SOME ASPECTS OF ARBITRATING

Transcription

1 Зборник на трудови од IV конгрес на математичарите на Македонија Струга, Македонија, 9-8 Стр Proceedgs of IV cogress of mahemacas of Republc of Macedoa ruga, Macedoa, 9-8 Pages OME APEC OF ARIRAIG ajaa Aaasova-Pacemsa, Lmoa Lazarova, ljaa Zlaaovsa Absrac Oe of he fudameal coceps uderlyg he heory of facal dervave prcg ad hedgg s ha of he arbrage hs cocep cera crcumsaces, allows us o defe he precse relaoshps amog prces ad hece her esablshme he mahemacal erpreao of hs cocep shows ha s ecessary o have owledge of moder heory of probably ad sochasc aalyss I hs paper we wll show ha here s a possbly of geg o rs prof o facal mare where he prces have radom characer Key words: prcg, facal mare,margal, arbrag, Iroduco Face s oe of he fases developg areas he moder bag ad corporae world hs, ogeher wh he sophscao of moder facal producs, provdes a rapdly growg mpeus for ew mahemacal models ad moder mahemacal mehods Amog he reasos o ge eresed facal mahemacs, he followg s oe: who has ever wodered, loog a he facal pages of a ewspaper, dsplayg he errac evoluos of quoaos o he oc Exchage, f hese were o govered by some models, lely o be probablsc hs queso was a he hear of he sudes coduced by Lous acheler, parcularly hs famous hess (9 ad he aswered he above queso erms of rowa Moo Laer, amuelso correced acheler ( 965 by replacg he rowa Moo by s expoeal ad he famous lac-choles formula bega ( 973 o play a esseal role he compuao of he opo prces Harrso ad Kreps (97 remared he exsece of a margale measure for he dscoued prce process mples he absece of he arbrage ce he 98s, we have wessed he exploso of probablsc models, alog wh facal producs, each ur becomg more ad more complex However, couous me case he absece of he arbrage s o loger a suffce codo for he exsece of a equvale margale measure A o-freeluch codo slghly sroger ha o-arbrage codo, was roduced by Kreps (98, who showed ha he exsece of a equvale margale measure f he dscoued prce process s bouded 374

2 I dscree-me case, he coverse saeme has bee proved by Dalag-Moro-Wllger (99 hs resul s referred o as he fudameal heorem of asse prcg Delbae ad chachermayer (994 wored ou a geeral verso of he fudameal heorem of asse prcg I hs paper we wll cosder a few versos of he proof of fudameal heorem of asse prcg We suppose ha he facal mare socs, fucos codos o suspese ad flucuao For s mahemacal descrpo s duced space of probably ( Ω, F, ( F, P, where: Ω s а space of elemeary eves for whch we suppose ha s fe; F s algebra of Ω subses (of all possble subses from Ω ; ( F s algebra flrao; P s probably rae (measure or probably he algebra flrao ( F we ca show le sream of formao avalable o all mare parcpas as of he me mome We cosder mare (, whch coss of d + asses followg sese: - s accou a ba ( o- rs (rs free asses d (,,, sor of asses ( rs asses he prce vecor a mare whch s cosdered has he mode d,,, acually (, mode, ha meas ha has he mode: ( d d (,,, (,,, he ba accou dyamcs ca be descrbe wh sochasc sequece ( whch has characer : ( : ( s F measurable, ha meas { ω Ω : ( ω x} { x} F al mome o he mare also occupes a specal place For we have a basc assumpo ha s sasfed F { O/, Ω} whch s rval algebra For dffere ypes of shares assume ha her dyamcs ca be,,, d descrbed also wh a posve sochasc sequeces ( ( ha have feaure ( : s { Ω : ( ω x} { x} F F measurable e ω hs shows ha here are dfferece bewee ba accou ad asses F measurable for sgfes ha ba accou prce s ow a he mome ha meas s predced O he oher sde F measurable for sgfes ha he values become ow afer all formao cludg he mome hs explas 375

3 ad why he ba accou ame rs-free asses, ad why he aco s called rsy asses We wll cosder he followg facors r Δ Δ, p ad for whch F we ca mmedaely coclude ha r are F measurable ad p are measurable From he facors r ad p we have ha Δ r, Δ p ( ad from here we ge oe represeao for he facal mare, whch s ow le represeao smple perce: ( + r ( + p (, ow, we wll precse he codos mare whch s observed Frs assumpo s he assumpo abou he deal suao o he mare he sese ha he operag expeses assocaed wh he rasfer of fuds from oe asse o aoher are cosdered eglgble Eve go o he exreme o assume ha hey do o exs Aoher mpora assumpo regardg he acos ad her properes amely we assume ha he aco besoeco produc he sese ha s possble o buy or sell ay shares how a small amou of socs Defo: he sochasc predcoed sеquece ( β, γ d ( γ ( ω,, γ ( ω where γ ad ( : γ are F measurable, β ( β ( ω ad β are also F measurable for ( s called vesme porfolo,, mare of asses o ( d oce ha varables β ( β ( ω ad γ (, ( ω γ ω (, γ may be posve, ull or egave whch meas ha vesor ca borrow from a ba accou or o sell socs O he oher had, he assumpo for F measurably meas ha he sze β ( ω ad ( ω γ whch descrbes he poso vesor me ( amou of moey ha has he sregh of he ba accou ad amou of shares you ow are deermed by formao avalable a he mome ad o (he ex poso s fully deermed oday Porfolo vesmes are also called vesme sraegy o be sressed hs dyamcs Defo: he value of vesme porfolo o (, mare of asses s he sochasc sequece ( where β + γ 376

4 Furher o, we wll use he shorer symbol γ ( γ, γ herefore we do o have o dffereae bewee cases where d ad d > o we have: β From hese wo defos we coclude ha Δ β Δ Δ + Δβ + Δγ where β Δ Δ s he chage of he ba accou ad he chage of he asses ad Δβ + Δγ s he chage of he asses porfolo srucure self aurally, ca ow be cocluded ha he real chages capal porfolo cosss oly of real value Δ ad Δ ad o he chage of he sze Δ β ad Δ γ Defo: Real (effecve prof from he possesso of he cosdered asses G for whch he porfolo, s descrbed wh sochasc sequece ( followg ca be appled G, G βδ Δ G ad hs meas ha cosss of he aure of he ba accou ad he aure of he asses chage he real capal a he mome s + G Defo: Porfolo of asses s called self- facg, F f s capal ca be preseed he followg mode + β Δ Δ, ( hs equao s equvale o Δβ + Δγ ( Obvous meag of he prevous codos s o chage he ba accou oly happeg because of chages he srucure of a pacage of socs ha eeps he aco ad vce versa I s clear ha he aco wh he porfolo we have oo may asses ad would be good o smplfy he srucure or o reduce hs large umber of asses For hs purpose, loo he relaoshp of hese wo cosdered sochasc processes ad e her quoe hs rao we ca see as he proporo of shares he ba accou, of course uder he assumpo ha we have for bass sze he hs ba accou Ad always s > he we ca observe hs same mare ad he sasfed ha ( 377

5 oher way amely, ow hs s see he mare fully descrbed slghly dffere model (, where: (, where,where ( approprae capal ( porfolo ( β, γ s equal o: β β β If he porfolo s self-face o he mare (, would be selfface ad o he mare (, because s sasfed he followg: Δβ + Δγ Δ β + Δγ Whe, Δ for self-face porfolo : + γ Δ o, ge o he self-face porfolo, sadardzed capal mees he equaly hs equao plays a ey role may of calculao ha s based o he cocep of mare o-arbrao he prevous aalyss of porfolo capal,, o he mare (, s good f you oe a few assumpos he frs s ha hs mare o flue ad sally moey or oher fuds ad ha here s o rasfer of expeses or hey ca be cosdered eglgble Hedgg Prce Compleeess I ecoomc sese, hedgg s he reduco of he sesvy of a porfolo o he moveme of a uderlyg asse by ag oppose posos dffere facal srumes Defo: Porfolo asses ( β, γ s called superor - ( x, f hedge, acually feror - ( x, f hedge f he followg codos are fulflled: x, x a he begg; f ( ω, f ( ω, for all ω a fe mome 378

6 β β γ γ x, f ( ω, ω where (, ( f he hedge s called perfec Hedge s a secury ool, whch eables guaraee come ad realzed capal parcular surace goal o he mare Hedge s vesme ha s uderae wh he am o reduce he rs, rese aoher vesme For fxed x >, we roduce: * H x f ; P : x, f ( ω ω class of all superor hedges (, { } ( ( x f ; P { : x, f ( ω } ( ω H class of all feror hedges *, Defo: f s crculag bod he value: * * C ( f ; P f { x : H ( x, f ; P O/ } s called superor prce (demaded prce of hedge crculag bod C* ( f ; P f{ x : H *( x, f ; P O/ } s called feror prce (offered prce of hedge crculag bod If we sell a corac wh he fal payme f ( ω hey would le o sell for maxmum prce A he same me we have o h ha f someoe bough for he prce ha we offer o we ca o wh oe had, pu he prce lower ha ha for he full corac erms ad ca o pu so grea ha we have o-rs yeld ( free-luch for he buyer geerally wll o o see * x [ C *,C ] ow, buyers ad sellers have a prce rs s compesao for hm pecal aeo deserves he case whe he upper ad lower prce whe he mach s me: C C Defo: (, secures mare s called -complee f each performg facal oblgao he sese ha here s perfec hedge ha me: f ω, ( ω 3 Arbrao ( I facal erms, here are ever ay opporues for mag a saaeous rs-free prof Prcsely, such opporues cao exs for a sgfca legh of me before prces move ad hus elmae hem he facal applcao of hs prcple leads o some elega modelg he ey words he defo of arbrage are saaeous ad rs-free prof y vesg eques somebody ca probably bea he ba, bu hs cao be cera If oe was a greaer reur he oe mus accep greaer rs 379

7 Defo: he self-facg sraegy realzes arbrage possbly (a he mome f: I order for he self-facg sraegy o acualze arbrage possbly ( he mome, he followg should be me:, ω he al mome ( ( ω, for > early cera (sure ha meas P ( > > We wll deoe wh F arb class of all arbral self-facg sraeges If F arb ad he: P ( P( > > I hs paper we wll gve some proofs of dffere aspecs, o he fudameal heorem of asse prcg heorem: (Dalag-Moro-Wlger We suppose ha he (, mare o flrag probably space ( Ω, F, ( F s esablshed of ba accou (, > ad d fe umber of asses (,,, ( We ALO suppose ha hs facal mare fucos he followg perod momes,,, < ad ha F O/, Ω, F F he hs facal mare (, s whou arbrage f ad oly f here s (a leas oe probably measure P ( margale measure equvale o he measure P, such ha relao o, he sequece of lowered prces s oe margale ha s: E P <,,, d;,,, E F,,, P Proof: Frs, we ca assume ha For self-facg porfolo we used he formula Δ γ Δ Icludg hs we have: Δ γ Δ From he prevous equaos we ca coclude ha he capal porfolo may be represeed as: 38

8 Δ + G, G γ Δ o show oly predcao s ow eough o show ha apply: F :, P G (, G ad ( ( γ Δ, ( ( ω, G, ( ( ω As sequeces ( are margales relao o he measure o have o apply: E G F E γ Δ F γ Δ E Δ F G + ecause : F measurable, for ad γ : F measurable, for ( ( ( ( For margale G we have: E( G E( G E( G Ad uque o-egave values wh zero mahemacal expecao s smlar o zero e G he equvalece of measures P ad P comes from he relaos: P P ( G P( G ( G > P( > G Ω be a probably space equpped wh a fe dscree- be a adaped d - Le (, F, P me flrao ( F,,,, F F ad le ( dmesoal processes Le R { : ξ H, H P} ξ where P s a se of all predcable d -dmesoal processes (e H s F -measurable ad H H Δ Δ, Pu _ A R L+ ; A s he closure of A probably, L + s he se of o-egave radom varables heorem: he followg codos are equvale: ( A L + {}; ( A L + {} ad A A (3 L {} A + (4 here s a probably P P wh d P L dp such ha s a P -margale 38

9 descrbes he evoluo of prces of rsy asses, ad H s he ermal value of a self-facg porfolo Codo ( s erpreed as he absece of arbrage; ca be wre he obvously equvale form R L + {} (or H H If Ω s fe he A s closed beg a polyhedral coe a fe-dmesoal space For fe Ω he se A may be o closed, whle R s always closed d L η R be such ha η lm f η < ha here are Lemma: Le ( d ( R η L such ha for all ω he sequece of η ( ω subsequece of he sequece of ( ω Lemma: Le L + {} s a coverge η K L be a closed covex coe L such ha + K he here s a probably P P E ξ for all ξ K wh d P L dp such ha Proof of a heorem: ( ( o show ha A s closed we proceed by duco Le uppose ha H Δ r ς as where measurable ad H H s F - r L + I s suffce o fd F -measurable radom varables whch are coverge as r L + such ha H Δ r ς as coverge Le Ω F form a fe paro of Ω Obvously, we may argue o each Ω separaely as o a auoomous measure space (cosderg he resrcos of radom varables ad races of σ -algebras Le H lmf H O he se Ω { H < } we ca ae, usg Lemma, F -measurable H such ha H ( ω s a coverge subsequece of ( ω for everyω ; r are defed correspodgly hus, f Ω s of full measure, he goal s acheved H O Ω { H } we pu G ad observe ha G Δ h as H y lemma we fd F -measurable G such ha ( ω H G s a coverge 38

10 subsequece of ( ω h ω G for every ω Deog he lm by G, we oba ha G Δ where h s o-egave, hece, vrue of (, G Δ G, here exss a paro of Ω o d dsjo subses Ω F, As ( such ha G o Ω Defe H H β G where β o G Ω he H Δ H Δ o Ω We repea he ere procedure o each Ω wh he sequece H owg ha H for all Apparely, afer a fe umber of seps we cosruc he desred sequece Le he clam be rue for ad le H Δ r ς as where are F -measurable ad r L + y he same argumes based o he elmao of o-zero compoes of he sequece H ad usg he duco hypohess we replace H ad r by H ad r such ha H coverges ( (3 rval (3 (4 oce ha for ay radom varable η here s a equvale probably P wh bouded desy such ha η L ( P Propery (3 s vara uder equvale chage of probably hs cosderao allows us o assume ha are egrable he covex se A A L s closed L ce A L {}, lemma esures he exsece of P P wh bouded + desy ad such ha E ξ for all ξ A, parcular, for ξ ±H Δ where H s bouded ad (4 ( Le A L+ ξ H F measurable hus, (, H E Δ F ξ As ( H F oba by codog ha E hus ξ H H E Δ, we 383

11 Refereces: [] Dalag R C, Moro A, Wllger W Equvale margale measure ad o-arbrage sochasc secures mare model ochascs ad ochasc Repors, (99 [] Dalbea F he Dalag-Moro-Wllger heorem [3] Par uloj, V I Perbarg, Elemes of Face Mahemacs, December 999 [4] Lecures Isue for Pure ad Appled Mahemacs, UCLA, Alber hrvaev, Esseals of he Arbrage heory, [5] R J Wllams, Iroduco o he Mahemacs of Face [6] Yur Kabaov, Chrsophe rcer, Laboraore de MAhemaques, A heachers oe o o-arbrage crera [7] Jelea Mlec, eorja Arbraza, 5 Uversy Goce Delcev p, Faculy of formacs, Republc of Macedoa e-mal: ajaapacemsa@ugdedum 384