nèect seicsth sÔgqronhqrhmatooikonomik mhqanik : apì...

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.

i

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

ii

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.

iii

• 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)

iv

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.

v

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

1

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]

2

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 .

23

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!

24

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

25

• 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)

26

• 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

27

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)

28

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)

29

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)

30

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

31

• 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�.

32

• 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

33

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)

34

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.

38

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

41

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

42

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)

43

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

44

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

45

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.

47

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�

48

∗ 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.

49

Reg

ime

Det

ecti

on

met

hods

Tur

bule

nt

peri

od

Nor

mal

co

nditi

ons

Exp

lana

tory

va

riab

les

Noi

se f

ilter

ing

- m

athe

mat

ical

m

etho

ds

- In

telli

gent

m

etho

ds

Beh

avio

ural

ana

lysi

s

- an

omal

ous

patte

rns

- be

havi

oura

l exp

lana

tory

va

riab

les

Filte

red

data

Fi

nanc

ial

data

-M

odel

ade

quac

y te

sts

-Exp

lana

tory

var

iabl

e si

gnif

ican

e te

sts

Mod

el id

enti

fica

tion

Is

the

mod

el

acce

ptab

le?

EN

D

YE

S

NO

Fun

dam

enta

l ana

lysi

s

Set o

f fu

ndam

enta

l var

iabl

es

abou

t the

com

pany

, the

se

ctor

, the

mar

ket a

nd

the

econ

omy

Secu

riti

es

mar

kets

-T

imes

erie

s -F

inan

cial

st

atem

ents

Sq 

ma10

:M

Ðanè

apr

osèg

gish

sthmo

ntel

opoÐhs

hqr

hmat

ooikon

omik¸ntim¸

n.

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

[1] F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal ofPolitical Economy 81 (1973), 673–654.

[2] T. Bollershev, Generalized autoregressive conditional heteroscedasticity, Journal ofEconometrics 72 (1986), 307–327.

[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.

[7] D. Kahneman, P. Slovic, and A. Tversky, Judgment under uncertainty: Heuristics andbiases, Cambridge University Press, Cambridge, 1982.

[8] H. Markovitz, Portfolio selection, The Journal of Finance 7 (1952), no. 1, 77–91.

[9] International Association of Financial Engineers, http://www.iafe.org.

[10] European Network on Intelligent Technologies for Smart Adaptive Systems, Descriptionof aims and scope of the iba-e committee: Finance, trade & services.

[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.

52

[18] A. Zapranis and A.-P. Refenes, Principles of Neural Model Identification, Selectionand Adequacy: with Applications to Financial Econometrics, Perspectives in NeuralComputing, Springer, 1999.

53