Forecasting Volatility of Dhaka Stock Exchange: Linear Vs Non-linear models

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1 Inrna. J. of Sc. and Eng., Vol. 3(:4-8, Ocobr 0, Maudul Ilam al. ISS: 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.

2 Inrna. J. of Sc. and Eng., Vol. 3(:4-8, Ocobr 0, Maudul Ilam al. ISS: 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 Mdan Sandard. dvaon.6380 Skwn Kuro Kolmogrov-Smrnov * 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

3 Inrna. J. of Sc. and Eng., Vol. 3(:4-8, Ocobr 0, Maudul Ilam al. ISS: 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 Sandard. dvaon Skwn 8.76 Kuro Kolmogrov-Smrnov * 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 ( 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

4 Inrna. J. of Sc. and Eng., Vol. 3(:4-8, Ocobr 0, Maudul Ilam al. ISS: 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 ( 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( Lnx Lnx0 Roo man uar Man abolu rn rror hl-u Lnx0 Lnx ARCH( GARCH(, GARCH(, GARCH EGARCH Rrcd GARCH 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.

5 Inrna. J. of Sc. and Eng., Vol. 3(:4-8, Ocobr 0, Maudul Ilam al. ISS: 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(, [] Ball, R..,Bollrlv,.(989. h Maur n Daly Exchang Ra: A Condonal Varanc al, Journal of Bun & Economc Sac, Vol.7, [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, [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., [8] Engl, R. F. (98. Auorgrv Condonal Hrocdacy wh Ema of h Varanc of Und Kngdom Inflaon, Economrca, Vol. 50, [9] Frnch K.R., Schwr W.S. and Sambaugh R.F. (987. Excd ock rurn and volaly, Journal of Fnancal Economc 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, [] 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] lon, D. B., 99. Condonal Hrocdacy n a Rurn: A nw Aroach, Economrca 59, [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, [6] Schwr, G. W. & P. J. Sgun (990. Hrocdacy n Sock Rurn, Journal of Fnanc45, IJSE Journal Vol. 3(0 8

Problem 1: Consider the following stationary data generation process for a random variable y t. e t ~ N(0,1) i.i.d.

Problem 1: Consider the following stationary data generation process for a random variable y t. e t ~ N(0,1) i.i.d. A/CN C m Sr Anal Profor Òcar Jordà Wnr conomc.c. Dav POBLM S SOLIONS Par I Analcal Quon Problm : Condr h followng aonar daa gnraon proc for a random varabl - N..d. wh < and N -. a Oban h populaon man varanc