nèect seicsth sÔgqronhqrhmatooikonomik mhqanik : apì...
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Panepisthmio Aigaiou
Tmhma Mhqanikwn Oikonomiac & Dioikhshc
Nèec t�seic sth sÔgqronh qrhmatooikonomik mhqanik : apì th stoqastik prosèggish stic
no monec mejodologÐec
NÐkoc S. Jwmaòdhc (M.Sc.)Upoy. Did�ktorac TMOD
1h et sia èkjesh proìdou didaktorik c diatrib c
9 MartÐou 2004
Trimel c epitrop :
Gi¸rgioc DoÔniac Aristof�nhc Dhm�khc Gi¸rgoc Li�gkouracAnapl. Kaj. TMOD Kaj. TMOD Epik. Kaj. TMOD
Prìlogoc
H ekpìnhsh thc didaktorik diatrib c me jèma: {Nèec t�seic sth qrhmatooikonomik mhqanik : apìth stoqastik prosèggish stic no monec mejìdouc} xekÐnhse to Dekèmbrio tou 2002, upì thn ar-qik epÐbleyh tou kaj. Aristof�nh Dhm�kh kai sth sunèqeia tou anapl. kaj. GewrgÐou DoÔnia.To eurÔtero pedÐo èreunac apoteleÐ h qrhmatooikonomik mhqanik � apìdosh sta ellhnik� touìrou financial engineering � kl�doc pou anaptÔqjhke kurÐwc thn teleutaÐa eikosaetÐa. Apì mÐa etu-mologik an�lush tou ìrou (engineer = mhqaneÔomai, epino¸), prokÔptei ìti h QM anafèretai sthneÔresh kat�llhlwn mejìdwn ergaleÐwn gia thn epÐlush problhm�twn qrhmatooikonomik c fÔsewc.Se antÐjesh, loipìn, me th diatÔpwsh mÐac (qrhmato)oikonomik c jewrÐac, h QM esti�zei se mÐapio praktik , problem-solving, prosèggish kai pragmateÔetai kurÐwc to prìblhma tou sunduasmoÔkat�llhlwn qrhmatooikonomik¸n proðìntwn (omolìgwn, metoq¸n, parag¸gwn, k.a.) prokeimènw naepiteuqjeÐ èna sugkekrimèno profÐl mellontik¸n tameiak¸n ro¸n (cash-flows) pou ja ekplhr¸sountouc qrhmatooikonomikoÔc stìqouc enìc organismoÔ. H diadikasÐa aut , bèbaia, proôpojètei thnepÐlush mÐac seir�c epimèrouc problhm�twn ìpwc p.q. thn eÔresh montèlwn pou na perigr�founth dunamik thc tim c qrhmatooikonomik¸n megej¸n (metoqèc, deÐktec, epitìkia, sunallagmatikècisotimÐec, k.a.), thn timolìghsh polÔplokwn proðìntwn (forwards, futures, swaps, options), k.o.k.Bèbaia, h anaz thsh kat�llhlwn ependÔsewn proc k�luyh tameiak¸n anagk¸n apoteloÔse anèka-jen èna apì ta zhtoÔmena thc qrhmatooikonomik c dioÐkhshc enìc organismoÔ. Wstìso, autì pouod ghse sthn eureÐa an�ptuxh tou kl�dou thc QM tan h ìlo kai megalÔterh emplok organism¸nse agorèc kefalaÐou kaj¸c kai h auxanìmenh poluplokìthta twn diajèsimwn qrhmatooikonomik¸nproðìntwn, kai sunep¸c twn enallaktik¸n trìpwn qrhmatodìthshc ependutik¸n eukairi¸n.
H qrhmatooikonomik mhqanik perilamb�nei s mera mÐa eureÐa gk�ma efarmog¸n, teqnik¸n kai jew-ri¸n pou eÐnai arket� dÔskolo na sumperil�bei kaneÐc se mÐa didaktorik diatrib . AnagnwrÐzontac toprìblhma autì, epilèxame sthn paroÔsa èreuna na esti�soume th prosoq mac sta ereunhtik� zht -mata pou prokÔptoun apì thn dhmiourgÐa susthm�twn pou analÔoun kai diaqeirÐzontai dÔo basikoÔctÔpouc qrhmatooikonomik¸n kindÔnwn:
a) ton kÐnduno thc agor�c (market risk) kai
b) ton pistwtikì kÐnduno (credit risk)
O kÐndunoc agor�c aporrèei apì th metablhtìthta thc tim c qrhmatooikonomik¸n megej¸n ta opoÐaephre�zoun to qartoful�kio enìc qrhmatopistwtikoÔ organismoÔ, en¸ o pistwtikìc kÐndunoc èqeina k�nei me ton kÐnduno pou sunep�getai h adunamÐa ekpl rwshc twn upoqre¸sewn tou antisum-ballomènou se èna qrhmatopistwtikì sumbìlaio (p.q. thn apoplhrwm enìc daneÐou). H eÔreshmejodologi¸n apotelesmatik c diaqeÐrishc twn parap�nw tÔpwn kindÔnou brÐskei s mera idiaÐteroendiafèron sto q¸ro twn meg�lwn ependutik¸n kai pistwtik¸n organism¸n (tr�pezec, qrhmatisthri-akoÐ oÐkoi k.a.), wc apotèlesma kurÐwc prìsfatwn metarrujmÐsewn sto kanonismì kajorismoÔ ke-falaÐou asfaleÐac kai katagraf c tou profÐl kindÔnou enìc organismoÔ1.
'Opwc diafaÐnetai apì ton tÐtlo, h en lìgw didaktorik diatrib ereun� dÔo t�seic sthn epÐlushproblhm�twn qrhmatooikonomik c mhqanik c. Apì th mÐa, thn eponomazìmenh klasik stoqastik
1Apì to 1996 ta kr�th-mèlh thc Eurwpaðk c 'Enwshc upoqre¸noun touc qrhmatopistwtikoÔc organismoÔc sthnapotÐmhsh tou kindÔnou thc agor�c qrhsimopoi¸ntac th statistik mèjodo thc Value-at-Risk. Bl. endeiktik� [6],kai to keÐmeno thc SumfwnÐac tou Basle [17], sthn opoÐa apofasÐsthke h tropopoÐhsh tou rujmistikoÔ plaisÐou twnqrhmatopistwtik¸n organism¸n.
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prosèggish (stochastic approach), pou basÐzetai sth jewrÐa twn pijanot twn kai thc statistik c,kai apì thn �llh tic eufueÐc no monec algorijmikèc mejìdouc (intelligent methods), �rrhkta sund-edemènec me thn epist mh twn upologist¸n. Apì th stigm pou h qrhmatooikonomik anadeÐqjhkese xeqwristì episthmonikì pedÐo, èpeita kurÐwc apì tic prwtopìrec ergasÐec twn H. Markovitz [8],W.F. Sharpe [16], kai F. Black & M. Scholes [1], sundèjhke �rrhkta me th jewrÐa pijanot twn kaitwn stoqastik¸n diadikasi¸n. Apì thn �llh, h exèlixh thc epist mhc twn upologist¸n od ghsesthn an�ptuxh ergaleÐwn kai algorÐjmwn pou emfanÐzoun {eufu } sumperifor�. H sÔgqronh qrhma-tooikonomik epist mh sunoyÐzetai telik� se dÔo t�seic pou epiqeiroÔn thn epÐlush parìmoiwn qrhma-tooikonomik¸n problhm�twn, th basik sullogistik twn opoÐwn perigr�foume sth sunèqeia.
H klasik stoqastik prosèggish basÐzetai kat� kÔrio lìgo se arqèc kai upojèseic thc klasik coikonomik c jewrÐac pou aforoÔn sth sumperifor� twn ependut¸n kai th leitourgÐa twn kefala-iagor¸n (orjologikoÐ ependut¸n, {tèleiec} agorèc, apotelesmatik di�qush plhrofori¸n, k.a.). Oiupojèseic thc klasik c prosèggishc mac epitrèpoun na qrhsimopoi soume to ploÔsio ulikì thc jew-rÐac twn pijanot twn kai twn stoqastik¸n diadikasi¸n (kÐnhsh Brown, martingales, l mma Ito,stoqastikìc logismìc, k.a.) gia th montelopoÐhsh thc abebaiìthtac se mÐa kefalaiagor�.
Oi eufueÐc mejodologÐec, se antidiastol me th stoqastik prosèggish, èqoun perissìtero empeirikìpar� axiwmatikì qarakt ra. Me ton ìro no monec mèjodoi, gÐnontai s mera antilhptèc proseggÐseic,ìpwc aut thc asafoÔc logik c (fuzzy logic), twn teqnht¸n neurwnik¸n diktÔwn (artificial neuralnetworks), thc mhqanik c m�jhshc (machine learning) kai tou genetikoÔ programmatismoÔ (geneticprogramming) [10]. Basik� shmeÐa afethrÐac thc efarmog c mÐac eufuoÔc mejìdou apoteloÔn a)ta dedomèna (data) pou phg�zoun apì tic diadikasÐec pou sunteloÔntai se mÐa qrhmatagor� seènan qrhmatopistwtikì organismì kai b) h gn¸sh, ek mèrouc eidik¸n kai empeirognwmìnwn, twnidiaiterot twn tou q¸rou efarmog c (domain knowledge). Oi no monec mèjodoi, ekkin¸ntac apìmÐa pio {empeirik } b�sh, epitugq�noun na katagr�youn susqetÐseic se polÔploka sust mata, ìpwcl.q. mÐa kefalaiagor�, twn opoÐwn h perigraf me th qr sh klasik¸n statistik¸n mejìdwn (p.q.an�lush palindrìmhshc) ja tan Ðswc adÔnath qwrÐc thn pragmatopoÐhsh enìc sunìlou perioristik¸nupojèsewn (p.q. thn Ôparxh grammik c sqèshc metaxÔ epexhghmatik¸n kai anex�rthtwn metablht¸n).
Sth di�rkeia tou pr¸tou ètouc tou didaktorikoÔ asqolhj kame me to na aposafhnÐsoume kai na kata-gr�youme pl joc paradoq¸n kai qarakthristik¸n thc klasik c prosèggishc, ìpwc epÐshc kai anoik-t� probl mata pou prokÔptoun apì thn efarmog aut c sth l yh qrhmatooikonomik¸n apof�sewn.Apì to meg�lo pl joc efarmog¸n, epilèxame na esti�soume thn èreuna mac sto eurÔtero plaÐ-sio twn susthm�twn diaqeÐrishc kindÔnou agor�c. Ta teleutaÐa epiqeiroÔn na analÔsoun kai naposotikopoi soun ton kÐnduno tou qartofulakÐou enìc qrhmatopistwtikoÔ organismoÔ pou aporrèeiapì th metablhtìthta twn tim¸n qrhmatooikonomik¸n megej¸n (timèc qreogr�fwn, qrhmatisthriakoÐdeÐktec, epitìkia, sunallagmatikèc isotimÐec, k.a.).
Par�llhla me thn klasik prosèggish, ègine mÐa ekten c melèth twn efarmog¸n eufu¸n algori-jmik¸n mejìdwn pou èqoun protajeÐ sthn bibliografÐa gia thn epÐlush qrhmatooikonomik¸n prob-lhm�twn. 'Opwc dh prokÔptei apì thn mèqri t¸ra melèth, oi eufueÐc mèjodoi parousi�zoun poll�pleonekt mata ènanti twn stoqastik¸n mejìdwn. ApodeiknÔontai idiaÐtera apotelesmatikèc sto naentopÐzoun polÔplokec (mh grammikèc) susqetÐseic metaxÔ diafìrwn metablht¸n, ìpwc kai �tupoucpar�gontec pou diamorf¸noun th dunamik qrhmatooikonomik¸n fainomènwn, ìpwc p.q. h diaqronik poreÐa twn tim¸n qreogr�fwn. To pleonèkthma autì eÐnai apotèlesma tou gegonìtoc ìti oi eu-fueÐc mèjodoi eÐnai sqediasmènec gia mÐa katexoq n empeirik prosèggish susthm�twn, dhl. ton
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entopismì montèlwn susqetÐsewn mèsa apì dedomèna (data) pou phg�zoun apì th leitourgÐa enìcsust matoc. Sto shmeÐo autì h {eufu c} prosèggish deÐqnei na upertereÐ thc klasik c, miac kaih teleutaÐa basÐzetai se perioristikèc upojèseic {tèleiac} agor�c orjologik¸n ependut¸n pousuqn� apèqoun apì thn pragmatikìthta. DiapÐstws mac eÐnai, wstìso, ìti mÐa amig c klasik empeirik prosèggish eÐnai ellip c gia polloÔc lìgouc. Apì th mÐa, oi upojèseic thc klasik cprosèggishc nai men mac epitrèpoun na qrhsimopoi soume ploÔsio ulikì kai {stibar�} apotelèsmatathc jewrÐac twn pijanot twn kai thc statistik c, wstìso teÐnoun na anafèrontai se exidanikeumèneckatast�seic pou sp�nia antikatoptrÐzoun tic pragmatikèc sunj kec mÐac agor�c. Apì thn �llh, h{pros lwsh sta dedomèna}, pou upagoreÔei h empeirik prosèggish, de lamb�nei upìyh oikonomikècarqèc kai qr sima sumper�smata thc klasik c prosèggishc sqetik� me th sumperifor� twn epen-dut¸n kai th fÔsh twn kefalaiagor¸n (mhqanismìc kajorismoÔ twn tim¸n, jesmikoÐ periorismoÐ,k.a.). JewroÔme ìti me to sunduasmì twn dÔo parap�nw proseggÐsewn mporoÔme na epitÔqoumemÐa kalÔterh anapar�stash kai katanìhsh twn qrhmatooikonomik¸n susthm�twn kai sunep¸c pioepituqhmèna sust mata prìbleyhc kai an�lushc tou kindÔnou thc agor�c.
Sta plaÐsia aut c thc filosofÐac, jewr same anapìspasto komm�ti thc èreun�c mac to pedÐo thcqrhmatooikonomik c jewrÐac pou anafèretai wc mpiqebioristik qrhmatooikonomik (behavioural fi-nance). H teleutaÐa epiqeireÐ mèsa apì thn katanìhsh thc sumperifor�c twn ependut¸n kai twnatelei¸n mÐac agor�c na ermhneÔsei di�fora {par�doxa} fainìmena kat� th diamìrfwsh twn tim¸n(problèyimec t�seic, epoqiakoÐ kÔkloi, èntonh apìkrish se etairikèc oikonomikèc anakoin¸seic). Oìroc {par�doxa} èqei epikrat sei gia na upodhl¸sei to gegonìc ìti tètoia fainìmena xefeÔgoun apìtic upojèseic thc klasik c prosèggishc perÐ orjologik¸n (rational) ependut¸n, tuqaÐac (random)exèlixhc twn tim¸n kai apotelesmatik c aporrìfhshc thc plhroforÐac pou diaqèetai se mÐa kefala-iagor� (efficient markets hypothesis). H mpiqebioristik qrhmatooikonomik basÐzetai se èreunec thcgnwstik c yuqologÐac (cognitive psychology) pou epibebai¸noun to gegonìc ìti ìtan oi �njrwpoièrqontai antimètwpoi me èna polÔploko kai abèbaio perib�llon, diamorf¸noun ektim seic, kajorÐzounprotim seic kai lamb�noun apof�seic me trìpo pou apèqei apì to klasikì oikonomikì prìtupo touèllogou orjologikoÔ par�gonta (rational agent). H mpiqebioristik qrhmatooikonomik epiqeireÐna ermhneÔsei poll� apì ta par�doxa fainìmena twn kefalaiagor¸n anazht¸ntac exhg seic sth mhorjologik (non-rational) pleur� thc sumperifor�c twn ependut¸n2.
Sthn prosp�jei� mac na sumbadÐsoume me tic trèqousec exelÐxeic, diamorf¸same èna nèo mejodologikìplaÐsio gia th montelopoÐhsh thc dunamik c tim¸n qreogr�fwn (omolìgwn, metoq¸n, parag¸gwn,k.a.) mèsw eufu¸n mejìdwn. To plaÐsio autì anagnwrÐzei ìti gia thn epex ghsh twn tim¸n apaiteÐ-tai o sunduasmìc jemeli¸douc kai mpiqebioristik c an�lushc. H pr¸th lamb�nei upìyh eurÔteroucmakrooikonomikoÔc par�gontec (timèc epitokÐwn, plhjwrismìc) kai qarakthristik� tou kl�dou thcagor�c (kladikìc deÐkthc, qrhmatisthriakìc deÐkthc, k.a.), en¸ h deÔterh epiqeireÐ na enswmat¸seimetablhtèc prìtupa (patterns) sthn poreÐa twn tim¸n (braquprìjesmec t�seic, makroprìjesmecepanaforèc, epoqiakoÐ kÔkloi), apotèlesma thc mh orjologik c sumperifor�c k�poiwn ependut¸n.To proteinìmeno mejodologikì plaÐsio anadeiknÔei th shmasÐa kai �llwn paragìntwn gia thn epituqh-mènh montelopoÐhsh tou mhqanismoÔ diamìrfwshc twn tim¸n, ìpwc p.q.
• thn arqik epexergasÐa (pre-processing filtering) twn dedomènwn2Qarakthristik thc metastrof c thc oikonomik c skèyhc se pio gnwstikoÔ-tÔpou arqèc eÐnai h aponom tou
brabeÐou Nobel Oikonomik¸n gia to 2002 ston Daniel Kahneman, pou se sunergasÐa me ton Amos Tverksyasqol jhke me proekt�seic thc gnwstik c yuqologÐac se oikonomikèc efarmogèc.
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• thn uiojèthsh mejìdwn gia ton entopismì tou {kajest¸toc} thc agor�c (market regime) kai
• th diexagwg elègqwn eparkoÔc poluplokìthtac montèlou (model adequacy tests) kai epex-hghmatik c isqÔoc metablht¸n (variables significance tests).
Na anafèroume ìti me b�sh to proanaferjèn plaÐsio montelopoÐhshc sunt�xame �rjro to opoÐo èginedektì proc parousÐash sto et sio diejnèc sumpìsio tou EurwpaðkoÔ DiktÔou Eufu¸n Teqnologi¸npou ja gÐnei sto Aachen thc GermanÐac (10-12 IounÐou)3.
KleÐnontac, anagnwrÐzoume ìti h parap�nw mejodologÐa paramènei sth arqik thc morf k�pwcgenik , qwrÐc na dÐnei safeÐc apant seic se erwt mata ìpwc p.q. poia eÐnai h periodikìthta twn epo-qiak¸n kÔklwn, pwc akrib¸c apokrÐnontai oi timèc se etairikèc kai �llec oikonomikèc anakoin¸seic,pwc anagnwrÐzoume metallagèc metaxÔ fusiologikoÔ kajest¸toc kai periìdwn {neurikìthtac} sticagorèc, poioi eÐnai oi akribeÐc èlegqoi ep�rkeiac montèlou kai epexhghmatik c isqÔoc metablht¸nk.a. Wstìso, oi arqikèc endeÐxeic gia thn isqÔ thc parap�nw mejodologik c prosèggishc, ìpwcprokÔptoun apì ta di�fora anagn¸smata, eÐnai arket� jetikèc. Ex�llou, ta epimèrouc {anoikt�}shmeÐa ja apotelèsoun antikeÐmeno èreunac sth metèpeita poreÐa tou didaktorikoÔ, apì thn opoÐaden apokleÐetai na prokÔyoun nèa sumper�smata pou ja odhg soun se tropopoÐhsh thc parap�nwprosèggishc.
Anafèroume ìti sth di�rkeia tou pr¸tou ètouc tou didaktorikoÔ ekpon jhkan oi parak�tw ereun-htikèc ergasÐec pou sqetÐzontai �mesa èmmesa me to jèma thc didaktorik c diatrib c:
• N. Thomaidis, G. Dounias, P. Vassilakis, “National and international sociopolitical andeconomic factors that influence the course of the national stock exchange”, submitted to the1st AFE Conference: Advances in Applied Financial Economics, Samos, Greece, 28-30 May2004.
• N. Thomaidis, G. Dounias, 2004, “Behavioural Finance: a new challenge to computationalintelligence”, to be presented in EUNITE-04, Aachen, Germany, June 10-12 2004.
• M. Michalopoulos, N. Thomaidis, G. Dounias, C. Zopounidis, 2004, “Using a fuzzy setsapproach to select a portfolio of greek government bonds”, to appear in the Fuzzy EconomicReview Journal.
• N. Thomaidis, G. Dounias, P. Zalimidis, 2003, “Measuring the Penetration of IntelligentTechnologies in Medical Business”, in Proc. of EUNITE-03, Oulu, Finland, July 10-12,2003.
• N. Thomaidis, G. Dounias, P. Zalimidis, 2003, “Intelligent tools and techniques for support-ing tourism services and applications, 2003, in Proc. of the 2nd Int. Conf. for SustainableTourism Development & Environment, Chios, Greece, 2-5 October, 2003.
Sth sunèqeia akoloujeÐ èna perÐgramma thc parousÐashc gia thn prìodo tou didaktorikoÔ pousunoyÐzei basik� sumper�smata kai k�poia pr¸ta eur mata pou proèkuyan apì th mèqri t¸ra ère-una. Sto tèloc tou teÔqouc parajètoume kai mÐa lÐsta endeiktik¸n phg¸n, qwrÐc wstìso na eÐnai
3European Network of Intelligence Technologies for smart adaptive systems (bl. www.eunite.org kai lÐstadhmosieÔsewn pio k�tw)
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exantlhtik . Anafèroume, plhroforiak�, ìti h paroÔsa èreuna qrhmatodoteÐtai apì to 2003 - 2004Prìgramma Upotrofi¸n proc 'Ellhnec tou KoinwfeloÔc IdrÔmatoc {Alèxandroc S. Wn�shc}4.
4bl. http://www.onassis.gr/greek/scholars/index.html.
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1 Dom thc parousÐashc
1. Perigraf tou q¸rou efarmog c
• TÐ eÐnai h qrhmatooikonomik mhqanik ?• DÔo basik� jèmata sthn qrhmatooikonomik mhqanik • An�lush qrhmatooikonomikoÔ kindÔnou• Epimèrouc probl mata
2. H klasik prosèggish sta qrhmatooikonomik mhqanik
• Genikèc arqèc, upojèseic kai qarakthristik�• P¸c h klasik prosèggish antimetwpÐzei ta epimèrouc probl mata?• Anoikt� probl mata
3. H Upologistik NohmosÔnh sta qrhmatooikonomik mhqanik
• Genikèc arqèc, upojèseic kai qarakthristik�• Mèjodoi upologistik c nohmosÔnhc• P¸c h upologistik nohmosÔnh antimetwpÐzei ta epimèrouc probl mata
4. Sumper�smata kai kateujÔnseic mellontik c èreunac
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2 Ti eÐnai h Qrhmatooikonomik Mhqanik ?
“Financial Engineering is the use of financial instruments such as for-wards, futures, swaps, options, and related products to restructure orrearrange cash flows in order to achieve particular financial goals,particularly the management of financial risk.” [9]
“Financial Engineering refers to the application of various [...] techniques tosolve practical problems in finance. Such problems include the valuation ofderivatives instruments such as options, futures and swaps, the tradingof securities, risk management and regulation of financial markets.” [9]
“Financial Engineering is the use of financial instruments to restructure anexisting financial profile into one having more desirable properties.” [5]
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3 DÔo basik� jèmata sth Qrhmatooikonomik Mhqanik
3.1 An�lush qrhmatooikonomikoÔ kindÔnou
An�lush - apotÐmhsh tou qrhmatooikonomikoÔ kindÔnou mÐac jèshc
3.1.1 Di�fora eÐdh kindÔnwn
• KÐndunoc Agor�c (Market Risk): Aporrèei apì th metablhtìthta qrhma-tooikonomik¸n metablht¸n (epitìkia, metoqèc)
• Pistwtikìc KÐndunoc (Credit Risk): Aporrèei apì thn pijanìthta ajèthshcthc upìsqeshc tou antisumballomènou se èna qrhmatooikonomikì sumbì-laio
GiatÐ eÐnai tìso shmantik ?
• An�ptuxh thc agor�c parag¸gwn (meg�loc ìgkoc sunallag¸n se par�g-wga, polÔploka sumbìlaia)
• Peript¸seic apotuqÐac meg�lwn qrhmatopistwtik¸n organism¸n apì th di-apragm�teush parag¸gwn
• Reustìthta qrhmatooikonomikoÔ perib�llontoc (kat�rreush tou sust -matoc isotimi¸n)
• Allag tou nomikoÔ kajest¸toc se pistwtikoÔc organismoÔc sqetik� meton kajorismì tou kefalaÐou {asfaleÐac} � Value-at-Risk (bl. parak�tw)
3.2 An�lush qrhmatooikonomik¸n ependÔsewn
Katanom kefalaÐou metaxÔ enallaktik¸n qrhmatooikonomik¸n ependÔsewn (qreì-grafa)
Prìblhma epilog c qartofulakÐou (portfolio selection problem)
• Exisorrìphsh metaxÔ apìdoshc kai kindÔnou
3
• Antist�jmish kindÔnou (hedging) - meÐwsh tou kindÔnou mÐac jèshc. KurÐ-wc mèsw parag¸gwn.
• Diaqronikèc EpendÔseic kai An�gkec se Kef�laio (Katan�lwsh) (Investment-Consumption problem)
4
3.3 An�lush qrhmatooikonomikoÔ kindÔnou
B mata pou akoloujoÔme:
1. Katagraf twn paragìntwn kindÔnou (risk factors) enìc qartofulakÐou:poreÐa metoq¸n, qrhmatisthriakìc deÐkthc, kampÔlh apodìsewn, k.a.
2. Prosdiorismìc thc dunamik c (poreÐac) twn paragìntwn kindÔnou - (mon-tèla prìbleyhc qronoseir¸n)
3. Prosdiorismìc thc sunarthsiak c sqèshc metaxÔ metabol¸n thc axÐac twntÐtlwn tou qartofulakÐou kai metabol¸n twn tim¸n twn paragìntwn -(timolìghsh tÐtlwn)
4. Prosdiorismìc thc katanom c kerd¸n/zhmi¸n tou qartofulakÐou
5. EÔresh enìc mètrou kindÔnou Value-at-Risk (VaR)
���������� �� ������������ �������� ������������� ���!"
Sq ma 1: O upologismìc thc VaR.
5
4 Epimèrouc Probl mata
4.1 Perigraf thc dunamik c twn paragìntwn kindÔnou
AnazhtoÔme montèla pou na problèpoun mellontikèc apodìseic twn paragìntwnkindÔnou (akrib c prìbleyh upì th morf katanom c)
4.2 ApotÐmhsh qreogr�fwn
Qreìgrafo:
• qrhmatooikonomik epèndush
• sugkekrimèno profÐl qrhmatoro¸n (cash flow stream)
• endeqomènwc abèbaiwn
Prìblhma apotÐmhshc
Dojèntoc enìc qreogr�fou me èna sugkekrimèno prìtupo plhrwm¸n
• PoÐa eÐnai mÐa {dÐkaih} tim gia to qreìgrafo (“fair” pricing)?
• PoÐa eÐnai h tim isorropÐac tou qreogr�fou sthn agor� (equilibrium pric-ing)?
– P¸c h agor� apotim� to qreìgrafo?– P¸c metab�lletai h tim k�je qreogr�fou an�loga me ton kÐnduno?
Prim kindÔnou (risk premium) = apìdosh qreogr�fou - apìdosh qwrÐckÐnduno (apìdosh agor�c)
– PoÐoi eÐnai oi par�gontec pou k�noun tic timèc qreogr�fwn (p.q. meto-q¸n) na summetab�llontai? (cross-subsection returns)
Di�fora eÐdh qreogr�fwn:
6
4.2.1 StajeroÔ eisod matoc (fixed income)
Eggu¸ntai èna stajerì profÐl qrhmatoro¸n (efìson krathjoÔn mèqri th l xh)
omìloga, èntoka gramm�tia, katajèseic
4.2.2 Metoqèc (equity)
H apìdosh twn metoq¸n eÐnai abèbaih.
Exart�tai apì polloÔc par�gontec ìpwc:
• th mellontik poreÐa to {profÐl} kindÔnou (kefalaiak di�rjrwsh) thcetaireÐac
• th poreÐa tou kl�dou thc agor�c (qrhmatisthriakìc deÐkthc)
• th genikìterh poreÐa thc oikonomÐac (epikratoÔsec timèc epitokÐwn k.a.)
4.2.3 Par�gwga (derivatives-contingent claims)
• Qrhmatooikonomik� sumbìlaia pou eggr�fontai epÐ enìc �llou qreogr�fou(upokeÐmenoc tÐtloc)
• Ta par�gwga sun jwc diapragmateÔontai se eidikèc agorèc kai èqoun h-meromhnÐa l xhc
• Oi apoplhrwmèc enìc parag¸gou sth l xh exart¸ntai apì thn telik tim th genikìterh poreÐa tou upokeÐmenou tÐtlou
• ParadeÐgmata parag¸gwn: options, futures
• H axÐa enìc parag¸gou phg�zei apì ekeÐnh tou upokeÐmenou tÐtlou.
• Oi apolabèc mÐac jèshc se par�gwga eÐnai epÐshc abèbaiec.
• All� up�rqei mÐa sunarthsiak sqèsh metaxÔ thc tim c k�je parag¸goukai ekeÐnhc tou upokeÐmenou tÐtlou.
7
4.3 Options - Sumbìlaia proaÐreshc*
O k�toqoc enìc (European-type) option èqei to dikaÐwma na agor�sei napoul sei mon�dec tou upokeÐmenou tÐtlou se k�poia hmeromhnÐa (expirationdate) kai tim (exercise price) pou kajorÐzetai ek twn protèrwn
O antisumballìmenoc se èna sumbìlaio option (ekdìthc � writer) èqei antÐs-toiqa thn upoqrèwsh na poul sei na agor�sei mon�dec tou upokeÐmenou tÐtlouefìson to dikaÐwma exaskhjeÐ
Gia thn agor� tou parap�nw dikai¸matoc o k�toqoc tou option plhr¸nei stonekdìth èna sugkekrimèno posì (prim dikai¸matoc) � eÐnai h tim diapragm�teushcpou dÐnetai sthn agor� (market/quoted price)
DikaÐwma agor�c (call option)
DikaÐwma p¸lhshc (put option)
Price at maturity
P r o
f i t /
L o
s s
K S 2 S 1
S 2 -K
In-the-money Out-of-the-money
At-the-money
Sq ma 2: To profÐl kerd¸n/zhmi¸n sth l xh mÐac jèshc agor�c se call option. K h tim ex�skhshckai S1, S2 dÔo pijanèc timèc tou upokeÐmenou tÐtlou sth l xh.
8
4.4 Futures - Sumbìlaia mellontik c ekpl rwshc*
O k�toqoc enìc future èqei thn upoqrèwsh na agor�sei na poul sei mon�dectou upokeÐmenou tÐtlou se k�poia hmeromhnÐa (expiration date) kai tim (exerciseprice) pou kajorÐzetai ek twn protèrwn
O antisumballìmenoc se èna sumbìlaio future (ekdìthc � writer) èqei antÐs-toiqa thn upoqrèwsh na poul sei na agor�sei mon�dec tou upokeÐmenou tÐtlou
Price at maturity
P r o
f i t /
L o s
s
K S 2 S 1
S 2 -K
S 1 -K
Sq ma 3: To profÐl kerd¸n/zhmi¸n sth l xh mÐac jèshc agor�c se future. K h tim ex�skhshc kaiS1, S2 dÔo pijanèc timèc tou upokeÐmenou tÐtlou sth l xh.
9
4.5 Prìbleyh metablhtìthtac (volatility)
Empeirikèc melètec èdeixan ìti h metablhtìthta twn qrhmatooikonomik¸n mege-j¸n all�zei me to qrìno akolouj¸ntac sugkekrimèna prìtupa.
OmadopoÐhsh metablhtìthtac (volatility clustering)→ Apokt� shmasÐa gia thnan�lush kindÔnou mÐac jèshc
high volatility
FTSE-100 returns (Jun 2000-Nov 2002
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Jun-00 Sep-00 Dec-00 Mar-01 Jul-01 Oct-01 Jan-02 May-02 Aug-02 Nov-02
low volatility
Sq ma 4: Oi kl�seic metablhtìthtac ìpwc apeikonÐzontai sthn hmer sia apìdosh tou FTSE 100.
FTSE-100 volatility (Jun 2000-Nov 2002)
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Jun-00 Sep-00 Dec-00 Mar-01 Jul-01 Oct-01 Jan-02 May-02 Aug-02 Nov-02
daily
vol
atili
ty (
%)
Sq ma 5: H hmer sia metablhtìthta tou FTSE 100.
10
5 H Klasik Prosèggish sth Qrhmatooikonomik Mhqanik � Basikèc arqèc ��������������� ������� �������� ����������� ��� !�"# $�%�&��'() �*"��+(�',�'�-% .�/0��%12�3�0�4�*+ (���������5� 6���7�00� ")1 %4���1,�(��*/1 �%�%!�8/19 %�"�4�/ �1 %4��/19��"�0�(�%"�*+ ���:��4%(-%�0)������#;<=>?<@AB
C�D���E��� �F�F��GD�D���� 6�3% &")"�1 H ("�8%("�*/1!�%!�*%(-�15� I�8%-% (����+ �0)������#5� �"�8%("�*/1 ��"%7�0/1 "��# Sq ma 6: H klasik prosèggish.
11
5.1 Oikonomikèc arqèc
5.1.1 Upojèseic sqetik� me th sumperifor� k�je ependut
• Ektim seic:Upokeimenikì mètro pijanìthtacAnajewr seic tÔpou Bayes
• Protim seicJewrÐa (Upokeimenik c) Qrhsimìthtac (Subjective Utility The-
ory) � sth morf tou Savage [15]
• Orjologik sumperifor� (infinite/full rationality)H agor� apoteleÐtai apì èllogouc ependutèc pou
– Lamb�noun k�je dunat plhrofìrhsh, pou afor� stic mellontikèc ap-olabèc qreogr�fwn, thn opoÐa kai epexerg�zontai orj¸c.
– Metatrèpoun thn parap�nw plhrofìrhsh se kat�llhlec ependutikècapof�seic
– Epilègoun to qartoful�kio me trìpo pou na megistopoieÐ thn ana-menìmenh wfèleia proexoflhmènwn mellontik¸n qrhmatoro¸n (discountedcash-flows)
– Oi ependutèc apotimoÔn k�je qreìgrafo me b�sh ta jemeli¸dh qarak-thristik� tou (fundamental analysis) - profÐl {kindÔnou} thc e-taireÐac, merÐsmata, genikìterh poreÐa kl�dou agor�c, k.a.
12
5.1.2 Upojèseic sqetik� me to perib�llon thc agor�c
Jesmikèc upojèseic
• Tèleiec agorèc: Den up�rqoun kìsth sunallag¸n �llec {tribèc}[kÐndunoc reustìthtac, {�noigma} prosfor�c kai z thshc (bid/ask
spread)]
• Short-selling : Poul�w èna qreìgrafo qwrÐc na to èqw upìthn katoq mou � isodunameÐ me mÐa jèsh epÐ tou qreogr�fou{arnhtik c} axÐac
Anaptugmènec agorèc
• SÔnolo èllogwn ependut¸n pou anazhtoÔn diark¸c eukairÐeckèrdouc
• Plhj¸ra proðìntwn (kurÐwc contingent claims)
Apotelesmatik leitourgÐa kefalaiagor¸n (efficient markets hypothesis)
• Mèsa apì th sumperifor� touc oi ependutèc antikatoptrÐzounstic timèc twn qreogr�fwn thn plhroforÐa pou lamb�noun
• Oi timèc antanakloÔn ègkura kai ègkaira th sunolik plhrofìrhshpou diaqèetai sthn agor�
• Proekt�seic sth dunamik thc tim c qreogr�fwn -TuqaÐo fainìmenopou exelÐssetai sto qrìno- stoqastik diadikasÐa (bl. kaiparak�tw)
13
Arbitrage
• Ti gÐnetai ìtan dÔo {parìmoia} ependutik� proðìnta up�rqounsthn agor�?
• JewreÐste dÔo tr�pezec A kai B pou daneÐzoun kai dèqontaiproc kat�jesh qr mata sto Ðdio epitìkio
• Upojèste ìti to epitìkio thc A eÐnai 10% en¸ ekeÐno thc B eÐnai12% (et sio epitìkio). P¸c mporoÔme na epitÔqoume kèrdoc ektou mhdenìc?
• Daneizìmaste apì thn A 10,000 Eur¸ sto 10% kai katajètoumeto posì autì sthn B sto 12%.
• Sto tèloc tou ètouc ja èqoume kerdÐsei 2% twn 10,000 Eur¸,dhl. 200 Eur¸, qwrÐc na èqoume kamÐa upoqrèwsh.
• To Ðdio ependutikì sq ma ja mporoÔse na pragmatopoihjeÐ semegalÔtero epÐpedo, p.q. 100,000 Eur¸ me kèrdoc 2,000 Eur¸
• Autì eÐnai mÐa morf arbitrage - epÐteuxh kèrdouc ek tou mhdenìc(qwrÐc th dèsmeush idÐwn kefalaÐwn)
• Se mÐa anaptugmènh agor� opoiesd pote eukairÐec gia arbitrage
q�nontai makroprìjesma. Mìlic k�poia eukairÐa gia arbitrage
parousiasteÐ oi ependutèc prospajoÔn na thn ekmetalleutoÔnpaÐrnontac kat�llhlec jèseic sthn agor�.
• Se kat�stash isorropÐac thc agor�c, oi timèc diamorf¸nontaikat� trìpo pou na apokleÐei to arbitrage.
• Sugkekrimèna, dÔo proðìnta me to Ðdio profÐl apolab¸n èqounthn Ðdia tim � Nìmoc thc MÐac Tim c (Law of One
Price).
14
5.2 Montèla apotÐmhshc se isorropÐa thc agor�c (equilibrium pricingmodels)
Exet�zoun to pwc h allhlepÐdrash èllogwn ependut¸n, mèsa apìto mhqanismì prosfor�c � z thshc, diamorf¸nei tic timèc se kat�s-tash isorropÐac thc agor�c
Kat�stash isorropÐac:
• sunolik prosfor�=sunolik z thsh
• k�je ependut c katèqei qartoful�kio pou megistopoieÐ thn ana-menìmenh wfèlei� upì ton eisodhmatikì periorismì
15
5.3 MontelopoÐhsh abebaiìthtac
JewrÐa pijanot twn & stoqastik¸n diadikasi¸n: basik� ergaleÐamontelopoÐhshc
• thc abebaiìthtac se mÐa kefalaiagor�
• thc dunamik c twn qrhmatooikonomik¸n metablht¸n (epitìkia,metoqèc, upokeÐmenoi tÐtloi)
Abebaiìthta ≡ Tuqaiìthta
5.3.1 Basikì plaÐsio montelopoÐhshc
Oi ependutèc de gnwrÐzoun exarq c thn telik kat�stash thc agor�c(timèc kleisÐmatoc qreogr�fwn). Kaj¸c o qrìnoc exelÐssetai lam-b�noun stadiak� plhrofìrhsh pou touc bohj�ei na apokleÐounk�poia sen�ria
It ≡ To sÔnolo thc plhroforÐac pou diajètei ènac ependut c thqronik stigm t
To It perilamb�nei:
• pareljoÔsec timèc qreogr�fwn
• dhmìsia plhroforÐa (public information/signal): epitokÐa, prooptikèc enìckl�dou, anakoin¸seic etairik¸n gegonìtwn (exagorèc, sugqwneÔseic, k.a.),politikèc kai oikonomikèc exelÐxeic, f mec k.a.
• Idiwtik plhrofìrhsh (private information/signal)
I = {It, 0 ≤ t ≤ T}, It1 ⊆ It2 gia t1 ≤ t2 (Ro plhroforÐac)
16
PoÐec oi sunèpeiec thc Upìjeshc Apotelesmatik¸n Kefalaiagor¸n?
• To sÔnolo plhrofìrhshc It antikatoptrÐzetai ègkura kai �mesasthn trèqousa tim diapragm�teushc (èstw Pt).
• Dedomènhc thc plhrofìrhshc It, h mellontik tim Pt+T exart�taiapokleistik� apì gegonìta pou ja l�boun mèroc kat� to di�sth-ma (t, t + T ].
• Tètoia gegonìta sumbaÐnoun kat� aprìblepto-tuqaÐo trìpo (una-
nticipated - random events)
• Sunep¸c, h mellontik tim Pt+T dojèntoc tou It eÐnai mÐatuqaÐa metablht sto q¸ro twn senarÐwn thc agor�c. TokalÔtero pou mporoÔme na upojèsoume gia thn Pt+T eÐnai mÐakatanom pijanìthtac � H dunamik qrhmatooikonomik¸n mege-j¸n stoqastikì fainìmeno.
• Basizìmenoi apokleistik� sto sÔnolo plhrofìrhshc It eÐnaiadÔnato na problèpoume diark¸c thn t�sh enìc qreogr�fou
• Basizìmenoi apokleistik� sto sÔnolo plhrofìrhshc It de m-poroÔme na epitugq�noume suneq¸c kèrdh
• Epomènwc,
17
Anamenìmenh (posostiaÐa) metabol tou Pt gia to [t, t+T ], dedomènou tousunìlou plhrofìrhshc It = 0
E
(Pt+T − Pt
Pt
∣∣∣∣ It
)= 0 ⇒
E (Pt+T | It) = Pt, ∀T ∈ (0,∞)
• H tim tou qreogr�fou akoloujeÐ mÐa diadikasÐa martingale!diadikasÐa qwrÐc problèyimec t�seic - ìpwc o tuqaÐoc perÐpatoc- fair game
• Ikanopoihtikì montèlo gia par�gontec kindÔnou ìpwc p.q. oimetabolèc twn tim¸n twn epitokÐwn me to qrìno all�
1. Oi ependutèc apaitoÔn k�poio prim gia na desmeÔsoun tì ke-f�laiì touc gia qrìno T (time premium = epitìkio qwrÐckÐnduno)Anamenìmenh apìdosh tou Pt gia to [t, t + T ], dedomènou tou sunìlouplhrofìrhshc It = rf
t,t+T
rft,t+T= epitìkio qwrÐc kÐnduno (risk-free-rate) gia to qronikì
di�sthma [t, t + T ]
Pt = E
(Pt+T
1 + rft,t+T
∣∣∣∣∣ It
)
H proexoflhmènh (discounted) tim tou qreogr�fou akoloujeÐmÐa diadikasÐa martingale!
2. Eidik� gia metoqèc, e�n akolouj soume thn Ðdia sullogistik enswmat¸nontac kai ta merÐsmata dτ , τ = t + 1, t + 2, ...,∞,
18
pou plhr¸nei h ekdÐdousa arq stic qronikèc stigmèc t + 1, t +
2, ...,∞ antÐstoiqa, tìte
Pt = E
( ∞∑i=1
dt+i
(1 + rf)i
∣∣∣∣ It
)
→ orjologik tim metoq c (rational price)
3. Epiplèon, oi ependutèc jewroÔntai eqjrikoÐ proc to rÐsko (riskaverse), epomènwc
• Anamenìmenh apìdosh tou Pt sto [t, t + T ] | It > rft,t+T
• H diafor� onom�zetai prim kindÔnou (risk premium)• Prim kindÔnou = Anamenìmenh apìdosh (dedomènou tou It)stouc susthmatikoÔc par�gontec kindÔnou pou ephre�zounthn poreÐa tou qreogr�fou
• Telik�,
∆Pt
Pt
∣∣∣∣ It = rft,t+T +Apìdosh stouc susthmatikoÔc par�gontec| It + ε
• ε→ tuqaÐa metablht - o aprìbleptoc par�gontac (innovation
/ surpise term) jìruboc (noise) sthn exèlixh thc tim c touqreogr�fou
• E(ε|It) = 0)
19
6 H klasik prosèggish sthn An�lush KindÔnou
6.1 Perigraf thc dunamik c twn paragìntwn kindÔnou
Klasikì montèlo: diadikasÐa Ito
dPt
Pt= µ(t, Pt)dt + σ(t, Pt)dWt (DiadikasÐa Ito) (1)
Pt: H tim tou par�gonta kindÔnou th qronik stigm t
Wt: h diadikasÐa Wiener kÐnhsh Brown. Basikèc idiìthtec:
1. Gia τ < t, h metabol Wt−Wτ eÐnai anex�rthth apì to sÔnolo plhrofìrhsh-c Iτ kai akoloujeÐ kanonik katanom me mèsh tim 0 kai diakÔmansh t− τ
(Wt −Wτ ∼ N(0, t− τ))
2. Sunep¸c, sto qronikì di�sthma t − τ eÐnai pio pijanèc mikrèc metabolècgÔrw apì to stajerì epÐpedo Wτ kai genik� ìso mikraÐnei to di�sthmatìso mikraÐnei to eÔroc twn metabol¸n
• Par�metroi µ(t, Pt) anatÐmhshc kai σ(t, Pt) metablhtìthtac →endeqomènwc stoqastikoÐ (dhl. exart¸ntai apì to epÐpedo touPt)
• Suqn� upojètoume µ kai σ stajer�. Sthn perÐptwsh aut mestoqastikì logismì paÐrnoume analutik lÔsh:
rt,T =
(µ− 1
2σ2
)(T − t) + σε
√T − t, ε ∼ N(0, 1) (2)
ìpou rt,T , log(
PTPt
)kai T to tèloc thc ependutik c periìdou
ParathroÔme ìti rt,T ∼ N((
µ− 12σ
2)
(T − t), σ2(T − t))
20
time
Fa
cto
r p
rice
PT=P
teµ(T−t)
time
Factor price
σ=10%
time
Factor price
σ=20%
Sq ma 7: H epÐdrash tou stoqastikoÔ tm matoc sthn poreÐa tou par�gonta gia a) σ = 0%, b)σ = 10% kai g) σ = 20%.
21
6.2 Ektim¸ntac th metabol thc diakÔmanshc me to qrìno
6.2.1 H prosèggish thc J.P. Morgan [11]
σ2t+1|It
= λσ2t|It−1
+ (1− λ)r2t (3)
rt: h trèqousa apìdosh epÐ tou par�gonta kindÔnou
It−1 = {rt−1, rt−2, ...}σ2
t|It−1: h (t−1) � ektÐmhsh thc diakÔmanshc thc apìdoshc dedomènwn
twn tim¸n apodìsewn pou diajètoume èwc th qronik stigm t− 1
λ: antistajmistikìc ìroc (eleÔjerh par�metroc)
• H ektÐmhsh thc tim c thc diakÔmanshc gia th qronik stigm t+1
prokÔptei apì thn ektÐmhsh tou epipèdou diakÔmanshc gia thqronik stigm t kai thn twrin metablhtìthta pou parathreÐtaisthn agor�.
• H par�metroc λ rujmÐzei to bajmì me ton opoÐo èntonec di-akum�nseic odhgoÔn se allag tou epipèdou metablhtìthtacsthn poreÐa enìc par�gonta kindÔnou
• Ac upojèsoume ìti kat� th di�rkeia twn dÔo teleutaÐwn sunedri�sewnparathreÐtai èntonh diakÔmansh sthn tim tou par�gonta pouodhgeÐ se mÐa uyhl tim gia to r2
t . Apì thn exÐswsh (3) anamè-noume mÐa aÔxhsh thc σ2
t+1|It.
• MÐa qamhl tim tou λ aux�nei th barÔthta tou par�gontar2t , me apotèlesma mÐa uyhl ektim¸menh diakÔmansh gia tic
epìmenec hmèrec. Antijètwc, gia mÐa tim tou λ kont� sto 1,
22
h epÐdrash mÐac endeqìmenhc uyhl c (parathroÔmenhc) metabl-htìthtac teÐnei na elatt¸netai taqÔtera.
6.2.2 Montèla tÔpou GARCH(Generalized Auto-Regressive Conditional Heteroscedasticity)
GARCH(1,1) (Bollershev [2])
σ2t+1|It
= γV + ασ2t|It−1
+ βr2t (4aþ)
me
γ + α + β = 1 (4bþ)
V - to makroprìjesmo epÐpedo metablhtìthtac
• H èndeixh �(1,1)� upodeiknÔei ìti h ektÐmhsh thc aurian c diakÔ-manshc σ2
t+1|ItbasÐzetai sthn pio prìsfath ektÐmhsh σ2
t|It−1kai
sthn pio prìsfath parathroÔmenh metablhtìthta r2t .
• Sth genik perÐptwsh tou GARCH(p,q) h ektÐmhsh thc au-rian c diakÔmanshc σ2
t+1|ItupologÐzetai apì tic teleutaÐec p
parathr seic thc r2 kai tic teleutaÐec q ektim seic thc diakÔ-manshc.
• To montèlo ekjetikoÔ stajmismènou mèsou ìrou eÐnai mÐa up-operÐptwsh tou (4) me γ = 0, α = λ kai β = 1− λ.
• Sta montèla tÔpou GARCH o suntelest c metablhtìthtac èqeithn t�sh na epanèrqetai makroprìjesma sth tim V .
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6.3 ApotÐmhsh qreogr�fwn
6.3.1 Qreìgrafa stajeroÔ eisod matoc
Basik jewrÐa tou epitokÐou (Krit rio ParoÔsac AxÐac)
Par�deigma
Omìlogo me l xh se N èth apì s mera
P =
N∑t=1
dtCt (5)
P ParoÔsa axÐa
Ct TokomerÐdio gia t < N , TokomerÐdio+Onomastik axÐa gia t =
N
dt Proexoflhtikìc par�gontac � Upìjesh suneqoÔc anatokismoÔdt = e−rtt
H (5) apoteleÐ mÐa tim isorropÐac tou omolìgou sthn agor�, hopoÐa apokleÐei th dunatìthta gia arbitrage!
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6.3.2 Metoqèc
Paragontik� montèla Meg�lo mèroc thc summetablhtìthtac twn meto-q¸n ofeÐletai sth koin epÐdrash enìc dedomènou arijmoÔ paragìn-twn.
Sunolikìc kÐndunoc mÐac metoq c:
• Susthmatikìc kÐndunoc (systematic risk)To mèroc tou kindÔnou mÐac metoq c pou ofeÐletai sthn èkjeshsto koinì sÔnolo paragìntwn kindÔnou
• Mh susthmatikìc kÐndunoc (idiosyncratic risk)'Eqei na k�nei me ta idiaÐtera qarakthristik� thc ekdÐdousacarq c. Diajètontac èna arkoÔntwc diaforopoihmèno qarto-ful�kio o kÐndunoc autìc exaleÐfetai!
Poluparagontikì montèlo K t�xewc
ri = αi + βi1rf1 + βi2rf2 + ... + βiKrfK+ εi (6)
r = α + Br + ε (7)
ìpou
• ri h (tuqaÐa) apìdosh thc metoq c sto tèloc thc ependutik c periìdou
• rfkH (tuqaÐa) apìdosh tou par�gonta k sto tèloc thc ependutik c per-
iìdou
• βik h par�metroc B ta (Beta) thc metoq c; mètro thc euaisjhsÐa thc tim cthc metoq c stic kin seic tou par�gonta k
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• εi h tuqaÐa sunist¸sa thc apìdoshc thc metoq c. Exart�tai apì ta idiaÐt-era qarakthristik� thc etaireÐac
Apì kataskeu c:
E(εi) = 0
Cov(rfi, rfj
) = 0 → gia k�je i, j, i 6= j
Cov(εi, rfj) gia k�je metoq i kai par�gonta j.
Apì upìjesh:
E(εiεj) = 0 gia dÔo opoiesd pote metoqèc i kai j → o mìnoc lìgocgia ton opoÐo metoqèc sumporeÔontai eÐnai h koin epÐdrash twnparagìntwn
Basik� apotelèsmata:
• Anamenìmenh apìdosh thc metoq cArbitrage Pricing Theory (APT) [12, 13]Upojètoume ìti up�rqoun sthn agor� qartoful�kia Pk, k = 1, 2, ..., Kkajèna apì ta opoÐa akoloujeÐ pl rwc tic kin seic tou par�gonta k (βPj =1, gia j = k kai 0 gia j 6= k).Epiplèon, k�je Pk eÐnai arkoÔntwc diaforopoihmèno ¸ste
∑i wiεi ≈ 0.
Tìte h anamenìmenh apìdosh k�je metoq c i, pou apokleÐei eukairÐecgia arbitrage, dÐnetai apì th sqèsh:
E(ri) = rf + βi1(r̄P1− rf) + βi2(r̄P2
− rf) + ... + βi1(r̄PK− rf)
ìpou, r̄Pkh anamenìmenh apìdosh tou qartofulakÐou Pk.
Epomènwc, kat� mèso ìro oi ependutèc den antamoÐbontai gia to mh susth-matikì mèroc tou sunolikoÔ kindÔnou mÐac metoq cSunolikì prim kindÔnou: βi1(r̄P1
− rf) + βi2(r̄P2− rf) + ... + βi1(r̄PK
− rf)
PrÐm lìgw par�gonta k: βi1(r̄Pk− rf)
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• DiakÔmansh apìdoshc metoq c
σ2i = β2
i1σ2P1
+ β2i2σ
2P2
+ ... + β2iKσ2
PK+ σ2
εi
Susthmatikìc kÐndunoc: β2i1σ
2P1
+ β2i2σ
2P2
+ ... + β2iKσ2
PK
Mh susthmatikìc kÐndunoc: σ2εi
• SundiakÔmansh apìdoshc metoq c i kai j
σ2ij = βi1βj1σ
2P1
+ βi2βj2σ2P2
+ ... + βiKβjKσ2PK
6.3.3 Sun jeic par�gontec kindÔnou
• Makrooikonomik� megèjh: epitìkia thc agor�c, plhjwrismìc,deÐktec oikonomik c an�ptuxhc [3]
• Jemeli¸deic (fundamental) metablhtèc: poreÐa agor�c, kl�dou,kefalaiopoÐhsh, k.a.
6.3.4 Prosèggish twn Fama & French [4]
DeÐqnoun ìti treic sugkekrimènoi par�gontec kindÔnou arkoÔn giana exhg soun tic mèsec apodìseic twn metoq¸n.
1. O par�gontac {agor�}=Mèsh apìdosh tou qartofulakÐou thc agor�c � apìdosh qwrÐc kÐnduno
2. O par�gontac {mègejoc} kefalaiopoÐhshc=Mèsh apìdosh enìc qartofulakÐou metoq¸n qamhl c kefalaiopoÐhshc �mèsh apìdosh enìc qartofulakÐou metoq¸n uyhl c kefalaiopoÐhshcKefalaiopoÐhsh (Market equity)=tim metoq c× arijmìc ekdojèntwn meto-q¸n
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3. O par�gontac Logistik /Pragmatik axÐa metoqikoÔ kefalaÐou (Book/Marketvalue of the equity)=Mèsh apìdosh enìc qartofulakÐou me metoqèc meg�lou B/M � mèshapìdosh enìc qartofulakÐou me metoqèc qamhloÔ B/M).
Telikì montèlo
E(ri)− rf = b1(+)
E(r1) + b2(−)
E(r2) + b3(−)
E(r3)
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6.4 Par�gwga
6.4.1 H mèjodoc tou replicating qartofulakÐou
• ProspajoÔme na broÔme èna qartoful�kio pou {antigr�fei} ticapoplhrwmèc tou sumbolaÐou (replicating portfolio)
• To qartoful�kio den eÐnai kat� an�gkh statikì. H sÔnjes tou anaprosarmìzetai qwrÐc ìmwc th qr sh idÐwn kefalaÐwn(dynamic self-financing trading strategy)
• B�sh tou Nìmou thc MÐac Tim c, h trèqousa axÐa tou sum-bolaÐou eÐnai Ðsh me thn trèqousa axÐa tou qartofulakÐou �eid�llwc up�rqei dunatìthta gia arbitrage!
6.4.2 Par�deigma statik c antigrafik c strathgik c
Sumbìlaio mellontik c ekpl rwshc � forward
ft trèqousa axÐa diapragm�teushc tou forward me l xh T
K tim ex�skhshc
T hmeromhnÐa l xhc sumbolaÐou
S trèqousa axÐa upokeÐmenou tÐtlou
Strathgik A: agìrase èna forward kai kat�jese Ke−r(T−t) sthn tr�peza
Strathgik B: agìrase ton upokeÐmeno tÐtlo
Kai oi dÔo strathgikèc èqoun telik axÐa Ðsh me ST
Sunep¸c (Nìmoc MÐac Tim c),
ft + Ke−r(T−t) = St
ft = St −Ke−r(T−t)
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6.4.3 Par�deigma dunamik c antigrafik c strathgik c
Sumbìlaio Dikai¸matoc - option
• Upìdeigma Black-Scholes [1]: H pr¸th epituqhmènh prosp�jeia gia thneÔresh thc tim c enìc dikaÐwmatoc agor�c EurwpaðkoÔ tÔpou (Europeancall option).
• To antigrafikì qartoful�kio apoteleÐtai apì ton upokeÐmeno tÐtlo kaiènan trapezikì logariasmì
• Oi jèseic twn dÔo qreogr�fwn anaprosarmìzontai se suneq b�sh.
• Gia thn agor� epiplèon metoq¸n daneizìmaste apì thn tr�peza en¸ opoiad -pote kèrdh apì th diapragm�teush metoq¸n katatÐjentai sthn tr�peza -(AutoqrhmatodotoÔmenh ependutik strathgik )
• O telikìc tÔpoc gia thn tim tou dikai¸matoc
C(St, t) = StN(d1)−Ke−r(T−t)N(d2) (8aþ)
ìpou
d1 =log
(SK
)+
(r + 1
2σ2)(T − t)
σ√
T − t(8bþ)
d2 =log
(SK
)+
(r − 1
2σ2)(T − t)
σ√
T − t= d1 − σ
√T − t (8gþ)
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7 Anoikt� Probl mata sthn Klasik Prosèggish
Poll� apì ta probl mata thc klasik c prosèggishc prokÔptounapì tic oikonomikèc kai pijanologikèc upojèseic
• AteleÐc agorècDuskìlwc reustopoioÔmena qreìgrafa (kÐndunoc reustìthtac),�noigma prosfor�c � z thshc (bid - ask spread), prom jeiec,jesmikoÐ periorismoÐ (èlleiyh dunatìthtac gia short - selling).
• Oi ependutèc den eÐnai p�nta orjologikoÐ:
– Den axiopoioÔn k�je dunat phg plhrofìrhshc (bounded
rationality)– Suqn� diamorf¸noun ektim seic pou de basÐzontai se jemeli¸d-
h an�lush thc etaireÐac all� sthn empeirÐa se f mec– StoiqeÐa thc yuqologÐac tou ependut prokaloÔn sumperi-
for� pou apèqei apì thn orjologik cognitive biases – (Kahneman and Tversky [7], bl. kai parak�tw)
– Anapotelesmatik leitourgÐa agor¸nOi plhroforÐec den enswmat¸nontai gr gora stic timècMèroc thc plhrofìrhshc den apokalÔptetai sqedìn potè(idÐwc h idiwtik plhrofìrhsh)H plhrofìrhsh den antikatoptrÐzetai kat� ègkuro trìpo(feeding the “bubble” [14], p. 204)
7.1 Perigraf thc dunamik c twn paragìntwn kindÔnou
• Jèseic se par�gontec kindÔnou <<mh dÐkaio>> paiqnÐdi
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• Probl mata gia to stoqastikì qarakt ra thc dunamik c twnparagìntwn kindÔnou
• Problèyima tm mata kai epoqiakoÐ kÔkloi (bl. kai tèloc thcdi�lexhc)
7.2 ApotÐmhsh qreogr�fwn
7.2.1 Qreìgrafa stajeroÔ eisod matoc
• Kumainìmeno epitìkioTo epitìkio all�zei me to qrìno - kampÔlh epitokÐwn (termstructure of interest rates)
• Abebaiìthta wc proc tic metakin seic thc kampÔlhc epitokÐwn(montèla prìbleyhc metakin sewn kampÔlhc epitokÐwn)
7.2.2 Metoqèc
Empeirikèc melètec èdeixan ìti oi diakum�nseic thc agor�c kai twnloip¸n jemeliwd¸n qarakthristik¸n den arkoÔn gia na exhg sountic apodìseic kai th summetablhtìthta twn metoq¸n.
Epiplèon qarakthristik� thc diamìrfwshc twn tim¸n:
• To prim megèjouc (The size premium)H apìdosh metoq¸n mikr c kefalaiopoÐhshc eÐnai megalÔterh ekeÐnhc meto-q¸n meg�lhc kefalaiopoÐhshc.
• Braquprìjesmh orm (Short-term momentum)Jetik braquprìjesmh autosusqètish apodìsewn tìso gia tic epimèroucmetoqèc ìso kai gia thn agor�.
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• Makroprìjesmh epanafor� (Long-term reversal)Metoqèc me {bebarhmèno} pareljìn apodìsewn prosfèroun makroprìjes-ma megalÔterec mellontikèc apodìseic kai antÐstrofa
• H probleptik ikanìthta qrhmatooikonomik¸n deikt¸nK�poioi qrhmatooikonomikoÐ deÐktec ìpwc
– hLogistik / Pragmatik axÐa metoqikoÔ kefalaÐou (Book/MarketValue of the equity)
– h Mìqleush (Leverage) (Upoqre¸seic / SÔnolo energhtikoÔ)– P/E (Price / Earnings) (Tim metoq c / Kèrdh an� metoq )
faÐnetai na exhgoÔn èna meg�lo mèroc thc diaforopoÐhshc stic apodìseictwn metoq¸n.Metoqèc etairei¸n me uyhlì B/M (“glamour” stocks) emfanÐzoun kat�kanìna qamhlìterec apodìseic apì ekeÐnec me qamhlì B/M (“growth” s-tocks ).
AntÐstoiqo fainìmeno parathreÐtai me to lìgo P/E
• Apìkrish se etairikèc oikonomikèc anakoin¸seicAÔxhsh metoqikoÔ kefalaÐou, sugqwneÔseic, allag kerd¸n kefalaiak cdi�rjrwshc, metabolèc epitokÐwn
H {èkplhxh} twn ependut¸n èpeita apì anakoin¸seic pou aforoÔn mi-a etaireÐa prokaleÐ proswrin� braquprìjesmh t�sh an�logou megèjouckai an�loghc kateÔjunshc (post-announcement drift) kai makroprìjesmhepanafor�.
7.2.3 Probl mata me ta paragontik� montèla
• Basik upìjesh twn paragontik¸n montèlwn eÐnai ìti ta {b ta}k�je metoq c kai ìqi ta idiaÐtera qarakthristik� thc etaireÐackajorÐzoun tic mèsec apolabèc. Gia par�deigma oi metoqèc axÐac
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kerdÐzoun uyhlèc apolabèc ìqi epeid èqoun uyhlì B/M deÐkth,all� exaitÐac thc meg�lhc {euaisjhsÐac} pou parousi�zoun s-ton par�gonta B/M. Empeirikèc melètec deÐqnoun ìti metoqèc mediaforetik� b ta kai Ðdio lìgo B/M den emfanÐzoun statistik�shmantikèc diaforèc stic mèsec apodìseic.
• Epeid èna paragontikì montèlo douleÔei kal� de shmaÐnei ìtimac bohj�ei na katano soume to mhqanismì kajorismoÔ apodìsewn.Oi Fama and French [4] isqurÐzontai ìti ta apotelèsmat� touceÐnai qr sima sto bajmì pou mporeÐ kaneÐc na exhg sei ta s-toiqeÐa ekeÐna thc sumperifor�c twn ependut¸n thc dom c twnagor¸n pou k�noun touc ependutèc na apotimoÔn ta qreìgrafakat� ton trìpo pou orÐzei to montèlo touc.
• Probl mata thc orjologik c prosèggishc me tic koinopoi seic- anakoin¸seic. EÐnai dÔskolo na exhg sei kaneÐc giatÐ ta primkindÔnou diamorf¸nontai kat� tètoio trìpo gÔrw apì tic h-meromhnÐec anakoin¸sewn kerd¸n, dedomènou ìti den up�rqounapodeÐxeic gia allag tou epipèdou tou susthmatikoÔ kindÔnoukat� tic hmeromhnÐec autèc. EÐnai dÔskolo na brei kaneÐc ènanpar�gonta pou ja exhgeÐ autì to fainìmeno.
• 'Alla probl mata: Upìjesh grammikìthtac, Statistikèc upo-jèseic mejìdwn upologismoÔ tou B ta (p.q. stasimìthta ermh-neutik¸n metablht¸n) - model specification bias
• <<Rebizionistik >> prosèggish (behavioural finance): exhg -seic gia ta parap�nw ja prèpei kaneÐc na anazht sei stic jesmikècatèleiec, sth mh orjologik sumperifor� ependut¸n, sthn anapote-lesmatikìthta twn agor¸n (bl. kai tèloc thc parousÐashc)
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7.2.4 Par�gwga
Oi atèleiec thc agor�c dhmiourgoÔn probl mata
• sto sqhmatismì kat�llhlwn antigrafik¸n strathgik¸n (jesmikoÐperiorismoÐ � short-selling, meg�lec prom jeiec apì th suneq allag tou qartofulakÐou)
• sthn ekmet�lleush eukairi¸n arbitrage kai sth diìrjwsh twntim¸n (jesmikoÐ periorismoÐ � short-selling, periorismènh orjologikìth-ta)
O Nìmoc thc MÐac Tim c den isqÔei p�nta
35
8 Upologistik nohmosÔnh (Computational Intelligence)
• SÔnolo mejìdwn/algìrijmwn pou ulopoioÔntai se H/U kai epiqeiroÔnna prosomoi�soun:
– stoiqeÐa thc anjr¸pinhc eufuòac (mhqanismì exagwg c sumperas-m�twn, sqhmatismoÔ empeirik c gn¸shc (expert knowledge))
– th dom tou anjrwpÐnou egkef�lou– genetikèc/exeliktikèc diadikasÐec
• Metexèlixh thc Teqnht c NohmosÔnhc (Artificial Intelligence)
• Basik diafor�: An�ptuxh mhqanism¸n exìruxhc gn¸shc/ mon-tèlwn pou perigr�foun to q¸ro efarmog c
36
9 Genik� Qarakthristik� Mejìdwn Upologistik c NohmosÔnhc
• Empeirikèc mèjodoi prosanatolizìmenec sto prìblhma (problem-
oriented)
– èqoun thn ikanìthta na majaÐnoun apì dedomèna tou q¸rouefarmog c (data-driven)
– par�goun gn¸sh montèla pou perigr�foun ton q¸ro efar-mog c
– h anapar�stash thc gn¸shc gÐnetai kat� kanìna me kanìnec dèndra → pio oikeÐa stouc eidikoÔc
– k�noun el�qistec upojèseic epÐ thc dom c tou q¸rou e-farmog c thc diadikasÐac pou par�gei ta dedomèna (data-
generating process)
• EÐnai tìso kalèc ìso kai ta diajèsima dedomèna
• Kat�llhlec gia q¸rouc meg�lhc poluplokìthtac: mh gram-mikìthtec - polÔplokec susqetÐseic, sumbibasmìc metaxÔ polu-plokìthtac kai katanoht¸n anaparast�sewn
• Dunatìthta diaqeÐrishc kai epexergasÐac posotik¸n/poiotik¸ndedomènwn
• Prosarmostikìthta stic allagèc tou perib�llontoc (adaptivity)(dunatìthtec autobeltÐwshc thc gn¸shc)
• Proseggistikèc mèjodoi: sun jwc katal goun se mÐa sumbibastik ikanopoihtik lÔsh (Ðswc upobèltisth all� antagwnistik ekeÐnhc twn eidik¸n)
• Suqn� apant¸ntai kai se ubridik morf (sunduasmìc epimèroucmejìdwn se èna sÔsthma)
37
10 Mèjodoi Upologistik c NohmosÔnhc
10.1 Teqnht� Neurwnik� DÐktua (Artificial Neural Networks)
SundesmologÐa apì neur¸nec kai sun�yeic metaxÔ neur¸nwn
Sq ma 8: H stoiqei¸dhc mon�da epexergasÐac tou ND.
Sq ma 9: Feed-forward Neural Network.
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To neurwnikì dÐktuo ekpaideÔetai rujmÐzontac ta b�rh twn sun�yewnmèqri na epitÔqei thn epijumht èxodo
To teqnhtì neurwnikì dÐktuo {majaÐnei} polÔplokec sunart seicme aujaÐreto bajmì akribeÐac
EntoÔtoic apoteleÐ gia ton eidikì èna {maÔro koutÐ} !
39
10.2 GenetikoÐ algìrijmoi - Genetikìc Programmatismìc (Genetic Algo-rithms - Programming, Evolutionary Computation)
• Stoqastik mèjodoc exereÔnhshc tou q¸rou twn dunat¸n lÔsewnenìc probl matoc.
• AparaÐthth mÐa sun�rthsh pou ja ektim� k�je dunat lÔsh(fitness function)
• AparaÐthto k�poio sq ma kwdikopoÐhshc lÔsewnp.q. duadik (binary) kwdikopoÐhsh - k�je lÔsh antistoiqeÐ seèna qrwmìswma (00010001)
10.2.1 B mata tou algorÐjmou
1. Epèlexe tuqaÐa èna arqikì sÔnolo pijan¸n lÔsewn (arqikìcplhjusmìc)
2. Upolìgise thn eurwstÐa (fitness) k�je lÔshc3. Ston trèqonta plhjusmì ef�rmose mÐa seir� genetik¸n telest¸n
mèqri na sumplhrwjeÐ ènac nèoc plhjusmìc:(aþ) Epilog (Selection) Me mÐa prokajorismènh pijanìthta epèlexe
dÔo mèlh tou plhjusmoÔ proc diastaÔrwsh (ìso megalÔtero tofitness tìso megalÔterh h pijanìthta epilog c)
(bþ) DiastaÔrwsh (Crossover) Me mÐa prokajorismènh pijanìthta dias-taÔrwse ta dÔo mèlh kai dhmioÔrghse dÔo apogìnouc (offsprings)
(gþ) Met�llaxh (Mutation) Me mÐa prokajorismènh pijanìthta {met�l-laxe} touc apogìnouc all�zontac tm mata thc kwdikopoÐhshc touc
4. Antikatèsthse thn trèqousa geni� me th nèa geni� lÔsewn.E�n h sunj kh termatismoÔ ikanopoieÐtai tìte term�tise tonalgìrijmo. Enallaktik� epanèlabe to b ma 2
40
• Sunj kec termatismoÔ tou algorÐjmou: ta mèlh tou plhjusmoÔeÐnai {arket�} ìmoia metaxÔ touc, o arijmìc twn gene¸n èqeixeper�sei k�poio kat¸fli, k.a.
• Oi genetikoÐ algìrijmoi endeÐknuntai gia probl mata beltistopoÐhsh-c sta opoÐa o q¸roc twn lÔsewn èqei exairetik� polÔplokhdom
• 'Otan h antikeimenik tou probl matoc den eÐnai saf¸c kajoris-mènh den èqei {kalèc} idiìthtec (asunèqeiec, gwniak� shmeÐak.a.)
• BasÐzontai sthn idèa thc prosomoÐwshc. Prosomoi¸noun -thn exeliktik diadikasÐa, thn {prosarmog } (adaptation) sthbèltisth lÔsh. Arg h sÔgklish tou algorÐjmou
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10.3 Asaf c Logik (Fuzzy Logic)-Asaf Sust mata Kanìnwn (FuzzyRule-Based Systems)
• MontelopoÐhsh asaf¸n poiotik¸n ennoi¸n (qwrÐc kajorismèno<<nohmatikì sÔnoro>>)ParadeÐgmata tètoiwn ennoi¸nAsafeÐc Ektim seic: Anodik /Stajer /Kajodik poreÐa thcagor�cAsafeÐc Protim seic: Apodìseic arket� megalÔterec tou 5%.'Ennoiec idiaÐtera dhmofileÐc sthn {argkì} twn eidik¸n
• Montèla diadikasi¸n pou perilamb�noun poiotikoÔc ìroucp.q. IF Dow Jones=slight increase AND FTSE=steady THEN
General Athens Index=Slight Increase
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10.4 Mèjodoi Mhqanik c M�jhshc (Methods for Machine Learning)
UlopoioÔn thn idèa thc m�jhshc apì paradeÐgmata-dedomèna (inductive
learning)
QrhsimopoioÔn dedomèna metablht¸n gia na parag�goun susqetÐ-seic upì th morf aitÐou - aitiatoÔ (causality relations)
Oi teleutaÐec dÐnontai upì th morf kanìnwn (IF - THEN) dèn-drwn.
EÔqrhsth kai katanoht anapar�stash susqetÐsewn diadikasi¸n(h orjìthta twn anaparast�sewn elègqetai pio eÔkola apì toucexperts)
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11 H Upologistik nohmosÔnh sth Qrhmatooikonomik Mhqanik
11.1 Oikonometrik prosèggish
Upojètoume ìti h data-generating process èqei th morf :
Yt = φ(X1t, X2t, ..., XNt) + εt
• X1t, X2t, ..., XNt � oi timèc twn epexhghmatik¸n metablht¸n(explanatory variables)
• Yt � h exarthmènh metablht metablht stìqoc (target vari-
able)
• εt � o tuqaÐoc par�gontac (sun jwc upì th morf gaussian
jorÔbou )
• φ � mh grammik �gnwsth nteterministik sun�rthsh
Poll� probl mata pou perigr�fhsan parap�nw ent�ssontai se autìto plaÐsio:
• Prìblhma perigraf c thc dunamik c twn paragìntwn - p¸c metab�l-letai h tim tou par�gonta kindÔnou me to qrìno?
• Prìblhma apotÐmhshc qreogr�fwn - poÐa eÐnai h sunarthsiak sqèsh apotÐmhshc pou qrhsimopoieÐ h agor�? - (prim kindÔnou= φ(X1t, X2t, ..., XNt))
• Prìblhma prìbleyhc metablhtìthtac - p¸c di�foroi par�gontecephre�zoun th dunamik tou epipèdou metablhtìthtac thc tim cenìc par�gonta kindÔnou
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GenetikoÐ algìrijmoi Exeliktik eÔresh thc sunarthsiak c morf c(functional form) thc φ
Neurwnik� DÐktua Ekm�jhsh - anaparagwg thc φ.
Mèjodoi Mhqanik c M�jhshc Montèla kanìnwn dèndrwn pou pros-eggÐzoun th φ
Asaf c Logik kai Sust mata Asaf¸n Kanìnwn (Fuzzy Rule-Based Systems)
Sust mata asaf¸n kanìnwn pou perigr�foun thn φ
• Qeirismìc kai apeikìnish asaf¸n ennoi¸n - metafor� thc apeikìn-ishc thc dunamik c enìc sust matoc apì to epÐpedo twn arijm¸nsto epÐpedo twn ennoi¸n (from reasoning with numbers to rea-
soning with words).
• exagwg proseggistik¸n sumperasm�twn (approximate reason-
ing)Dunatìthta montelopoÐhshc abebaiìthtac (probabilistic / pos-
sibilistic rule-based systems)
AntÐ na upojèsoun k�poia sugkekrimèn-h sunarthsiak sqèsh, oi mèjodoi UN{majaÐnoun} kai anapar�goun thn φ pouqrhsimopoieÐ h agor�! Pio realistik an-timet¸pish
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11.2 Teqnhtèc Agorèc Pollapl¸n Paragìntwn (TAPP) (Multi-AgentArtificial Markets)
• Enallaktik prosèggish sthn montelopoÐhsh qrhmatooikonomik¸nfainomènwn � akoloujeÐ sullogistik antÐstrofh thc oikonometrik -c (bottom-up)
• Oi TAPP prosomoi¸noun se èna teqnhtì perib�llon H/U ticprwtarqikèc diergasÐec pou sunteloÔntai se mÐa kefalaiagor�:
– Epimèrouc ependutikèc apof�seic (Sullog plhrofori¸n - Diamìr-fwsh ektim sewn, Protim seic - St�sh apènanti sto rÐsko, Epilog QartofulakÐou)
– Entolèc agor�c � p¸lhshc qreogr�fwn– Sunolik Prosfor� kai Z thsh, Ekkaj�rish thc agor�c (Market clear-
ing)
• DÐnoun thn eukairÐa ston analut na dei pwc diamorf¸nontai oitimèc sthn pr�xh.
• BohjoÔn sthn ermhneÐa diafìrwn fainomènwn pou parathroÔn-tai sth dunamik twn tim¸n stic pragmatikèc agorèc.
11.2.1 Enallaktikèc proseggÐseic TAPP
• TAPP kai Neurwnik� DÐktuaK�je neur¸nac anaparist�nei th diadikasÐa apìfashc enìc ependut . Topo-jet¸ntac polloÔc neur¸nec se topologÐa diktÔou dhmiourgeÐ kaneÐc mÐateqnht agor�. K�je neur¸nac dÐnei entolèc p¸lhshc agor�c, sug-kekrimènhc posìthtac qreogr�fwn. Mèsw enìc mhqanismoÔ exisorrìphshcthc sunolik c prosfor�c me th z thshc kajorÐzetai h tim kleisÐmatoctou diapragmateÔsimou qreogr�fou. H teleutaÐa apoteleÐ kai th metabl-ht exìdou tou diktÔou, thn opoÐa sugkrÐnoume me thn pragmatik tim thcagor�c
46
• TAPP me stoiqeÐa GenetikoÔ ProgrammatismoÔO genetikìc programmatismìc eis�gei to stoiqeÐo thc exèlixhc sth sumper-ifor� twn ependut¸n.ExereÔnhsh nèwn ependutik¸n strathgik¸nExèlixh montèlwn problèyewn
11.3 JewrÐa Sumperifor�c tou Ependut
Ta asaf sÔnola qrhsimopoioÔntai gia na ekfr�soun dunatìthta(possibility) kai protÐmhsh.
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12 Sumper�smata - KateujÔnseic mellontik c èreunac
• Oi no monec mèjodoi kat� kanìna uperèqoun twn sumbatik¸n s-tatistik¸n montèlwn sthn anapar�stash polÔplokwn susqetÐsewnkai thc mh-grammik c dunamik c
– H ikanìtht� touc ìmwc na anaparistoÔn me meg�lo bajmìakrÐbeiac polÔplokec mh-grammikèc susqetÐseic eÐnai tautì-qrona kai phg problhm�twn
– To dÐlhmma tou overfitting- Kat� pìso to sÔsthma ana-parist� to {mhqanismì} tou fainomènou kai ìqi to jìrubo?
– DÔo om�dec diagnwstik¸n tèst gia thn epituqÐa tou telikoÔmontèlou (bl. kai sq ma 11):1. Tèst shmantikìthtac epexhghmatik¸n metablht¸n (expla-
natory variable significance tests)2. Tèst ep�rkeiac montèlou (model adequacy tests)
– Tètoiou eÐdouc tèst èqoun protajeÐ gia ta neurwnik� dÐktua(bl. Zapranis and Refenes [18])
• H Ôparxh mh orjologik¸n ependut¸n stic pragmatikèc kefala-iagorèc dhmiourgeÐ probl mata sthn klasik (jemeli¸dh) prosèg-gish sthn an�lush twn tim¸n qreogr�fwn
– Oi timèc twn qreogr�fwn teÐnoun na mh antikatoptrÐzountic problèyeic gia ta jemelei¸dh qarakthristik� thc kl�douoÔte touc susthmatikoÔc par�gontec kindÔnou
– Problèyima tm mata kai epoqiakoÐ kÔkloi∗ Uyhl metablhtìthta se sqèsh me ta jemeli¸dh qarak-thristik�
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∗ Braquprìjesmec t�seic∗ Makroprìjesmec epanaforèc∗ T�seic èpeita apì etairikèc anakoin¸seic gegonìta (kèrd-h, merÐsmata, aÔxhsh metoqikoÔ kefalaÐou, exagor�, k.a.)(post-announcement drift) � Sun jwc, braquprìjesmht�sh sthn Ðdia kateÔjunsh me to gegonìc kai makroprì-jesmh epanafor�
– H enswm�twsh {mpiqebioristik¸n} paramètrwn sth montelopoÐhshtwn tim¸n qreogr�fwn dÔnatai na aux sei thn akrÐbeia ana-par�stashc tou fainomènou
– Wstìso, apaiteÐtai èreuna gia thn exakrÐbwsh kai ton en-topismì protÔpwn (patterns) pou dhmiourgeÐ h Ôparxh mhorjologik¸n ependut¸n
– MÐa nèa prosèggish sth montelopoÐhsh thc dunamik c twntim¸n qreogr�fwn proteÐnetai sto sq. 10.
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Reg
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Det
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Tur
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Nor
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50
ffdfdf
Specification testing
Model adecuacy testing
Test the residuals for randomness
Variables significance testing
Evaluate the statistical significance of explanatory variables and remove insignificant ones
Consider Ÿ adding new explanatory
variables Ÿ changing the functional form
No
Model selection
Model specification Fitness evaluation
END
Is the model of adequate complexity?
Are all variables
significant? No
Sq ma 11: MÐa diadikasÐa prosdiorismoÔ montèlwn upologistik c nohmosÔnhc. Upojètoume ìti k�jemèjodoc prosdiorÐzei èna montèlo tou tÔpou Y = φ̂(X1, X2, ..., Xn), ìpou φ̂ eÐnai mÐa ektÐmhsh thcsqèshc φ me b�sh ta diajèsima dedomèna.
51
References
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[3] N.F. Chen, Roll R., and S.A. Ross, Economic forces and the stock market, Journal ofBusiness 59 (1986), 383–403.
[4] E. Fama and K. French, Common risk factors in the returns on stocks and bonds, TheJournal of Financial Economics 33 (1993), 3–56.
[5] L. Galitz, Financial engineering: Tools and techniques to manage financial risk, IrwinProfessional Publishers, 1995.
[6] Ph. Jorion, Value at risk: the new benchmark for controlling market risk, McGraw-Hill,1997.
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[9] International Association of Financial Engineers, http://www.iafe.org.
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[11] RiskMetrics Group, J.P. Morgan, RiskMetrics, 4th ed., http://www.jpmorgan.com-/RiskManagement/RiskMetrics.
[12] S.A. Ross, Return, risk, and arbitrage, Risk and Return in Finance (I. Friend andBicksler. J., eds.), Ballinger, Cambridge, MA.
[13] , The arbitrage theory of capital asset pricing, Journal of Economic Theory 13(1976), 341–360.
[14] J. M. Samuels, F. M. Wilkes, and R. E. Brayshaw, Financial management and decisionmaking, International Thomson Business Press, 1998.
[15] L. Savage, The foundations of statistics, John Wiley & Sons, 1964.
[16] W.F. Sharpe, Capital asset prices: A theory of market equilibrium under conditions ofrisk, Portfolio Selection 19 (1964), no. 3, 425–442.
[17] Basle Committee On Banking Supervision, Amendment to the capital accord to incor-porate market risks, Tech. report, http:www.bis.org, 1996.
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[18] A. Zapranis and A.-P. Refenes, Principles of Neural Model Identification, Selectionand Adequacy: with Applications to Financial Econometrics, Perspectives in NeuralComputing, Springer, 1999.
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