r w+ ( a+ f + ( b-f - - -( br-( a+ bf - -( ar+ ( a+ bf - ( whch, how ha h comonn modl a (nonlnar rrcd GARCH(, modl. III. EMPIRICAL RESULS o valua h rformanc of h lnar and nonlnar modl n dcrbng ock ndx fuur volaly, w comar hr ou-of aml forca wh our bnchmark modl (. h o-aml forca comaron ar carrd ou a follow. Fr, w rrv h la 60 obrvaon for forca comaron. Scondly, all h modl ud n forcang ar mad ung h fr 664 obrvaon. Such a chm rovd 60 on- ahad forca. h objcv o valua forcang caably of dffrn modl durng h lowland hgh volaly rod on h ba of rror maurmn crra. W ummarz h forca rformanc by condrng h roo man uard rror(rmse, man abolu rcnag rror (MAPE, hl- U and LIEX lo funcon whch ar dfnd a follow: whr, RMSE: MAPE: BM hl-u - ( - ( - ( BM h bnchmark forca. - LIEX [ x{ -a( - } + a( - - ] ( (3 (4 (5 In abl 3, w roo man uar, man abolu rcnag rror, Lnx0 ar mall for movng avrag modl, hl-u mall for ar( bu nd b movng avrag, and Lnx0 ar mallr for random walk modl bu nd b movng avrag.so, an xamnaon of abl 3 rval ha whn h lnar modl h movng avrag modl domna all of h modl ung RMSE, MAPE of all h modl, lnar and nonlnar, h movng avrag modl o all h modl followd cloly by random walk modl and rrcd GARCH. hu, all rror maurmn crra clarly dnfy h lnar cla modl and movng avrag modl a uror o all lnar and non-lnar modl. Roo man uar Man abolu rn rror hl-u abl III Error maurmn of lnar and nonlnar modl for DSE-0 Indx. RW HM S R MA ES Ar( 6.969 6.350 3.83 5.344 5.87 3.485 38.986 4.94 4.546 4.95 3.999 34.973 0.08 07 0.9 0.093.86 0.08 Lnx0 0.038 0.3 0.4 0.03 0.33 0.79 Lnx0 Roo man uar Man abolu rn rror hl-u Lnx0 Lnx0 0.00 0.434 0.300 0.078 0.483 0.375 ARCH( GARCH(, GARCH(, GARCH EGARCH Rrcd GARCH 6.6 5.99 5.98 5.90 6.9 5.943 39.489 37.555 37.758 36.7 38.69 37.594 59 393 384 05 459 635 0.357 0.9 0.56.38 0.493 0.5 0.3 0.599 0.33 0.65.00 0.3 IJSE Journal Vol. 3(0 7
Inrna. J. of Sc. and Eng., Vol. 3(:4-8, Ocobr 0, Maudul Ilam al. ISS: 086-503 hn w forca la 60 obrvaon and comar wh acual valu. W fnd ha h forcad valu almo clor o h acual valu whch ndca h movng avrag modl forca wll for DSE-0. Fg 3. Comaron grah of Movng Avrag Modl & GARCH Modl IV. COCLUSIOS h uro of h ar o xamn h rlav ably of varou modl o forca daly ock ndx fuur volaly. Undrandng and modlng ock volaly moran nc volaly forca hav many raccal alcaon. Inv dcon and a rcng modl dnd havly on h amn of fuur rurn and rk of varou a. h xcd volaly of a cury rurn alo lay an moran rol n h oon rcng hory. h x lnar modl condrd hr ar: ( random walk, ( horcal avrag, (3 movng avrag, (4 ml rgron, (5 xonnal moohng, (6 auorgrv modl. I found ha among lnar modl of ock ndx volaly, h movng avrag modl rank fr ung RMSE, MAPE, hl-u, Lnx lo funcon rror crra. W alo xamn fv nonlnar modl. h modl ar ARCH, GARCH, EGARCH, GARCH and Rrcd GARCH modl. W fnd ha lnar modl domna nonlnar modl ulzng dffrn rror ac and movng avrag aar o b b modl for forcang ock ndx fuur volaly followd cloly by random walk modl. REFERECES [] Akgray V. (989. Condonal hrocdacy n m r of ock rurn: vdnc and forca, Journal of Bun 6(,.55-80. [] Ball, R..,Bollrlv,.(989. h Maur n Daly Exchang Ra: A Condonal Varanc al, Journal of Bun & Economc Sac, Vol.7,. 60-68. [3] Baua, C. C. (998. Sock Mark Volaly n h Phln, Collg of Bun, Admnraon, Unvry of h Phln, Dlman, Quzon Cy, Phln. [4] Bra, A., Bubny, E. & Park, H. (993. ARCH Effc and Effcn Emaon of Hdg Rao for Sock Indx Fuur, Advanc n Fuur and Oon Rarch, 6,.33-38. [5] Bollrlv,, Engl. lon, D.B. (994. ARCH Modl,Handbook of Economrc, Vol. IV, Elvr Scnc B.V. [6] Brokwll, P. J. & Dav, R. A. (996. Inroducon o m Sr and Forcang, Srngr, w York. [7] Chu, C.C. & Bubny, E. L.(990. A Lklhood Rao of Prc Volal: Cy of London, Scoh Journal of Polcal Economy, 36.,.90-95. [8] Engl, R. F. (98. Auorgrv Condonal Hrocdacy wh Ema of h Varanc of Und Kngdom Inflaon, Economrca, Vol. 50,.987 007. [9] Frnch K.R., Schwr W.S. and Sambaugh R.F. (987. Excd ock rurn and volaly, Journal of Fnancal Economc 9,. 3-9. [0] Glon, L.R., Jaganahan R. & Runkl D.E.(993. On h rlaon bwn h xcd valu and h volaly of h nomnal xc rurn on ock, Journal of Fnanc 48,.779-80. [] Gujara, D..(003. Bac Economrc, 4 h d, McGraw-Hll. [] Hll J., Jan A. &Wood R., A. Jr. (988. Inuranc: Volaly Rk and Fuur Mrcng, Journal of Porfolo Managmn 4/:. 3-9. [3] lon, D. B., 99. Condonal Hrocdacy n a Rurn: A nw Aroach, Economrca 59,. 347-370. [4] Poon, Sr-Huang (005. A raccal Gud o Forcang Fnancal Mark Volaly, John Wly & Son, Ld. [5] Porba, J. M. & L. H. Summr (986. h Prnc of volaly and Sock Mark Flucuaon, Amrcan Economc Rvw 76,. 43-5. [6] Schwr, G. W. & P. J. Sgun (990. Hrocdacy n Sock Rurn, Journal of Fnanc45,.9-55. IJSE Journal Vol. 3(0 8