Forecast of Stock Index Volatility Using Grey GARCH-Type Models

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1 Send Orders for Rerns o rerns@benhamscence.ae he Oen Cybernecs & Sysemcs Journal, 015, 9, Oen Access Forecas of Sock Index Volaly Usng Grey GARCH-ye Models L-Yan Geng 1, and Zhan-Fu Zhang 1 School of Economcs and Managemen, Shjazhuang edao Unversy, Shjazhuang, , Chna; Dearmen of Elecrcal and Engneerng, Sfang College, Shjazhuang edao Unversy, Shjazhuang, 05113, Chna Absrac: hs aer negraed he genec algorhm (GA) and grey forecasng (GM(1,1)) model no hree GARCH-ye models and roosed he GAGM-GARCH-ye models. he GM (1,1) model was used o modfy he error erms of he GARCH-ye models o mrove he volaly forecasng erformance of he radonal GARCHye models. Meanwhle, as for he shorcomngs n arameers esmaon of GM (1,1) model, he GA was adoed o fnd he omal grey arameers of GM(1,1) model. Usng he sock daa of Chna sock marke, he aer comared he erformance of he GAGAM-GARCH-ye models n ou-of-samle volaly forecasng wh hose of he GM-GARCH-ye, RGM-GARCH-ye, and GARCH-ye models. I s ndcaed by values of he evaluaon crera ha he GAGM-GARCH-ye models have beer volaly forecasng erformances relave o he oher hree yes of GARCH-ye models. Keywords: Genec algorhm, Grey GARCH-ye models, Volaly forecasng, ARCH model, GM-GARCH-ye Models, Omal grey arameers. 1. INRODUCION Volaly s one of he moran varables n fnancal economy sudy. Invesmen orfolo, asse rcng, rsk managemen and moneary olcy formulang, all deend on volaly. herefore, s necessary and moran o model and forecas volaly of he fnancal marke. o dae, here are varous models o analyze and forecas fnancal volaly. Among hem, GARCH-ye models develoed from ARCH model [1] are more oular han he oher yes of volaly models. Furhermore, he hree GARCH-ye models: GARCH [], EGARCH [3], and GJR-GARCH models [4], are wdely used by researchers n modelng and forecasng volaly. Fnancal me seres usually conans known and unknown nformaon due o he comlexy of fnancal marke. So, s dffcul for he radonal GARCH-ye models o descrbe he unknown nformaon n error erms seuence. Grey forecasng (GM (1,1)) model roosed by Deng s manly used for a sysem wh he unceran nformaon [5]. I shows advanages such as hgh shor-erm forecasng recson, less samles, and smle calculaon [6]. seng used he forecasng roery of GM (1,1) model o modfy he error erms of GARCH model and roosed GM-GARCH model [7]. Laer, seng and Wang rovded GM-EGARCH [8] and GM-GJR-GARCH models [9], ulzng GM (1,1) model o modfy he error erms of EGARCH and GJR-GARCH models. he resuls ndcaed ha he nroducon of GM (1,1) model mroved he shor-erm forecasng accuracy of he GARCH-ye models o a ceran degree. However, due o he heorecal shorcomngs, GM (1,1) model may roduce larger forecas error when fore Address corresondence o hs auhor a he 17# Eas Road, Second Norh Rng, Shjazhuang, Chna. Poscard: ; el: ; E-mal: genglyan_8117@163.com casng he error seuences whch are hghly volale. Genec algorhm (GA) suggesed by Holland s a owerful omzaon algorhm and has been wdely aled o varous omzaon roblems. Wh he advanages of selforganzng and self-adaon, GA can fnd he global-omal soluon whou rang n he local-omal ons. Based on hese characerscs, was ndcaed ha GA can enhance he forecasng accuracy of GM (1,1) model [10, 11]. hs aer used GA o esmae he grey arameers of GM (1,1) model o ncrease he accuracy n error seuences forecasng, and hen mroved he forecasng erformance of he GARCH-ye models. he man srucure of hs aer s arranged as follows. Secon 1 nroduces he research background and objecve of hs aer. Secon summarzes he heory of GARCH-ye and GM-GARCH-ye models. Secon 3 descrbes he heory of GA brefly and desgns he rocedure of GA-based arameer omzaon for GM (1,1) model. he emrcal research on wo socks n Chna sock marke s resened n Secon 4. Fnally, he conclusons and suggesons are rovded n Secon 5.. GREY GARCH-YPE MODELS.1. GARCH-ye Models Generalzed auoregressve condonal heeroskedascy (GARCH) model nroduced by Bollerslev s he exenson of ARCH model, whch assumes ha he curren condonal varance s assocaed wh he as condonal varances and he as random error. GARCH (, ) model wh Gaussan dsurbance can be exressed as:! = " v N(0,1) (1)! = " + &# $ % + &' j () X/ Benham Oen

2 94 he Oen Cybernecs & Sysemcs Journal, 015, Volume 9 Geng and Zhang Where, ω denoes he uncerany of he condonal varance. And α, β j denoe he shor-erm and long-erm nfluence on he condonal varances, resecvely. hese arameers should sasfy he resrcons: "! + # j " < 1 (3)!,",# j $ 0 (4) he GARCH model has he ably of descrbng he henomenon of volaly cluserng and he dsrbuon of fa al exsng n he fnancal asses reurns, bu canno exlan he asymmerc feaures of he reurns. Nelson roosed exonenal GARCH (EGARCH) model o caure he leverage effecs of he asses rce varyng on condonal varance by addng an asymmerc erm no he GARCH model. Dfferen from GARCH model, EGARCH model defnes he condonal varance as he logarhm form, whch has no resrcons on he arameers! and! j. he condonal varance euaon of EGARCH (, ) s wren as: ( $ ln! = " + # % % +! % & + ' $ % -. + )! -./ j ln (5) %, Where, he arameer γ reflecs he asymmery of he reurns. γ >0 reresens ha he osve reurn has a bgger mac on he volaly, γ <0 reresens he negave reurn havng bgger mac on he volaly and ha here s no asymmery when γ =0. Glosen e al. added anoher asymmerc erm no he GARCH model for accoun of he asymmery of he reurn behavor. hey called he roosed model as GJR-GARCH model. he condonal varance euaon of GJR-GARCH (, ) s reresened as:! = " + (# $ % + & S % $ ) ' % + '( % j (6) Where, S - s a dummy varable. S - =1 when! " < 0, and S - =0 when! " # 0. γ =0 ndcaes no asymmerc effec, whle γ! 0 ndcaes he resence of asymmerc effec. Addonally, he arameers of EGARCH model should mee he followng resrcons: "! + # " + 1 " $ < 1 (7)!,",# j $ 0,! + " # 0 (8).. GM-GARCH-ye Models Accordng o he GARCH-ye models, he curren condonal varance! essenally deends on he as error erms " (! < ), bu hs s no conssen wh he acual! suaon. In he acual fnancal marke, wh he excluson of he as rce, he error erms are also affeced by he unceran facors, such as he economc, olcal, envronmenal and oher comlex facors. hese facors cause he changng of errors all he me. hus, he curren error may have an mac on he curren condonal varance!, whle he radonal GARCH-ye models may jus neglec hs on. Grey forecasng model s he core model of he grey sysem heory. Usng he accumulaed generang oeraon (AGO) o rerocess he orgnal daa, grey forecasng model fnds and grass he develomen law of he sysem, and hen forecass he fuure sae of he sysem uanavely. GM (1,1) model s he commonly used grey forecasng model. o srenghen he mac of he curren error on he curren condonal varance, GM (1,1) model was used o connuously modfy he suared error erm seuences of he GARCH-ye models (GM-GARCH-ye models). ha s, he one-se-ahead forecased error values obaned from GM (1,1) model were u no he condonal varance euaons for enhancng he forecasng ably of he GARCHye models, as shown n he leraure [7-9]. he condonal varance euaons of he GM-GARCH-ye models are wren as: ( )! = " + &# $ % + ˆ$ + &' j (9) ( $ ln! = " + # % + ˆ$ % +! % & + ' $ % + ˆ$ -. + )! -./ j ln (10) %,! = " + [# ($ % + ˆ$ ) + & S % ($ ' % + ˆ$ % )] + '( j (11) Where, ˆ! reresens he random error forecass obaned by usng he GM (1,1) model. 3. GM-GARCH-YPE MODELS WIH GA 3.1. Genec Algorhm Genec algorhm (GA) s a global omzaon algorhm, followng he mechancs of bologcal evoluon. I mmcs he henomena of reroducon, mang and muaon whch occur n he rocess of naural selecon and naural nherance. Based on he naural law of survval of he fes, GA roduces he referred ndvdual generaon by generaon and fnds he omal ndvdual by usng he genec oeraors such as selecon, crossover and muaon. he deals on he genec oeraors can be found n he leraure [1]. GA has s own characerscs aar from all he feaures of evoluonary comuaon: (1) GA drecly deals wh he code se of he decson varables raher han he acual value self. Durng he search rocess, GA neher laces any consrans on he connuy of he omzed funcon nor reures exsence of he dervave of he omzed funcon. () GA looks for omal soluon by usng mul-on search or grou search, whch shows hgh mlc arallel-

3 Forecas of Sock Index Volaly Usng Grey GARCH-ye Models he Oen Cybernecs & Sysemcs Journal, 015, Volume 9 95 sm. (3) GA s an adave search echnue. he selecon, crossover and muaon are oeraed n a robablsc manner, ncreasng he flexbly of he search rocess. A he same me, GA can converge o he omal soluon wh a large robably. Accordngly, GA shows good ably n he global search and omzaon. (4) GA has beer general adaably and exensbly snce GA akes he objecve funcon value as he search nformaon whou reuremen regardng he behavor of he funcon. Meanwhle, GA mroves he search effcency by allowng users o concenrae on he hgher fness degree from he search range o he search sace. 3.. he Omal Grey Parameers by GA Grey arameers a and b are moran for he GM (1,1) model. he forecasng erformance of he GM (1,1) model deends on he accuracy of he arameers soluon o a and b. In GM (1,1) model, he esmaors of a and b are obaned by he leas suares mehod under he assumon ha he random error seuences are normally dsrbued. However, due o beng affeced by varous comlex facors, he error seuences do no follow he normaly dsrbuon. As a resul, he arameer esmaors by he leas suares mehod may be based and non-conssen. Moreover, when esmang he wo arameers, he leas suares mehod should sasfy he resrcon of ˆx (1) (1) = x (1) (1) = x (0) (1), whch may cause larger sysem error and mac on he forecasng accuracy of GM (1,1) model. o mrove he forecasng ably of GM (1,1) model n error erms seuence forecasng, GA s aled o search for he omal grey arameers of GM (1,1) model. he general rocedure of GA-based arameer omzaon o GM (1,1) model for forecasng he error seuences can be summarzed as follows: Se 1: Prerocessng he daa. ransform he orgnal error erms seuence ε (0) =( ε (0) (1), ε (0) (),, ε (0) (n)) wh! (0) () " R,,,,n no he non-negave seuence u (0) = u (0) (1),u (0) (),!,u (0) (n)} {, where u (0) () =! (0) () + mn(! (0) ()),,,,n. (1) Se : Inalzaon oulaon. Inalze he arameers of GA, conssng of he oulaon sze, he number of evoluonary generaon, he crossover rae and muaon rae. Se 3: Defnon objecve funcon. he objecve funcon s defned based on he creron of mnmzng he mean suares error: 1 F = (13) n 1/ n "(û (0) ()! u (0) ()) Where û (0) and u (0) are he forecased and acual error values, resecvely. Se 4: Evoluon oeraon. Calculae he objecve value of each ndvdual n he oulaon and search for he omal soluon by he ses of selecon, crossover, muaon and evoluon. Se 5: Evoluon sos. Reea Se 3 o Se 4 unl he number of evoluonary generaon s me, when he omal grey arameers a and b are obaned. Se 6: Model consrucon and forecasng. Usng he obaned arameers a and b, he GA-based GM (1,1) model (called GAGM (1,1)) s consruced as: " % û (1) () = $ u (0) (1)! b 'e!a(!1) + b, =,, n. (14) # a & a hen, he error values forecased by GAGM (1,1) model are ransformed no he orgnal error forecass: ˆ! (0) () = 1" e a # ( )%! (0) (1) " b $ a & (e "a("1) " mn(! (0) ()) =,, n. (15) ' Fnally, ˆ! (0) () s added n he condonal varance euaons of he GARCH-ye models. 4. EMPIRICAL RESEARCH 4.1. Daa Descron wo sock ndces of Chna sock marke were examned: HuShen 300 Index (HS300) and HangSeng Index (HSI). he daly radng rces of he wo sock ndces were exraced from Sna webse durng July 4, 011 o July 10, 014, ncludng 1338 observaons. he connuously comounded logarhmc reurns were calculaed by usng he daly closng rces: r = ln P! ln P!1, Where P,, P -1 are he daly closng rces for day and -1, resecvely. he descrve sascs of he daly reurn seres of HS300 and SZCI can be found n able 1. I s clear from able 1 ha for he wo ndces, he means of he reurn seres are close o zero, sgnfcanly smaller han he corresondng sandard devaon whn he consdered erod. hus, he condonal mean of he reurn seres can be assgned as zero. he J-B es of he reurn se- able 1. Descrve sascs of daly reurn seres of HS300 and SZCI. Indces Mean Max. Mn. Sd. Dev. Skewness Kuross J-B es LB(0) LB (0) LM(0) HS ab ab ab SZCI ab a ab ab Noe: JB es s he Jarue-Bera normaly es for he dsrbuon of he reurn seres. LB(0), LB(0) are he Ljung-Box es for he 0 h order seral correlaon of he reurn and suared reurn seres, resecvely. LM (0) s he Engle s (198) LM es for heeroscedascy of he reurn seres. a, b denoe sgnfcance a he 5% and 1% levels, resecvely.

4 96 he Oen Cybernecs & Sysemcs Journal, 015, Volume 9 Geng and Zhang res rejecs he null hyohess of normaly a 1% and 5% levels, resecvely. Besdes, negave Skewness and hgh Kuross beng found n he reurn seres, s shown ha he dsrbuon of he reurn seres s negavely based and faaled. LB(0) sasc for seral correlaon suggess ha he reurn seres of SZCI has seral correlaon a he 5% sgnfcance level, whle he reurn seres of HS300 has no sgnfcan seral correlaon. LB (0) and ARCH (0) ess suor he rejecon of he null hyohess of heeroskedascy a 1% and 5% levels, ndcang ha he srongly ARCH effec exss n he wo reurn seres. I s reasonable o consruc he GARCH-ye models. 4.. Emrcal Resuls he daly reurns of HS300 and SZCI, normalzed n he range from 0 o 1, are classfed no wo secons: he frs 1000 daly reurns are used for model ranng, and he remanng 337 ones for model evaluaon. he rollng forecasng mehod s used o forecas he sochasc error of he GARCH-ye models. When esmang he grey arameers of GM (1,1) model, he conrol arameers of GA are se as: oulaon sze=30; crossover rae=0.95; muaon rae=0.08, number of evoluonary generaon=50. For HS300 and SZCI, GAGM (1,1) model forecass he nex error value of he GARCH-ye models usng he egheen mos recen errors. he forecased error values were added no he varance euaon of he GARCH-ye models and he arameers of he GARCH-ye models were esmaed by usng he maxmum lkelhood esmaon (QMLE) mehod. hen, he consruced GAGM-GARCH-ye models were emloyed o one-se-ahead forecas volaly. o comare he forecasng resuls of he roosed models, hree yes of GARCH models were aled for forecasng volaly of HS300 and SZCI wh he same daa samles. hey are GM-GARCHye, RGM-GARCH-ye, and GARCH-ye models. Fnally, he volaly forecass were ransformed no he orgnal volaly forecass. he ou-of-samle forecasng erformances of each ye of models were evaluaed by four sascal ndces: he roo mean suared error (RMSE), he mean absolue error (MAE), he logarhmc error sasc (LL), and he Lnear Exonenal ndex (LINEX). hese ndces are defned by: RMSE =!1 #( ˆ"! R ) (16) MAE =!1 # ˆ"! R (17) LL =!1 '# $ ln( ˆ" )! ln(r )% & (18) LINEX =!1 ({ex $ %"( ˆ#! R )& '! "( ˆ#! R )!1} (19) Where, s he number of volaly forecass.! ˆ s he suare roo of he volaly forecass. R s he roxy of he acual daly volaly. In hs sudy, he range-based ex os volaly R s aken as a roxy of he acual daly volaly, exressed as: R = k (log(p,h )! log(p,l )) "100 (0) Where, P,h and P,l are he nraday hgh and nraday low rces, resecvely. k s he calbraon arameer beween he range-based uncondonal varance and he reurn-based uncondonal varance. he four sascal ndces measured he forecasng errors of he evaluaed models. he model wh smaller ones showed beer volaly forecasng ably. able lss he comarson of he resuls of he four yes of models n forecasng volaly of HS300 and SZCI. I can be seen from able : Frsly, for HS300 and SZCI, GAGM-GARCH-ye models generae smaller RMSE, MAE, LL, and LINEX comared o he oher yes of models, whch show a beer erformance han he oher yes of models n forecasng volaly. Among he hree GAGM- GARCH-ye models, comared wh GAGM-GJR-GARCH model, GAGM-GARCH model generaes smaller RMSE and LINEX bu larger MAE and LL, ndcang ha he forecasng ably of GAGM-GARCH model s somewha mxed comared wh ha of he GAGM-GJR-GARCH model. GAGM-EGARCH model shows he wors volaly forecasng erformance accordng o he four evaluaon crera. Secondly, wh he exceon of LINEX of RGM-GARCHye models, GM-GARCH-ye models roduce smaller RMSE, MAE, and LL han RGM-GARCH-ye and GARCH-ye models, suggesng ha on he whole, GM- GARCH-ye models ouerform he RGM-GARCH-ye and GARCH-ye models n volaly forecasng. hrdly, for HS300, RGM-GARCH-ye models rovde beer volaly forecasng resuls n erms of RMSE, LL, and LINEX. Whle for SZCI, GARCH-ye models seem o rovde beer volaly forecasng resuls n erms of RMSE, MAE, and LL. Hence, s dffcul o deermne ha whch one of hese wo yes of models s beer n volaly forecasng erformance. he volaly forecasng resuls of he GAGM-GARCHye models for HS300 and SZCI are shown n Fgs. (1) and (). As shown n he wo fgures, he hree GAGM- GARCH-ye models can forecas he man volaly varyng rend of he wo sock ndces, where GAGM-GARCH model rovdes sueror volaly forecass n he smaller flucuaon sage and GAGM-GJR-GARCH model rovdes sueror volaly forecass n he larger flucuaon sage. CONCLUSION o enhance he forecasng erformance of he GARCHye models,.e., GARCH, EGARCH, and GJR-GARCH models, n hs aer, GAGM-GARCH-ye models were roosed, whch combned genec algorhm (GA) and GM(1,1) model wh hree GARCH-ye models. GM(1,1) model omzed by GA was used o modfy he error erms of he GARCH-ye models. he resuls of he emrcal sudy on HS300 and SZCI ndces of Chna sock marke show ha he roosed models have sueror erformances n ou-of-samle volaly forecasng han he GM- GARCH-ye, RGM-GARCH-ye, and GARCH-ye models. he GAGM-GARCH and GAGM-GJR-GARCH models erform beer han he GAGM-EGARCH model, bu he forecasng erformance of he GAGM-GARCH model s

5 Forecas of Sock Index Volaly Usng Grey GARCH-ye Models he Oen Cybernecs & Sysemcs Journal, 015, Volume 9 97 able. RMSE, MAE, LL, and LINEX of four yes of volaly models for HS300 and SZCI. Indces Models RMSE MAE LL LINEX GAGM-GARCH GAGM-EGARCH GAGM-GJR-GARCH GM-GARCH GM-EGARCH HS300 GM-GJR-GARCH RGM-GARCH RGM-EGARCH RGM-GJR-GARCH GARCH EGARCH GJR-GARCH GAGM-GARCH GAGM-EGARCH GAGM-GJR-GARCH GM-GARCH GM-EGARCH SZC GM-GJR-GARCH RGM-GARCH RGM-EGARCH RGM-GJR-GARCH GARCH EGARCH GJR-GARCH Fg. (1). HS300 volaly forecass by GAGM-GARCH-ye models. Fg. (). SZCI volaly forecass by GAGM-GARCH-ye models.

6 98 he Oen Cybernecs & Sysemcs Journal, 015, Volume 9 Geng and Zhang somewha mxed comared o he GAGM-GJR-GARCH model. In addon, he GM-GARCH-ye models, as a whole, roduce sueror volaly forecass comared o he RGM-GARCH-ye and GARCH-ye models. Whle, he RGM-GARCH-ye and GARCH-ye models show dfferen volaly forecasng ables accordng o dfferen evaluaon crera, whch need o be furher suded. CONFLIC OF INERES he auhors confrm ha hs arcle conen has no conflc of neres. ACKNOWLEDGEMENS hs work was suored by he Scenfc Research Foundaon of he Mnsry of Educaon of Chna for Young Scholars Inellgen Forecasng Mehods for Fnancal Volaly and Is Emrcal Research (No. 11YJC790048). REFERENCES [1] R. F. Engle, Auoregressve condonal heeroskedascy wh esmaes of he varance of uned kngdom nflaon, Economerca, vol. 50, , July 198. []. Bollerslev, Generalzed auoregressve condonal heeroscedascy, Journal of Economercs, vol. 31, , Arl [3] D. Nelson, Condonal heeroskedascy n asse reurns: A new aroach, Economerca, vol. 59, , March [4] L. R. Glosen, R. Jagannahan, and D. E. Runkle, On he relaon beween he execed value and he volaly of he nomnal excess reurn on socks, Journal of Fnance, vol. 48, , December [5] S. F. Lu and N. M. Xe, Grey Sysem heory and Is Alcaons. Scence Press, Bejng, 014. [6] A. Samved and V. Jan, A grey aroach for forecasng n a suly chan durng nermen dsruons, Engneerng Alcaons of Arfcal Inellgence, vol. 6, , March 013. [7] C. H. seng, S.. Cheng and Y. H. Wang, New hybrd mehodology for sock volaly redcon, Exer Sysems wh Alcaons, vol. 36, , March 009. [8] C. H. seng, S.. Cheng, Y. H. Wang and J.. Peng, Arfcal neural nework model of he hybrd EGARCH volaly of he awan sock ndex oon rces, Physca A: Sascal Mechancs and s Alcaons, vol. 387, , January 008. [9] Y. H. Wang, Nonlnear neural nework forecasng model for sock ndex oon Prce: hybrd GJR-GARCH aroach, Exer Sysems wh Alcaons, vol. 36, , January 009. [10] Y. S. Lee and L. I. ong, Forecasng nonlnear me seres of energy consumon usng a hybrd dynamc model, Aled Energy, vol. 94, , February 01. [11] L. C. Hsu, A genec algorhm based nonlnear grey bernoull model for ouu forecasng n negraed crcu ndusry, Exer Sysems wh Alcaons, vol. 37, , June 010. [1] N. Gordn, A genec algorhm aroach for SMEs bankrucy redcon: Emrcal evdence from Ialy, Exer Sysems wh Alcaons, vol. 41, , Arl 014. Receved: Seember 16, 014 Revsed: December 3, 014 Acceed: December 31, 014 Geng and Zhang; Lcensee Benham Oen. hs s an oen access arcle lcensed under he erms of he Creave Commons Arbuon Non-Commercal Lcense (h://creavecommons.org/- lcenses/by-nc/4.0/) whch erms unresrced, non-commercal use, dsrbuon and reroducon n any medum, rovded he work s roerly ced.