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2 I denoes any nformaon avalable a me (Informaon Se a me ). Defne e { ( )} u rocess by θ e { ( )} { u ( θ )} { µ ( )} =,, y θ u θ rocess s en defned o follow an ARCH model f e condonal mean euals zero, E ( u ( θ )) =, bu e condonal varance vares roug me, θ = Var u θ E u. ( ) ( ) ( ( )) ( ) = θ. Modelng e condonal varance Numerous aramerc secfcaons for e me varyng condonal varance ave been roosed n e leraure. e frs model s e ARCH() model nroduced by Engle (98). e condonal varance s a lnear funcon of e as suared nnovaons : = a au, a >, a,,,. In emrcal alcaons of ARCH() models a long lag leng and a large number of arameers are ofen needed. us, Bollerslev (986) generalzed e ARCH() model and nroduced e General Auoregressve Condonal Heeroskedascy GARCH(,) model. e condonal varance s a lnear funcon of e as suared nnovaons and e as condonal varances: for a >, a = = a au β, = a,,,, β, =,,. In emrcal nvesgaons e esmae of β s very close o uny. Engle and Bollerslev (986) referred o a model sasfyng a β = as an negraed GARCH rocess, denoed IGARCH(,). = Under an IGARCH rocess e uncondonal varance of u s nfne, so neer u no u sasfes e defnon of a covarance saonary rocess. GARCH models are suable o caure some caracerscs of fnancal markes. ey eleganly caure e volaly cluserng n asse reurns frs noed by Mandelbro (963): large canges end o be followed by large canges of eer sgn, and small canges end o be followed by small canges. However, e srucure of a GARCH model moses an moran lmaon. GARCH models assume a only e magnude and no e osvy or negavy of nnovaons deermnes e feaure of because s a funcon of lagged and lagged u and so s nvaran o canges n e algebrac sgn of e u s. On e oer and, asse reurns end o be leokurc (eavly aled). Denoe as e erm "nnovaon" s used nsead of e "resdual" and exresses e unredcable ar of a fnancal seres.

3 ( θ ) u ( θ ) ( ) z θ e sandardzed rocess, wll ave condonal mean zero and me nvaran condonal varance uny. If e condonal dsrbuon for z s furermore assumed o be me nvaran w fne four momen, en e uncondonal dsrbuon for u wll ave faer als an e dsrbuon for z. For nsance, for e ARCH() model ( ) ( ) w condonally normally dsrbued errors, ( E ( u ) 3a ) E u 3 a =, f 3a <, E( u ) and = oerwse, bo of wc exceed e normal value of ree. E ( u ) Fnancal markes are caracerzed by e leverage effec, frs noed by Black (976). e leverage effec refers o e endency for e canges n e sock rces o be negavely correlaed w canges n sock volaly. I.e. volaly ends o rse n resonse o bad news (reurns lower an execed) and fall n resonse o good news (reurns ger an execed). Nelson (99), roosed e followng model for e evoluon of e condonal varance of u : ( ) u u u log = a π E δ = s model s referred o as exonenal GARCH, or EGARCH model. In s model, deends on bo e magnude and e sgn of lagged resduals. e δ arameer allows for e asymmerc effec. If δ = en a osve surrse as e same effec on volaly as a negave surrse. If < δ <, a osve surrse ncreases volaly less an a negave surrse. If δ <, a osve surrse acually reduces volaly wle a negave surrse ncreases volaly. For δ < e leverage effec exss. Snce EGARCH descrbes e log of, e wll be osve regardless of weer e π coeffcens are osve. us, n conras o e GARCH model, no resrcons need o be mosed on e model for esmaon. We can exress e nfne movng average reresenaon of e model as e rao of wo fne order olynomals. us, an ARMA rocess rovdes a smler arameerzaon of e form. We denoe as EGARCH(,): u u u log( ) = a a a E δ ( β log( )). = e famly of ARCH models s remarkably rc. Anoer roue for nroducng asymmerc effecs s o se: = a [ a I( u > ) u a I( u ) u ] β, were I () denoes e ndcaor funcon. e model nroduced by Zakoan (99) s called resold ARCH or ARCH(,). Glosen, Jagannaan and Runkle (993) nroduced e GJR(,) model w e followng form: = ( > ) = I f > u u, oerwse zero. ( ) = I f u, oerwse zero. u 3

4 ( u < ) u = a a u δ β, I = were I () denoes e ndcaor funcon. e leverage effec s suored f δ >. Good news as go an mac of a and bad news as go an mac of a δ. Engle (99), roosed e Asymmerc ARCH or AARCH(,) model: = a ( ) au u β = δ, were a negave value of δ means a osve reurns ncrease volaly less an negave reurns. aylor (986) modeled e condonal sandard devaon funcon nsead of condonal varance. Scwer (989) modeled e condonal sandard devaon as a lnear funcon of lagged absolue resduals. e aylor/scwer GARCH(,) model s defned as = a a u = β. Hggns and Bera (99) nroduced e Non-lnear ARCH or NARCH(,) model: γ = a a u γ = γ β. Geweke (986) and Panula (986) nroduced e log-arch(,) model: ( ) = a a log( u ) log( ) log β. Senana (995) nroduced e Quadrac ARCH or QARCH(,) model of e form = u δ uu = = = a au δ β. Dng, Granger and Engle (993) nroduce e Asymmerc Power ARCH or APARCH(,) model: γ = a a γ ( u u ) = γ δ β, wc ncludes seven ARCH models as secal cases. Dng, Granger and Engle (993) esmae e Sandard & Poor's 5 (ereafer S&P 5) reurns by e APARCH(,) model and e esmaed ower γ / for e condonal eeroskedascy funcon s.3, wc s sgnfcanly dfferen from (aylor/scwer model) or (GARCH model). Non-radng erods Informaon a accumulaes wen fnancal markes are closed s refleced n rces afer e markes reoen. If, for examle, nformaon accumulaes a a consan rae over calendar me, en e varance of reurns over e erod from e Frday close o e Monday close sould be ree mes e varance from e Monday close o e uesday close. Fama (965) and Frenc and Roll (986) ave found, owever, a nformaon accumulaes more slowly wen e markes are closed an wen ey are oen. Varances are ger followng weekends and oldays an on oer days, bu no nearly by as muc as would be execed f e news arrval rae were consan. ARCH, GARCH, aylor/scwer GARCH, GJR, ARCH, NARCH and logarch

5 . Modelng e condonal mean e condonal mean ( θ ) = ( y ) µ sould be modeled n order o ncororae E nformaon from emrcal regulares of asse reurns. Non-syncronous radng Accordng o effcen marke eory, e sock marke reurns emselves conan lle seral correlaon. Moreover, wen g freuency daa s used, e non-syncronous radng n e socks makng u an ndex nduces osve frs order seral correlaon n e reurn seres. o conrol s Scoles and Wllams (977) suggesed a frs order movng average form, wle Lo and Macknlay (988) suggesed a frs order auoregressve form. Nelson (99) wroe as a raccal maer, ere s lle dfference beween an AR() and an MA() wen e AR and MA coeffcens are small and e auocorrelaons a lag one are eual. Rsk reurn radeoff Many eores n fnance are deal w e radeoff beween e execed reurns and varance, or e covarance among e reurns. Accordng o e Caal Asse Prcng Model (CAPM) e excess reurns on all rsky asses are rooronal o e nondversfable rsk as measured by e covarances w e marke orfolo. Meron (973) n Ineremoral Caal Asse Prcng eory sowed a e execed excess reurn on e marke orfolo s lnear n s condonal varance. e ARCH n mean or ARCH-M model, nroduced by Engle e al. (987), was desgned o caure suc relaonss. In e ARCH-M model e condonal mean s an exlc funcon of e condonal varance: were e dervave of e (.,.) ( θ ) g[ ( θ ) θ ] µ =,, g funcon w resec o e frs elemen s non-zero. e mos commonly emloyed secfcaons of e ARCH-M model osulae a lnear relaons n or /, e.g. g[ ( θ ), θ ] = µ µ. A osve as well as a negave relaons beween rsk and reurn could be conssen w e fnancal eory. We exec a osve relaons f we assume a raonal rsk averse nvesor wo reures a larger rsk remum durng mes e ayoff of e secury s rsker. Bu we exec a negave relaons under e assumon a durng relavely rsker erods e nvesors may wan o save more. In aled researces, ere s evdence for bo relaonss. Volaly and Seral Correlaon LeBaron (99) found a srong nverse relaon beween volaly and seral correlaon for Sandard & Poor, CRSP value weged ndex, Dow Jones and IBM reurns. He nroduced e Exonenal Auoregressve GARCH model or EXP-GARCH n wc e condonal mean s a non-lnear funcon of condonal varance: µ 3 ( θ ) = µ e y µ, As LeBaron saed, s dffcul o esmae µ 3 n conuncon w µ wen usng a graden ye of algorm. For s reason, µ s se o e samle varance of e seres. Asse reurn mnus e rsk free neres rae. As an aroxmaon o e rsk free neres rae we usually use e ree mon reasury Bll reurn. 5

6 LeBaron found a µ s sgnfcanly negave and remarkably robus o e coce of samle erod, marke ndex, measuremen nerval and volaly measure. 3. Modelng e Dynamc Srucures of e Greek Sock Marke: Alyng an ARCH model e daa se we wll analyze s e General Index of Aens Sock Excange (ereafer GI). ere are oally 98 observaons from 3 July 987 o 3 July 999. Defne y = log( ) as e connuously comounded rae of reurn for GI a me ( =,...,98), were s e daly closng rce of GI. In e followng lnes, we esmae a model o examne several ssues revously nvesgaed n e economcs and fnancal leraure namely a) e relaon beween e level of marke rsk and reured reurn, b) e asymmery beween osve and negave reurns n er effecs on condonal varance, c) fa als n e condonal dsrbuon of reurns d) e conrbuon of non-radng days o volaly e) e nverse relaon beween volaly and seral correlaon and f) e non-syncronous radng. We use e model develoed by Nelson (99), assumng an Auoregressve Movng Average reresenaon for ln ( ) n e condonal dsrbuon of reurns, we assume a e. o allow for e ossbly of non-normaly 6 u z are..d. draws from e Generalzed Error Dsrbuon (GED). e densy of a GED random varable normalzed o ave a mean of zero and a varance of one s gven by v z λ ve f ( z ) =, v v λ Γ( ) v < z <, < v, were Γ (). denoes e gamma funcon, and Γ( ) v ( 3 ) v λ Γ. v e v s a al-ckness arameer. Wen v =, z as a sandard normal dsrbuon. For v <, e dsrbuon of z as cker als an e normal (for v =, z as a double exonenal dsrbuon) and for v >, e dsrbuon of z as nner als an e normal (for v =, 3, 3 ). us, we model e log of e condonal varance as: ( Ψ L... Ψ L ) ln( ) = a z, ( L... ) L were L s e lag oeraor. o accoun for e conrbuon of non-radng erods o marke varance, we assume a eac non-radng day conrbues as muc o varance as some fxed fracon z s unformly dsrbued on e nerval [ ] We are graeful o GrSocks.com for rovdng e daa. Oer dsrbuons a ave been emloyed are e Normal-Posson mxure dsrbuon of Joron (988), e -dsrbuon of Bollerslev (987), e Generalzed -dsrbuon of Bollerslev, Engle and Nelson (99), e Power Exonenal dsrbuon of Balle and Bollerslev (989), e normal-log normal mxure of Hse (989) and oers.

7 of a radng day does. If, for examle, s fracon s one en, an on a ycal Monday would be er cen ger an on a ycal uesday. us we relace e consan erm a w: a = a ln( N δ ), were N s e number of non-radng days beween radng days and, and a and δ are arameers. Fama (965) and Frenc and Roll (986) ave found a nonradng erods conrbue muc less an do radng erods o marke varance, so we exec a < δ. o accommodae e asymmerc relaon beween sock reurns and volaly canges we sould use a funcon g ( z ) nsead of z. e g ( z ) mus be a funcon of bo e magnude and e sgn of z. One coce, a n ceran cases urns ou o gve well beaved momens, s o make g ( z ) a lnear combnaon of z and z : g( z ) = θ z z E z. < z <, ( z ) θ z, g ( ) s lnear w sloe θ. us, ( ) Over e range g s lnear w sloe, and over e range < z g z allows e condonal varance rocess o resond asymmercally o rses and falls n sock rce. Fnally, we model e log of condonal varance as ( Ψ L... Ψ L ) ln( ) = a g( z ). ( L... L ) e reurns are modeled as: µ y = µ µ µ µ e 3 y u, were e condonal mean and varance of u a me are and resecvely and µ, µ, µ, µ 3 and µ are arameers. e µ y erm allows for e auocorrelaon nduced by dsconnuous radng n e socks makng u an ndex. e µ erm allows e radeoff beween e execed reurns and varance. e 3e µ µ y erm allows for e nverse relaon beween volaly and seral correlaon of reurns. As we ave already saed, s dffcul o esmae µ n conuncon w µ 3 wen usng a graden ye of algorm. For s reason, µ s se o e samle varance of e seres, 7 ( y y) = µ =. In order o maxmze e lkelood funcons, we use e Evews 3. obec LogL. e maxmum lkelood arameer esmaes were comued usng e Maruard algorm as e BHHH algorm fals o converge. For a gven ARMA(,) order, e { z } =, and { } =, seuences can be easly derved recursvely gven e daa { y } =, and e nal values,..., max(. ). Also, ln,..., ln( max. ) were se eual o er uncondonal execaons ( ) ( )

8 ( δ ) a N ( δ ) ln,, a ln N max(, ) as: L = ( θ ) = ln( f ( y I ; θ )) = = y v ln λ = µ µ µ µ 3e λ µ. s allows us o wre e log lkelood y v ( v ) ln( ) ln[ Γ( v )] ln( ). o selec e order of e ARMA rocess for ln ( ), we use e Scwarz Creron(SC) (Scwarz (978)), ( l k ln( n) ) SC = n, were k s e number of esmaed arameers, n s e number of observaons, and l s e value of e log lkelood funcon usng e k esmaed arameers. e model w e lowes SC value s cosen as e mos arorae. Hannan (98) sowed a e SC rovdes conssen order esmaon n e conex of lnear ARMA models. e asymoc roeres of e SC n e conex of ARCH models are unknown. We do no use e Akake Informaon Creron (Akake (973)) as ends o coose e model w e ger number of arameers. able 3. lss e SC values for e varous ARMA orders of e model. able 3.. Scwarz creron for exonenal-e-garch(,) n Mean model. MA order() AR order() e ARMA(,) gves e SC lowes value. Nelson aled a smlar model n daly reurns for CRSP value weged marke ndex for July 96 o December 987 and seleced e ARMA(,) order. able 3. gves e arameers esmaes, e esmaed sandard errors and e -sascs of e Exonenal E-GARCH(,) n Mean model: µ y = µ µ µ µ 3e y u ln ( ) = a ln( N δ ) ( Ψ ) ( ) L ΨL u u u θ E L L able 3.. Parameers esmaes for ex-egarch(,) n Mean model Coeffcen Sd. Error z-sasc Prob. α

9 δ θ Ψ Ψ µ µ µ µ ν e esmaed correlaon marx of e arameer esmaes s resened n able 3.3. ese are comued from e nverse of e sum of e ouer roduc of e frs dervaves evaluaed a e omum arameer values. able 3.3. Esmaed correlaon marx for arameer esmaes for ex-egarch(,) n Mean model (only lower rangle reored) α δ Θ Ψ Ψ µ µ µ µ 3 ν α δ -.8 θ Ψ Ψ µ µ µ µ ν Le now examne e emrcal ssues rased n e revous Secon. a) Marke Rsk and Execed Reurn: e esmaed rsk remum s osvely correlaed w condonal varance, w µ =. 85 beng sascally sgnfcan. s agrees w e sgnfcan osve relaon beween reurns and condonal varance found by researcers usng GARCH-M models (Cou (987) and Frenc, Scwer and Sambaug (987)), bu conracs w e fndngs of Nelson (99) wo used a smlar model and of oer researcers no usng GARCH models (Pagan and Hong (988)). b) e asymmerc relaon beween reurns and canges n volaly, as reresened by θ s nsgnfcan. Accordng o e leverage effec, θ sould be negave, as we sould exec e volaly o rse (fall) wen reurns surrses are negave (osve). Esodes of g volaly sould be assocaed w marke dros. Bu lookng a e los of e daly condonal sandard devaon of reurns and e log value of e GI (Fgure 3.), we fnd a g volaly esodes are assocaed bo w marke eaks and dros. c) Fa als. I s well known a e dsrbuon of sock reurns as more weg n e als an e normal dsrbuon (muc ger kuross an 3), and a a socasc rocess s ck aled f s condonally normal w a randomly cangng condonal varance (lke GARCH rocesses). In our case e model 9

10 generaes ck als w bo a randomly cangng condonal varance and a ck aled condonal dsrbuon for u. e esmaed v s aroxmaely.39 w a sandard error of abou., so e dsrbuon of z s sgnfcanly cker aled an e normal. d) e esmaed conrbuon of non-radng days o condonal varance s rougly conssen w e resuls of Frenc and Roll (986). e esmaed value of δ s abou.38, wc sascally sgnfcan, so a non-radng day conrbues more an a rd as muc volaly as a radng day. e) e nverse relaon beween volaly and seral correlaon for GI daly reurns as reresened by µ 3 erm s sascally sgnfcan. us e condonal mean s a osve non-lnear funcon of condonal varance. f) e µ erm euals. sows a e osve non-syncronous radng effec exss n e consrucon of e GI. 9.5 Fgure 3. e Log Value of e General Index of Aens Sock Excange and e Daly Condonal Sandard Devaon of Reurns 3/Aug/87-3/Jul/99 8% Log Values of GI /3/87 /3/88 8/3/88 /3/89 8/3/89 /3/9 8/3/9 /3/9 8/3/9 /3/9 8/3/9 /3/93 8/3/93 /3/9 8/3/9 /3/95 8/3/95 /3/96 8/3/96 /3/97 8/3/97 /3/98 8/3/98 /3/99 e log value of e General Index e Daly condonal Sandard Devaon of Reurns 5. Concluson e aer resened e mos moran eorecal regulares a govern e dynamc srucure of fnancal me seres and ess er valdy n Aens Sock Excange. e Greek Sock Marke s examned by alyng an ARCH model on e log-reurns of e General Index of Aens Sock Excange from 3 July 987 o 3 July 6% % % % 8% 6% % % % Daly Condonal S. Devaon n ercen

11 999. e ARCH model fs well o Greek Sock Marke daa and rovdes emrcal evdence on eorecal regulares. Some of e concluson are: e exsence of a osve (non-lnear) rade-off beween sock reurns and volaly, e absence of leverage effecs, e ck aled sock reurns dsrbuon, e nformaon accumulaon n a slower rae wen e marke s closed an wen s oen, e exsence of osve non-syncronous radng effecs and e exsence of a long-erm memory aern n sock reurns. I would be neresng o develo alcaons of e model n e felds of orfolo rsk managemen and fnancal dervaves rcng. Moreover we sould es e effcency ganed, f any, n modelng w dsrbuons oer an normal. ese uesons are e scoe of furer researc. References Akake, H. (973). Informaon eory and an Exenson of e Maxmum Lkelood Prncle. Proceedngs of e second nernaonal symosum on nformaon eory. B.N. Perov and F. Csak (eds.), Budaes, Balle, R.. and Bollerslev. (989). e Message n Daly Excange Raes: A Condonal Varance ale. Journal of Busness and Economc Sascs, 7, Black, F. (976). Sudes of Sock Marke Volaly Canges. Proceedngs of e Amercan Sascal Assocaon, Busness and Economc Sascs Secon, Bollerslev,. (986). Generalsed Auoregressve Condonal Heeroskedascy. Journal of Economercs, 3, Bollerslev,. (987). A Condonal Heeroskedasc me Seres Model for Seculave Prces and Raes of Reurn. Revew of Economcs and Sascs, 69, Bollerslev,., Engle, R.F. and Nelson, D.B. (99). ARCH Models. In Caer 9 of Handbook of Economercs, Volume, Nor-Holland. Box, G. E. P. and ao, G. C. (973). Bayesan Inference n Sascal Analyss. Readng, Mass. Cou, R.Y. (987). Volaly Perssence and Sock Reurns: Some Emrcal Evdence Usng GARCH. Journal of Aled Economercs, 3, Dng, Z., Granger, C.W.J. and Engle, R.F. (993). A Long Memory Proery of Sock Marke Reurns and a New Model. Journal of Emrcal Fnance,, Engle, R.F. (98). Auoregressve Condonal Heeroskedascy w Esmaes of e Varance of U.K. Inflaon. Economerca, 5, Engle, R.F. (99). Dscusson: Sock Marke Volaly and e Cras of 87. Revew of Fnancal Sudes, 3, 3-6. Engle, R.F., Davd M. Llen, and Russell P. Robns (987). Esmang me Varyng Rsk Prema n e erm Srucure: e ARCH-M Model. Economerca, 55, Fama, E.F. (965). e Beavour of Sock Marke Prces. Journal of Busness, 38, 3-5. Fama, E.F. (97). Effcen Caal Markes: A Revew of eory and Emrcal Work. Journal of Fnance, 5, Frenc, K.R. and Roll, R. (986). Sock Reurn Varances: e Arrval of Informaon and e Reacon of raders. Journal of Fnancal Economcs, 7, 5-6. Frenc, K.R., Scwer, G.W. and Sambaug, R.F. (987). Execed Sock Reurns and Volaly. Journal of Fnancal Economcs, 9, 3-9. Geweke, J. (986). Commen. Economerc Revews, 5, 57-6.

12 Glosen, L.R., Jagannaan, R. and Runkle, D. (993). On e Relaon Beween e Execed Value and e Volaly of e Normal Excess Reurn on Socks. Journal of Fnance, 8, Hannan, E.J. (98). e Esmaon of e Order of an Arma Process. Annals of Sascs, 8, 5, 7-8. Harvey, A.C. (98). e Economerc Analyss of me Seres, Oxford. Hggns, M.L. and Bera, A.K. (99). A Class of Nonlnear ARCH models. Inernaonal Economc Revew, 33, Hse, D.A. (989). Modelng Heeroscedascy n Daly Foregn-Excange Raes. Journal of Busness and Economc Sascs, 7, Joron, P. (988). On Jum Processes n e Foregn Excange and Sock Markes. Revew of Fnancal Sudes,, 7-5. LeBaron, B. (99). Some Relaons Beween Volaly and Seral Correlaons n Sock Marke Reurns. Journal of Busness, 65,, Lo, A. and MacKnlay, C. (988). Sock Marke Prces Do No Follow Random Walks: Evdence From a Smle Secfcaon es. Revew of Fnancal Sudes,, -66. Mandelbro, B. (963). e Varaon of Ceran Seculave Prces. Journal of Busness, 36, Margora, F. and J. Panareos, (). Auoregressve Condonal Heeroskedascy Models and e Dynamc Srucure of e Aens Sock Excange, Aens Unversy of Economcs and Busness, Dearmen of Sascs, ecncal Reor,. Meron, R. C. (973). An Ineremoral Caal Asse Prcng Model. Economerca,, Nelson, D.B. (99). Condonal Heeroskedascy n Asse Reurns: A New Aroac. Economerca, 59, Pagan, A.R., Hong, Y. (988). Non-Paramerc Esmaon and e Rsk Premum. W. Barne, J. Powell, and G. aucen (eds.), Nonaramerc and Semaramerc Meods n Economercs, Cambrdge Unversy Press. Panula, S.G. (986). Commen. Economerc Revews, 5, Scoles, M. and Wllams, J. (977). Esmang Beas from Nonsyncronous Daa. Journal of Fnancal Economcs, 5, Scwarz, G. (978). Esmang e Dmenson of a Model. Annals of Sascs, 6, 6-6. Scwer, G.W. (989). Sock Volaly and e Cras of 87. Revew of Fnancal Sudes, 3, 77-. Senana, E. (995). Quadrac ARCH Models. Revew of Economc Sudes, 6, aylor, S. (986). Modelng Fnancal me Seres. Wley and Sons: New York, NY. Zakoan, J.M. (99). resold Heeroskedasc Models, manuscr, CRES, INSEE, Pars.

RELATIONSHIP BETWEEN VOLATILITY AND TRADING VOLUME: THE CASE OF HSI STOCK RETURNS DATA

RELATIONSHIP BETWEEN VOLATILITY AND TRADING VOLUME: THE CASE OF HSI STOCK RETURNS DATA Mchaela Chocholaá Unversy of Economcs Braslava, Slovaka Inroducon (1) one of he characersc feaures of sock reurns