A Novel Application of the Copula Function to Correlation Analysis of Hushen300 Stock Index Futures and HS300 Stock Index

Transcription

1 A Novel Applcaon of he Copula Funcon o Correlaon Analyss of Hushen3 Sock Index Fuures and HS3 Sock Index Fang WU *, 2, Yu WEI. School of Economcs and Managemen, Souhwes Jaoong Unversy, Chengdu 63, Chna 2. School of Economcs and Managemen, Souhwes Unversy, Chongqng 475, Chna Absrac Sock ndex fuures s grounded n hedgng, wh correlaon beween sock ndex fuures and sock ndex spo s mporan due o s sgnfcance n asse prcng and venure managemen. In hs paper, we apply he sac and me varyng Copula funcons, by whch correlaon of HS3 sock ndex fuure and HS3 ndex s analyzed n deal. Our research shows ha, jus before and afer he offcal launchng of HS3 sock ndex fuures, dfference beween he average rae of reurn of HS3 ndex s smooh, bu flucuaon of he former s much obvous as compared wh he laer. Furhermore, o model he correlaon of HS3 sock ndex fuure and HS3 sock ndex, he -Copula funcon s he bes sac funcon, whle he low-al correlaed Roaed Gumbel Copula funcon s he mos suable me varyng funcon. Keywords - HS3 Sock ndex fuure;, HS3 sock ndex; me-varyng Copula I. INTRODUCTION Afer four years daa smulaon and esng, Chnese sock ndex fuure came o s offcal launchng n Aprl 6, 2. A frs, Chnese sock ndex fuure marke has grea shock on he spo marke, and hen urned o be maure as me goes by. For he correlaon beween sock ndex fuure marke and sock ndex spo marke, scholars have developed hree vewpons. Harrs and Damodaran ake he opnon ha he sock ndex spo marke can ge much nformaon when he sock ndex fuure marke s launched [-3]. The hgher nformaon ransmsson speed, he hgher he effcency and he hgher nvesor s reacon speed. As a resul, he much flucuaon of sock ndex spos marke. Bessembnder hnk ha sock ndex fuure marke has he capably of hedgng [4], falls of sock ndex spo marke wll lead o sell-off among he nvesors, and cause a new round of falls n he sock ndex spo. Bu hs knd of vcous crcle can be reduced by he exsence of sock ndex fuure marke, and flucuaon n sock ndex spo marke can be lmed oo. Ohers hnk ha sock ndex fuure marke has no nfluence on sock ndex spo marke, n whch he posve effecs and he negave effecs are coexsng and cancel each oher ou. To fnd an effecve ool o scale he correlaon beween sock ndex fuure marke and sock ndex spo marke, Sklar defned he Copula funcon n 959. Snce he Copula funcon s no only a margnal dsrbuon funcon, bu also a jon dsrbuon funcon ha akes value belong o,, s he brdge ha connecs he sngle marke margnal dsrbuon and he muldmensonal marke jon dsrbuon. In 999 Nelsen proved ha a Copula funcon can be unquely denfed by a margnal dsrbuon and a jon dsrbuon, and vce versa [5]. Furhermore, seres of Copula funcon famly are proposed. For example, he Clayon Copula and he Gumbel Copula funcons have he characerscs of al asymmery, much smlar o he propery of rae of reurn n fnancal producs, and hus wdely used n research. In 26, Paon ransfer a ARMA(,)funcon no a me varyng Copula funcon by usng correlaon coeffcen, and found he me varyng Copula funcon s suable for daa fng [6]. In Chna, he Copula famly s appled o some research, for nsance he hydrology and he fnance felds, and has been proved well for Chnese fnancal daa fng [7-]. However, n he research of correlaon beween sock ndex fuure marke and sock ndex spo marke, he GARCH analyss s mosly used, and few works are done on he Copula funcon. In hs research, he Copula funcon s appled o Hushen3 sock ndex, and focuses on wo opcs: () Flucuaon abou Hushen3 sock ndex durng he perod before and afer s launchng; (2) Correlaon beween he HS3 sock ndex fuure marke and he marke of Hushen3 sock ndex. II. BACKGROUND AND PRELIMINARY OF COPULA FUNCTION A. Copula Funcon Copula s he funcon ha connecs sngle varable margnal dsrbuon and mulvarable jon dsrbuon. Here, he Sklar heorem s defned as below []. The Sklar heorem: For a jon dsrbuon,,,, n N, here s a Copula funcon sasfes Eq. (): F x,, xn,, xn CF x,, Fnxn,, FN xn () funcon F x x x DOI.53/IJSSST.a ISSN: x onlne, prn

2 Where 2 x, x,, xn are he varables, F x s he margnal dsrbuon, C s he abbrevaon of Copula funcon. By usng he propery of Copula funcon, we can ge he followng ransfer equaon. C F x,, F x,, F x n n N N,, n n h f x x f x (2) Eq. (2) descrbes he scalng propery of Copula funcon clearly. For a Copula funcon, f he nvolved varables are ndependen o each oher, he Copula funcon s ndependen, whch means he jon dsrbuon s he produc of he margnal dsrbuons. On he oher hand, f he varables are dependen o each oher n some degree, he Copula funcon descrbes nner relaonshp among he varables. Therefore, he Copula funcon can be regarded as margnal dsrbuon and jon dsrbuon under dfferen condons, and hus makng self a suable model n he analyss of fnancal research. B. Parameer Esmaon Mehod Presenly, hree mehods are wdely employed n he research of marke correlaon. () The maxmum lkelhood esmaon (MLE)[3]. In F x and he hs mehod, he margnal dsrbuon Copula funcon are combned o consruc a lkelhood funcon. By usng he prncple of leas logarhm, he parameers are esmaed. (2) The non-paramerc esmaon. Ths mehod s also called as he Genes and Rves esmaon mehod [2], and s mosly cooperaed wh he Archmedean Copula funcon. In hs esmaon, one coeffcen,.e. Spearman or Kendal, should be calculaed n advance. Consequenly, he coeffcen and he Archmedean Copula funcon are used o calculae he Copula parameers. (3) Sem-parameer esmaon. There are wo seps n hs mehod. Frsly, he margnal dsrbuon s calculaed by usng he kernel densy esmaon mehod. Then, he MLE mehod s adoped o esmae he Copula parameers. C. Idea of Ths Paper The paper wll esmae he relevan srucure by hree seps: frsly, we deermne he margnal dsrbuon of HS3 sock ndex and HS3 sock ndex fuure respecvely by GARCH (, )- model [4-6], whch s f for he daa of fnance. Then wo resduals of above models have a probably negral ransform no obedence o he value n he range of o of he unform dsrbuon and become sequences of U and V; secondly, he paper esmaes he parameers of sac Copula ;n he end,he parameer of me-varyng Copula s also esmaed. () Confrm he margnal dsrbuon. I s a very mporan o deermne he margnal dsrbuon of wo varables, and usually he margnal dsrbuon s assumed a approprae margnal dsrbuon. The yeld of sock ndex s me-varyng, skewness, hgh-peak and fa-al, whch characers s descrbed well by -dsrbuon, and GARCH model s able o capure he volaly characers of fnancal me seres, so s suable o selec GARCH(,)- model for descrbng he margnal dsrbuon of HS3 sock ndex fuure and HS3 sock ndex. R 2 v v 2 ~ v where R s yeld of HS3 sock ndex for, s he uncondonal mean, assumes ha he resduals obey dsrbuon whch degree of freedom s v,,, are nvarable parameers o be esmaed, s all nformaon a - and s represenave of me. The las oucome of equaon 3 s resdual seres whch should no be used o Copula funcon drecly, because of he demand ha varable of Copula mus obey unform dsrbuon range o, so he probably negral ransform s necessary. (2) Selec he opmal sac Copula funcon. The canddae funcons n hs paper are nne Copula funcons, whch nclude ellpc Copula (normal Copula and -Copula), Archmedean Copula and SJC Copula. There s a parameer n ellpc Copula whch has a equaon wh Kendall sn( / 2 ), we can esmae he parameers (nclude v n -Copula) of normal--copula funcon and - Copula by maxmum lkelhood esmaon (MLE). For he parameer esmaon of Archmedean Copula, Genes e al (993) dscussed he mehod of esmaon parameers for Archmedean Copula by Kendall, here are hree relaonshps beween Kendall and hree parameers n Archmedean Copula a followng able: TABLE I THE RELATIONSHIP BETWEEN KENDALL AND PARAMETERS OF ARCHIMEDEAN COPULA funcon Clayon Gumbel Frank Kendall /( 2) / 4[ D ( ) ]/ Remark: D ( ) d. exp( ) The paper seleced he bes fng effec beween Copula funcon and emprcal Copula by calculang he maxmum gap beween Copula funcon and emprcal Copula wh Kolmogorov-Smrnov es (K-S es). (3) Parameer esmaon for me-varyng Copula funcon. We assume ha parameer s consan n he above analyss; nex he parameer s allowed o change. Paon proposed me-varyng normal-copula, me-varyng - Copula, me-varyng Roaed Gumbel Copula (me-varyng RG-Copula) and me-varyng Summarzed Joe-Clayon Copula (me-varyng SJC Copula). He seleced a procedure smlar o ARMA (, ) as ral of change for.the paper (3) DOI.53/IJSSST.a ISSN: x onlne, prn

3 adops me-varyng normal Copula o esmae parameer, as follow s he correlaon for me-varyng Normal Copula evoluon equaon: (4) [ ( u ) ( v )] x e Where s a modfed logsc ransform whch s x e o ensure s always n he [-, ], and (.) ndcae he nverse dsrbuon of sandard normal dsrbuon. A. Daa III. DATA AND RESULT To examne he correlaon srucure beween sock ndex fuure and sock ndex, we esmae a seres of Copula funcon usng daly daa from Aprl 6, 28 o Ocober 23, 24. The daase comes prmarly from Onlne Daa Secon of WIND's webse and ncludes HS3 sock ndex fuure and HS3 sock ndex. The daa are arranged no four sequences, hey are HS3 sock ndex from Aprl 6, 28 o Ocober 23, 24, HS3 sock ndex from Aprl 6, 28 o Aprl 5, 2, HS3 sock ndex from Aprl 6, 2 o Ocober 23, 24 and HS3 sock ndex fuure from Aprl 6, 2 o Ocober 23, 24, he lengh of me for HS3 sock ndex Fuure s shorer because s brh a Aprl 6, 2. Now he daa we gahered s Prce, however, he daa n model s yeld, so a ransform s essenal. r *((log( P) log( P )), r and P are logarhmc yeld and closng prce a me separaely. B. Descrpve Sascs We can fnd ha he dsrbuon of HS3 sock ndex s smlar o shaped dsrbuon wh hgh-peak and fa-al from Fgure, herefore s assumed as dsrbuon. Table II repors he sascal resuls of four seres, Accordng o able II we can ge he followng concluson: frsly, here s a no sgnfcan dfference beween before Aprl 6,2 and afer ha me on he yeld of HS3 sock ndex; nex, here are many smlary such as negave mean, hgh-peak, fa-al, non-normal and seady abou HS3 sock ndex fuure and HS3 sock ndex, he Pearson correlaon on hem s up o.9448, whch represens ha hey had hgh correlaon. Frequency HUSHEN Fg. Hsogram of HS3 sock ndex C. Margnal Dsrbuon We can fnd ha s good dea o apply GARCH(,)- (equaon (3)) o he margnal dsrbuon of HS3 sock ndex and HS3 sock ndex fuure, and we ge able III by MLE on he above wo seres. TABLE II THE DESCRIPTIVE OF HS3 STOCK INDEX AND HS3 STOCK INDEX FUTURE me name mean Sandard devaon skewness kuross JB ADF HS (***) -3.4(***) 28-2 HS (***) -2.69(***) 2-24 HS (***) -2.88(***) 2-24 HS3 Fuure (***) -23.7(***) Noe: (***)ndcaes % confdence level sgnfcanly TABLE III OUTCOME OF ESTIMATE FOR MARGINAL DISTRIBUTION (GARCH(,)-T) sample LLR K-S P HS (.268) HS Fuure (.283) Noe: s mean,,, Kolmogorov-Smonov es..769 (.85).67 (.2).393 (.258).63 (.58).9496 (.39).9676 (.358) (.39) 6.97 (2.42) are consan, s he degree of freedom for dsrbuon, LLR s lkelhood Rae; K-S s DOI.53/IJSSST.a ISSN: x onlne, prn

4 TABLE IV ESTIMATION FOR STATIC COPULA ype LL AIC kappa [aul,auu] Normal-Copula [, ] Clayon-Copula [.8794, ] Roaed-clayon-Copula [,.8669] Placke-Copula [, ] Frank-Copula [, ] Gumbel-Copula [,.837] Roaed-gumbel-Copula [.8453, ] -Copula [.9489,3.79] [.7567,.7567] SJC Copula [.8273,.8552] [.8552,.8552] Noe: LL s he log lkelhood, AIC s Akake sasc, kappa s parameers n model, [aul, auu] are respecvely he lower and upper al dependence coeffcen. In general, GARCH model s very su for descrbng he cluser effec of volaly, mean equaon and varance equaon are boh esmaed well. The yelds of HS3 sock ndex and HS3 sock ndex fuure are negave, whch ndcae ha major nvesors are defc n hs perod. From he able III, he degree of freedom for HS3 sock ndex fuure s 6.97, whch s bgger han ha of HS3 sock ndex who s 3.757, hrough he resul; we can consder ha HS3 sock ndex fuure has hcker al han HS3 sock ndex, so ges more exreme evens and rsk. K-S sascs and P-value n able III s based on probably negral ransform o margnal dsrbuon, he resul of Lkelhood Rae(LLR) and K-S sascs ndcaes ha GARCH(,)- s approprae for boh HS3 sock ndex and HS3 sock ndex fuure, he only dvergence s he degree of freedom. D. Copula Esmaon In order o sudy he relaonshp beween HS3 sock ndex and HS3 sock ndex fuure, he paper selecs nne sac Copula as he ndex of descrbng dependency srucure for wo seres, he resul s n able IV. From able 4, we fnd ha -Copula s he bes fng one among nne dfferen Copula funcon, and s esmaed o.9489, ha s a small dfference wh real correlaon whch s I llusrae ha fng s effecve and srong correlaon beween HS3 sock ndex and HS3 sock ndex fuure; and here s a hcker symmery al. In recen years scholars pon ou ha he relaonshp of dfferen marke s changng wh dfferen relaed srucures n dfferen whle, ha s because of dynamc and mevaryng fnancal marke. Ths paper selec he wo mos common me-varyng Copula funcon (me-varyng normal Copula and he me-varyng Roaed Gumbel Copula funcon) as represenave of me-varyng ellpse Copula and Archmedean, he resuls are as follows: Copula funcon s more suable for he daa. The mevaryng Roaed Gumbel Copula s superor o me-varyng normal Copula, me-varyng Copula has furher comparave fng effec correspondng sac Copula, he concree resuls as below Normal copula (a)tme-varyng normal Copula VS sac Copula Roaed Gumbel copula me-varyng consan me-varyng consan Type TABLE V ESTIMATION FOR TIME-VARYING COPULA Tme-Vary Normal Copula Tm-Varyng RG Copula In Table V, he LL value of me-varyng Copula s less han ha of sac Copula, whch ndcaes ha me-varyng LL (b)tme-varyng RG Copula VS sac Copula Fg. 2 Tme-varyng Copula VS sac Copula The horzonal lne n fgure 2 refers o parameer of sac Copula and wave lne s of me-varyng Copula, we can observe ha he flucuaons n fgure (b) are more DOI.53/IJSSST.a ISSN: x onlne, prn

5 volen han ha of fgure (a). Therefore, we can consder here s sgnfcanly rgh-al nfluence n he daa. IV. CONCLUSION Jus before and afer HS3 sock ndex fuure s offcal launchng, flucuaon of he former s much obvous as compared wh he laer. bu he dfference s no remarkable, so volaly cluserng of HS3 sock ndex fuure wll no owe o he offcal launchng of HS3 Sock Index Fuure. In nne sac Copula, -Copula s he mos suable for he daa, whch has a hgh and symmercal al dependence. Tme-varyng Copula ges beer fng effec comparng o sac Copula, especally for me-varyng Roaed Gumbel Copula, whch has log lkelhood superor o oher Copula funcons, consequenly, we wll use hs mehod o more. REFERENCES [] Harrs, F. H. deb, McInsh, T.H, and Wood, A common facor componens versus nformaon shares. Alernave approaches o prce dscovery research [J], Journal of Fnancal Markes, 22, 5, [2] Harrs, Lawrence. S&P5 cash sock prce volales [J]. Journal of Fnance, 989, [3] Damodaran, Aswah, Kose John, Crocker Lu. The deermnans of organzaon from changes--evdence and mplcaon from real esae [J]. Journal of Fnancal Economcs, 997, 45:69-92 [4] Bessembnder, H, P. J. Segun, Fuure Tradng Acvy and Sock Prce Volaly [J]. The Journal of Fnance, 992, Vol.47, [5] Nelsen R B. An nroducon o Copulas [M], New York : Sprnger,998 [6] Andrew J. Paon, Copula Based Models for Fnancal Tme Seres,Deparmen of Economcs and Oxford Man Insue of Quanave Fnance,26. [7] Zhang yaong, he lnk Funcon (Copula) Technology and he Fnancal rsk analyss [J], Sascal Research, 22(4), 48-5 [8] We Yanhua, Zhang shyng, Guo Yan, Research on Fnancal marke of correlaon and relevan paern [J], Journal of Sysems Engneerng, 24(4): [9] Lu Qongfang, Zhang Zongy, Research on he correlaon beween he sock prce of Real esae and Fnancal ndusry[j],journal of Indusral Engneerng and Engneerng Managemen,2(), [] Peng Xuanhua, Fu Qang, Mul-scale Sudy on Tme-varyng Dependency Srucure beween he Sock Index Fuures and THE acual [J], Sysems Engneerng, 2(5), 4-22 [] Sklar A. Funcon dereparon an dmensons eleurs marges[j] Publcaon de Insu de Sasque de Unverse de Pars,959,8: [2] Genes C,Rves L. Sascal nference procedures for bvarae archmedean copulas [J].Journal of he Amercan Sascal Assocaon, 993, 88:34-43 [3] Johansen, S. and K. Juselus. Maxmum Lkelhood Esmaon and Inference on Conegraon---wh Applcaons o he Demand for Money [J] Oxford Bullen of Economcs and Sascs, 99:52:69-2 [4] Bollerslev T.Generalzed auogressve condonal heeroskedascy[j].journal of Economcs,986,3: [5] Taylor S J. Modelng fnancal me seres [M]. New York: Wley,986. [6] Engle R F. Auoregressve heeroskedascy wh esmaon of he varance of U.K. nflaon [J].Economerca, 982, 5:987-8 [7] Guo Yanfeng, Huang Dengsh and We Yu, Prce Dscovery and Volaly Spllovers n he Index Fuures and Spo Markes n Chna [J], Managemen Revew, 29, (8):3-22. [8] Soll, H.R. and Whaley, R.E. The Dynamcs of Sock Index and Sock Index Fuures Reurns [J].Journal of fnancal & Quanave Analyss, 99, 25(4): [9] Wang K, Chen Y H, Huang S W. The dynamc dependence beween he Chnese marke and oher nernaonal sock markes: A mevaryng copula approach [J]. Inernaonal Revew of Economcs and Fnance,2. DOI.53/IJSSST.a ISSN: x onlne, prn