The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

Suepr-Hegdnig ndaAr btraige rPcingi n Mairekst iwh traTnsactoni Cstos na Trdaind Congtrasitsn

teSanfR J.sackhe

eDembcre3,9169Insttuitfu rMa hemttai Hkumobdt-lniUveristta zu eBlri nnUetr dneLi den 6 1n0909B elrni Ge,many rfax:+ 4 (903)2834 466 amil j:ashkce@mthametaik.uh-eblrn.de ihtp:t//ww.samhtematk.hui-berinld.e~/jsackh/ eucrrenlt ayt aml:i ajschekproba.j@ssueiuf. htrp:t/ww/.probw.jausisu.erf~/asjche/k

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

AbsrtcatTheabrirate gripcng irinpcilpeh s baeneu es td odeive rpric eelrtaino slke ite hlaBk-cchSlose orfmla aun Heatdh-Jarow-Mrorontmod els in the oncetx otff rcitoinesl masrktes ndau consnraitnde tadrng. Tiehe selratonism y ora aym not bego doa propximtionsat oeraliyt. venEi f,t eyh reac rtainlye no etforncedby r aelwor ld arbtiage becauresof tarnsaciotncosts an d rtaind gonstcraits.n Rceetly,n erulsst nose ervla spectsa of ht eoubdn ismpledi y tbe hasenbc oe afbrtrige ia realnistcim deol hsvaebe en stabelihed sb yeDrmoyd an Rodcafellkra( 1991);lE Krouai nd auQneez (991)2;vitaCnc anidK rataazs(1 993);D emroydan doRckafllea r(1959; Jouini)an Kdllaa l(1959,b), aand vCtaini andc aKartzas (1969).U ing shtsee ersltus andcont rbuiint sgom eenwo nes thi,sp aep rttemats pt puo togetter h\araleisictar ibrtgeap rcinigt heoyr" in ad scretei-tim emarke tikle htatin ( Hrrisonaand lPskia; 198)1.U dernt rnaacstoi ncsosta ndtr daingc ontrainst,s sverea alpecsstcha ngef nudmaetanll ycopmrad et oteh lacsicsa tlehroy.Th e -to1- 1eraltonibetw enep rceifu cntoinas land matirnalg meeausre cas nb meantainideo nyli fterhei sa umernari ewthiut tornaastcionc sotsan tdadinr congtsrinat. Asv rsien oo tfe fhndauenmtla hetoemro f nane (prescetnvalue pr ncipil) cen ba emantainei odnylo thf eetsof admssiblie traind stragtgeie si s caoe nnad trasncaion toctssare lin ae. rIntead sofl okiog fnr exoacts loutonisto the rpeens valutee qautoi (nmartiganl eeasurme) ones shuld lookof ro paprximato soeutlinso to th perseetn vluea equatoin,w ih the mterti cdpenedni gont ehs trctuur eof htetr asanctionc oss.

tKywoedrs:rbaitraeg oubnsd s,uerph-edgni,g ilearn pogrrmmiagn d,uaityl tehor,ytr asanctinoc ots, smiinmx arpolbmse JE cLaslsicat ion E4:3 C,41,C 61Mathe mtiasc ubjSetc Clsas caiitn:

oCedritsThs paperi i psrtaof an npuuilsed dochtrol thasies I'.dl ke to tihan myk usprveior, srPfosesroU e wKcuherl,f o rhi gsuiance addn upsoprt.

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

Suep-Hregdng aidnArb trageiPr cini gi narMketsw thiTransa ctin Coost sna drTadignConstr intsa I1tnordutcointSratig niwht he tlack-ScholeB sFormlua(Bla kc nd Sahcles;o1973 ),th teheroyo Maftheamical Ftnanci hae sebe nomdinaetdby B rwonin-amoiontridvn eomdlesa dno terhmo elsd hat atrein hreenlty ocmplte (eaHrirsno nadKre s;p 9179;Har irsn and olPiska;9811 Cox; nd aubRisnein;t19 5; H8eta eh atl.;1992.)T hsi\ cmpoletema krts eteory" hsugestg tso wtinghs t oteho piotns rtaerd:t hat i tis ossipleb ot udplicte ah(edeg an) dyervitaievcl iam by cleer tradvig inn het nduelryng;ia dn tht ahteo pital mehdgings rtaetyg i st ooctinunosu

ly aapt dnoe' stsaek in te uhdnreliny tg toeho ptinos d'lte a(detlahedg-in).g cAdemiaclly sapakeign,amkret asera ssmeu do betco pmltee ndeurco nitnuos utrdina. Ig ntihs theroy p,irces ofd eirvtivaea sess atred teemrnei dybarb trage.i f oIn eates khit tsehry loiteallr,y it s ii sobth rpactiacly lnadth oretically mesiealingd.In practic,e mots ediratvvis are notedup lcibla, eesepcilal yf tianrsactoni osts care onsidercde .Th termenedos usucces sofn ncaial eridvaives tamkret silek ht ehicaCgoB aor df oTadre or he tonLod nInetnatronil ainaFcial nuFuret sxchaEne gshwo shta dertvativiesare f ar rom beinf gerudndna. toMeroev, rapporixmtea dltaeh-edgignmay rpveo evy rcstlyo cmopared ot ohet redghig nprnciilps (Benseaid t ea.; 1l92). 9n Ithoey,rth very peinrcpielo f abrirtgeay eldsii enuqailtise mona gasset picers. ivGn ehe tripesc f ote hnudrleiyn gescurties,it hsi ladesto bounds o then ripe ocf teh ediravtiv.e eWll p,rctationier dson' tatk thiset hoeryl tirelal.y oMid caiton osft e hlacB-kcholSse-modl are secuecssfllyuu sde sa a ehivle tc doe enp irecfu cntioaln snad edhgng istrtaegesifo r mny atpey sf optooinc ntracto.sV reisns oor modi ctioan sof hte modles f Coxo etal .(1985 )a ndH aeh etta l.( 1929) areu se dotv lae iutneestrr ta derevatiievs .t sIeems taht rpce irelatonsi erivddeb yt int gcetairnar bitaregfr-ee,complet emodls eotthe r ae wolrdl rea n iacf gtod appoorixmtainso Now., we ave theh a letst aacaedimcallyun stasfiyni sigtuatoint ath hett eohryu se tod etch taih scompl\et market eomel dappraoch"d eson toex plan iwh yt worki isnpr atcci.eI n othr werod,s lBak-Sccolhs ewros, kut nbo tecausb ofe teharbi tager picingrp rnciipe. lAs attiticaslop tinov luaaitn ohtero ytathr garedsth e rpolebmo fdning a\g oo" pdrci feucntoialn a as robple ofm setiamitn oorlt reign,resp cteviley c,uod expllainwhy cerain modteslt etbetrt anh other.s Ana tlrentavieto looking or fnoesin gel pric ef ohted eivrtiave erlaitve otth enduerylnigi s to cnoisdr ehet rpic ebuond stht aaeren orcfedb yarb triaeg .Ths ii svey arpealpni gfrom otbha nconomiecan da rpoabilistbi ciewpoivnt th: essumapiot thatn amrktse area brtraieg-reef s weiaka d nolds hn priactcei 3

alm

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

sotall t e htime.A ribratg ebouds nneithr deeenpd n opererfnees cnr oo pronbbailiites{ w hic hra neevre nokw nfor sure{ btu noy lo tne het so fall posisleb sencraio. seRcnety,l rseult so nsevearlas pets of arbicrtageb onus hade beve nstabelishe bydDe rmody na dRcokaellfar(19 91; E) lKraoui nd Quanee (199z);2C vtiani anc darKtazas( 1939); Dremoyd adn Rcokfealarl 1995)(; Juoinian dKalal (1995al,)b, andCvi antci ad Knratazas(1 969)) wh,ch biuidl a sbsis far ao\rea lstic irbairaget pircin gheotr".yP eviorsluy ar,btrigeab ondus ofr tsoc kopitnos erw feond ut boe nuinteresitn gfom rn aacaedicmand us lees sfrm ao rpacicalt viwpeoint,as t h esperuhe-dgig nsrttaegyfo a Eruoreapncall on a tsck ois o but oney shra ofeth e udnreliny (gaviDsa n dlCrk; a995). I1n xedinc meomarkets, howveer, here tae rhnduresdo

f\erdiavtvise on"on enduerlyign:t e mhoneymark e rtte. Coarerpsnoinglyd,a bitrraeg buonsdar qeiuet tgih tfo r arnga efo scueirtis (eaJcshk; e1995). Coputmtaio nf orbaitrge bouansd sies entiallsyw osr tcasea nlysas ain adss ch couud be usledby re ugltaroy bodiesas taolo o astessst eh risknessi of annica inlsttiuito's pnotfolris.o hTispaper g ivsea ompacc revitew f oteh raitragebthe oryo f icnmopetl mareetksa dn deevlpso nw ecocnetps n i aisdcetert mies ettni. Sectgio n int2rdoceu sotatnio. Sentioc 3n devlope\scalsicals results" nia incnompleetm raekt wti dhiidvnd-epyinagse ucriiets .Wileheq uivlean tarmitngae measures alr tee hke concypet n ihet lacsiscalt eohr,ys etcino 4s hos what tteh ylooe shts irole toc nsistonet ripecf ucnitonalsif trans actoni ostscar intreodcue. dSectoni5 pr oevs agener liazatinoo fth eFu\damenntl aheoTerm foF inanec" ot mraekstwi tht ranascitno osts acd tnadinr cogsnratits.nSe ciot 6 nropivds ea hacartericatizn of oht ecnosisent ptirecfu ctninoal apspaerin ignt eh sctiones4 na d a5\sgod-o t unfctinoas".l2.1No atitno

A 2iDscrte eTiem MrkaetThe ollowinfgs tuepa d notatnonia erfa riylcl os te o(Hrrisona an Pdlska;i 191),8 wit hhe etxtnsione hat tesuritice psy aidvdeidn and tradisn ggneeates trrnasatcio cnosts. eWco sined r adicserte, int seetof tardnigda et 0;s1;::: ;T] a d n aine steto scefnarois We. htink of! 2 a aspo sisle bath p(!= ( ! 1:;::; !T ) ) There.i sn (anuknwon pr)boabliti myaseuer (P(f g! )0)a dn hetn autar llrtaiton=Ff F tgt2f;:0:;T g:,tF= ( rpoj;ss 2 1;::: t;,) rojp (t)=! !1 (;:::; t! . We a)lso hav ael tartin oG= (tG ) t,G tt tFath errepesntsthe i fnroatmino knwn oo taengt.sI twillb eonvcenint toeassu e F0 m G== f;;0 . Stgcohsati ccash os arewG- dapted aprcoeses sZwith Z= 0 and 0Z t=Z t? Z?1ti s th emaonu taip atdtim et .(oFr ay nporecs s Xwed e neX=:tX? X?t1. We h)ve ams cueirtieswi htp ice 4r

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

prcoeses Ss= (S1 t;:::; S t )m2ft;:0:;: Tgand c uulatmiv devideidn ropcsses D= (eD1;: ::; m )D2ft;:0::T;g 1 wihtt h einetrprtetaoint at Sh ti sht ee-xdviidnde rpice vetor at timcet nd consatan to nthet iem intrvae tl; t+ )1 .tDi i the disvdenid (r oocpuon) pid bay esurcty ii at imte .tD?1:= 0 S.an dD a er -aGdpate.dNote th ta Gn eedno bt ethe a-gelba rgeenartedby San d. DInvesort samyhave nfiomrtaoin rof mropescss teah tre nao ttraedd T.rading starteiegs raeG pr-dectaileb pocresesswi t thh einerpretattin toht it ia sht nuemeb rf soeurctiie soft ypei hel odve trh ietervna (lt? 1; t] .At ti e m intvstorse

stll hildo,trec eivedi vdiedn t stDon t eihr oldhngs,i 2obsevr theev aue tl t of tShie prroftoiol ta time-tex -ividdedn rpiec, adsjuts htierpor folio tfomr t ott+1 at ac sot o (ft+ 1? t)S t.De ne ( )+ t:=+t? t 1. eWw lil asl wriot et+:=t+1 tor eimnd ourselev that ts1 is+hoscn ewthi he kntwledgo eo fG tand ujs tfate timrtep icres ad dnviiendds are obsrvede . ts di eendfo t 2rf0;::;: T+1 gwhee 0 2r Ris he itnitail protfo

li oadn T 1 2+ (LT )G3i shett rmienalti emport flioo.The ets fo traingd trstaeges i Pight me brserittedc For. xemalpe i, (0nos hrt-selolni gi sencuitryi )o r S ji i joncst(l imis tn voloue).m i ' seaxc td eniitn woill d ir erfm osetciont os etcin.o If onhtignelse is s ai,dis omsesubs eto fth eset of G- redpitcalbep ocerses shtt cantaoin ast elat sthez ero tradng itsratgey.De ne teh vlua proecse st V()=:t+S t, whcih siG-a datpd.4 ehT caes ho w C () enegrate dby atardng itsrteay gsig ien byv tC () =t Dt (?+ )ttS? At )(; () C1? 1:=0. umulaCtvi terasantcino csto Ast( ) re aassmedu o tb

Aet ()= tac t(?1:= 0. AW eoulc dcnsoiderm+ t )= taci (X+ i ); tt=1i()2linaer ratnsaciotn cost s:atict(x )= ait+x+ bi xt? l,iner apuls xd coesst:tcit (a)x= atix++ibtx? +1fix=60g it c,or

tiwil l eb clearf om the rcnotxt. 2e oF rwt vectoro sxa d y let xn dyeonett e scahar lrpoucd. 3t etLL(G ) tendote ht set oefGt -emsuraabe lrnadmovar aibes.l 4Th s di iesr rof (Harrmsoinand lPisak; 911) fo8rt adrngi srattegiseth a tae rnt oeslnancingf.1I thni spaper xm endteost h em'htc moponetnof a vec or.t f wIe mena xeopnetiatnoi,

5n

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

earlitsc itransacton ioscs: tacit (t): s idscontiiuousn t 0a( becasueof ex doscts,) ha scocnae svape fhorsm la ltarnscatiosn b(ceuse oa fdcereasig ranestf or ighhr eolvues)m, an convex sdaphe or bigf tanrscatiosn (bceaue osf daditionl aamrket ipmat cocss for vtery bg iratnsctiaon). Heres, tcit, aati,bt, iandc t arieG-a adted prpcoesse.sWe have th efollowi gndi eerce equnatoinf o rV Vt(: )= t( S+Dt?)C t( )?At ( ) (:3 We) aclls ef- lanncnig f iC( ) 0-Pas..We od otn geernalyl asusmeS 0 (.Tinhk f theo aluv oef a interenstrat e wspa ontcrca, tor fxaeplme. We) as thytas euciryti ca nactas a unmearirei fS i> a0n dD P-0.sa .n this cIae sw de nee te chrrosepnoidn dgsciuonitg i pnocers s:=tS 0 Sti=

.2.2Supe -Hrdgeng and irbAtirae PgicrngieD e Z n i0fZ i incseraisng,which de nes n oadrerngio thne steof ca h owss.Co rrseopnindgl,y w edene aonnngaeitvepr iecfun tcinaol0 a asfunc iotnla tha is notnnegtave oin al ilcreanisg nrpceosses.W ecllaa t radni gtrsateyg 2as upr-hedeigng trastgye orfa c saho wZ if teh proesscC ( )? Z is ncieraisg nnad T+= 0P a-..sL e t (Z )+ denote thes e tof htee strasegiest. e deW en th eabrtirge appeur oundbfor te prihc oefa c sh awoZ realivt etoth earkemt(; G; P D; S;),t heset of tr dina gtrsateieg s, nad tarsnacion tocsst atc:)(as (Z) inf=?f0C( ) 0j= g0: 2 (+ Z) (4)eW cal alsolution of ()4a le sat osct upse-hrdgeing starteyg forZ . 'Z asrbtrage ilwoe rboundi s( )=Zs puf 0 C ) ( 0j 0g=2:+ (Z )?5)(Obviuosly,Z ) (?=?( Z)h old. Osna nietprob baliiy tspceaa n cysaho w i bounsedd. S, oifth er is a numerairee,ny aash cowcan b sepeurhe-dgd. Mero foemarll:yLe mma I1 fanum raeriee ixtss( escruti iy an)d conains tth ecoenf p reidcatlbej i 0 k; 0=8k= 6gi hent+ (Z ) is onnmetypfor a llc sah owsZ . 6

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

RearmkU nil

ke Demodr and Ryockaeflal (r911)9 we d ono mtode lhe txisteecneo af monte yn tieh den itoniof Y\s ita laset a soog ds Z a ("\Y=? Z is inceasing").r o Tmodel teh xeistnceeof ash,cin cule an dasst Se 1; D1 10an di mpsoethe ertsriciotn 01. T oodelm te hexsietnc eof anac ocntu wih tdi reen btrroowin and gledinn ratge s randR, include S11t= S0 2St 2= S0 tYs1= tY(+1r )s;D 01; 1( R+s; D)2 0;=1san dipome s10, 20. maAkrt iesc llae dstonrgyl rbairategf-er ie fhtee irsno ratidg sntaregty wth i 0 0,= Ct ( ) P0-.s. foa rll ta 2 0;:f:: ;Tg a nd P (C t( )> ) 0> f0ros om e. Stuh ac trstega wouyldprom is\meaye sobme ropt in het ufture"at aezro cot. sThem rket ia casledl( eakly) arbwitrge-freeai f( 0)= 0 .If 0)< (, 0ht ceroerponsdignt rdian sgrtteagyr eprsenetsa\ sur earibrtae grp o now"t Such.st ategirse aer clled aabriratg opeoprutntiei sf toehrs t type adns ceno dytp, enit hsi rdoe r(ngeIsroll 1;978, .p3)5. he Tontoinof rbiatarge i (narHisonr adn lPskia;1 91) c8orrspenods ot he trts tpey.A sw ec noised rhe tecson dyte opnl5y ew as ytathNA h ods li ftehm rkat ei s(wakeyl ar)birate-fgree. fI smooene anwtsto byuZ f r oap rci (eZ> ()Z )het senl lZat he tprice Z( ), akingt hetoblig tion ao tay p Zn the fituur, ferom (Z) set up a laestc sot sper-uedhinggs rtateyg,pokcetth e di eence rZ ( ) (? )Z wh,ci ihs a nabitrargepr to,iscen ew stratwi htn tohig n (0 0= )and C ( ) ovecrst eh bolgaiiot nZ ni ay nscearno. iAn nalogoua asrgmenut ohldsf o rth erabitrae lgwor ebuno. dIf smoeon eants wo tsll e Zofra prcei(Z )< Z ()b u Zy so,w weli lrceivee Z i nthe utfue,ro fteh rt typse des oon.t oSi ti somerco venient notcon idser ht eecosn dtype Al.o,sa rbtiarego pproutnties ifothe rt stye pthta o rot narbitrga epoortupnitis eofthe s cenodt pey d voirtuall y\not oc"cu rni prctiae.c5A ribrate gofthe esondct ye corrpespodn st oopimizttioanov r closed seest wihel htt

Jauts iactoin fothe N mea7

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

et su ap raditg nstraegt wyohes aschow i scoever bdyZ ((Z C+( ) ))0, ockep tth di eerecne(Z )?Z( ) .El aKrou andi uQene (19z92 co)nsdire upsre-edghni gf ocntingont ecalmsi (apoys a t ht teeminarlt ime oTny) lan cdlla tehi pmilde rpicingbou dn suybin gricp end sellian prgci, eersepticely. veBsaid ne tal. (192) n9me taehe sobunds anumactfruig costn an dreevne, ruespeticvley.De romd anyd Rckofallae r(911, 1995)9c noides rdteerminstici cahsow sna calldt ehc oresporningd obndu smiuted polgn nad hsort pirce,re sectivpey. lCivanic ant dKaatzars( 9913,996)1 cnosiedrt he puperb unod and cla itlhe ding gprce. Jouiini an Kdlaal l1995b(a) us,et h naeem raitragbebou dsn .I thni subsecstoni ews tta soem earhtre bvoiou saftcs aout babitrraeg oubdns. aCll fmpapin ga evcor stpce a Mtoa onhter vcetros pacel niea ri scnle af i f x)(= f ( x )froa l lx2 M an d 0.>2.3omSePr portiese of th Arbetraie gBondus

Lmeam 2If tehre ra lineea rrtanactsin cootssa d ni sac oe nhte andnreal inea rn scaile.Ina r alietic mosdelm ihgtn o tb

ea onc,efo rxeample if s,orth-esllngi s iallweo dut brsterctei dn iize.s

Lema 3m f iIs lneia inrsc lae het nhet mxiaml arbaitrga erp t o (0)? si ieher t0 r o1.tehn is ocven xnad i cosncae.v

eLmma4 fI thefu ctnios x n7 t!cia(tx) raecon vx eandt e hst ei csnovxLete+ (Z ) d eone tht eetsof tr dang isrtategesi tat hupse-hedgerZ on he tt ntievra t+l1; ]T:+ (Z )= f 2j C(( ? ) Z) 0s8s 2f t+ 1:;:: ;Tg; T+= g0: teDn (e)6 t(Z):= ni fVft ( )+ At( ) j = 0tg+

2.4 A brtirag eBund Poorecssseand (tZ ):?=t?( )Z.T e huppe and rowlerarb itrge aobnudpro cesess ra Ge-aadtpd. (Oesbevert ht? aCt( )=Vt ( )+A( ) undtr te 0=.)82t Z( )

In

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

the rpesence f troanasctoni csts oro shot-serling lcosnrtiatnsit i senecssrayt oc osindre he initial tortpfloi onit heop imizttioan rpolebm( seeal s oDe(rodyma ndR okafcllaer 1;99)5.)D eent (;Z ) x

=in fftV ()+2 +( ) tZtA )( j t= gx

(7)ast eh uctsoized mrsik-free uperp obnd valud ifr oa invensto rith twe hoptrfloo x ita tie tm t. (;Z x)=? t (?;Zx ) s tihecu stomizd eirsk-ree lfoer bwuno dvlad foi rn inaesvtrowi h the ptorftlooi x attime . (t tZ() vlauesthe re minaing cas how Zt(1+;::: ;T Z),w hcih orcesprnodst ourocon evtnin thotap ircsear ee x-divdiend.)? (0;0 )x= 0( 0 x;)i ths mexamil arsi-fkere ortfpoloiim rpvomeetn rp o tpsoibse wliht anin iitalportf looi x .Reamr Fkroma mtaehamtica pelrpsctivee neo coldu acll 0(;x a) arbnitrgea ro t pof allrx sinc ehit si asc ompletly riske-fer peo r.tI nec nomois,c owheev,ran rabtiaregh sa ac onntoaito nfobe nigp ssiolb weihtot inutial iwealt. hI fhete rar enot arsnactio nostcsa ndn o rtaidngc onstainrs,t hitsdi tsincito nsi ot nneecssry aand t Z;( ) x= (tZ;0) f r alo x. Delmrdyo an Rocdkfalela (r195)9cal lt h anaeoguelof (:;x)a n (d: x;)cus toizmd lengo ndashort pr ies.c

3Te Idhel MaakreItnthis secti o newd velepo ady naicma brtraie gripcng thieoryi disnrect eime. Itedas aer weo dot H(raisron ad Knepsr;197 9;aHrirons adnPli ks; a9811;De rody andm ocRkaelfalr; 991; 1El Karui aondQ enuze 1;92;9 vCitnia acd Karnazts; 19a93) T.ehmain pino wtll be toisho wt (Z t) (Z )= uspE Q=s Q2P s=+t1t TX s i n fE QQ2Ps t=+1t

T XZsj t];GZ j Gts];(8 )9()whic hac neb sene s aap reest vnlau epinrcipe lfo a rospiblys niocmlpte eutbo threiws eidel maarket.deIl maakretm ensa

:There raen o trnsactaonic sot,s9

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

teher aer on retsicrtoinson tra din gstatregie (tshat is, i ths seet foal lG-rpdietcbale porescss eitwh+T=0),an d al inlvetsro osbesvre ht esmaes te f inofrmotian oG()a ndre ecve ith easm eacsh ow fsrmoa ivge nstrtaey g(i..e t,erheare no ta xs). 1e e aWlso asusm etehe ixtesne ofca numerair Se 1>0 a d nde ent S= 0St1 .= i sav etco rpasec .etL D dneote theset f oal l-adaptGd preceosess,hewe Pr in-disitngusihalbepr coesessar e dintei de C . !:D i s lanier mapaping A. fnuciotanl: D R! i scaled al onsictsetn ricep ssytm if esi lieanr,sino nneatgiv (e ),0a d n( C ())= V (0 )f ro lal2 . eLt

denteo teh esto focnisstne ptrceis ytesm. As ropbaibilt ymeasur e Qhtat i de nsedon ;( T G) na id sabsoutleyl oncinutuso iwthr epsct to Pe( Q Pj GT) sica leld maatrngial meeasreui fthe dis coutndegain p ocres Xs~= SS+D 6 i a (G;s )-Qmratignle:aE Qt1 (S++1t+ tD+ )1 Gtj=]t St:1()0L te P dneoe the stt oefma trinage mlesuarse.

X~X 0V (+ S)=( V+ is)a Q- amtirnagel.Lemm 5a Fr oallQ 2 anP dlla2C()

1()1rPoo.fChe c ekqauton i(11) b tykina dgie rncee. sTe heltfha nd sdi ie sa P Xsia ma tinrgal feromar itgale benacuse thestoc hasictsu m i(negrta) aln yamrintalg eX ad nan prydiecatblei tngrenda 2

The.rom e 6hTee rs ia noe-t-ooen rlationebet ween cnsostiet pnrce syisemtsa dn artmignae melausrs oen(; GT by)(Z )= EQZ a]dnQ (A )=(0(:;::; 0;1=A T)): usmmtiaon nidex P. e WiwllfXerquentl dyropt hvaeue isl0 at imet0 . Thne( ) whostes1= ss t2f0;;1::;:TgTX

6P Xdneote tshe rocepss(21)(1 3)

10

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

Proof .2 igvn. eThatis nnnoegatie avdnl iena rnsuees thrt a Qedn ed b y(1) 3sia emasure To. se etah t Q P re,cal lhat it in fscat afu nctinoalo n he tqeuiavenlec cassls iendcue byd -indiPsingutsiabilihy. thTi esnsresu tha Qt(A) d ned bye 1()3 is erzofor a ll2 ad n -Pull sntse . ADe nek=1fk 1g=1t2(f;0]Tg 1f! g; i2.e, h.ld toe humenrirea throuhgout0 T;.]pA ptl yte hocsinstecnye uatqinoto and osbrve Qe ()= 1,t ha is t is a pQrobabliiytme sauer. Nw doe n tehe tadrnig stategyr htt satrts awit nhtohin,gbu s y Xshraes of esuricty 6k 1 a= ttie t man desllst eseh shreasa tti e mt 1+( j= X1fj=gk1 f=ts1g+) .I teneragte she ctah sw: o site cmah swo C( )

tX?Skt t+1 XS(k+t1+ tkD+1 )Ths traiindg tsrtega isy -predGicablt feoran yt -meaGsuablerra nod mvriaabel X. e haveWTX

( )C=(

kX t1+ (t+1+S~Dk+1 t)? t S kt) = XtS+1 k:

~~Si ne is consicsent,tE QX S kt1+]= 0 8 X2L( Gt ):;encHe S k,s ai1 isa armtngilea nyaway,as it si ocstnat.~n(;GQ -)mrtainale for agl l 6k=1 S Q.2 P ivge.n bOviousy, (1l)2d en esa onnneatgvi leiearn uncfiotnl an o.DE uqaiotn 1(1,)l mmea5 ndaV ( T)= 0mply that iEQ C()]= V0( ) for a ll 2ndaQ 2P . Tis ehnures tsath e dnd eyb( 1) 2sico sisnentt.I teramisnto s ohwtha (1t2)P 13()a r enievsr maepinpgs .Strta iwht nada 2QP a d nd ene(Z )= E Q Z]T A.pply 13( to)ge t~ Q= (0(;:::;;0 A1=T ))~ nad bsorve teat Q(Ah)= Q ()Afor al lA2 G T On .th ethero ahnd, star twihta an2 dedne bQ y1(3.) Q 2 Psasho w abovne. Aplyp 1(2)to ge X Z t): ()4) 1~Z (= )((; 0::;:0;1T TX

S oew hveat osh o twaht (ZX)= ((0:;::; 0; 1

ZT )1)115)

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

holds fo( arn yocsisntnetp icre ysset. Tmh ed ierence fot hec sh oa Z wandthe cash ow n tihe igrh hatnd idseo f( 1) ca5n eb egernaetdby a radtignst ratey 2 gtht atadrseo nl yi nte hnmuerarei ( k=; 08 6= 1k) adnh s a zaro eniitalicos (V0 t( )= 0 ).Ap plyin gte hocdnitio nC(:)()=V0 (:) t os ohsw tat (h1) h5losdfo any 2 . Trhs cimolepts eth epoofr (.Not teah t(1) a3dn(12 )a e ron gtneearlylin evrs eamppinsg utbo lynfor consi tenst rpceisy stesm) 2Th.e markte i asbitraregf-reeif nd anlyo fi consaisettn pirecsys temexis st .n Itihs csea thear biragte ounbs dfo te hpirecof a asc oh Zwcan becom pteu ds ati smxaimlaa n mdniial pmersnet valu,ere septcievyl,tak ne ovrea llconsi stntepr ci systeesm: 0Z( )= mxa(Z )=m xa QE 2 2PQT

XTheore m 7Pesrnet Vlue aPircnipelZ];]Z:(16) 1()7mi n0(Z )= m i nZ ( )= 2QP E 2QTXTh eotpmia lvauls aeer ataitned .Fr tohe prblom e16( th)simeans th t a atrdingas tratge y+ (2 ) Znd a aartimnage mealures Q2 P eixs tsuc hhtta

?0C()= QEhodls.X

Z]Proof (1.8) 0 Z( )= ifnf?C 0 )( (C j()? Z t) 08t 1 P-a. .s: g Writ2ig nte hLagrangina:=nif upsf C? (0? () C( )? Z g ):( 91 2 )2; D w0ehr ie san onnegtiae vinelra fuctinona ln D.oN w eochxngea nf ian sdpu:D2; 0

us

p

) (Z (+) nf fi?0C () (? Cg):{z} 2|()

(02

( ) i) szeor if adn nly oi 2f, ohertwis ei tsi?1 ( snci iesa ectorvspa c aendC isl niera.) m=xa (Z (2)) 1 122

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

adnb y heoret (6) minSec i sine, ts i nati-edmeniionsa lvctoresp cea adn hte s(tonrg )dalityut heoemro f lnier arpogrmamnigapp lei: 1.sT e primalh( 81 h)s a naiet soultoi nf and oniy lif th feasebielset nit h edula(22) i s nonmepy.t . 2nIt ish csae, asadledpoin (t; o) tfe Lagranhgia nexitss:= ma xE Q2PQT

X]:Z(22)

?0( C=)V (0)= (Z=)EQX

Z]:

Th eequtaionf orfo lowl frosm(Z )= (??Z).

2

heoreT 8 Fmo rna ysoctastich cahsowtZ Z(= Q)2P

uspE Q

ss=t+ 1

TtXs Z Gjt] 8P-a.ts.+Z ()and TX(2)3

hold ansd hetere ixtst

2Q2 Ps uc hhtta(4)2V t (= t ()Z=)1 QE Z jtG] t8P-.a.sTh eaalonguos ersult ohlsdfor t eh aritbrgael owe rbundopr oces t sZ(). R mare Pkricinpllay, hert coued bel i dernteop itml a+(Z2 fo) erveryp air (; t) !if w(e nok whtat he tnifin 6)(i sattanide(t; !)-wise). S toihst hoeerms tats e 3afts:c1 . hteduali yt gp ai zsreo 2; .if nnad ups rea attined; a3. tere hxiestsa pia (r; ) Qhtat i ssmultanieosluyoptim a flo rall (; t).!Pr oo. fWede e nhte ropecsst XN ( ) t=t Vt ( )+ Z;wihc ih a s-Qspuemrraingatel of rla lQ2 Pa d

mnatrngial en liema 5 pmlsua d erceainsg pocers.) FrosmN t E QNTjGt] and VT ( )=0 olfowlstV t

(

+2(Z). N( i ths

e )QET X t1

Z+ jt]G8t -aP..13s

C

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

nsoider th esaddl peinto(;Q of) the revipos tuehoemrV ( )= EQ0T

Z]X;

ro,equvialnetl, yN 0 ()=EQ NT ( ). A t]cheicanll meam7 mpliesi tat N (h) s i Q -martanigale. Henct Vt e(

)=QET X+t1ZGt] 8jP-a.ts.

2Thea susmptin ohat ti snte alilowst o ybass mapny ethnccai dlfciutlies(as de alt withby El K raouia nd Queezn(1 99)2and vCitain cnad Kraatazs( 1993))a dnd evelop he mtia indes afo uspr-ehegdnig ndaar bitarg peiring in a scrurpiisgln yisplm eadne lgane tay.wTh e ase amproaphcwi llal ow ul sotg o fa reybnod(E l Kraoiu nda uQenez; 9129)a d (nCvtainica d Kanrazas;t 9193) in ters mo form reelasitcim oed lssaumptinso ni th nexe tectiso.

n 4MatrnigaleM asuree,s Supremraingtle Maeasure sand ConistsetnP rce iSsyetsmGivnea rpbabioltyi mesareu Q, Z ) (=ETQX Pna a nodnegatnive -aGdpate procesds(2)5Z]de nesa onneganivet inelarp ric feucntinoal n othe seto fasc ohwsD . I nacf, atn yonnneatige vinlea functirona lon D acn e brpersentede tht way. aWec al l linaare unfctonial: D ! R aocnsitesntp ric syesetm i fis nnoegative nnad

C 0 ( )+(C ( ) )0for lalamxfC0 () (+ ( C))=g: 20

2:(26)( 72)

inSc ehe zeto trraidg sntategy irs in,(26 ) sieq uvalentito Letd noee thet ste f oocnsstenit picr eyssetm. shTep ira(; Q )i casledl conssitnte f ii de tensa con sstiet npicr sestemy (b y(5)2). 7fI X(t )2 0;T] ts a isueprmratnigael ndaE XT] =0 XhentX sia mrtinaagle.14

Th

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

erome9 Udenr te fhllooiwg ncoditnios ntehr es i 1-ta-o1c rorepsondenecbetewenc nsoitesntpr cie systmse adnab olutesylc ntonuousi prboablity imaesuer Qs o (; GT ) nscuhth ta( S1 1; )Qis consis etn:tth re isea nume airre D1( 0; S1> 0); tradig innt ih sseurict ysi nrustrieced;ttransa cion ctotss n ihts siceurity arez ro; ande n ithe thore seucrtiise radint gsino t restictedr withr seect tp ionfrmoatin. Toht means caontian sa tlaet thsestrate igse that tast withrn ohtin,gb yu s1hra oe fecuristyk >1 a ttmei int ht eeent v Aand esl lhtse sheresa at tiemt+1 ( j( ) != A 1!()f1=jg 1fskt+=g1 ),f o arll s 6= 1k,t,1a n d A 2G t.Poro.f Tihs teoreh ms ia idrct generaeizlatoni of hetore m6 nd ahe proto fropceds en i aisilma wrya .heTfou rhtc oditinn saoys tat hteh raditg nstrategie sneedd inet ht praoo aref contiand en i .(21 )and(13) d ne ienvrese apminpg: Tseh cndoiiot that tnardni gi then nmeruari eisun estricret dsis cuint eo enstreuthe ex sitnceeo a frtadingst aretyg htt haas zreoin itail P V0 (=), g0neretea tseh csaho wosc Ct ()= (0?;Z1;:: :;? Z?T1; Z) fr arobirtra Zy, nd artade osln iy nte nuhmraere i( k= 0 8k 6= )1 .Thatt ehe arren otran ascton icsto shen wrtadngii nhe nutemairre neuressth at C( )=?C (? . A)plpy theco nsisency tcodintinos o bttohand? nd aboerve shatt (5)1h ldso fo arllZ 2 D nad2 . Ta e ak 2 nd aed n e Qby(13). Qi sa eamsur end aQ . TPo se etaht Q s ia prbobailty imesaur (e( Q)= )1,ap lpyt h ecnossitecync noidiont(2 )6t o the ollfoingwtwo t raind sgrattgiese:=pu\ t$1in t the numeorare ait tme 0i,sell at tie mT"=\b;rrow o1 fr$omthe nu ermire aatt ime0 p,y abak ctat im T"e (.;Q )isco nisstnt ebcaesu (e3)1 ad (1n2) d neei nvrse eampinpgs.T akes cuha Q tha t (; )Q isconsis tent de. edn by 2() 5i consisstnte by d entiin. o2emRak. Trh eink lebweten conistsnt priec systees and prmobailiby mtaserue sn o;( GT ) brake ifsth ereare d i rent beorrwion gna leddinng rteas o rtere hrea trnasationc cost in the snmuerarei Th.n epiar (;s )Qa re enedd teo egernteaal lc nsositnetp ricef ncuiontals (ouinJ inadK lall;a 199b).5 In the retso tfihs ectsino ew ssuma etht ahteco ndiiotns oftheo emr9 a er etm. heT ni istju sti e tdo dene t e shetQ ofc onissetntp orabbliit yemsuares s thasoe emsaures

tht carresoond tpo caonsitentsp rice ystse.

mLemam1 0 or Fla lQ 2P nadal Glp-redictbalepr oecssse0 V( )

X+~ S= V ()+X

( )C+X

A( )

28)

(51

The arbitrage pricing principle has been used to derive price relations like the Black-Scholes formula and Heath-Jarrow-Morton models in the context of frictionless markets and unconstrained trading. These relations may or may not be good approximations to

is aQ-mrtianaleg

. P~rof.oChe ck qeation (u8) b2y takingd i eercnse.B deyn itoinof ,PS isa -mQatringle far oall Q2 . 2PLemm 11a Eevyrmart iganl meeauresis cosisntetn P( )Q.Proo.fS nci e0 i1plies m PC () (aA)Q sia nincr esiagn rocesp,sQlemmSinca eV( )tha t(V)+ is-s purmaetingral feo arl l2 .P=, 0 TQEP

hXC )(i

?C0()8 2 Q 82 P:

2W ealcl P aQ useprmatirgnlae maesue rf ihet idcousntde gani rpoces~ S iss Q-sauermpratigane.l Deno(e thit sse ty P b>).Lemm 1a2 I fhsro-tslliegn s foribddien ni al lsecuirtiseex cep the tumnreaier,the envey sruepmarritnaleg mesure as consiisetnt .I tnhis ascew haveeP >PQ.

A(P ) si a incneraings porcess,k 80k6=1, andS 11, ~Proof .Snce i P (C i)s Q-supermaartignal efroa llle mm 10 iapliesm hta V ( )t+ 2 Q> P. iSne cTV( )= 0,EQ

hXC( )

i?0C( ) 2 8Q 82 P:

P2>P si boiovsu.Lemma 31I fhteer ae nort arsnatconi cotss an

codnatis nta elst ahte onc eof nonngetiaev repicdabtl epocresse, thsen eveyrc nsoistne ptobrablitiy emasrue sia su ermprtaingale emsurae .(QP<.)

Xu ins t(X2 tG; XP 0 ) o secufrty kiat t mi et ndasell ti tat mie t+ 1 .~ Fort hsiwe hav e T ( )= XCStk+1 .T h conseisenct yoncdition( 62)k] 0 fro allk,,tX 2 t G So S.i asQ superm-ratniglea~~mplieis QEX St 1+ofr al Ql2 .Q2Th epontii shat,tund r traesnatcinoco ts ansd tadirn regstrciionts, Qi tshe aproprpitea es totco nsier adso poped ts o P .mEpricaliyl, Ps ofiet enmpy wtihelQ is ont J(schka eet la;. 196)9. Tehreotiaclly,Q i seacxltyth e estwe nee dor fhe ptresnetva ue prlniipcel. 16Porf. oAs int hepr ofo oft heore 6, cmosnidr etardng itrastgeesi hta tuyb