GARCH option-pricing model with analytical solution when interest rate and risk premium change randomly

Transcription

1 Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8 Noedde Lahoel (sa) Mokha ok (sa) GARCH opopcg model wh aalycal solo whe ees ae ad sk pemm chage adomly Absac We vesgae he GARCH opopcg model wh adom ees ae ad adom sk pemm. he developed aalycal solo geealzes he fomla of lack & choles (97) adjsed wh skewess ad koss of sadadzed cmlave es. I fac he hypohess of cosa ees ae ad sk pemm cosdeed he GARCH opopcg famewok seems ealsc. I becomes dspesable o solve hs weakess of GARCH opopcg models by lfg ha hypohess. he sdy of mecal ad empcal effecs of ew sae vaables complees hs wok. eywods: GARCH opopcg ees ae sk pemm aalycal appomao pefomace hedgg. JL Classfcao: C C G G. Iodco he famly of GARCH models occped a mpoa place he empcal asse pcg ad he facal sk maageme. he sccess of GARCH models fg asse es ad he fale of deemsc volaly models fg opo pces leaded he seaches o hk abo odcg he GARCH model opopcg famewok. he fs applcao hs doma was he oe of Da (995) ha esablshed he fodao of opopcg de GARCH. Da Rchke & (5) deved a opopcg heoy eedg he sadad GARCH models o clde jmps. he sccess of GARCH models opopcg s de o he fac ha he opopcg heoy s fleble becase ca be adaped o ay GARCH specfcao. Also he GARCH pocesses ae lked p wh sochasc volaly models. Ideed Nelso (99) showed ha vaae GARCH pocesses ca be sed o appomae sochasc volaly models. Da (996) geealzed hs esablshed fac povg ha he esg bvaae dffso models ae lms of he GARCH models. I ealy he oaly of sded GARCH specfcaos cosdeed a cosa sk fee ees ae ad cosa sk pemm. hs hypohess appeas ealsc gve he esls of empcal sdes elaboaed o sdy he dyamcs of ees ae ad Noedde Lahoel Mokha ok 8. Hll & Whe (987) e & e (99) Heso (99) aksh Cao & Che (997) ad chöbel & Zh (999) ae eamples of sochasc volaly opo valao models. Ohe seaches lke Heso & Nad () Da Rchke & (5) ad Chsoffese Heso & Jacobs () esablshed he same obsevao. sk pemm. A aal eeso ca coce he volao of he wo hypoheses. I hs pape we popose addessg a possble espose o hs qeso g o he GARCHM pocesses of gle Lle & Robs (987) o descbe he ew adom vaables (ees ae ad sk pemm). hs ew appoach cosdes he GARCH opopcg model whe ees ae ad sk pemm ae wo sochasc sae vaables goveed by a GARCHM pocess whch he elaoshp bewee he codoal mea ad vaace s olea. he facos movag he ees of hs seach ae: o he oe had he sochasc volaly models cosde moe ha oe sae vaable he valao pocess whle he GARCH models famewok we cosde a sgle sae vaable (he delyg asse e). hs ca make p a lm of GARCH models compaavely o sochasc volaly models. O he ohe had he se of olea pocesses o descbe he ew sae vaables s jsfed by seveal woks lke Lahoel (7) fo he descpo of ees ae ad Lo & eo (999) fo he descpo of sk pemm. I hs wok we esmae he dyamc elaoshps bewee he sae vaables lookg fo a sable fomlao allowg sdyg he mechasms of coelaos bewee hose vaables a mlvaae famewok. hs pocede allows he se of VAR model wh mlvaae GARCH eo (VAR MGARCH). Ideed he mlvaae appoach was aded wh he VAR models (gle & oee 995). he mlvaae appoach of GARCH models becomes he moe wdespead facal ecoomec o aalyze he shock asmssos ad o sdy he smlaeos vaably of volales of dffee vaables. he behavo of he ees ae was eamed by ambh & Mosse () Joes () ad Lahoel (7) amog ohes. he sk pemm was ake o acco by eo (999) ad Lo & eo (999) amog ohes.

2 Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8 he calclao of opo pce he GARCH model famewok s caed o sg mecal mehods whch eed a mpoa me of applcao. Fo hs easo he empcal sdes o he GARCH opopcg models ae lmed. I fac he lmao ca be cased by he dffcly of calclaos o pehaps by he qe specfcao of he dyamc of delyg es. heefoe he seach of aleave pocedes becomes ecessay. Hake (997) poposed a appomao of he GARCH opopcg model by eal ewoks. A coce appoach poposed by Heso & Nad () cosss developg a aalycal solo fo he opea opos de GARCH. hs mehod s based o he chaacesc fco of cmlave es. I s a eesg appoach b eeds o esolve he chaacesc fco aalycally whch o always possble wh all GARCH specfcaos. Moe geeally Da Gahe assevlle & moao (6) developed a aalycal appomao fo he GARCH opopcg model sg he dgewoh aso of he skeal omal desy fco. We sgges he e o adap hs appoach o he poposed model. he obaed fomla s smla o he oe of lack & choles (97) adjsed by skewess ad koss of sadadzed cmlave es de GIRR pocess. he obaed aalyc appomao allows geg hedgg paamees favog he effecve ealzao of sks aached o opo pofolos. Ideed wh he model of lack & choles (97) we ca alk oly abo a sac sgfcao sce hs fomla assmes ha volaly ad ees ae ae cosa. ch a hypohess does pem o have eal hedgg paamees b oly some sac compasos o hese vaables. Also wh GARCH models we ca o oba eal meases of sk of opoal pofolo. hs fac s ofe allocaed o he oappopao bewee he heoecal fodao of valao model ad he make ealy. We ogaze he es of he pape as follows: seco oe we pese he aalycal famewok of he GIRR model ha pems o esl esablshg a appomaed fomla o valae he opea calls (seco wo). I he hd seco we make he mecal sdy of he obaed model. he sdy of empcal pefomace wll be eaed seco fo. I hs seco we compae he GIRR model ad he GJRGARCH of Glose Amog he mecal sdes esece we qoe Da & moao (999) ad Rchke & evo (999). Che & Hag (7) appled hs appoach o F de opos. Jagaaha & Rkle (99) sg daa o &5 de obseved o he CO. Fally he las seco cocldes.. pecfcao of he GIRR model ce lack & choles (97) he seach o he modellg of he delyg asse e dyamc was vey esve. he moe mpoa developmes hs sese coce especally coosme opopcg models. Coceg he dsceeme opo valao we always qoe he GARCH models. eaches ha vesgaed he empcal aspecs of GARCH model pefomace ae o meos. he moe kow ae Am & Ng (99) gle & Msafa (99) Da (996) Hadle & Hafe () Heso & Nad () ad Chsoffese & Jacobs (). he poposed GARCH opopcg models cosde ha he compod codoal es R =Log( / ) whee s he delyg asse pce a me ca be modelled as: R () wh s he skfee ees ae s he cosa pce of sk h z wh z N ad h s he codoal vaace of es me vayg ad goveed by a GARCH pocesses. I hs pape wh he ew fom of R we oba a veco of sochasc pocesses y ' R whose specfcao ca employ a mlvaae GARCH model (MGARCH) ode o cape he dyamc evolo of he vaacecovaace ma. hee es a lage mbe of MGARCH specfcaos b he e we wll se he oe of gle (). We choose hs model becase ca be esmaed fom a seqece of vaae GARCH models. I s also elavely easy o mpleme fo he pacoe o he pofessoal becase does mpose heavy hypohess of dsbos as he MGARCH model of olleslev (99) fo sace. Hypohess. Dsceeme pefec make We cosde a dsceeme ecoomy wh possble bea sale of asses ad wh ll asaco coss ad aes. he ceay s chaacezed by a pobably space whee s he se of eal mbes s a be ad s a objecve pobably mease. hs pobably space eqpped wh a flao F... ha follows a sadad owa moveme. We y o specfy he pce of a opea call opo wh ske may ad wg o a delyg asse ha does pay ay dvded. he pce o specfy s so fco of a se of sae vaables of

3 Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8 : he pce he delyg asse ; he sk pemm ad he ees ae. Coseqely he eqao () descbg he codoal es becomes: R. () C R If we oce he pemm of a opea call opo he kowledge of C s closely bod o he oe ad of he veco of sae vaables ' C wll be accodgly moe plasble as. Howeve o defe he pocess ha descbes he me vayg of each sae vaable we move owads he dsceeme pocesses of Makov. Coseqely we ca cosde a class of GARCH models. hese pocesses seem hs o be moe adaped o descbe he volaly flece o he asse es... he model of he ees ae. I hs model we poslae ha he ees ae cve s fco of a aloe sae vaable whch s he shoem ees ae. Howeve he shoem ees aes ae chaacezed by a hgh pessece ad by he pesece of a codoal heeoskedascy. Coseqely a ealsc modellg of ees aes ms cosde hese wo esseal feaes. Hypohess. he ees aes he ees aes o he peods coespodg o opo maes ae goveed by a volaly pocess of he fom: ;... () wh s a cosa posve whch s he mea level of ees ae h z ; whee z foms a seqece of depedely ad decally dsbed vaables (d) wh vaace. I hs model whee o a sgle delyg asse we coespod seveal soces of hazad ha ae o pefecly coelaed (he dmeso of he veco of whe oses s o eqal o he mbe of sky asses) he make s hs complee. Howeve he ees ae cosdeed as a adom vaable s o clealy he case of hs make compleeess. I Coceg he logem ees aes we ca bld fo hem a em sce of ees aes by cosdeg a pacla dyamc fo he shoem ees aes. hs qeso was sded he wok of Lahoel (7). Meo (97) geealzes he coosme model of lack & choles (97) by assmg ha he ees ae s sochasc. He calls a sae vaable whch s he zeocopo bod pce whose cemes ae coelaed wh hose of delyg pce o dvesfy he sk bod o sochasc chaace of he ees ae. fac he ceay geeaed by he adomess of hs vaable ca be dvesfed by a zeocopo bod. I hs way havg assmed ha he ees ae cve s fco of a sgle sae vaable (he shoem ees ae ) we cosde ha pces of bods of dffee may pese a coelaed evolo whe he shoem ees ae flcaes. Ohewse hese pces ca be lkeed o he ae of a shoem zeocopo bod. Coseqely s specfed by he saaeos shoem ees ae dg all he may of he bod. I fac sce he sesvy of he opo pce o he vaaos of he ees ae s elavely weak we sppose ha he em sce of ees ae depeds o a sgle sae vaable as follows:. () o deeme he aalycal esso of sgges he followg poposo: oposo. ce of a zeocopo bod we Le a zeocopo bod wh fal flow eqal a y dae. If he shoem ees ae s modelled by eqao () he pce of he zeocopo bod whch s fco of he may ad ees ae s gve by:. 5. (5) oof: see Apped A... he skpemm bod o he volaly of es. he hypohess of cosa skaveso coeffce cosdeed he eqao () of delyg asse es shold be volaed. Howeve he qeso ha shold be esolved s o specfy he dyamc of hs ew sae vaable. I hs espec we poslae o say always he aoegessve model famewok wh eo codoally heeoskedasc. I fac Lahoel (6) cosdeed he followg hypohess: Hypohess. he skpemm of es he skpemm bod o he volaly of delyg asse pces s goveed by a GARCH pocess of he fom: ; h z (6) wh s a cosa paamee ad z s a seqece of adom vaables havg a d dsbo. he veco of sae vaables ca be lkeed o a VAR specfcao wh mlvaae GARCH eo.

4 Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8.. coomec specfcao. Cosde a veco y of sochasc pocesses wh N compoes. We ca cosde ha he veco y depeds o he pas fomao cosced by s delayed vales l he sa. I he followg gvg he veco of kow paamees we ca we: y wh H / z (7) s a veco of codoal mea of y wh z d ad / H s a posve defe ma of dmeso NN sch as H s he codoal vaace ma of y. We ca followg gle () decompose he ma H as follows: / / H D R D wh D dagh h (8)... NN whee R s he ma of mevayg codoal coelaos gve by: / / dagq Q dagq R (9) whee Q s a symmec posve defe ma of dmeso NN defed by: Q ' Q Q. () dagq s a dagoal ma coag he elemes o he dagoal of Q Q s he codoal vaacecovaace ma ad ad ae oegave paamees sasfyg. Coceg he esmao of he paamees gle & heppad () ad gle () se a sedo Mamm Lkelhood (o Qas Mamm Lkelhood: QML) as a esmao mehod whch s a wosep esmao appoach. I he fs sep we fd he vale of ˆ ha mamzes he fco L V L gve as: ' Log. () V D I he fs esmao we apply a vaae GARCH model o he codoal vaace of each vaable. I hs way we oba he volaly coeffces of each vaable ake dvdally. We adop he vaae GJRGARCH specfcao of Glose Jagaaha & Rkle (99) he es of he wok. I hs model he codoal vaace s gve by: h h h z h ma z.. I he secod phase of esmao he volaly coeffces obaed he fs sep ˆ ae maaed cosas ad seve o codo he lkelhood fco L C sed o esmae he paamees of he coelaos ˆ. hs fco s gve by: L C ' ' Log R R. (). he aalycal appomao fomla of he opea call Jaow & Rdd (98) elaboaed a geeal heoecal famewok o develop a opo valao aalycal appomao fomla. hey poposed a echqe o appomae he pobably dsbo called he e dsbo o a aleave dsbo called he appomag dsbo. I he sascal leae hs echqe s called he geealzed dgewoh sees aso. Usg a smla appoach we ca deve a aalycal appomag fomla o valae he opea call opo de he GIRR model. oposo. Aalycal appomao of a opea call opo Le Log / he cmlave e shavg a mea m ad a sadad devao. Le m / he sadadzed cmlave es. I he poposed GIRR famewok he pemm of a opea call opo wh ske pce ad may ca be appomaed wh he followg fomla: C appo whee C C C () m. N U NU C () C C 5 / 6 m. U U NU 5 / m. 5 U U U (5) N U (6) wh ad U Log / m / (.) ad N(.) epese especvely he desy ad he cmlave fcos of a sadad omal adom vaable. he ems ad ae especvely he skewess ad koss of he sadadzed cmlave es. oof: see Apped.

5 Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8 he applcao of hs aalycal appomao eeds kowg he essos of he fo fs momes of he cmlave e fo all may. he qeso s o oba hese momes as fcos of he model paamees ad he may. Howeve fo all ee m{ } we have : m Log / m R m (7) wh ; ; H / z ad. m. Nmecal pefomace of he GIRR model.. mlao of delyg asse pce. o smlae he pocess of es ad afe ha he delyg asse pce we eed o f vales fo he GIRR model paamees. We assme ha =.7 ad =.5 ad we f he same vales fo he codoal vaace paamees wh all sae vaables (e ees ae ad sk pemm): =.5 =.7 =. ad =.5 fo R o o. he paamees of shocks ad coelaos bewee sae vaables ae fed as: =. =.8 R =. R =.7 ad =.5. We oba fally delyg asse pces fg he al vale a =. he obaed esls ae epoed Fge below: Moe Calo Aalycal.7.6 Moe Calo Aalycal 6.5 Udelyg asse pce 5 Desy of pobably Obsevaos Res Fg.. Udelyg asse pce smlaed wh he GIRR model I s clea ha he GIRR model s a good appomao of he asse pces dyamc. aclaly he aalycal pces ed o cofse hose of Moe Calo... Appomao of he opea call pce wh he GIRR model. As show by eqao () he aalycal appomao of he opea call opo pce s composed of a em smla o he lack & choles fomla ad wo adjsme ems fo he skewess ad koss of sadadzed cmlave es. Howeve s ecessay o eame he flece of he paamees o he opea calls.... Iflece of skewess ad koss o he call pces. I s a mae of sdy o aalyze he flece of skewess ad koss o he o pea call pces hogh a compaso wh em C of fomla (). We cosde he dffeece bewee he delyg asse pce ad he ske pce. We epese he dffeece of pces (adjsed pce o adjsed pce (C )) as fco of he moeyess ad may. We ask he followg qeso: whch s he flece of paamees ad o he call opo pces? Usg he paamee vales fed a he op we oba he followg smlaos: he aalycal essos of he momes de he GIRR model ae avalable fom ahos po eqes: oeddelahoel@yahoo.f. he fsode codoal vaace h s eqal o he saoay va h / volaly pessece. he qaes ace gve by: ad R ae lls. 5

6 Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8.6. May = days May = days May = 6 days.5 emm dffeece (C NC) Moeyess (Log(/)) emm dffeece (C NC) May.5 Moeyess (Log(/)) Fg.. Dffeece bewee coeced pce by skewess ad koss ad o coeced pce Fge allows o come o aga he followg emaks: he dffeece bewee coeced ad ocoeced pces s posve fo he hemoey ad oofhemoey opea call opos. he opo pces ae oveesmaed ad he oveesmao s moe mpoa fo he oofhemoey opos. he eahemoey call pces ae deesmaed becase he pemm dffeece s egave. he dffeece as fco of he may level s a ceasg fco fo hemoey ad oofhemoey opos ad a deceasg fco fo eahe moey opos. he pemm dffeece becomes ll fo hemoey ad oofhemoey opos whe he may deceases.... he ypcal opea call pce de he GIRR model. I s possble ow o povde a dea o he appeaace of he opo pce gve by GIRR model as fco of moeyess. Fge daws he evolo of hs pce fo dffee vales of he may: 6 5 May = days May = days May = 6 days 6 5 GIRR pemm GIRR pemm Moeyess (Log(/)).5 May.5 Moeyess (Log(/)) 6 Fg.. volo of he opea call opo pce de he GIRR model he eamao of Fge allows o dedce ha he GIRR opea call pce s a ceasg fco of moeyess ad may.. dy of he GIRR empcal pefomace o he CO he objecve of hs seco s o sdy he compaave empcal pefomace of he GIRR valao model ad he GARCH model wh cosa ees ae ad cosa sk pemm. he compaso of he valao model pefomaces allows mpovg he make dyamc kowledge becase hs aalyss ses eal make daa. Howeve gve ha he pefomace of a valao model depeds o he coec specfcao of es dyamc he ees of hs sdy cosss esg he eal accodace ad coheece of he pocess paamees. I s a mae of esablshg a elao bewee he codoal volaly paamees efleced mplcly he opo pces ad he chaacescs of he mesees of delyg asse es. I hs coe we ca cay o he ess of model pefomace valao

7 (sac pefomace) ad hedgg (dyamc pefomace). We focs he empcal sdy o he & 5 de opos egoaed o he CO (Chcago oad Opos chage)... Daa. We se daa o &5 de pces colleced evey mes coveg he peod fom Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8 6/9/6 o 8//6. he ees ae s appomaed by he ae o he easylls of he same may as he sded opos ( moh mohs ad 6 mohs). able smmazes he pcpal descpve sascs of & 5 de es dg he peod de sdy. able. ascs of &5 de es (6/8/ 8/5/) Mea adad devao kewess oss Q() Q²() J Q() ad Q²() ae sascs of aocoelao Ljgo es of ode of es ad sqae es especvely. J s he Jaqeea sasc esg he ll hypohess of omal dsbo of es. he descpve aalyss esls do allow o affm ha he empcal dsbo of &5 de es s assmlaed o a omal dsbo. hs coclso s jsfed by he vales of skewess ad koss... smao of he DCCGARCH model. We esmae a DCCGARCH model whch he vaables R ad appea as edogeos vaables. he esmaed paamees ae egoped able. able. smao of scal paamees of GIRR ad GJRGARCH models aamees GIRR GJRGARCH < 6 6< 8 < 6 6< 8. (.5).6 (.).8 (.) (.5) (.) (.) (.7) (.7) (.8) Iees (.78) (.667) (.87) Rsk pemm.557 (.8).87 (.67).7 (.7) Re (.) (.89) (.89) (.8) (.58) (.658) Iees (.7) (.96) (.78) Rsk pemm.799 (.).798 (.66).87 (.76) Re (.) (.97) (.) (.75) (.6) (.) Iees...7 (.9) (.) (.6) Rsk pemm. (.88). (.7).55 (.9) Re (.5) (.58) (.588) (.8) (.) (.) Iees... (.88) (.888) (.876) Rsk pemm. (.6).9 (.568).7 (.6) Re (.) (.7) (.9) (.5) (.) (.99) 7

8 Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8 able (co.). smao of scal paamees of GIRR ad GJRGARCH models aamees GIRR GJRGARCH. (.7).886 (.66) R.7 (.9).76 R (.55).5998 (.88) < 6 6< 8 < 6 6< 8.78 (.).8765 (.5).9 (.).7 (.9).598 (.76).7 (.6).8658 (.).99 (.).7 (.6).66 (.88) LogL he GJRGARCH model s esmaed sg he oesep mamm lkelhood mehod. We f fo he ees ae he vale obaed by he GIRR model fo he coespodg class of may. he esls obaed able ae compaable wh he sadad coclsos he leae o GARCH pocesses. smaos of he paamees ad have appomaely he same mpoace as he esg leae. he volaly pessece dedced fom he esmaed paamees s also mpoa accodace wh he leae. he sadad eos (vales paeheses) dcae ha esmaed paamees ae sgfca. he eos of he GIRR model ae smalle ha hose of he GJRGARCH model ecep fo he paamee fo a may bewee 6 ad 8 days. he LogL vale (obaed a he opmm of he loglkelhood fco) dcaes ha he GJRGARCH model s moe effce ha he GIRR model. he sk pemm s coelaed egavely wh he ees ae ( < ) wha s logcal becase he sk pemm s deceasg wh ees ae... dy of GIRR sac pefomace valao.... Isample appoach. hs appoach cosss gve he paamees esmaos esg he eal coheece of he models. Howeve we adop he followg mehod: we jec he scal paamee esmaos obaed a me he volaly pocess. he epeo of hs opeao fo all obsevaos of he sded peod gves s a mpled volaly sees. Afe we calclae he vaao of he obaed sees ad he coelao bewee he ecosced mpled volaly vaaos ad he delyg asse es. he fod vale fo he coeffce of vaao s compaed o he paamee ad he secod coeffce vale s compaed o zeo becase he coelao bewee volaly ad es s assmed o be ll he GARCH model famewok. Whe we have a weak dffeece bewee hese vales he cosdeed specfcao fo he model s coec. able. Ieal coheece of coelao ad vaace paamees aamees esmaed fom es aamees obaed fom mpled volaly of es GIRR GJRGARCH GIRR GJRGARCH May < able shows clealy he eal coheece of paamees he sded models. hs coheece appeas evde fo he mpled coelao bewee shocks of vaace ad & 5 es. compag he vales of he coeffce of he mpled 8 volaly obseved pces o he vales of ecosced vaaces lghes he coheece. Howeve he GIRR model poves moe cohee especally whe s a mae of esmao of he mpled vaaces.

9 ... Oofsample appoach. I hs appoach he mpoa qeso s: gve he scal paamee esmaos ad hose of he vaace wha s he degee of he model pefomace he opo pce foecass? o aswe hs qeso we se paamees esmaed a o calclae he pce a of he opo belogg o he same caegoy of may fom whch we esmaed he paamees vales. I fac he make opeaos ae o capable o kow he paamee vales saaeosly fo ha hey se he esmaos calclaed a he pevos dae. he Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8 able. Oofsample foecasg eos delyg asse pce sed o calclae he opo pemm s ha of he ce sa. he calclaed pce wll be compaed o he oe obseved a he same me ad havg he same may. he dffeece bewee he wo pces s he foecasg eo of he adoped model. y epeo of he pocede fo each obsevao dg he sded peod fo hee ypes of may we calclae he oo of he measqaed eos (RM) he mea of he sadadzed absole eos (MA) ad mea of he peceage of valao eos (M). he obaed esls ae: GIRR GJRGARCH < 6 6< 8 < 6 6< 8 RM MA M he foecasg pefomace of he models mpoves wh may gve ha he eos dow whe may ceases. he absole valao eo (MA) s sable fo all caegoes of may whch dcaes he absece of valao bas ed o he may. he mea of he peceage of valao eos (M) shows ha he wo cosdeed models ed o oveesmae he opos fo all may levels ecep fo he GJRGARCH model fo a may speo o 6 days. Coceg he compaso of he wo models s clea ha he GIRR model s moe effce fo opo valao. I fac he odco of a adom ees ae ad a adom sk pemm he GARCH model mpoves s pefomace fo e adjsme ad opo valao. hs esl s eced becase of he ealsc hypohess of cosa ees ae ad cosa sk pemm... Dyamcal pefomace opoal pofolo hedgg.... he hedgg saegy. o es he capacy of he GIRR model o hedge opo posos we adop a delaeal hedgg saegy. We sdy he case of a ade yg o hedge a sho poso o a call opo (age opo C ) of may ad ske pce. he logc of hedgg saegy foces he ade o ake poso o he delyg asse a popoo a he dae o wap p agas he sk of he pce. he delaeal hedgg pcple ad he adomess of he volaly lead he ade o ake Fges dawg he mevaao of calclaed ad obseved call pces as well as fges dawg he mevaao of he valao elave eos ae avalable fom aho po eqes: oeddelahoel@yahoo.f poso a me o a call opo C a popoo C. hs call opo has a may decal o he age opo C b wh ske pce. Fhemoe he adomess of he ees ae leads he ade o ake poso o a zeocopo bod a popoo a me o wap p agas he ees ae sk. he eplcao pofolo vale of he age opo a he dae s gve by: (8) C C wh s he emade of lqdes (cash) o he ade a he dae. Dffee popoos ca be obaed sg he followg fomlae: C CC (9) C () C C V V (). ()... Vaace mmm of hedgg pofolo. he vaace mmm ceo of he hedgg pofolo es s boowed fom Nad (996). Accodg o hs logc he hedgg pofolo s peodcally adjsed ode o ealze he sesvy o he fs ode vaaos (dela). I able 5 we egop he mea vaaces of hedgg pofolo vale chages o he sded peod accodg o GJRGARCH ad GIRR models: 9

10 Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8 able 5. Mea vaaces of hedged pofolo vale chages de he GIR GJRGARCH ad models GIRR GJRGARCH May OM AM IM OM AM IM OM AM IM < Compag he models we oe ha he GIRR model has he moe lle vaace fo all caegoes of may ad moeyess. he mea vaace of he pofolo vale chages deceases wh he may. I also deceases wh he moeyess ecep whe may s less ha days s moe mpoa fo AM opos. Fo he model ad wh may bewee ad 6 days he AM opo pofolo vaace s weake. he edco peceage of vaace s moe mpoa whe we compae GIRR wh ha whe we compae GIRR wh GJRGARCH. Coclso I hs pape we poposed a opea opopcg model (GARCHochasc Iees Rae ochasc Rsk emm: GIRR) whch geealzes he GJR GARCH model of Glose Jagaaha & Rkle (99). hs model allows capg as well as egave skewess ad ecess of koss of e dsbo he o saoay chaace of ees ae ad sk pemm. I fac all esg GARCH opopcg models cosde cosa ees ae ad cosa sk pemm hypohess. eveal ahos sch as Chsoffese & Jacobs () sgges he volao of hs ealsc hypohess. I hs way o eply o hose pospecs he GIRR model cosdes ha all sae vaables ae goveed by GARCH pocesses. o ake coelaos bewee dffee sae vaables o acco we cosdeed he mlvaae GARCH model of gle () (DCCGARCH) o descbe he codoal vaacecovaace ma. o have a aalycal opopcg fomla we sed he geeal heoecal famewok elaboaed by Jaow & Rdd (98). hs echqe allows o appomae a kow pobably dsbo (e dsbo) sg a aleave dsbo (appomag dsbo). Howeve dog he appomao of he sadadzed cmlave e dsbo by he omal dsbo we maaged o esablsh he aalycal appomao of he opea call opo pemm. he moeyess vale s gve by: Log( /). Fo he AM opos we cosdeed moeyess bewee.5 ad.5. I he followg he pah of he pocess descbg he delyg asse pce dyamc was cofoed wh he Moe Calo smlao. Resls show ha he pocess callg adom ees ae ad adom sk pemm seems well adaped o descbe he facal e dsbo havg a lepokocy. We also sded he mecal pefomace of he aalycal appomao fomla valg he opea call opo elaboaed de GIRR model. Howeve sce hs fomla s coeced by skewess ad koss of cmlave es we esed he mpac of he adjsme paamees o he opea call pces. he wo adjsme ems flece sgfcaly opea call vales eve f we vay he moeyess ad he may. he cofoao of he poposed GIRR model wh he oe cosdeg cosa ees ae ad cosa sk pemm showed he peece of hese sae vaables he delyg asse e pocess. I fac he sdy of empcal ad compaave pefomace of models opo valao ad hedgg favos GIRR model compaed wh he GJRGARCH model of Glose Jagaaha & Rkle (99). Lookg fowad he effo made hs acle ca be followed. We ca qoe some sbseqe eseach pospecs: he fs developme coces applyg he poposed GIRR model o ohe daa (fo sace daa o he MON). he secod deco ca coce compag he GIRR model wh coosme models lke aksh Cao & Che (997). Ohe aleave sggess cldg he asaco coss. Howeve he make ealy shows he esece of asaco coss joed o opeaos bewee he ades. he cosdeed dffcly coces esolvg he poposed models. I s heefoe ecommeded o o mecal mehods of esolo. Fally s mpoa o cosde applyg he GIRR model o ohe opos. We ca sdy he Ameca opos he eoc opos he echage opos o bae opos. Howeve he majoy of opos echaged o he makes ae Ameca opos.

11 Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8 Refeeces. Am. ad V. Ng (99) ARCH pocesses ad opo valao Mascp Uvesy of Mchga.. aksh G. C. Cao ad Z. Che (997) mpcal pefomace of aleave opo pcg models Joal of Face lack F. ad M. choles (97) he pcg of opos ad copoae lables Joal of olcal coomy olleslev. (99) Modellg he coheece sho omal echage aes: a mlvaae geealzed ARCH model Revew of coomcs ad ascs Chsoffese. Heso. ad Jacobs. () Opo valao wh codoal skewess FA Maasch Meegs ape No. 96. Avalable a RN: hp://ss.com/absac= Chsoffese ad Jacobs. () Whch GARCH models fo opo valao? Maageme cece Da J.C. (995) he GARCH opo pcg models Mahemacal Face Da J.C. (996) Cackg he smle Rsk 9 Decembe Da J.C. Gahe G. assevlle C. & moao J.G. (6) Appomag he JGRGARCH ad GARCH opo pcg models aalycally Joal of Compaoal Face 9 ().. Da J.C Rchke. ad Z. (6) Jmp sag GARCH: cg ad hedgg opos wh jmps es ad volales FR of Clevelad Wokg ape No. 69 Avalable a RN: hp://ss.com/absac=798.. Da J.C. ad moao J.G. (999) Ameca opo pcg de GARCH by a Makov cha appomao wokg pape Hog og Uvesy of cece ad echology.. gle R.F. () Dyamc codoal coelao: a smple class of mlvaae GARCH models Joal of sess ad coomc ascs 95.. gle R. ad oe F.. (995) Mlvaae smlaeos geealzed ARCH coomec heoy 5.. gle R. Lle D.M. ad Robs R.. (987) smag me vayg sk pema he em sce: he ARCHM model coomeca vol gle R. ad Msafa C. (99) Impled ARCH models fom opos pces Joal of coomecs gle R.F. ad heppad. () heoecal ad empcal popees of dyamc codoal coelao mlvaae GARCH Mmeo UCD. 7. Glose L.R. R. Jagaaha ad D.. Rkle (99) O he elao bewee he eced vale ad volaly of he omal ecess e o socks Joal of Face Hake M. (997) Neal ewok appomao of opo pcg fomlas fo aalycally acable opo pcg models Joal of Compaoal Iellgece Face Hadle W. ad Hafe C. () Dscee me opo pcg wh fleble volaly esmao Face ochascs Heso. (99) A closedfom solo fo opos wh sochasc volaly wh applcaos o bod ad cecy opos Revew of Facal des Heso. ad. Nad () A closedfom GARCH opo pcg model Revew of Facal des Hll J. ad Whe A. (987) he pcg of opos wh sochasc volales Joal of Face 8.. Jaow R.A. ad Rdd A. (98) Appomae opo valao fo abay sochasc pocesses Joal of Facal coomcs Joes C. () he dyamcs of sochasc volaly: vdece fom delyg ad opos makes Joal of coomecs ambh J. ad Mosse.C. () he effec of ees ae opos hedgg o em sce dyamcs FRNY coomc olcy Revew Decembe. 6. Lahoel N. (6) valao des opos avec pme de sqe vaable defves/caf6_lahoel.pdf 7. Lahoel N. (7) La cosco d e sce pa eme des a d éê à l ade d modèle makove à de égmes avec ees GARCH IFC Hammame sa. 8. Lo O. ad eo. (999) he shape of he sk pemm: vdece fom a sempaamec GARCH model Cowles Fodao fo Reseach coomcs Yale Uvesy. 9. Meo R.C. (97) he heoy of aoal opo pcg ell Joal of coomc ad Maageme cece 8.. Nad. (996) cg ad hedgg de opos de sochasc volaly: a empcal eamao Wokg pape Fedeal Reseve of Alaa.. eo. (999) empaamec weak sme egessos wh a applcao o he ske adeoff CRD wokg pape 99 Uvesy de Moéal.. Rchke. ad evo R. (999) cg opos de geealzed GARCH ad sochasc volaly pocesses Joal of Face 5 77.

12 Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8. chöbel R. ad Zh J. (999) ochasc volaly wh a OseUhlebeck pocess: A eeso opea Face Revew 6.. e. ad e J. (99) ock pce dsbos wh sochasc volales Revew of Facal des Y. Che M. ad Hag C. (7) A applcao of closedfom GARCH opo pcg model o F opo ad volaly Naoal awa Uvesy. Apped A. oof of he poposo I geeal he pce of zeocopo bod s calclaed by pdag s eced vale de a physcal pobably mease as follows:. If we p dow he s a omal dsbo wh mea ad vaace V ad afe ha:..5 d d d f g dow y we oba: 5.. Apped. oof of he poposo Fom he eqao / m we ca have: m. I follows ha: U m Log / /. Ude he physcal pobably mease he call pce s a magale: C ma. I follows ha: U d f m m C. Followg Jaow & Rdd (98) he e desy of pobably of ca be appomaed by:!! d d F d d F f whee F(.) s he cmlave fco of he adom vaable ad sce d d d d d d d d 6 ad we oba fally:

13 Ivesme Maageme ad Facal Iovaos Volme 5 Isse 8 f 6.!! I comes ha: C m d m U I!! U I m 6 d. d U I o calclae he hee ems I I ad I we ms se he popees coceg he sadad omal adom vaable. We oba: I U m d d m.5 d m.5.5 d U U m.5 d d. U U I I Makg he chage of vaable Y fo he fs em I ad log he sadad omal adom vaable we oba: I N ad I NU U I he followg:. m. N U NU I. 5 U U Usg esls of appedes A ad he same ype of calclao allows wg I ad I as: I m. U U NU m.5 U U U NU 5 I. d U d