Copula Effect on Scenario Tree

Transcription

1 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 Copula Effec on Scenao Tee K. Suene and H. Panevcu Abac Mulage ochac pogam ae effecve fo olvng long-em plannng poblem unde unceany. Such pogam ae uually baed on cenao geneaon model abou fuue envonmen developmen. In he peen pape he cenao model developed fo he cae when enough daa pah can be geneaed bu due o olvably of ochac pogam he cenao ee ha o be conuced. The popoed aegy o geneae mulage cenao ee fom he e of ndvdual cenao by bundlng cenao baed on clue analy. The K-mean clueng appoach modfed o capue he neage dependence. Such geneaon of cenao ee can be ueful n cae when dffcul o conuc he adequae cenao ee fom he ochac dffeenal equaon o me-ee model and he ampled pah can be obaned by amplng o eamplng echnque. Whle geneang he nal fan of ndvdual cenao he copula employed fo modelng he dependence beween ochac vaable n a mulvaae ucue. I allow o model nonlnea dependence beween non-ellpcally dbued ochac vaable. Whle nvegang he copula effec on he cenao ee ucue we wll y o anwe he queon: doe he copula feaue ae capued n he appoxmae epeenaon of unceany n he fom of cenao ee. The popoed cenao ee geneaon mehod mplemened on ampled daa of dcoun bond yeld. The Gauan copula and Suden -copula ae employed whle geneang he e of ndvdual cenao n he mulvaae ucue. Index Tem Copula K-mean clueng Mulage cenao ee conucon Sochac pogammng. I. INTRODUCTION The concep of cenao uually employed fo he modelng of andomne n ochac pogammng model [] [] n whch daa evolve ove me and decon have o be made ndependenly upon knowng he acual pah ha wll occu. Such daa ae uually ubjec o unceany o ome knd of k. Fo nance he andom vaable ae he eun value of each ae on an nvemen n pofolo managemen poblem and he nvemen decon mu be mplemened befoe he ae pefomance can be obeved. Each cenao can be vewed a one ealzaon of an undelyng mulvaae ochac daa poce. The modelng of andomne Manucp eceved Ocobe 3 7. K. Suene wh he Bune Infomac Depamen Kauna Unvey of Technology Sudenu 6-3 Kauna LT-44 Lhuana (coepondng auho o povde phone: ; fax: ; e-mal: kna.uene@ud.ku.l. H. Panevcu wh he Bune Infomac Depamen Kauna Unvey of Technology Sudenu 6-3 Kauna LT-44 Lhuana (e-mal: hepan@f.ku.l. employee he e of avalable pa daa wh he am of buldng ubmodel fo each ndvdual ochac paamee. Thee ubmodel ae ued o geneae a e of cenao ha encapulae he conen depcon of pahway o poble fuue baed on aumpon abou economc and echnologcal developmen. Thu he faco dvng ky even ae appoxmaed by a dcee e of cenao o equence of even. Th poce known a cenao geneaon. Scenao can be geneaed ung vaou mehod baed on dffeen pncple: condonal amplng amplng fom gven magnal and coelaon momen machng pah baed mehod opmal dcezaon a n [3] [7]. Sochac pogammng (opmzaon ha been appled n he followng aea: Manufacung poducon capacy plannng; Eleccal geneaon capacy plannng; 3 Ae lably managemen; 4 Pofolo elecon; Taffc managemen; 6 Machne chedulng. In hee applcaon decon mu ofen be aken n he face of he unknown. A good appoxmaon may nvolve a vey lage numbe of cenao wh pobable. A bee accuacy of unceane decbed when cenao ae conuced va a mulaed daa pah ucue alo known a a cenao fan. Bu he numbe of cenao lmed by he avalable compung powe ogehe wh a complexy of he decon model. To deal wh h dffculy we can educe he dmenon of he nal cenao e by conucng he mulage cenao ee ou of. The decon on he numbe of age on he ze of me peod and on he banchng cheme vey mpoan fo a good epeenaon of he unceany n he fom of cenao ee whch npu no he mulage ochac pogam. The dealed decpon of boh cenao fan and cenao ee wll be gven n Secon III. In he peen pape we concenae on he geneaon of cenao ee when he undelyng ochac paamee have been deemned and he daa pah of he ealzaon can be geneaed. The cenao ee can be conuced ou of ampled pah by employng ome clafyng mehod uch a clueng analy. Whle bundlng he cenao o he clue he neage dependence have o be capued. An appoach mla o ou wok noduced n he acle [8] bu whou a dealed clueng algohm. Due o h he K-mean clueng mehod modfed o ea popely he neage dependence and mplemened whle conucng he cenao ee fom mulaed daa pah. Such geneaon of cenao ee can be ueful n cae when dffcul o conuc he adequae cenao ee fom he ochac dffeenal equaon o me-ee model and he ampled pah can be obaned by amplng o mulaon (Advance onlne publcaon: 7 Novembe 7

2 Sochac pogammng (opmzaon combne model of opmum eouce allocaon and model of andomne heeby ceae a decon makng famewok (ee Fg. []. Wheea deemnc opmzaon poblem ae fomulaed wh known paamee eal wold poblem almo nvaably nclude paamee whch ae unknown a he me a decon hould be aken. Tha why he deemnc Opmum decon model and conan IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 echnque. The popoed cenao ee conucon algohm appoach expanded. allow ncopoang a copula-baed dependence meaue [9] In ochac pogam he f elemen he objecve [] o decbe he dependence beween ochac vaable n funcon whch ogehe wh he conan decbe he coe a mulvaae ucue. Due o aumpon of ung he of he poblem ha ha o be olved and vae wh each Peaon coelaon coeffcen he uefulne of uch ndvdual applcaon. The econd elemen he cenao coelaon eced. The man advanage of employng geneao and ued o decbe he unconollable (k copula ha hey allow o model he nonlnea dependence faco affecng he elevan yem uch a nflaon ae beween non-ellpcally dbued ochac vaable. The nee ae GPD faco ha ae no unde conol of he copula funcon ha been noduced n fnance by Embech decon make. The uncean elemen ae modeled a McNel and Saumann [9]. To ou knowledge he copula andom vaable o whch he pobably heoy can be ll ae no vey popula n geneang he cenao ee. appled. A concep of cenao ued o epeen of how he Accodng o h we popoe o appoxmae he mulvaae fuue mgh unfold. Some knd of pobablc model o ochac poce by a cenao fan wh mulvaae ucue mulaon can be ued o geneae a bach of cenao. The ung copula. Then he cenao ee conuced ou fom model of andomne wh he fne and dcee ealzaon he ampled pah ung he modfed K-mean clueng ae called cenao geneao. The oucome of uch yem algohm. Numecal expeence epoed fo conucng uncean even when he value of all he decon vaable ae mulvaae cenao ee of dcoun bond yeld employng fxed. Scenao can alo be ued n decpve model whee wo epaae Gauan and Suden copula. a e of mahemacal opeaon ae defned ha can pedc The e of he pape oganzed a follow. The cenao how a mahemacal yem wll behave e.g. Makov model. geneaon model noduced n Secon II. The mahemacal The man feaue of ochac pogammng mulage model con of wo man componen: model fo he fomulaon. Depe ch nvolvemen of he fuue eveyhng unvaae magnal dbuon of uncean faco and a amed o make a well hedged decon n he peen. The model of he dependence ucue employed n he noon of aude adoped ha a decon wll be popely made n he copula. Gauan copula and Suden -copula ae peen only akng no accoun a lea ome o exen he condeed. The mulaon algohm of modelng he copula oppoune fo modfcaon o coecon a lae me. baed dependen daa gven. Secon III dcue how he Decon a lae me can epond o he nfomaon ha ha copula funcon can be ncopoaed whle geneang become avalable nce he nal decon. Thu dung he cenao. Secon IV decbe how he mulaed daa pah me he decon alenae wh obevaon: nal decon can be anfomed o he cenao ee ung clue analy. obevaon ecoue decon obevaon The K-mean clueng algohm modfed o bundle he ecoue decon. Th equence doen go on ndefnely bu me-dependen daa. Secon V demonae he numecal he numbe of age can be lage enough. Decon ha ae example of cenao ee geneaon baed on dcoun bond aken have no effec on he pobably ucue. Thu we have yeld daa. Fnally ome concludng emak ae gven. a mulage poblem. The numbe of age ued n modelng he unceany; we wll fomalze h lae n em of he mulage cenao ee. II. SCENARIO GENERATION COMPUTATIONAL PROCEDURE In he pape [] he geneal cenao geneaon pocedue fo mulage ochac pogam gven. We append h pocedue wh addonal ep (Sep payng he mpoan aenon o he dependence modelng among k faco. The followng ep (ome o all have o be pefomed whle geneang cenao: Sochac pogammng (opmzaon Model of andomne cenao geneao Fg.. Sochac pogammng paadgm Collecng hocal daa of ochac paamee aumpon of a model emaon/calbaon of paamee fo a choen model. Choong he appopae model o decbe he dependence ucue among ochac paamee. 3 Geneaon of cenao accodng o he choen model o dcezaon of he dbuon ung appoxmaon of acal popee. We wll conde all hee ep n deepe manne. A. Modelng Paadgm To olve a ochac decon makng poblem we need knowledge abou he pobably dbuon of all andom (Advance onlne publcaon: 7 Novembe 7

3 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 vaable among he unconollable npu. In pape [] he Smulaed Dependen Value auho popoe fou ype of poblem concenng he level of 4 he avalable nfomaon: 3 Full knowledge of undelyng pobably dbuon ; Known paamec famly ; 3 Sample nfomaon ; 4 Low nfomaon level. Thee fou goup ae no cly dnguhable. Dffeen - nfomaon level can be appled o he dnc paamee of he model. The mo popula modelng paadgm ae []: - Economec Model and Tme See (ARMA -3 GARCH VAR model ; -4 Geomec Bownan Moon (Dffuon Pocee ; XNomal( Afcal Inellgence (Neual Newok ; Sacal Appoache (Sacal appoxmaon Foecang Momen Fng ; Samplng. I vey mpoan ha he ample we ue o epeen he (b Dependence beween X and X wh ρ =. 7 ochac paamee n he fom of cenao would be Fg.. Dffeen dependence ucue conen wh empcal daa. Theefoe one ha o pecfy he ochac pocee fo k paamee and emae he coeffcen doe no capue any non-lnea dependence and uually ued aumng he ellpcal hape of nomal paamee of uch model ung empcal daa. dbuon n applcaon. We nclude Fg. a movaon fo he dea of h pape. I how andom vaae fom wo B. Dependence ucue modelng dbuon wh dencal andad Gauan magnal In h pape we concenae on geneaon of cenao dbuon: cae (a and cae (b depc bvaae ucue of epeenng he ealzaon of mulvaae ochac poce X and X wh lnea coelaon coeffcen ρ =. 7. whoe componen ae coelaed. We defne uch cenao a Howeve he dependence ucue beween X and X necoelaed cenao meanng ha hey coelae hough he componen of mulvaae ucue. Hocally meaung and modelng of dependence ha ceneed on coelaon. The modelng of dependen vaable pefomed employng he Peaon coelaon max o decbe he mulvaae ucue. Many applcaon how ha elaonhp among ochac vaable may be vey complex and lnea dependence can eflec hee elaonhp adequaely. The eaon ha he Peaon coelaon XNomal( Smulaed Dependen Value XNomal( (a Dependence beween X and X wh ρ =. 7 XNomal( qualavely que dffeen. I elae ha n cae (b exeme value have a endency o occu ogehe. Th example how ha he dependence beween andom vaable canno be dnguhed on he gound of coelaon alone. Addonally n eal applcaon ae fo dbuon o follow he c phecal aumpon wh a conan dependence aco he dbuon mpled by coelaon. To ovecome he lmaon of coelaon he pacone can daw on copula funcon. I vey poweful echnque whch allow o epeen jon dbuon by plng he magnal behavo embedded n he magnal dbuon fom he dependence capued by he copula elf. Th upeoy of ung copula eleae he modelng emaon and mulaon of dependen andom vaable. Le defne he copula elf. A funcon C he d -dmenonal copula f fulfll he followng popee [3]: The doman of C [ ] d ; C gounded and d -nceang ; 3 The magn C k of C afy C k ( u = u k = d fo all u n [ ]. Le conde d andom vaable Y Y K Yd wh mulvaae dbuon F and unvaae magn F ( y F ( y K F d ( y d. Skla heoem whch he foundaon fo copula ae ha any jon dbuon can be wen n a copula fom. (Advance onlne publcaon: 7 Novembe 7

4 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 Skla Theoem (99. Gven a jon dbuon funcon dbuon funcon wh max Co of lnea coelaon F ( y y K y d fo andom vaable Y Y K Yd wh coeffcen. The man popey of uch dependence ucue magnal ( F F K F d F can be wen a a funcon of ha Gauan copula doe no have nehe uppe no lowe al dependence. magnal: Smulaon pocedue fo Gauan copula pefomed a follow: F ( y y K yd = C( F ( y F ( y Fd ( yd τ (a conve Kendall au co o he lnea coelaon τ whee copula C ( u u K u d a jon dbuon wh coeffcen co ung fomula co = n( π co / (he unfom magnal. Moeove f each F connuou C elaonhp beween he lnea coelaon coeffcen co unque. S S and Speaman ho co co = n( π co The dependence ucue can be epeened by a pope / 6 and copula funcon. Moeove he followng coollay aaned conuc he lowe angula max Α = [ a ] ha hold fom Skla heoem. Co = AA ; Coollay. Le F be an d - dmenonal dbuon (b geneae ndependen andad nomal vaable ε funcon wh connuou magn F ( y F ( y K F d ( yd = d and fom a column veco ε ; and copula C. Then fo any u = ( u u u d n [ ] d : (c conuc a jon pobably deny funcon akng he max poduc ε = Aε ; (d e u C( u u K ud = F( F ( u F ( u Fd ( ud = Φ( ε ; (e e x = Φ ( u. whee F he genealzed nvee of F. A he eul x = d ae dependen vaable baed on A good many copula ae avalable wh dffeng Gauan copula. Fg. 3 how fou cae plo of chaacec ha lead o he dffeen elaonhp among andom value fom a bvaae Gauan copula fo vaou vaable geneaed. Noe ha copula dffe no o much n he level of coelaon coeffcen o lluae he ange of degee of aocaon hey povde bu ahe n whch pa of dffeen dependence ucue. Thu he famly of Gauan he dbuon he aocaon onge: he behavo of copula paameezed by he lnea coelaon max. copula n he gh and lef al can be ued o dnguh The Suden -dependence ucue noduce an among jon dbuon ha poduce he ame oveall addonal paamee compaed wh he Gauan copula coelaon. namely he degee of feedom. Suden -copula can be wen a In h pape we wll conde wo copula: Gauan copula and Suden -copula. Thee copula do no have a mple cloed fom bu ae mpled by well known mulvaae dbuon funcon: mulvaae Gauan and mulvaae Suden dbuon epecvely. The dffeence beween hee wo copula ha he Suden -dependence ucue uppo jon exeme movemen egadle of he magnal behavo of ochac vaable compaed wh he Gauan copula. A complee copula-baed jon dbuon can be conuced ung aeed ank-ode coelaon and magnal dbuon. Example of ank-ode coelaon ae Speaman ho and Kendall au coelaon whch ae ued o decbe he dependence elaon of a monoonc naue: ndcae he endency of wo andom vaable o nceae/deceae concomanly (pove dependence o conawe (negave dependence. The Gauan o nomal d -copula gven by C Ga Co ( d ( u Φ Φ ( u Φ ( u Φ ( u = K Co whee Φ denoe he andad unvaae nomal dbuon funcon and Φ d Co denoe he andad mulvaae nomal d U U ( ( u ( u ( u d C Co v ( u Co v v v v ho = U ho = U = K U U ho = U ho = U Fg. 3. Gauan dependence ucue wh dffeen coelaon coeffcen d (Advance onlne publcaon: 7 Novembe 7

5 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 d Suden' -copula al effec whee Co v ae he paamee of -copula Co v denoe he jon dbuon funcon of he d -vaae Suden -dbuon wh v degee of feedom v he nvee of unvaae Suden -dbuon wh v degee of feedom. Suden -copula ha he addonal paamee v compang wh Gauan copula. Inceang he value of v deceae he.8.6 endency o dcove exeme co-movemen. Smulaon pocedue fo Suden -copula pefomed a follow:.4. τ (a conve Kendall au co o he lnea coelaon τ coeffcen co = n π co / (he co ung fomula ( elaonhp beween he lnea coelaon coeffcen co = n π S / 6 and Α = ha hold S and Speaman ho co co ( co conuc he lowe angula max [ ] Co = AA ; (b geneae ndependen andad nomal vaable ε = d and fom a column veco ε ; (c conuc a jon pobably deny funcon akng he max poduc ε = Aε ; (d geneae a andom vaae γ χ v ; (e calculae ε = v ε γ ; (f u = v ( ε ; (g x = Φ ( u. A he eul x ae dependen vaable baed on Suden -copula. Fg. 4 how fou cae plo of andom value fom a bvaae Suden -copula wh degee of feedom equal fo vaou level of coelaon coeffcen o lluae he ange of dffeen dependence ucue. Thee plo demonae ha -copula dffe fom Gauan copula (ee Fg. 3 even when he componen have he ame coelaon. U U ho = U ho = U U U a ho = U ho = U Fg. 4. Suden dependence ucue wh degee of feedom equal bu dffeen coelaon coeffcen Degee of feedom Fg.. Tal dependence fo -copula Coelaon coeffcen The man dffeence beween he condeed copula funcon n meaung he dependence beween he occuence of exeme value. Bvaae al dependence coeffcen meaue he engh of dependence n he uppe and lowe quadan al of a bvaae dbuon. The uppe al dependence coeffcen a follow []: λ ( ( Y Y = P Y > F ( α Y > F ( α U lm. α Analogouly he lowe al dependence coeffcen λ ( ( Y Y = P Y F ( α Y F ( α L lm. α The uppe (lowe coeffcen quanfe he pobably o obeve a lage (mall Y when Y lage (mall. If λ U λ L ( ] hen wo andom vaable Y and Y ae ad o be aympocally dependen n al. And f λ U = λl = hen vaable ae ad o be aympocally ndependen n al. Fuhemoe gven he adal ymmey popey of ellpcal dbuon he lowe and uppe al dependence coeffcen concde. In he wok [] wa hown ha al dependence coeffcen equal o zeo confmng he aympoc ndependence n al of he Gauan copula. Suden -copula al effec fom boh degee of feedom and coelaon coeffcen depced n Fg.. One can ee ha he onge he lnea coelaon coeffcen and he lowe he degee of feedom he onge al dependence. Calbang he copula paamee o he eal daa he acve eeach aea n he cuen ac leaue [ 4 6]. Mo popula appoache ued n emaon of copula ae Exac Maxmum Lkelhood mehod (EML Infeence Funcon fo Magn mehod (IFM Canoncal Maxmum Lkelhood mehod (CML and ohe. In h wok we won go no he deal of paameezng he copula. Dung he dcezaon poce of d-dmenonal dbuon funcon F one can enghen he dependence n dffeen pa of dbuon hough he choce of copula. (Advance onlne publcaon: 7 Novembe 7

6 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 Indeed he aumpon of nomaly fo he magn can be hozon conng of T age. The ochac poce emoved and ( F F K F d may be fa-aled dbuon ξ = { ξ } T defned on ome fleed pobably pace = (e.g. Suden Webull Paeo and dependence may be ( Ω S F P. The ample pace Ω defned a chaacezed by a Nomal o ohe choen copula. Tha he d dependence ucue beween ochac vaable can be Ω : = Ω Ω KΩ T whee Ω R. Noe ha he ample modeled ndependenly of magnal dbuon. pace ae aken a fne dmenonal. In h cae we conde d -dmenonal pace bu n ohe applcaon poble o vay he dmenonaly. Fo nance hee daa may coepond o he obeved eun of d fnancal ae a dffeen me momen. The σ -algeba S he e of C. Geneaon of Scenao The fou man cenao geneaon appoache ae [3]: Samplng. Samplng appoache ae Mone Calo (Random Samplng Impoance Samplng Booap Samplng and Condonal Samplng. Fo each amplng mehod he man pncple o ake a ample fom a pobably dbuon funcon o ha fo a gven value we have an aocaed pobably whch gve he cenao value and banch pobably. Sacal Appoache. The man pncpal o deemne he value of pacula acal popee of gven daa. The mo popula acal momen o popey machng appoach wheeby we do no aume knowledge of a andom vaable pobably dbuon funcon. Inead we decbe he dbuon by acal momen o ohe popee e.g. mean vaance pecenle. 3 Smulaon. I an appoach fo cenao geneaon whee ome undelyng mahemacal poce mulaed: andom numbe ae ncopoaed no he andom componen of an equaon and he eul ecoded. Scenao geneaon by mulaon eul he e of mulaed daa pah wh equal pobably. To educe he numbe of pah omeme pah ae bounded by ome mehod. Sochac Poce Smulaon Eo Coecon Model Veco Auoegeve model ae mo popula mulaon ued fo ochac pogammng. 4 Ohe mehod. I can be mehod fom ohe feld uch a Afcal Neual Newok Clueng o can be combnaon of ome cenao geneaon mehod. None mehod of cenao geneaon appoved a opmal bu he goal hould be he adequae epeenaon of unceany. To emembe he dea of h pape we am o ncopoae a copula n geneaon of cenao. In he pape [7] he momen machng mehod wa ued o geneae copula baed coelaed cenao. In ou pape we wll ue he combnaon of ome mehod allowng u o employ copula funcon fo modelng dependen ochac vaable: he mulaon and clueng appoache ae combned o conuc he cenao ee. In he nex econ he ochac pogammng noaon gven o make he decpon of cenao geneaon moe fomal. III. STOCHASTIC PROGRAMMING NOTATION In mulage ochac pogam he undelyng mulvaae ochac daa poce ha o be dcee n me. = T and a me Mahemacally we have a me ndex { } even wh agned pobable by meaue P and { } T = F a flaon on S. The decon ae of wo ype: he nal decon x aken a nal me momen and he ecoue decon x > aken a T ecoue age. Thu he decon a age he andom vaable x : n Ω R ; decon ae e a fne dmenonal bu of pobly vayng dmenonaly. In he ochac pogammng model he obevaon and decon ae gven a a equence x ( ξ x ( ξ x K ( ξt x T whee x = { x } T a decon = poce meauable funcon of ξ. The conan on a decon a each age nvolve pa obevaon and decon. I mean ha decon x a meauable wh epec o F F. Thu followng [8] he decon poce ad o be nonancpave. I mean ha he decon x = x ( x ξ aken a any > doe no decly depend on fuue ealzaon of ochac paamee o on fuue decon. A he me when nal decon mu be choen nohng abou he andom elemen n ou poce ha been pnponed. Bu n makng a ecoue decon we have he evealed cuen nfomaon unl h momen and he edual unceany ll he end of me hozon. Moe nfomaon on mulage ochac pogam can be found n leaue [] [] [8]. We connue on wh ochac noaon of cenao geneao. The d-dmenonal pobably dbuon funcon of ξ y = y yd f y he d-dmenonal cumulave dbuon funcon denoed by F ( y. The jon dbuon F povde a complee nfomaon concenng he behavo of ξ. The magnal pobably dbuon funcon and cumulave dbuon funcon of each elemen ξ a pon y = d denoed by f ( y and F ( y epecvely. The pmay am of cenao geneao o epeen he dbuon f n a = ( d ξ ξ K a pon ( K denoed by ( eaonable way. In ochac pogammng he undelyng pobably dbuon f eplaced by a dcee dbuon P caed by a fne numbe of aom ξ ( ξ ξt = ( d ξ ξ K ξ = S wh pobable p P( ξ S = = = p and p =. The aom ξ = S of he (Advance onlne publcaon: 7 Novembe 7

7 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 decon. The mulage ee eflec he neage dependency and deceae he numbe of node whle compang o he cenao fan. The ucue of mulage ee (ee Fg. 7 a = alo decbed by a ole oo node and by banchng no a fne numbe of cenao a wa n pevou cae. The age ae conneced wh pobly o ake addonal decon baed on newly evealed nfomaon. age age Such nfomaon can be obaned peodcally (evey day week monh o baed on ome even (expaon of = = = =T nvemen pofolo. The dncon beween age whch Fg. 6. Scenao fan coepond o he decon momen and me peod eenal becaue n paccal applcaon mpoan ha he dbuon P ae called a cenao. Naually he hocal numbe of me peod would be geae han he daa n conjuncon wh an aumed backgound model ae coepondng node. The ac lnkng node epeen vaou ued o geneae he cenao applyng uable emaon ealzaon of andom vaable. The numbe of banche fom mulaon and amplng pocedue. each node can vay dependng on poblem pecfc Whle appoxmang he mulvaae ochac dbuon equemen and no defnely conan hough he ee. One F employng copula he e of d-dmenonal necoelaed cenao ξ = ( ξ ξt = ( of he aege o ue an exenve banchng a he d ξ ξ K ξ = S begnnng of me hozon and a elavely poo banchng a he la level of ee. Each pah hough he ee fom oo o geneaed. Aumng ha all cenao concde a = he one of leave coepond o one cenao.e. o a pacula nal oo node fomed and hu he mulaed daa pah ae equence of ealzaon of andom coeffcen. called a a cenao fan (ee Fg. 6. The ucue of mulaed The algohm of anfomng he cenao fan o he daa pah can be dvded no wo age. The f age mulage cenao ee of pecbed ucue decbed n uually epeened by a ngle oo node and he value of he nex econ. andom paamee dung he f age ae known wh ceany. Movng o he econd age he ucue banche no ndvdual cenao a me = a hown n he Fg. 6. If uch cenao fan ued a npu no he mulage ochac pogam he model of -age poblem a all σ -feld F = T concde. The -age mulpeod ochac pogam ha he followng popee a n [8]: Decon a all me nance = T ae made a once and no fuhe nfomaon expeced. Excep fo he f age no nonancpavy conan appea. Dependng on he condeed poblem uch popee can be egaded a dadvanage. Ou am o ceae a mulage cenao ee whch can be ued fo mulage model. Mulage fomulaon chaacezed by obune ably of oluon: mla ubcenao eul n mla IV. K-MEANS CLUSTERING: FAN TO TREE Whle conucng he mulage cenao ee fom he cenao fan he fan of ndvdual cenao modfed by bundlng cenao baed on he clue analy. The dea of bundlng cenao o he clue depced n he Fg. 8. Value Value Tme (a Inal cenao fan Tme (b level clueng Value Value age age M age = = = =T- =T Tme (c nd level clueng Tme (d 3 d level clueng Fg. 7. Mulage cenao ee Fg. 8. Illuaon of 4-age ee conucon (Advance onlne publcaon: 7 Novembe 7

8 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 I aumed ha a e of ndvdual cenao fo he ene cene of clue. me hozon ( me momen aleady geneaed (ee In he followng we dcu how he appoach fom clue Fg. 8a. The cenao fan of cenao chemacally analy appled o goup mla cenao. The cenao fan lluaed. A me = all hee cenao (whch ae he uually con of lage numbe of cenao ha why he ame fom he oo node of he ee. The aegy o conuc heachcal mehod can fal. We don alo eque he mehod he cenao ee wh wo banche fo each decon momen. ha n fndng he clue would be opmal by ome meaue. If wo banche ae deed fom he cuen cenao ee node In he leaue he clueng mehod uually ae ued fo hen wo clue have o be fomed. Le aume ha we have able daa; hu we have o make ome modfcaon n ode hee decon dae.e. we ae plannng o make decon a o clue he me dependen daa uch a cenao. One of he and 8 me momen. Wh h nal eng we ae eady o man faco o delneae he ucue of cenao ee he conuc he mulage cenao ee. Thu a he me banchng cheme. Le aume ha K banche ae deed momen wo clue ae fomed by he f eaon of ome fom each cenao ee node: he ee homogeneou. I clueng algohm. The eul dplayed n Fg. 8b. The mean ha K clue wll need o be fomed. Thu he cene of each clue compued whch epeen he K -mean clueng algohm [9] choen o conuc he one-level node a me =. Nex a pevou ep fomed cenao ee fom he e of mulaed pah (cenao fan. clue ae dvded no wo ubclue. I eul ha a me Clueng con n paonng of a daa e no ube o = we have fou clue epeenng wo-level node nce ha he daa n each clue hae he common abue. Th he cene ae calculaed (ee Fg. 8c. Such aegy of mlay ofen defned by ome dance meaue. Afe a bundlng cenao o he clue connue fo all defned dcuon of he knd of equemen we ae ung we decon momen. Fg. 8d depc he hd level of ome decbe he modfed K -mean clueng algohm. clueng algohm nce wo moe ubclue ae fomed a Gven a fan of ndvdual cenao ξ = ( ξ ξt me = 8. The conuced cenao ee ha 4 age and 8 K = S and he numbe K of deed clue C cenao. K C The compued node (clue cene ae denoed by black needed o fnd he clue cene ξ k k = K uch pon n he geneaed cenao ee (ee Fg. 9. Jonng he ha he um of he -nom dance quaed beween each black pon by lne we ge he gaphcal epeenaon of k cenao ξ and neae clue cene ξ mnmzed: cenao ee. Such aegy of bundlng cenao o he clue can connue ll he end of me hozon eached. K The dcued echnque allow o poduce he ee wh uch k ξ ξ mn. k chaacec: k= ξ C The pojecon of andom vaable neae he me hozon le ccal han hoe fo he nea fuue Whle clueng he cenao he man dea ae: becaue numbe of cenao gow malle down he In cuen age M new ubclue have o be ee and he cene ha epeen he cenao clue fomed fom clue fomed n pevou age ae calculaed fom a malle ample ze. ( M. Tha why called mul-level clueng. I allow o model exeme even becaue a evey Cenod hould be calculaed only a age ndexed age he mulaed cenao n all of he clue ae me momen bu dance meaue hould evaluae no dcaded and a he nex age all mulaed all cenao. cenao n all of he clue ae ued o calculae he Ohe conan ha ae ued o ealze vaou equemen fo new fomed clue can be added. The pobable of each node hould be evaluaed. Value Tme Fg. 9. Gaphcal epeenaon of 4-age cenao ee Accodng o he dea gven above he modfed K-mean clueng algohm gven a follow. A he begnnng he decon momen ae e coepondng o he age ndex T. Then eae: ( Sep : Seng nal cene. Le ξ k k = K be he clue cene. Some mehod can be ued o chooe nal clue cenod poon omeme known a eed. I mgh be choen o be he f K cenao nce he cenao ae ndependenly geneaed; o K cenao by andom. Sep : Clue agnmen. Fo each cenao ξ agn ξ o he clue k C uch ha cene k ξ neae o ξ n he (Advance onlne publcaon: 7 Novembe 7

9 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 -nom whch modfed o explo he whole equence of coelaed mulaed daa pah: Real nee ae Inflaonae d k ( = T k ξ ξ ξ ξ. = Tem ucue of eal nee ae Inflaon expecaon of nveo If cenao ae all n he ame phycal un hen he Eucldean dance mec uffcen o uccefully goup mla daa. I poble o apply ohe dance mec uch a Manhaan dance Maxmum nom Mahalanob dance o goup mla cenao only ome modfcaon have o be done o employ he whole mulaed daa equence. Sep 3: Clue updae. Compue ξ k a he mean of all k cenao agned o he clue C : { ξ } k ξ E. k = ξ C Th fomula can be eplaced by ohe emae uch a medan mode o ele. Sep 4: Repea. Go o Sep unl convegence.e. no cenao move he goup. Sep : Calculaon of pobable. Pobably of ξ k equal he um of pobable of he ndvdual cenao belongng o he elevan clue k C. Sep 6: Modfcaon. Modfy ξ ( ξ ξ k ξ wh ξ f k ξ. C K T ξ = by eplacng Sep 7: Repea. Go o Sep f nex age ndex ex. The clueng pocedue a ove ung each of clue fomed n cuen eaon. Th algohm poduce a epaaon of cenao no goup. The gven algohm le o ea popely he neage dependence explong he whole equence of mulaed cenao pah. A he end he cenao ee conuced conng of node ξ k wh he pobable and he banchng cheme. V. COMPUTATIONAL EXPERIMENT The cenao ee geneaon appoach appled o conuc cenao ee ou of ampled cenao povded by Hbbe Mowbay and Tunbull (HMT ochac ae model []. The followng ae ome geneal popee ncopoaed n he cenao geneao: Mean-eveon. I aumed ha a long-em aonay equlbum level ex o whch he ae eun poce wll end ove me. Auoegeon. The me ee do flucuae aound a cean equlbum level and a each ep he poce eac Yeld on nomnal dcoun bond Fg.. A cacade ucue of HMT model fom a pevou devaon wh one me lag. The dependence ove me (neempoal dependence condeed. 3 Volaly. In h pape vaance wll be condeed conan ove me. 4 Coelaon. Noe ae coelaed. The geneal model mulvaae mean eveng model of fnancal eun. HMT model compoed of a numbe of componen pa ha ae dven by a e of ochac dve. Thu he dependence among vaou k faco (conempoaneou ae modeled. The alenave mehod copula baed dependence meaue choen o model he elaonhp among ochac paamee. On he echncal level he Mone-Calo mulaon eleced o geneae vey lage numbe of plauble cenao becaue of flexbly and nuve peenaon. We ue h model o geneae he ample whch con of a fne numbe of cenao epeenng ealzaon of dcoun (zeo-coupon bond yeld. A cacadng ucue a chaacec of HMT model: eal nee and nflaon ae ae mulaed whch hen dependng on he elaonhp ucue aumed nfluence he ealzaon of dcoun bond yeld (ee Fg.. In HMT model peened hee he economc elaonhp beween nflaon nflaon expecaon eal nee ae and nomnal nee ae explcly condeed. The model poduce he em ucue ha ha cloed-fom oluon fo bond pce o ha he ene em ucue fo any fuue pojecon dae can be quckly geneaed: he analycal expeon ae avalable fo dcoun bond pce. In HMT model he man ochac dve nflaon ae and eal nee ae ae decbed by wo faco Onen-Uhlenbeck poce n connuou me: d d ( = α ( ( ( d + σ dz ( ( = α ( µ ( d + σ dz ( whee he ho-em ae (denoed by eve o a long-em ae (denoed by ha elf ochac. The long-em ae eve o an aveage mean eveon level µ. The auoegeve paamee α α ae mean eveon peed. The econd em on he gh-hand de epeen he unceany n he poce: he andad devaon σ and σ denoe he volaly dz ( he hock o he eal (Advance onlne publcaon: 7 Novembe 7

10 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 ho-em ae poce whch dbued nomally ( be ued wh efeence o HMT wok []. dz he hock o he eal long-em ae poce whch dbued nomally. Th model allow he pobly of negave eal ae. The pcng equaon whch ued n geneang he pce P of dcoun bond a me ha pay one un n eal em (poeced fom nflaon a me T gven by ( T exp[ A( T B ( T ( B ( T ( ] P = ( whee A ( B ( ( B ae funcon of he paamee fo eal nee ae movemen. The expeon can be found n pape []. Of coue once we have obaned pce fo eal dcoun bond hen poble o calculae he connuouly compounded yeld a me fo mauy T R ( T = log( P ( T ( T. ( The mplemenaon exece wa olved n uch way: ung ho me neval he dcee poce appoxmae he connuou poce. In fnancal applcaon he long-em modelng equed and he ue of neval uch a monhly enough appopae. The model peened hee nclude a andad Bownan moon whch n he dcee fom epeened by ε ε. So we ge: = α к = α к ( ( µ + σ ε + σ ε. We eaange he la equaon o how ha h poce an auoegeve poce: = = = α к = α + ( ( µ к ( αк αк ( α µ α + к к + σ ε + σ ε ; + σ ε + σ ε ; In mulaon afe he dcezaon pocedue we ge he dcee ample aken fom hee ochac equaon and epeenng plauble cenao fo uncean vaable ove he plannng peod. Smulaon pocedue of wo faco Onen-Uhlenbeck poce gven below: (a geneae ε N( mulply by ; (b mulae long-em eal nee ae ung econd equaon of (3 fomula ; (c mulae ho-em nee ae ung f equaon of (3 fomula ; (d deemne mauy me T fo dcoun bond ; (e ue analycal expeon ( fo he eal po ae and eal fowad ae a any em T ; (f calculae dcoun bond eun a me fo mauy T ung ( fomula. Exacly he ame model ucue ued o model he behavo of he ho-em nflaon ae (denoed by q a wa ued fo eal ho-em nee ae bu wh paamee fo nflaon poce. A em ucue fo nflaon expecaon P q ( T can be nfeed fom he cuen nananeou nflaon ae q and q ung pcng equaon (. Combnng a em ucue of eal nee ae and nflaon expecaon poble o deve a em ucue fo nomnal nee ae: Pnom ( T = P ( T Pq ( T. A wa dcued n cenao geneao he fnancal vaable have o be pojeced n uch way a o eflec he appopae nedependence beween hem. I eaonable o conde he cae when nee ae and nflaon ae move ogehe: ho-em eal nee ae coelae wh ho-em nflaon ae and long-em eal nee ae coelae wh long-em nflaon ae. Inedependence among hee vaable ae denfed hough Wene pocee dz ( = 4. The cumulave dbuon funcon of Wene poce F y ( y; = exp = Φ π Z y and + + = α + к = α µ + к ( αк ( α к + σ ε + σ ε. (3 = 4. Thu o model he dependence o dz ( N( d beween ochac dve we conuc he jon dbuon F by lnkng hee magnal dbuon hough he copula funcon. We ge Equaon (3 how ha he ho ae a weghed + aveage beween he cuen level and he long ae. The long ae + elf a weghed aveage of he long-em mean µ and cuen value. Equaon (3 can be ued n ode o emae he paamee of he model. In h pape we don deal wh emaon pocedue and he paamee wll y y y y ( 3 y Φ Φ Φ y y3 y4 = C Φ F 4 whee C he 4-dmenonal copula funcon. In h wok wo copula funcon ae condeed: Gauan copula and Suden copula. The mulaon algohm ae gven n Secon B. (Advance onlne publcaon: 7 Novembe 7

11 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 τ τ Table I. Dmenon of cenao fan A h momen le aume ha a max Co = [ co ] Node Tme peod Scenao τ co j = 4 of Kendall au coelaon ha Scenao fan of 4 4 aleady been aeed whch denoe ank-ode coelaon dcoun bond yeld beween wo andom vaable. In HMT model Kendall au coelaon coeffcen e equal o. beween ho-em In Fg. a he dependence ucue deemned by he eal nee ae and ho-em nflaon ae and e equal coelaon max Co fo Gauan copula n Fg. b he o. beween long-em eal nee ae and long-em dependence condoned by he coelaon max Co and nflaon ae. by degee of feedom υ = fo Suden -copula. The lowe HMT model ued o mulae 3 7 yea coupon al dependence lmed by he lowe bound on he level of bond yeld ove a hozon of yea wh me ncemen of nflaon and eal nee ae. The uppe al dependence one monh. The nal paamee ae e wh he efeence o dffe wh epec o he employed copula. he Hbbe e al. wok. The condon abou he envonmen Each cenao fan of 3 7 yea coupon bond yeld ae aumed a follow: nflaon level.% long-em geneaed by cenao geneao decbed by dmenon. nflaon level.83% cuen 3-monh T-bll nom % and The dmenon of he pacula cenao fan gven n Table I. cuen -yea T-bond yeld.. The lowe bound on he The cenao fan lluaed ung he funnel of doub level of nflaon and of eal nee ae ae placed o enue plo (ee Fg. eulng fom unceany n he fuue ha negave ae don appea. A he oupu of h cenao value. In he followng analy -yea and -yea dcoun geneao he daa coned of a fne numbe of cenao bond eun ae condeed. The funnel of doub gaph ( S = epeenng he ealzaon of dcoun bond dplay he h h h 7 h 9 h 99 h pecenle value yeld. In Fg. he dependence ucue beween mulaed and he mean ample value (lgh dahed lne. The pead value of eal nee ae and nflaon ae ae dplayed n he aound medan expand a he me nceae cayng a me momen = yea. cean k of unceany ha nceae wh me bu end o ablze a he end of me hozon whch he effec of mean.4 eveon value. The aumpon of avodng negave value of. nomnal nee ae deemne ha he expeced value of dcoun bond yeld df up ove me. VaR (Value-a-Rk. ype concluon ha n ( p % of he cae he yeld.8 hghe o equal o VaR p value (vecal ax whee < p <.6 a pecenle value. The pead of -yea dcoun bond Inflaon Rae.4 Y Coupon bond yeld Y Coupon bond yeld Real Inee Rae (a Gauan dependence ucue.4. (a Gauan dependence ucue..8 Y Coupon bond yeld Y Coupon bond yeld Inflaon Rae Real Inee Rae (b Suden dependence ucue Fg.. Dependence ucue beween mulaed value (b Suden dependence ucue Fg.. Scenao fan of Y and Y Coupon bond yeld (Advance onlne publcaon: 7 Novembe 7

12 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 yeld le han he pead of -yea dcoun bond yeld becaue of he effec of mean eveon. Some of acal Table IV. Dmenon of cenao ee K= K=3 chaacec mean value dpeon and 99 h pecenle of -yea and -yea dcoun bond eun ae calculaed fo he evaluaon of geneaed cenao (ee Table II Table III. The mulaed cenao fan amed o anfom o he 3-age ee -age ee Node 7 3 Scenao 4 6 Node 3 Scenao 9 8 cenao ee wh dffeen numbe of age and wh Table V. Chaacec of cenao ee when K = dffeen banchng faco employng he clueng algohm dcued n Secon IV. The numbe of age depend on he Gauan Decon momen n numbe of decon momen. The banchng cheme of he dependence = = = = cenao ee nfluence he numbe of clue. Fo nance Mean we chooe he numbe of cenao equal o K = and K = 3 Dpeon..9 whch geneaed pe cenao ee node. Two ype of cenao ee ae geneaed fo he analy: 3-age cenao Pecenle. 4.8 ee wh decon a = and -age cenao ee wh 99 h Pecenle. 4.7 decon a =. Table IV how he dmenon of cenao ee fo he cae K = and K = 3. I how ha he dmenon of cenao fan noably educed whle anfomng he cenao fan o he cenao ee. Mean Dpeon Pecenle 99 h Pecenle In he followng analy we am o nvegae how Mean dependence ucue affec he value of age vaable and he ucue of cenao ee. The mean value dpeon Dpeon.4.7 and 99 h pecenle of -yea and -yea dcoun bond eun ae calculaed fo he evaluaon of geneaed cenao ee (ee Table V Table VIII. Table II. Chaacec of cenao fan Gauan Decon momen n dependence = = = = Scenao Mean fan of Y Dpeon coupon bond eun Pecenle % 99 h Pecenle Scenao Mean fan of Y Dpeon..9.. coupon bond eun Pecenle % 99 h Pecenle Table III. Chaacec of cenao fan Suden Decon momen n dependence = = = = Scenao Mean fan of Y Dpeon coupon bond Pecenle eun % 99 h Pecenle Scenao Mean fan of Dpeon Y coupon Pecenle bond eun % 99h Pecenle Y coupon bond eun % Y coupon bond eun % 3-age ee -age ee 3-age ee -age ee Pecenle h Pecenle Mean Dpeon Pecenle h Pecenle Table VI. Chaacec of cenao ee when K = 3 Y coupon bond eun % Y coupon bond eun % Gauan Decon momen n dependence = = = = Mean Dpeon.6. Pecenle h Pecenle Mean Dpeon Pecenle h Pecenle Mean Dpeon..8 Pecenle h Pecenle.9.7 Mean Dpeon Pecenle h Pecenle age ee -age ee 3-age ee -age ee (Advance onlne publcaon: 7 Novembe 7

13 dae ae defned. I hold fo boh Gauan copula and Suden -copula coelaed daa. The nal fan of ndvdual cenao and he conuced cenao ee how ha he daa obaned unde Suden -dependence have malle mean value epeenng he malle bond eun han daa obaned unde Gauan dependence. Thee nfeence can be appoved fom he gaphcal epeenaon of conuced cenao ee (ee Fg. 3 Fg. 6. Table VII. Chaacec of cenao ee when K = Y coupon bond eun % Y coupon bond eun % Suden Decon momen n -dependence = = = = Mean Dpeon.4.9 Pecenle h Pecenle Mean Dpeon..6.. Pecenle h Pecenle Mean Dpeon.4.7 Pecenle h Pecenle. 4. Mean Dpeon Pecenle h Pecenle age ee -age ee 3-age ee -age ee IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 % 6 % % Y Coupon bond yeld when K= % % 6 % % Y Coupon bond yeld when K= % 6 % % Y Coupon bond yeld when K=3 % (a Gauan dependence ucue % 6 % % Y Coupon bond yeld when K=3 Table VIII. Chaacec of cenao ee when K = 3 Y coupon bond eun % Y coupon bond eun % Suden Decon momen n -dependence = = = = Mean Dpeon.6. Pecenle h Pecenle Mean Dpeon Pecenle h Pecenle Mean Dpeon..9 Pecenle h Pecenle.4.46 Mean Dpeon Pecenle h Pecenle age ee -age ee 3-age ee -age ee The emak on obaned eul ae a follow. I un ou ha fo a lage banchng faco K he daa of dcoun bond eun become moe dvee he neval beween he and he 99 h pecenle become wde bu he mean value eman he ame. The ame effec obeved when moe decon % % (b Suden dependence ucue Fg age cenao ee of Y Coupon bond yeld wh decon a ={} yea % 6 % % Y Coupon bond yeld when K= % % 6 % % Y Coupon bond yeld when K= % % 6 % % Y Coupon bond yeld when K=3 % (a Gauan dependence ucue % 6 % % Y Coupon bond yeld when K=3 % (b Suden dependence ucue Fg age cenao ee of Y Coupon bond yeld wh decon a ={} yea (Advance onlne publcaon: 7 Novembe 7

14 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 Y Coupon bond yeld when K= % % 6 % 6 % % % Y Coupon bond yeld when K=3.4.3 Pobably.. % % (a Gauan dependence ucue Y Coupon bond yeld when K= % % 6 % 6 % % % % % Y Coupon bond yeld when K= % % 6 % 6 % % % % % Y Coupon bond yeld when K= % % % % %.3. % % % Value of cenao ee node (b Suden dependence ucue Fg. 7. Value of Y Coupon bond yeld wh he pobably Y Coupon bond yeld when K=3. (a Gauan dependence ucue % Value of cenao ee node.4 (b Suden dependence ucue Fg.. -age cenao ee of Y Coupon bond yeld wh decon a ={} yea % (a Gauan dependence ucue Y Coupon bond yeld when K=3 Pobably equal o -3. Bu h no he cae fo cenao ee whee pobably of each cenao condoned by clue ze. Fo nance le ake he cenao ee wh K = 3 banchng faco and decon a = {} yea. Fg. 7 depc he dbuon of Y Coupon bond yeld unde dffeen dependence ucue a me =. I how he elaonhp beween value of andom vaable (node of cenao ee and pobably. Y Coupon bond yeld when K=3 % % VI. CONCLUDING REMARKS 6 % 6 % % % In he peen pape we decbed he pocedue baed on boh mulaon and clueng o geneae he cenao ee ou of daa pah. The compuaonal expemen howed ha he ze of geneaed cenao ee much malle han he dmenon of cenao fan and nevehele hey ae good appoxmaon wh epec o he Eucldean dance ued o meaue he me-dependen daa pah. Anweng o ou queon doe he copula feaue ae capued n he appoxmae epeenaon of unceany n he fom of cenao ee we conclude ha dffeen dependence ucue wh he ame coelaon coeffcen beween ochac vaable affec he ucue of mulage cenao ee. The accomplhed analy of cenao ee how ha cenao ee geneaed fom dependen daa baed on Suden -copula ae moe exeme han geneaed fom dependen daa employng Gauan copula. The effec of ung Suden -copula a dependence meaue beween eal nee ae and % % (b Suden dependence ucue Fg. 6. -age cenao ee of Y Coupon bond yeld wh decon a ={} yea Scenao ee wh a hghe banchng faco le o model moe exeme cenao. Ung of Suden -copula a dependence meaue beween eal nee ae and nflaon ae ha effec o oban lee value of dcoun bond yeld compang wh he cae when he Gauan copula ued. In he cenao fan each cenao ha he ame pobably. If cenao ae geneaed hen pobably of any cenao (Advance onlne publcaon: 7 Novembe 7

15 IAENG Inenaonal Jounal of Appled Mahemac 37: IJAM_37 8 nflaon ae o deceae value of dcoun bond yeld. I eul fom he feaue ha ung Gauan copula he exeme even ae ndependen o we don ge eally exeme cenao. Fuue eeach on h opc nclude he mpovng clueng appoach by addng ome conan on clue ze on clue chaace. Then he evaluaon of qualy/ably of cenao geneaon mehod fo a gven ochac pogam can be condeed. [9] L. Kaufmann and P. J. Roueeuw Fndng Goup n Daa: An Inoducon o Clue Analy. Canada: Wley-Inecence 99 ch. 3. [] J. Hbbe P. Mowbay and C. Tunbull A ochac ae model & calbaon fo long-em fnancal plannng pupoe Techncal Repo Bae&Hbbe Lmed U. REFERENCES [] L. Y. Yu X. D. J and S. Y. Wang Sochac pogammng model n fnancal opmzaon: uvey AMO Advanced Modelng and Opmzaon vol. ( 3 pp. 6. [] J. Dupačová J. Hu and J. Šěpán Appled Opmzaon 7: Sochac Modelng n Economc and Fnance. Dodech Holland: Kluwe Academc Publhe ch. 3. [3] S. Ma Scenao geneaon fo ochac pogammng Whe Pape Opk Syem U 6. [4] J. Dupačová N. Göwe-Kuka and W. Römch Scenao educon n ochac pogammng Mahemacal Pogammng vol. 9(3 3 pp [] K. Høyland and S. W. Wallace Geneang cenao ee fo mulage decon poblem Managemen Scence vol. 47( pp [6] H. Hech and W. Römch Geneaon of mulvaae cenao ee o model ochacy n powe managemen n IEEE S. Peebug PoweTech Poceedng Rua. [7] G. Pflug Scenao ee geneaon fo mulpeod fnancal opmzaon by opmal dcezaon Mahemacal Pogammng vol. 89 pp. 7. [8] J. Dupačová G. Congl and S. W. Wallace Scenao fo mulage ochac pogam Annal of Opeaon Reeach vol. pp. 3. [9] P. Embech A. McNel and D. Saumann Coelaon and dependency n k managemen: popee and pfall n Rk managemen: value a k and beyond M. A. H. Dempe Ed. UK: Cambdge Unvey Pe pp [] K. Aa Modellng he dependence ucue of fnancal ae: a uvey of fou copula Reeach Repo SAMBA//4 Nowegan Compung Cene Noway 4. [] N. D. Domenca G. Bbl G. Ma and P. Valene Sochac pogammng and cenao geneaon whn a mulaon famewok: an nfomaon yem pepecve Techncal Repo CARISMA U 3. [] J. Dupačová Sochac pogammng: appoxmaon va cenao Apoacone Mahemaca Se. Communcaone vol pp [3] P. Embech F. Lndkog and A. McNel Modellng dependence wh copula and applcaon o k managemen n Rachev S. (Ed. Handbook of Heavy Taled Dbuon n Fnance Eleve ch. 8 pp [4] W. Hu Calbaon of mulvaae genealzed hypebolc dbuon ung he EM algohm wh applcaon n k managemen pofolo opmzaon and pofolo ced k PhD The Floda Sae Unvey. [] S. Daul E. DeGog F. Lndkog and A.J. McNel The gouped -copula wh an applcaon o ced k RIS 6 3 pp [6] O. Roch and A. Alegea Teng he bvaae dbuon of daly equy eun ung copula. An applcaon o he Spanh ock make Compuaonal Sac & Daa Analy ( 6 pp [7] M. Kau and S. W. Wallace Shape-baed cenao geneaon ung copula Sochac Pogammng E-Pn See vol Avalable: hp://edoc.hu-beln.de/docvew/abac.php?lang=ge&d=769 [8] Sochac Pogammng Communy Home Page ponoed by he Commee on Sochac Pogammng Onlne Reouce. Avalable: hp:// (Advance onlne publcaon: 7 Novembe 7