ֻ13जֻ9௹ܵ ॓ ࿐ ࿐ Б 2010୍9ᄅ JOURNALOFMANAGEMENTSCIENCESINCH ০ੱٟഝଆ༯֥০ੱڄགࡎ۬ྙൔൌᆣ࣮ 郑振龙1,柯 鸿2,莫天瑜1(1.༰ն࿐ࣁವ༢,༰361005;2.ֻ၂Էြᆣಊ,ധᎪ518028)ᅋေ:在仿射利率期限结构动态模型(affineDTSM)框架下,利率风险价格主要有4种设定形式:完全仿射模型(CAM)a实质仿射模型(EAM)a扩展仿射模型(EXAM)和半仿射模型(SAM),经过前人理论和实证的证明,EAM优于CAM,EXAM和SAM均优于EAM.然而,EXAM和SAM的孰优孰劣无法单从理论上的比较得出结论,同时亦鲜有相关的实证研究对其进行比较.因此,文中运用卡尔曼滤波估计法,在三因子CIR模型的基础上对SAMaEXAM和EAM进行实证比较,实证结果表明EXAM要优于SAM.此外,稳健性检验表明,EXAM虽然已为目前最优的利率风险价格形式,但其仍然不够完善.ܱՍ:利率仿射模型;利率风险价格形式;扩展仿射模型;半仿射模型ᇏٳোݼ: ໓ངѓ്:A ໓ᅣщݼ:1007-9807(2010)09-0004-120 ႄ ቆ(RiccatiODEs).෮ၛ,ཌྷбࢠఃDTSMط,ᄝٟഝଆॿࡏ༯ؓ০ੱ௹ཋࢲܒ֥ൌᆣᆌؓ০ੱ௹ཋࢲܒଆ(dynamicsterm࣮ࣼэ֤ٳၞႿԩ,طᆃ္൞ٟഝଆંstructuremodels,DTSM),࿐ඌࢸބြࢸၘࣜࣉྛᄝ࿐ඌࢸߎ൞ြࢸٳ൳ߋ႒֥ᇶေჰၹ.ਔྸ؟࣮aѩؿᅚԛ၂༢ਙଆ,২ೂ,ٟഝଆٟؓഝଆࣉྛൌᆣቋࠎЧ֥ٚم,ь൞০(affineDTSM)aؽՑۚථଆ(quadratic gaussi Ⴈ൬ၭੱ౷ཌ֥૫ϰඔऌটކҕඔ.ႮႿ૫ϰan)a٤ٟഝෛࠏѯੱଆ(nonaffine stochasticඔऌൈЇওਔ০ੱޘࢩ૫ބൈࡗਙ֥ྐ༏,volatility)ၛࠣЇও๋ᄁࠇࠏᇅሇߐ֥ଆၹՎ০Ⴈ૫ϰඔऌॖၛൈ֤֞ሑэਈᄝڄག(jumpsaregimeswitching)֩.֩ᄝᇙ؟ଆᇏ,Ч໓෮ܱᇿ֥൞ଢభႋႨቋູܼ֥ٟٗഝଆᇏྟקࡎҩ؇Qބགྷൌҩ؇P֥ҕඔ,ࠧ,ൈ֤ॿࡏ֞ሑэਈXᄝڄགᇏྟҩ؇Qބགྷൌҩ؇P֥.ٟഝଆؓݖӱ.DTSM3۱ቆӮ҆ٳᇏ֥ਆ۱ࣉྛਔཋקಖط,ྸ؟ൌᆣіૼ,ᄝԮ֥০ੱڄགࡎ۬:ྙൔ(২ೂࡼ০ੱڄགࡎ۬ഡקູටൈѯੱބ1)ටൈ০ੱrФഡקູሑэਈX֥ཌྟݦඔ;2)ڄགᇏྟҩ؇Q༯,ሑэਈX֥ටൈ၂۱ӈඔ֥Ӱࠒ)֥ഡק༯,ٟഝଆمൈ၍ੱაටൈٚҵФഡקູX֥ཌྟݦඔሙಒֹ૭ඍሑэਈXᄝགྷൌҩ؇Pބڄགᇏྟ.ᄝᆃਆ۱ࡌק༯,ᅏಊࡎ۬ॖၛٚьֹіൕҩ؇Q֥ݖӱ,ऎุіགྷ:ູᄝࢠݺֹކఃູPޘࢩ૫০ੱ௹ཋࢲܒ֥ൈ,ީಏمൈؓໃটt(t)=exp[A(t)-B(t) Xt],طA( )aB( )ᄵڛՖႨඔᆴٚمٳၞࢳ֥वิӈັٳٚӱ൬ၭੱ֥эࣉྛࢠݺֹყҩ;ࠇمࢠݺֹ ൬۠ರ௹:2009-12-10;ྩרರ௹:2010-05-28.ࠎࣁཛଢ:ݓࡅሱಖ॓࿐ࠎࣁሧᇹཛଢ(70971114);࢝ტ҆ݓ࠽ࣁವາࠏႋ࣮ؓႋࠤሧᇹཛଢ(2009JYJR051);ڞࡹസሱಖ॓࿐ࠎࣁሧᇹཛଢ(2009J01316).ቔᆀࡥࢺ:ᆢᆒ(1966!),ଳ,ڞࡹฉದ,Ѱൖ,࢝൱,Ѱൖള֝ഽ.Emai:lzlzheng@
ֻ9௹ᆢᆒ֩:০ੱٟഝଆ༯֥০ੱڄགࡎ۬ྙൔൌᆣ࣮!5!ކ௹ཋၮࡎ֥ൈэྟ֩.֩֝ᇁٟഝଆԛགྷᆃݖӱູ၂໙ี֥ჰၹႵޓ؟,২ೂ,০ੱڄགࡎ۬ഡקྙdXt=( Q-KQXt)dt+!StdWQt(2)ൔ҂ܔਲࠃ,ටൈ০ੱrࠇྸ൞ሑэਈX֥٤St(i,i)=∀i+# iXi,St(i,j)=0,ཌྟݦඔ,ીႵॉ੮๋ᄁބࠏᇅሇߐᆃུၹ,֩ i#j,1∃i,j∃n(3).֩ၹՎ,ေڿࣉٟഝଆॖၛՖ؟ٚ૫ೆ൭,طఃᇏ,WQt൞ڄགᇏྟҩ؇Q༯ѓሙ֥҃ᄎ,Ч໓෮ܱᇿ֥൞০ੱڄགࡎ֥۬ഡקྙൔ໙.ี Q,∀,#i൞n∀1ཟਈ,KQ,!൞n∀n֥इᆔ,St০ੱ֥ڄགࡎ۬,൞৵ࢤڄགᇏྟҩ؇Qބ൞n∀n֥ؓ࢘इᆔ.གྷൌҩ؇Pቋᇗေ֥൷୧.ሑэਈXᄝᆇൌൗD[4]uffeބKanᆣૼਔᄝൔ(1)a(2)ބ(3)ࢸ֥ݖӱ,აڄགᇏྟൗࢸ൞҂၂ဢ֥.২֥ࡌק༯,ഺჅ௹ཋູ a֞௹ᆦڱ1֥ਬ༏ௐᅏೂ,ᄝڄགᇏྟൗࢸᇏ,০ੱݖӱ֥၍ੱ൞ಊ,ᄝtൈख़֥ࡎ۬Pt( )ࡼڛՖೂ༯ྙൔཌྟ,֥൞ᄝᆇൌൗࢸಏॖି൞٤ཌྟ֥.ᄜt+ ೂPt( )=EQtexp(-(%rsdt),ਆ۱ൗࢸᇏ֥नᆴ߭گ؇ࠇӉ௹नᆴඣt)ॖି൞҂၂ဢ.֥ڄགࡎ۬ྙൔഡק҂ᆞಒ,ࡼᆰ=expA( )-B( ) Xt(4)ࢤ֝ᇁհ༂ܙ࠹ਆ۱ҩ؇༯০ੱݖӱ,Ֆط()ᇁհ༂ิ౼ᆇൌྐ༏.ၹՎ,ູਔ۷ሙಒֹ૭ඍఃᇏ,A( )൞1∀1֥ѓਈ,B( )൞n∀1֥ཟਈ,০ੱᄝڄགᇏྟҩ؇ބགྷൌҩ؇ᇏ֥ݖӱૌ൞ഺჅ௹ཋ ֥ݦඔ,ѩڛՖೂ༯֥ӈັ,ݓຓ࿐ᆀૌၘࣜᄀটᄀᇗ൪ൌᆣᇏ෮࿊ᄴ֥০ੱٳٚӱቆڄགࡎ۬ഡקྙൔ,ѩؿᅚԛਔ۷ࡆਲࠃa۷ࡆdA( )Q=-∃ B( )+ކ֥ྍ၂ս০ੱڄགࡎ۬ഡקྙൔd .n֞ଢభູᆸ1,ંࢸᇶေؿᅚਔ4ᇕ০ੱڄ 2&[! B( 2)]i∀i- 0གࡎ۬ഡקྙൔ:ປಆٟഝଆi=1(5)(CompletelydB( )A=-KQ B( )-ffineModel)aൌᇉٟഝଆ(EssentiallyAffined Model)aঔᅚٟഝଆ(ExtendedAffineModel)nބ϶ٟഝଆ1[! B( 2)]2&i#i- x(Semi AffineModel),ၘࣜᆣૼ,ঔi=1ᅚٟഝଆބ϶ٟഝଆ൞ቋႪ֥ਆᇕ০ੱڄགႮႿ֞௹ᆦڱ1֥ਬ༏ௐᅏಊ,ᄝః֞௹ൈࡎ۬ྙൔ(D[1][2]uffee,Duarte,Cheridito֩[3]).֥ࡎ۬Pt(0)сྶ֩Ⴟ1,ڎᄵ߶ԛགྷڄགส০ಖط,ᄝൌᆣൈಯಖ߶მ֞၂۱໙ี:EXAMބࠏ߶,෮ၛᆃ၂ӈັٳٚӱቆߎсྶડቀшࢸ่SAMଧ۱۷ݺ?ଢభູᆸഉໃؿགྷݓଽຓႵ໓ᅣࡱ:A(0)=B(0)=0.ؓᆃਆᇕ০ੱڄགࡎ۬ྙൔࣉྛᆞൔ֥бࢠླေᇿၩ֥൞,DuffeބK[4]anᄝ֝ൔ࣮,طᆃᆞ൞՜ӮЧ໓֥ᇗေჰၹᆭ၂.(4)a(5)ൈ,ѩીႵؓሑэਈXᄝགྷൌҩ؇PЧ໓ᄝၹሰCIRଆ֥ࠎԤഈ,০Ⴈव༯֥ݖӱቔޅ֥ཋק.ၹՎ,ંഈ҂ંࡌغણੲѯمؓEAMaEXAMބSAMࣉྛൌᆣбࢠקሑэਈXᄝགྷൌҩ؇༯֥ݖӱູޅᇕྙ࣮.ൔ,ᆺေડቀਔൔ(1)a(2)a(3)֥ࡌק,ൔ(4)a(5)ିܔӮ৫.ط০ੱڄགࡎ۬൞৵ࢲགྷൌҩ؇ބڄགᇏྟҩ؇ຸ၂֥൷୧,ᆺေކഡקਔ1 ٟഝଆॿࡏ༯֥ᅏಊקࡎބ০০ੱڄགࡎ֥۬ྙൔ,ࣼିܔ֝ԛሑэਈXੱڄགࡎ۬ؿᅚઝᄝགྷൌҩ؇༯֥ݖӱ,ࣉط֤֞۲ࣜ࠶эਈೂӻႵ௹൬ၭੱa৵࿃گ০൬ၭੱ֩ᄝགྷൌҩ؇ ٟഝଆॿࡏ༯,ටൈ০ੱrtФഡקູ༯֥ݖӱ,Ֆطؓᆃུࣜ࠶эਈࣉྛܴҳrt= 0+ xXt(1)ყҩ.ఃᇏ, 0൞1∀1֥ѓਈ; x,Xt൞n∀1֥ཟਈ.ෙಖ০ੱڄགࡎ֥۬ഡקྙൔ҂႕ཙൔطࡌקሑэਈXᄝڄགᇏྟҩ؇Q֥(4)a(5)֥,֝০ੱڄགࡎ֥۬ഡקಯླေ
!6!ܵ ॓ ࿐ ࿐ Б2010୍9ᄅડቀ၂༢ਙ֥ࡌഡ่ࡱ,২ೂླડቀส০ࡌഡaิԛEAM֥ଢ,֥ᇶေ൞ູਔᄝѯੱᆭླ࣐ਈሙಒֹख़߂ሧᆀؓ০ੱ֥ڄག؇aေຓ,ႄೆఃၹ(ೂོੱ)ট႕ཙڄགၮԏ֥ିൈख़߂০ੱ၂ࢨइބؽࢨइ֥э߄ݖӱэ߄,Ֆطख़߂ڄགၮԏनᆴཬaٚҵն֥֩֩.ଢభંࢸᇶေؿᅚਔၛ༯4ᇕ০ੱڄགห.ྟࡎ۬ഡקྙൔ.ಖط,EAMಯಖթᄝૼཁ֥҂ቀᆭԩ:ູਔ ປಆٟഝଆ(CompletelyAffineMode,l۷ݺֹख़߂ڄགၮԏ֥หᆘ,сྶᄝ၂קӱ؇ഈCAM)٢ఙ০ੱٚҵ֥ൈэྟหᆘ,ࠧଆᇏᇀഒေႵ[5]FisherބGillesิԛਔೂ༯֥ڄགࡎ۬1۱ၹሰڛՖোරVasicekProcess.ᆃ࣮ᆀᄝྙൔ࿊ᄴଆൈ߶૫ਢָ֞ေख़߂ڄགၮԏߎ൞০ੱ%ٚҵൈэྟ֥ᒹᒸֹ.t=St&1(6)ఃᇏ,&1іൕn∀1֥ཟਈD[1]uffeeᄝิԛEAM,ުႨQMLܙ࠹,ᄝ.CAMิԛ,ުФ࿐ᆀૌܼٗႨ.২ೂ,ChD[7]aiބSingleton֥ׅٟࣜഝଆࠎԤഈؓEAMenބ[6]SބCAMࣉྛਔൌᆣбࢠ.ࢲݔіૼ,EAMି۷ݺcottaDaiބ[7]Singletona[8]JongaLamoureuxބW[9]itte,֩ၛࠣఘࣂູᆸݓଽ֥ն҆ٳ࿐ᆀֹᄝགྷൌҩ؇ᇏყҩໃট൬ၭੱ౷ཌ֥э߄.ൈD[1]ႨCAM֥ڄགࡎ۬ഡקࣉྛ۲োൌᆣ࣮uffee္ᆷԛ,ၹሰ.ಖطEssentially GaussianModelؓໃট൬ၭੱ၂ࢨइ,CAMႵሢࠞն֥ಌཊ,২ೂ,م૭ඍ০ੱڄགၮԏ֥ൈэྟaྸ؟ҕඔႮགྷൌҩ؇ބ֥ყҩིݔቋݺ,ؓໃট০ੱѯੱ֥ყҩིڄགᇏྟҩ؇֥ݖӱ܋Ⴈݔಏ҂ೂၹሰCIRଆ..֩[11]෮ၛᄝႨCAMࣉྛൌᆣ֥ݖӱᇏ,ࣜӈ߶ԛDaiބSingletonࢹ[1]Duffee֥ڄགࡎགྷمൈሙಒކ০ੱ֥ޘࢩ૫ྟᇉބൈࡗ۬ഡקٚൔࣉྛ࣮,္ؿགྷਔဢ֥໙ี:ਙྟᇉ֥໙ี,ေહڄགၮԏކ҂ݺطمࣉEAMᄝCIRॿࡏ༯مൈሙಒყҩໃট০ੱྛࢠݺֹყҩ,ေહ།വਔఃҕඔ(২ೂ,ކ֥၂ࢨइބؽࢨइ.௹ཋࢲܒྙሑބקࡎ֥ҕඔ)֥ކིݔটڿ ঔᅚٟഝଆ(ExtendedAffineMode,lڄགၮԏ֥ކིݔEXAM).ູਔڿࣉEAM֥ಌཊ,Cheridito֩[3]ิԛD[7]aiބSingleton০ႨCAMؓఃิԛ֥ׅٟࣜഝଆਔEXAM֥ڄགࡎ۬ྙൔ(CanonicalAffineModel)ࣉྛਔൌᆣ࣮,ఃࢲݔіૼCAMޓݺֹख़߂ਔӻႵ௹൬&1+&2Xt%t=(9)ၭੱڄགၮԏ֮नᆴaۚٚҵ֥ห,ྟಖطޘࢩ૫St০ੱ௹ཋࢲܒྙሑ֥ކ༂ҵಏޓնఃᇏ,&1൞n∀1ཟਈ,&2൞n∀nइᆔ;ൈ,ሑ.эਈᄝڄགᇏྟҩ؇ބགྷൌҩ؇༯֥ݖӱनAhn֩[10]ؿགྷ,CAMمሙಒख़߂০ੱ่֥ࡱѯੱ֥э߄ડቀFeller่ࡱၛ֤StޚնႿ0..Ֆંഈඪ,EXAMбEAM۷ऎ၂Ϯ,ྟၹ ൌᇉٟഝଆ(EssentiallyAffineMode,lEAMູᆺေؓҕඔࣉྛFeller่ࡱཋᇅ,Ќᆣٚҵޚ)D[1]նႿ0,ᄵંࡌഡሑэਈ֥ݖӱູଧᇕuffeeᄝCAM֥ࠎԤᆭഈࣉྛڿࣉ,ิԛਔEAM֥ڄགࡎ۬ྙൔྙൔ(Їওनٚ۴ݖӱ),ሑэਈᆭࡗିཌྷ႕ཙ֥֞ؓٚڄགၮԏэ߄.ѩ,ᄝEXAM%t=St&-1+St&2Xt(7)ᇏ,ਆ۱ҩ؇༯֥၍ཛᇏ֥෮Ⴕҕඔ(ЇওӉఃᇏ-,&1іൕn∀1ཟਈ,&2іൕn∀nइᆔ,Stі௹नᆴa߭گ؇aሑэਈᆭࡗ֥ཌྷܱܱ༢)ൕn∀n֥ؓ࢘इᆔ,ఃؓ࢘ჭູॖၛႚႵ҂֥ᆴ.ᆃ֤ࣼᄝൌᆣ࣮ൈ,၂1(∀i+# iX-t),۱ҕඔ҂Ⴈൈ૭ඍਆ۱ҩ؇֥ݖӱ,Ֆط-Si(i,i)= (∀i+# (8)iXt)>0ᄝંഈॖၛሙಒֹൈख़߂ਆ۱ҩ؇༯֥0,ڎᄵݖӱ.
ֻ9௹ᆢᆒ֩:০ੱٟഝଆ༯֥০ੱڄགࡎ۬ྙൔൌᆣ࣮!7!Cheridito֩[3]ᄝิԛEXAM֥קၬ,ުᄝ૫ϰඔऌ,๙ݖवغણੲѯܙ࠹ԛEAMaEXAMDaiބ[7]Singleton֥canonicalٟഝଆࠎԤഈ,ބSAMᄝၹሰCIRଆࠎԤഈ֥۲۱ҕඔ,ѩႨMLEܙ࠹бࢠਔEXAMaEAMބCAM3ᇕڄགбࢠ3ᇕڄགࡎ۬ؓ௹ཋࢲܒޘࢩ૫ྟᇉބൈࡗࡎ֥۬Ⴊਜ,ࢲݔؿགྷ,EXAMಒൌିܔᄝ҂႕ཙਙྟᇉ֥ކၛࠣყҩି৯,ၛՎቔູڄགࡎ০ੱޘࢩ૫ྟᇉކ֥౦ঃ༯ڿࣉ০ੱൈࡗਙ۬Ⴊਜ֥ѓሙ.ྟᇉ֥ކ. ϶ٟഝଆ(Semi AffineMode,lSAM)ဢ,ູਔڿࣉEAMᄝCIRଆᇏ֥ႋႨ,D[2]uarteิԛਔSAM-1%t=!&0+St&-1+St&2Xt(10)ఃᇏ,&0ູn∀1ཟਈ,ఃژݼაEAMཌྷ.SAMؓEXAM֥ڿࣉุགྷູၛ༯ࠫׄ.൮༵,ᄝᆃᇕഡק༯,ڄགၮԏaགྷൌҩ؇༯֥၍ཛ൞ሑэਈ֥٤ཌྟݦඔ.ᆃ൞აᆭభ֥3ᇕڄགࡎ۬ഡקྙൔ෮҂.֥ఃՑ,ູਔख़߂ڄགၮԏ֥ཬनᆴaնٚҵห,ྟླေڄགၮԏᇏ֥۲ҕඔିܔڿэژݼ.EXAM֥ٚم൞ᄝఃᇏ၂۱ሑэਈ֥ڄགၮԏᇏႄೆਔਸ਼ຓ၂۱ሑэਈ֥႕ཙ,ၛڄགၮԏᇏ֥۲۱ҕඔ֥ژݼॖၛڿэ;طSAMᄵᆰࢤᄝڄགၮԏᇏႄೆሑэਈሱദ֥नٚ۴,Ֆط1 ০ੱڄགࡎ۬ؿᅚઝ֤۲ҕඔژݼିܔڿэ. ၹሰଆ֥࿊ᄴD[2]uarteᄝิԛSAM,ުႨMLEؓSAMaᄝAffineDTSMॿࡏ༯,Ч໓࿊ᄴၹሰCIREAMࣉྛਔܙ࠹ބбࢠ,ؿགྷᄝն؟ඔଆᇏ,ଆቔູбࢠڄགࡎ۬Ⴊਜ֥ࠎԤ.Ⴍః൞؟ၹሰCIRଆ,SAMିᄝ၂קӱ؇ഈڿᄝၹሰCIRଆ༯,ටൈ০ੱაሑэਈࣉEAMؓ൬ၭੱ౷ཌ֥ყҩି৯.֥ݦඔܱ༢ູ3rt=&X2t,i Ч໓֥ൌᆣ࣮ٚمi=1طሑэਈᄝڄགᇏྟҩ؇༯֥ݖӱॖၛ2ཿູ.1 Ч໓֥ൌᆣၹ1۳ԛਔ4ᇕڄགࡎ֥۬ؿᅚݖӱ.۴ऌ∋QQ1k100D[1][2]uffeeaDuarteၛࠣCheridito֩[3]֥ൌᆣॖdXt=∋Q2-Q0k20Xtdt+ᆩ,EXAMބSAM൞ଢభູᆸቋႪ֥ਆᇕ০ੱ∋QQ300k3ڄགࡎ۬ഡקྙൔ,ಖطഉم֤ᆩEXAMބ(1Xt,100SAMᆭࡗଧ၂ᇕ۷Ⴊ,طᆃᆞ൞՜ӮЧ໓֥ᇶေჰၹ 0(2XdWQtt,20.Ч໓ၛ༯ࡼؓEXAMބSAM֥Ⴊਜࣉྛൌ00(3Xt,3ᆣбࢠ,ၛҀݓଽຓᄝᆃ၂ॶ֥ॢϢ,ູರު֥ݖӱᄝڄགᇏྟҩ؇༯թᄝຸ၂ࢳ෮ࣉ၂࣮҄ิ܂ҕॉ.ླေ֥ҕඔཋᇅ൞Ч໓࣮֥ᇶေٚم൞০Ⴈ০ੱ௹ཋࢲܒ֥∋QQi∋0,ki∋0,i=1,2,3
!8!ܵ ॓ ࿐ ࿐ Б2010୍9ᄅՎൈᅏಊࡎ۬ބ৵࿃گ০൬ၭੱॖၛіൕູ(-), )2(1-ei) Q(-), )iP()i+ki)(1-e)+2)(-), )iiet( )=exp(A( )-B( ) Xt)R ))2i=ki+2(2it( A()=-+B( ) X ڄགࡎ۬ഡקྙൔ֥࿊ᄴఃᇏЧ໓࿊ᄴEAMaEXAMބSAM3ᇕڄགࡎ۬A( )=ྙൔࣉྛбࢠ࣮.ᇶေଢ֥൞бࢠEXAMބSAM32∋Q&i1Q֥Ⴊਜ,طEAM֥࿊ᄴᄵ൞ູਔิ܂၂۱ࠎሙ.{[ln(2)i)+(ki-)i) -i=1(22iູьႿбࢠ,ࡼ3ᇕڄགࡎ۬ഡקྙൔ༯֥Q(-), )(-), )ii ln(()i+ki)(1-e)+2)ie)]}ڄགၮԏཿ֞1۱ൔሰᇏ,ѩؓҕඔቓ၂ུ֩ࡎB( )i=֥эߐ&0(1)Xt,1+&1(1)+&2(1,1)Xt,1+&2(1,2)Xt,2+&2(1,3)Xt,3 et,x=&0(2)Xt,2+&1(2)+&2(2,1)Xt,1+&2(2,2)Xt,2+&2(2,3)Xt,3&0(3)Xt,3+&1(3)+&2(3,1)Xt,1+&2(3,2)Xt,2+&2(3,3)Xt,3ఃᇏ,ીႵ༯߃ཌ֥іൕEAMaEXAMބSAMູЌᆣሑэਈᄝགྷൌҩ؇༯֥ݖӱႵ܋Ⴈ֥ҕඔ,Ⴕ၂่༯߃ཌ֥սіSAMህႵຸ֥၂ࢳѩޚնႿ0,Ԣീࡆ∋QQi∋0,ki∋0,i=1,ҕඔ,Ⴕਆ่༯߃ཌ֥սіEXAMህႵ֥2,3֥ҕඔཋᇅᆭຓ,ླေٳљؓEAMaEXAMބҕඔ.SAM֥ҕඔቔၛ༯ཋᇅі1 ҕඔཋᇅTable1LimitsoftheparameterestimationsEAMkQ-&i2(i,i)∋0,i=1,2,3;∋Q∋1(2,i=1,2,3i2i&2(i,j)∋0,1∃i,j∃3,i#j;∋Q+&i1(i)∋1(2,i=1,2,3;2i∋Q+&(1)EXAM11P-1(K)∋Q2+&∋0;∋Q∋1(2,i=1,1(2)2,3i2i∋Q3+&1(3)SAMkQ-&i2(i,i)∋0,i=1,2,3;∋Q∋1(2,i=1,2, ඔऌ૭ඍ-01-04ᇀ2008-05-05܋163۱ྒ௹,2771۱ඔऌটჷ:Windඔऌ९,ၿྛࡗݓᅏ൬ၭੱඔऌႨႿဢЧ௹ଽކ,2008-05-06ᇀ2008-౷ཌ.12-01ರ܋29۱ྒ௹֥ඔऌႨႿဢЧ௹ຓཌྷбࢠྐႨᅏط,ݓᅏ൬ၭੱսіڄགҩ.০ੱa҂൳ఒြྐႨඣ֥э߄ۄಠ,ି۷ݺֹս ܙ࠹ٚم֥࿊ᄴ:KalmanFilterі০ੱ֥э߄,ൈႮႿݓᅏӮࢌਈնaੀD[14]uffeeބStantonބ[8]Jong๙ݖбࢠ۲ᇕྟࢠݺ,ၹՎ࿊ᄴݓᅏ൬ၭੱ౷ཌඔऌࣉྛކ,ܙ࠹ٚم֥Ⴕིྟުؿགྷ,ႨKalmanFilter൞ି۷ݺֹࡨഒႮႿඔऌ҂৵࿃֩ၹ෮ᄯӮ֥࣮ٟഝଆቋႵིa༂ҵቋཬ֥ܙ࠹ٚم.ၹ༂ҵ.Վ,Ч໓࿊ᄴKalmanFilterمࣉྛܙ࠹,ऎุܙ࠹ൈࡗॴ؇:Ֆ2005-01-04ᇀ2008-12-҄ᇧབྷڸ.01;ൈࡗࡗۯ:1ྒ௹,ሹ܋192۱ྒ௹.ఃᇏ,2005Ч໓ҐႨቋཬ߄җҵٚބ֥ٚم࿊ᄴሑ
ֻ9௹ᆢᆒ֩:০ੱٟഝଆ༯֥০ੱڄགࡎ۬ྙൔൌᆣ࣮!9!эਈ֥Ԛᆴ2m)B( )in3 ൌᆣࢲݔX&RA( 0( )--0 (( +( X0)))s..tX0∋ ҕඔܙ࠹ࢲݔR0( )іൕᄝ0ൈख़a֞௹ರູ ֥৵࿃گ০൬ၭ,ੱֆູ໊Ϥٳҕඔܙ࠹ࢲݔі2..ׄі2 ҕඔܙ࠹ࢲݔTable2TheresultsofparameterestimationsҕඔщݼҕඔEAMSAMEXAM1ؓඔරಖᆴ(2∗()()()∋Q1()()()∋Q2()()()∋Q3()()()()()()()()()()(1()()()(2()()()(3()()()!!12&0(1)!()!!!13&0(2)!()!!!14&0(3)!()!!!&1(1)!!()!!&1(2)!!()
!10!ܵ ॓ ࿐ ࿐ Б2010୍9ᄅ࿃і2Table2ContinueҕඔщݼҕඔEAMSAMEXAM!!&1(3)!!()&2(1,1)()()()!!&2(2,1)!!()!!&2(3,1)!!()!!&2(1,2)!!()&2(2,2)()()()!!&2(3,2)!!()!!&(1,3)2!!()!!&2(2,3)!!()&2(3,3)()()() ᇿ:ওݼ()൞ҕඔ֥ѓሙ༂.Վຓ,ູิۚކࣚ؇,Ч໓༵ࡼ൬ၭੱඔऌ٢ն100Пުᄜࣉྛҕඔܙ࠹,ܣі1ҕඔ෮ൡႨ֥൬ၭੱ࠹ෘ܄ൔູRA( )B( )( )=-+X/( ( )t) ۴ऌі2۳ԛ֥ܙ࠹ࢲݔ,ॖၛ൮༵֤֞ೂ֥႕ཙ.༯֥၂ུࢲં.൮༵Ֆ2ॖၛुԛ,EAMބSAMؓ؋௹০൮༵,Ֆֻ2ྛ֥ࠞնරಖᆴ,ुሹุط,ੱ֥ყҩொ༂ࢠն,ቋնൈٳљղ֞۱ϤٳEXAMؓEAM֥ڿࣉིݔေૼཁႪႿSAM.ׄބ۱Ϥٳ.ׄཌྷбࢠط,EXAM֥ყҩఃՑ,ֻ3ྛ֥(2∗սіवغણੲѯҩਈٚӱொ༂֥໗קྟࢠ఼,ؓ҂֞௹ರ֥০ੱ֥ყҩ֥༂ҵٚҵ,ॖၛᄝ၂קӱ؇ഈսіޘࢩ૫൬༂ҵनᆴ၂Ϯ҂ӑݖ۱Ϥٳׄ.ၭੱ౷ཌྙሑ֥ކིݔ.طEXAM֥(2∗ᄝ3ᇕ ဢЧ௹ଽ၂ࢨइყҩ༂ҵଆᇏ൞ቋཬ,֥ᆃඪૼEXAMؓଆކႪ؇Ֆ3ॖၛुԛ,EXAM֥ဢЧଽ၂ࢨइ֥ิ,ۚ҆ٳჰၹ൞ႮႿఃႪ߄ਔ০ੱ௹ཋࢲܒRMSEေૼཁཬႿSAMބEAM֥.طSAM֥ؓޘࢩ૫ކིݔ.EAM֥ڿࣉᄵ҂ն.Վຓ,ં൞SAMߎ൞EXAM,ૌбEAM๙ݖ2ބ3֥ٳ༅,ॖၛ֤֥֞ࢲં൞:෮؟ԛ֥ҕඔ֥ѓሙ༂(1ޓཬaPᆴཁᇷֹEXAMؓဢЧଽ০ੱ௹ཋࢲܒ၂ࢨइ֥ކӱၳႿ0.ᆃඪૼሑэਈ֥ڄགၮԏ,Ԣਔ൳ሑ؇,ေႪႿSAMބEAM,ࠧ,EXAM۷ሙಒֹख़߂эਈሱദ֥ѯੱ႕ཙຓ,ಒൌߎ൳֞ఃၹਔ০ੱ֥ڄགၮԏ֥э߄ݖӱ.(SAMбEAM؟ԛ֥ҕඔႵҕඔ12a13a14,EXAMбEAM؟ԛ֥ҕඔႵҕඔ15a16a17a19a20a21a23a24a25.
ֻ9௹ᆢᆒ֩:০ੱٟഝଆ༯֥০ੱڄགࡎ۬ྙൔൌᆣ࣮!11! ဢЧ௹ຓ၂ࢨइყҩ༂ҵ.֮Վຓ,3ᇕଆؓ1ᇀ4୍ࡗ֥০ੱބӉ؊൮,༵๙ݖ4ބ5ؿགྷ,EXAMؓဢЧ௹০ੱ֥ყҩིݔࢠݺ.ຓყҩིݔ֥ڿࣉѩીႵପહૼཁ,ఃყҩི๙ݖၛഈٳ༅,ॖၛ֤֥֞ࢲં൞:ᄝဢЧ௹ݔაEAMࠎЧোර.طSAMؓဢЧ௹ຓ֥ყҩຓ,EXAMބSAMؓEAM֥၂ࢨइყҩིݔڿࣉ༂ҵ֥ѯбࢠն,ႵൈႪႿEXAMބEAM,Ⴕིݔѩ҂ૼཁ,ሹุط,EXAM֥၂ࢨइყҩൈಏбૌᄦ,Ֆ5ᇏॖၛޓૼཁुԛིݔࢠູ໗ק.Ֆ6a7ॖၛुԛ,3ᇕଆᆃ၂ׄ.ౠཟႿۚܙ০ੱ֥ѯ.ੱطEXAMყҩؽࢨइఃՑ,3ᇕଆؓ؋؊০ੱ֥ყҩ҂൞ޓི֥ݔ,ཁಖေݺႿఃਆᇕଆ,ᆃ၂ׄՖݺ,൞ཌྷбࢠط,ಯಖ൞EXAM֥RMSEࢠ8a9֥RMSE္ॖၛޓૼཁֹुԛ.
!12!ܵ ॓ ࿐ ࿐ Б2010୍9ᄅ ሹ ࢲ݂߭ၛഈбࢠਔ3ᇕڄགࡎ۬ഡק֥ൌᆣࢲݔ,Rt++t( )-Et(Rt++t( ))=∀ +#1, levelt+ᆃ,ࡼ۲۱бࢠࢲݔቓ၂۱ཬࢲ.ູਔٚьᄇ #2, slopet+#3, convexityt+∗t++t,ࡼ۲ᇕбࢠࢲݔ݂ବೆі(0 5)+Rt(8)+Rt(15)Ֆіlevelt=3֥бࢠॖၛؿགྷ,EXAMᄝڿࣉ০ੱ3གྷൌҩ؇၂ࢨइaؽࢨइݖӱކ֥ൈ,ߎslopet=Rt(15)-Rt(0 5)ڿࣉਔ০ੱڄགᇏྟҩ؇ݖӱ֥ކ.ၹՎ,convexityt=Rt(0 5)+Rt(15)-2Rt(8)EXAM֥ڄགࡎ۬ഡקྙൔ,ᄝޓնӱ؇ഈေႪ(11)ႿSAM.ٚӱቐшіൕଆႅݣ֥ဢЧଽყҩ༂ҵ.і3 ൌᆣࢲݔбࢠೂݔEXAM֥ڄགࡎ۬ഡק,ॉ੮ਔ෮Ⴕ֥Table3ComparisonoftheempiricalresultsܱႿ൬ၭੱ౷ཌྙሑ֥ྐ༏ؓڄགၮԏэ߄֥႕ࠞնරಖᆴEXAMႪႿཙ,ପહഈඍٚӱᇏ֥༢ඔႋھ൞҂ཁᇷ֥,PᆴSAMႪႿEAMႋھࢠն.ၹՎ,Ч໓࿊ᄴਔഺჅ௹ཋ୍ູa5൬ၭੱ౷ཌྙሑ֥EXAMႪႿ୍ބ10୍֥൬ၭੱູսі,ٳљؓഈඍٚӱࣉྛޘࢩ૫ކ༂ҵٚҵSAMႪႿEAM݂߭,ఃᇏ+tᆷקູ1۱ྒ௹.ဢЧଽ၂ࢨइყҩEXAMႪႿі4۳ԛਔ݂֥߭ࢲݔ.༂ҵSAM࣍රႿEAMՖі4ؿགྷ:൮,༵ં൞ଧ၂ᇕڄགࡎ۬ഡဢЧຓ၂ࢨइყҩ3ᇕଆקྙൔ,ః݂߭ٚӱ֥ն҆ٳ༢ඔཁᇷ҂ູ༂ҵૼཁљ0),ᆃඪૼყҩ༂ҵಯಖა൬ၭੱ౷ཌྙሑ֥3ဢЧଽຓؽࢨइEXAMႪႿ۱ၹሰᆭࡗթᄝሢޓཁᇷ֥ܱ༢.ၹՎ,ෙಖؿགྷყҩ༂ҵEAMႪႿෙಖбSAMބEAM֥၂ࢨइყҩིݔ۷ ࣉ၂֥҄ဒݺ,൞ᄝ3ၹሰCIRଆॿࡏ༯,ఃಯಖمປᄝु֞EXAM֥ڿིݔ֥ൈ,ߎླေॉಆଵও൬ၭੱ౷ཌྙሑ֥෮Ⴕྐ༏.੮ਸ਼၂۱໙ี:ڄགࡎ۬ഡק൞ڎጫਔڄགၮఃՑ,ෙಖі3֥ࢲݔ҂ದડၩ,ಯԏэ߄֥෮Ⴕଽಸ?ః൞ڎປಆࢳथਔ؟ၹሰಖՖ၂۱ҧ૫ඪૼਔEXAMಒൌႪႿSAMބCIRଆ֥໙ี?ູਔҩEXAM൞ڎॉ੮ਔ০EAM.ၹູՖі4ᇏ֥טᆜR2,ुՖEAM֞ੱ௹ཋࢲܒ֥෮Ⴕྐ༏ၛࠣ൞ڎࢳथਔ؟ၹሰSAMᄜ֞EXAM֥݂߭ކႪ؇թᄝ־ࡨCIRଆ֥໙ี,ؓၛ༯ٚӱࣉྛဢЧ௹ଽ֥֥൝.)Duarte[2]ҐႨোර֥ϷمбࢠਔEAMބsemi affinemode,lဢؿགྷ༢ඔཁᇷ҂ູ0.
ֻ9௹ᆢᆒ֩:০ੱٟഝଆ༯֥০ੱڄགࡎ۬ྙൔൌᆣ࣮!13!і4 ဢЧଽყҩ༂ҵؓlevelaslopeބconvexity֥݂߭ࢲݔTable4Theregressionresultsofin sampleforecasterrorsregressingonleve,lslopeandconvexityESSSEMIEXT =ྩᆞ∀ ()()()#, ()()() ()()() ()()() =5ྩᆞ∀ ()()()#, ()()() ()()() ()()() =10ྩᆞ∀ ()()() ()()()#, ()()() ()()() ᇿ:ওݼ()іൕpᆴ.ੲѯܙ࠹,бࢠਔEXAMaSAMބEAM3ᇕڄག4 ࢲඏეࡎ۬ؓ০ੱэ߄၂ࢨइaؽࢨइ֥ဢЧଽຓ֥ყҩིݔaၛࠣ۲ሱؓޘࢩ૫ྟᇉ֥ކି৯.ൌᆣ০ੱڄགࡎ۬,൞৵ࢤགྷൌҩ؇ބڄགᇏྟࢲݔіૼ,ંՖ၂ࢨइყҩaؽࢨइყҩߎ൞ޘҩ؇֥൷୧.০ੱڄགࡎ۬ഡקྙൔ֥Ⴊਜ,߶ᆰࢩ૫ྟᇉ֥ކིݔഈ,EXAMૼཁႪႿࢤ႕ཙ֞ଆކ֥ሙಒӱ؇.SAM,طSAMؓEAM֥ڿࣉᄵ҂൞ޓն.Ч໓ၛၹሰCIRଆູࠎሙ,๙ݖवغણಖط,Ч໓๙ݖ၂ࢨइყҩ༂ҵؓlevela
!14!ܵ ॓ ࿐ ࿐ Б2010୍9ᄅslopeބconvexity3ᇕၹሰ֥݂߭ؿགྷ,EXAMಯ၂ࢨइყҩି৯ູսࡎ.֥ၹՎ,๙ݖڄགࡎ۬ഡಖمປಆࢳथEAM֥໙,ีᄝ؟ၹሰCIRଆקটڿࣉ؟ၹሰCIRଆؓ০ੱ၂ࢨइ֥ყҩ,ॿࡏ༯,ؓؽࢨइყҩି৯֥ิۚ,ಯಖ൞ၛ།വಯಖླေࣉ၂֥҄ധ࣮.ҕॉ໓ང:[1][J].JournalofFinance,2002,57(1):405-443.[2][J].ReviewofFinancialStudies,2004,17(2):379-404.[3]CheriditoP,FilipovicD,:Theoryandevidence[J].Jour nalofFinancialEconomics,2007,83(1),123-170.[4]DuffieD, factormodelofinterestrates[J].MathematicalFinance,1996,6(4):379-406.[5]FisherM, AffineModelsoftheTermStructure[R].NewYork:FederalReserveBoard,1996.[6]ChenRR,[J].JournalofFixedIncome,1993,3(3):14-31.[7]DaiQ,[J].JournalofFinance,2000,55(5):1943-1978.[8] sectioninformationinaffineterm structuremodels[J].AmericanStatisticalAssocialtionJournalofBusiness&EconomicStatistics,2000,18(3):300-314.[9]LamoureuxCG,:Theinformationinthedataviewedthroughthewindowofcox,ingersol,landross[J].JournalofFinance,2002,57(3):1479-1520.[10]AhnDH,DittmarRF,:Theoryandevidence[J].ReviewofFinancialStudies,2002,15(1):243-288.[11]DaiQ,,time varyingriskpremia,andaffinemodelsofthetermstructure[J].JournalofFinancialEconomics,2002,563(3):415-441.[12][J].JournalofFinancialEconomics,1977,5(2):177-188.[13]CoxJC,IngersollJE,[J].Econometrica,1985,53(2):385-407.[14]DuffeeG,[R].:HaasSchoolofBusiness,2004.[15]DaiQ,[J].SocFinancialStudies,2003,16(3): long,KEHong,MOTian ,XiamenUniversity,Xiamen361005,China;,Shenzhen518028,ChinaAbstract:ThereexistsfourprimaryspecificationsofinterestriskpriceundertheframeworkofaffineDTSM:Completelyaffinemodel(CAM),essentiallyaffinemodel(EAM),extendedaffinemodel(EXAM),semi af fineModel(SAM).IthasbeenprovedbothintheoryandempiricalstudiesthatEAMissuperiortoCAM,
ֻ9௹ᆢᆒ֩:০ੱٟഝଆ༯֥০ੱڄགࡎ۬ྙൔൌᆣ࣮!15!helptodetermineabettermodelbetweensemi ever,therobusttestsuggeststhatEXAMisnotperfectenoughtocapturealltheinformationinthetermstruc :interestrateaffineDTSM;marketpriceofinterestrisk;extendedaffinemode;lsemi affinemodelڸ1 ၍ཛູሑэਈཌྟݦඔൈ,ሑэਈ่֥ࡱनᆴ-dt)t) (ei(T---(dej+dk)(T-)}]ა่ࡱٚҵఃᇏေႨवغણੲѯࣉྛܙ࠹,൮༵ေ࠹ෘሑэਈ֥G0=!*diag(∀*)!* ,่ࡱनᆴა่ࡱٚҵ.G=!*diag(#*(+,i))!* iࡌഡሑэਈ֥ݖӱູ{f(j,k)}іൕֻ(j,k)۱ჭູf(j,k)֥इᆔ.dX=K(,-X)dt+!SdW,tttt2 Semi AffineModel༯,ሑэਈ่֥ࡱनᆴބ่ࡱS=∀+#XttٚҵFisherބG[6]illes۳ԛਔAffineModel༯่ࡱनᆴބ่ࡱٚҵ֥၂ϮྟൔሰᄝSAM༯,ሑэਈᄝགྷൌҩ؇༯֥၍ཛ൞٤ཌྟ֥,مႨభ૫֥ٚمᆰࢤ่֤ࡱनᆴބ่ࡱٚҵE(XTX)=∋(T-t)X+D(T-t)K,|tt֥ࢳ༅ൔ.ၹՎ,ᆃҐႨDu[2]arte֥ٚم࣍ර่֥ࡱVनᆴބ่ࡱٚҵ.Duarteіૼ,֒ൈࡗࡗۯཬႿ1۱ᄅൈ,ar(XTX)=T-s)F(t,s)∋(T-s)ds|t%T∋(tᆃᇕ࣍ර෮Ӂള֥༂ҵޓཬ,ࠫެॖၛޭ҂࠹.ఃᇏᄝSAM༯,ሑэਈ֥၍ཛॖၛཿູF(t,s)=∀+#E(XTX),|t−(X)=K,+!S!-1&0-KX,∋( )=(-K )e,S(i,i)=∀+# XiiK=KQD-!(&( )=%t∋(s)ds1(1)# 1,,,&1(1)#b n) -0ᇿၩ(-K ) !SS-&,eіൕؓᆜ۱इᆔ(-K )e,ᄝMatlabᇏ2ႨK,=KQ,Q+!(&(1)∀ ,,,&(1)∀ )expmଁіൕ.111nطDu[1]ffee۳ԛਔ֒Kॖؓ࢘߄ൈ,่ࡱनᆴބ่ؓ∀+# XቓXڸ֥࣍၂ࢨীᅚष,֤֞iitࡱٚҵ֥ࢳ༅ൔ.∀+# X~∀# (X-X)+# X+itiiiitK=NDN-1,DіൕK֥หᆘᆴइᆔ,D(i,i)=∀+# XiiitѩקၬX*=N-1X,ᄵX*֥ݖӱॖၛіൕູࡼীᅚष࣍රᆴսೆ၍ཛ,ѩކѩোཛॖ֤tdX*=D(,*-X**)dt+!S*dW−(X)~K,+!A(X)!-1&t0-ttttS*=∀+#*X*-1;,*=N, (KX-!B(X)!-1&;0)tttఃᇏ,A(X)aB(X)൞ؓ࢘इᆔ,ఃؓ࢘ཌჭູ!*=N-1!;#*=#Ntt๙ݖ֝ॖ֤∗-# XXitA()=∀+# X+i,itiitE)(T-t))(XT|X(-K(T-t))=(1-e),+-KeX2∀+# X,iitttV# Xar(XT|X)=NVar(X*tT|X*)N tB=it(X)i,jt2∀+# XViitar(X*T|X*)={(d+d-1k)G0(j,k)∀tjᆃဢ,ࣼॖၛ֤֞SAM٤ཌྟ၍ཛ֥ཌྟ࣍ර,Ֆn-t)(d+d (1--(Tejk)-1)}+&[,*{(d+dijk)G(j,k)∀iط০Ⴈᆃ۱࣍රᆴࣼॖၛႋႨFisherބG[9]illesބi=1nDu[1]ffee֥ٚمট֤ሑэਈ่֥ࡱनᆴބ่ࡱٚҵ.-(T-t)(dj+dk) (1-e)}]+&[(X*-,*)∀t,ii(༯ሇֻ25်)i=1 {(d+d-d-1)∀G(j,k)∀jkii∗ᇿၩDuffee[1]۳֥Var(X*|X*)Ⴕ༂,ࡼൔᇏ֥ଖ۱ࡆݼཿູӰݼਔ;طDuarte۳֥ൔሰ္ٕਔဢ֥հ༂.rt
ֻ9௹ේ Ւ֩:ࠎႿෛࠏളӁݦඔ֥վॻקࡎଆࠣႋႨ!25!LoanpricingmodelbasedonstochasticproductionfunctionandapplicationSUICong,CHIGuo taiSchoolofManagement,DalianUniversityofTechnology,Dalian116024,ChinaAbstract:Thispaperestablishestheformulaofloanpricingwithestimatedvalueofparametersinthestochas ,:Firstly,thismodelascertainstheloanrateofnewloanonthepremisethatthetechnicalefficiencyisoptimalbasedonthestochasticproductionfunctionandensurestheoptimaltechnicalefficiencyofloanpri cinganddealswiththecontradictoryproblemthattheloanpricingnotonlyoverlaysthecostandriskbutalsoisacceptedbyclients,,throughtheloanpricingformulabasedonstochasticproductionfunction,theloanrateofoptimaltechnicalefficiencycanbefixedandthemethodthattheoutputisfixedinthestochasticproductionfunctioncanbeprovidedwhenthetechnicaleffi ,combinedwithstochasticfrontier,theloanpri textDatabaseandElsevier,etc.,:loanpricing;stochasticproductionfunction;technicalefficiency;stochasticfrontieranalysis(ഈࢤֻ15်)RA( ,.)B( ,.)=-+Xt++t|t t++t|t3 वغણੲѯܙ࠹҄ᇧVBR VB+(2,t++=t|t.іൕ෮Ⴕླေܙ࠹֥ҕඔ,ᄵҩਈٚӱॖၛ t++t|t ∗іൕູ4)࠹ෘყҩ༂ҵet++=Rt|tt++-Rtt++t|tR( ,.A( ,.)B( ,.))=-+X+∗t tt, 5)Ⴈt=+t֥ྐ༏۷ྍሑэਈ∗t, іൕޘࢩ૫ކ֥༂ҵ.Xt++XVBV-1t|t++=tt+++t|tt++t|t R,t++et|tt++tवغણٚӱ֥םս҄ᇧູ:1)Ⴈቋཬ߄җҵٚބ֥Ϸم֤ሑэਈ֥ԚVBB Vt++t|t++=VV-1tt++-Vt|tt++t|t R,t++t|t t++t|tᆴX0|0,ѩູڮჍ.ॖି֥ᆴ.๙ݖ҂؎םս1ᇀ5֥҄ᇧ,ቋᇔॖၛ֤ࠞնරಖ2)࠹ෘX่֥ࡱनᆴބ่ࡱٚҵtᆴ.ࠞնරಖᆴݦඔູXTt++=(I-e(-/+t)),+e(-K+t)X,t|tt|tVLog likelihood=-&(log(VR+e V-1,t|t-+))ttR,t|t-+ettT=NV*N |tT|tt=+t3)࠹ෘRt( ,.)่֥ࡱनᆴބ่ࡱٚҵ
