金融经济学讲义 35页

中华证券学习网www.1000zq.comFinancialEconomics
 84817008spirits77@263.netwww.redrival.com/sinavc/teachTextbook:LeroyStephrenF.,JanWerner.2001.PrinciplesofFinancialEconomics.CambridgeUniversityPress.("!$#)References:HuangChi-fu,RobertH.Litzenberger.1988.Foun-dationsforFinancialEconomics.North-Holland,Newwww.1000zq.comYork.(('*))%&+,)465678:9$:?%.-2004-0/213-;6<=">ProcessPlanned:1-2chaptersperweek.中华证券学习网中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comPartIEquilibriumwww.1000zq.com中华证券学习网1中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comChapter1EquilibriuminSecurityMarket1.1NotationsandrepresentationTimeisdividedintodate0,whichrepresentsthecer-tainty,anddate1,whichrepresentsuncertainty.Atdate1,thereareSpossiblestates.Securityjisidentiedbyitspayovector(rowvector)x=(xwww.1000zq.com;


;x)2(Rs)T,jj1jswherexjsdenotespayoofunitsofconsumptioninstatesatdate1.Itspriceatdate0isp2R+.TherejareniteJnumbersofsecurities.Thepayomatrixis01x1X=@...A2RJS.AportfolioisidentiedbyaxJcolumnvectorh=(h;:::;h)T2RJ,wherehisthe1Jjholdingsofthesecurityj.PortfoliopayoisdenotedbyhTX.中华证券学习网Therepresentativeagent'sconsumptionisc02Rat1中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comdate0andc2RS.Hispreferencesareindicatedbya1continuousutilityfunctionu(c;c):RS+1!R.We01assumethattheutilityfunctionisalwaysdierentiablethroughthebook.Hisendowmentisw02Ratdate0andw2RS.ThereareIagents.1Utilityfunctionuisincreasing(orstrictlyincreasing)00atdate0ifu(c0;c1)>u(c0;c1)(oru(c0;c1)>u(c0;c1))00wheneverc0>c0(orc0>c0)foreveryc1.Itisincreasing0(orstrictlyincreasing)atdate1ifu(c0;c1)>u(c0;c1)000(oru(c0;c1)>u(c0;c1))wheneverc1>c1(orc1>c1)foreveryc0.Itis(strictly)increasingifitis(strictly)increasingatbothdate0anddate1.ForaSdimensionvectore2RS,itssthelementsis1andallotherelementsare0.Forexample,e1=(1;0;:::;0).esiscalledtheArrowsecuritywww.1000zq.comorstateclaimwiththeclaimtooneunitofconsumptiononoccurrenceofstatesatdate1.Matrixcomparison,seep6inthetextbook.Equivalencyrepresentation,seep6inthetextbook.1.2CompletesecuritymarketDenition1.1TheassetspanMisM=fz2RS:z=hTXforsomeh2RJg.中华证券学习网Denition1.1saysthattheassetspanissome(maynot2中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comall)portfolio'spayo.Denition1.2ThemarketiscompleteiifM=RSandincompleteiifM6=RS.Denition1.2saysthatthepayomatrixcangeneratethewholeJSdimensionsspace,whichimpliesthatalltheportfolio'spayocanbegeneratedbythepayomatrix.Denition1.3Securityjisredundantifthereexistsaportfoliov2RJsuchthatx=vTXandvisnotajvectorlikeej;816j6J.Denition1.3saysthatasecurityisredundantifitspayocanbegeneratedbyotherportfolio.Denition1.4LiftheleftinverseofXwww.1000zq.com,LSJXJS=IS,whereISistheunitmatrix.,r(X)=S.Ifex-ists,L=(XTX)1XT.RiftherightinverseofX,XJSRSJ=IJ,r(X)=J.Ifexists,R=XT(XXT)1.Proposition1.1Thereexistsnoredundantsecurity,TherightinverseofXexists.Proof:Bythedenition,thereexistsnoredundantsecurityiifalltherowvectorsofthepayomatrix中华证券学习网arelinearlyindependent,whichisequivalenttothat3中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comtherankofXisJandsothattherightinverseofXexists.Theorem1.1Marketsarecompleteiiftheleftin-verseofXexists.Proof:Step1.It'sobviouslythatM2RS.SomarketsarecompleteifRSM.ThenmarketsarecompleteisequivalenttothattheequationTz=hX(1.1)hassolutionoftheunknownvectorh,8z2RS.0Step2.Letr(X)=S0for0agentj16j6Iandj6=i.Letfhigand(ci;ci),01jjjfhgand(c;c)areequilibriumportfolioandcon-01sumptionallocation.BecausecipTh6!ipT(hi+h)0000andci6!i+(hiT+hT)X,portfolioallocationfhi+hg1100andconsumptionallocation(cipTh;ci)arealso001budgetfeasibleandstrictlypreferredforagentibe-causetheutilityfunctionisstrictlyincreasingatedatewww.1000zq.comj0.Bythesameanalogy,portfolioallocationfh1hgandconsumptionallocation(cj+1pTh;cj)I100I101arealsobudgetfeasibleandstrictlypreferredforeveryPotheragentj.And(hi+h)+(h1h)=0,0jI1016j6I;j6=iwhichimpliesmarketclearing.Thisiscontradictandtheconclusionisobtained.Theorem2.3Ifagent'sutilityfunctionisstrictlyin-creasingatdate1andthereexistsaportfoliowithpositiveandnonzeropayoandthedate0consump-中华证券学习网tiondoesnotentertheagents'utilityfunction,then12中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comthelawofonepriceholdsinanequilibrium.Proof:Whenthedate-0consumptiondoesnotentertheutilityfunction,theagenti'sconsumptionportfo-liochoiceprobleminequilibriumismaxu(c1)(2.3)c1;hs:t:pTh6!;(2.4)0c6!+hTX:(2.5)11Phi=0:(2.6)iThefollowingisonthesameanalogytothatinthetheorem2.2.Leth^beaportfoliowithpositiveandnonzeropayo.Bycorollary2.1,thezeropayohaswww.1000zq.comanypriceifthelawofonepricedoesnotholds.Thenlethisaportfoliowithzeropayo(hTX=0)and00pTh=pTh^forarbitraryagenti(16i6I)and0pTh=pTh^foragentj16j6Iandj6=i.Letfhigand(ci;ci),fhjgand(cj;cj)areequilib-0101riumportfolioandconsumptionallocation.BecausepT(hi+h^h)=pTh6!iandci+h^TX6!i+0011(hiT+h^hT)X=!i+(hiT+h^T)X,portfolioal-01locationfhi+h^hgandconsumptionallocation0(c;ci中华证券学习网+h^TX)arealsobudgetfeasibleandstrictlypre-01ferredforagentibecausetheutilityfunctionisstrictly13中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comincreasingatedate1.Forthesamereason,port-folioallocationfhj1(h^h)gandconsump-I10tionallocation(cj;cj+1(h^TX)arealsobudgetfea-01I1sibleandstrictlypreferredforeveryotheragentj.PAnd(hi+h^h)+(h1(h^h)=0,0jI1016j6I;j6=iwhichimpliesmarketclearing.Thisiscontradictandtheconclusionisobtained.2.2.1Notestoexample2.4.3onp18ofthetextbooku60andthezeroportfolioofcoursegeneratethemaximumutility.portfolioh=(1;1;0)Tandh=(0;0;1)Tgenerate12thesamepayoof(1;1)butwithdierentprice,pThwww.1000zq.com1andpTh,ifp+p6=p.2123Thelawofonepricedoesn'tholdevenifthereexisttheequilibriumprices(pricetomeettheaboveprob-lem).Thereasonisthattheutilityfunctionisnotstrictlyincreasing.2.2.2Notestoexample2.4.4onp18ofthetextbookTheproblemis中华证券学习网maxu(c1;c2)=lnc1+lnc2(2.7)c1;c2;h1;h214中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.coms:t:h1+h260;(2.8)c161+h1+2h2;(2.9)c261h12h2;(2.10)c1>0;c2>0:(2.11)Thesolutionfortheaboveproblemish1=2h2h2>0Thelawofonepricedoesn'tholdevenifthereexisttheequilibriumprices(pricetomeettheaboveproblem).Thereasonisthateverypayoisintheformofh1+2h2:(h1+2h2)www.1000zq.comwhichcannotbepositiveandnonzero.2.3consumptionandportfoliowithstatepricesAtheoreminthefunctionalanalysis:AfunctionalF(x);x2RLhasarepresentationintheformofF(x)=fTxforsomef2RL.Ifthemarketiscompleteandthelawofonepriceholds,thenq(z)islinearfunctionalbytheorem2:2.So中华证券学习网q(z)=~qz(2.12)15中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.com,where~q=(~q1;


;q~S).Becausezsisthepayoinstates,~qcanbeinterpretedasthepriceofunitpayoinstates,whichisthestatepriceofs,orthepriceofArrowsecurityinstates.Asaresult,thepriceofsecurityj,pjisTTpj=q(xj)=~qxj;(2.13)orinmatrixnotation,p=Xq:~(2.14)Ifmarketiscomplete,thentheleftinverseofXexists(L)andpremultiplying(2.14)byitleadstoq~=Lp(2.15)Withthepayopricingfunctionalq(z):M!R,theagent'sconsumptionportfoliochoiceproblemcanbewww.1000zq.comrewrittenasmaxu(c0;c1)(2.16)c0;c1;hs:t:c06!0q(z);(2.17)c16!1+z:(2.18)z2M:(2.19)Ifmarketiscomplete,M=RSandtheconditionof(2.19)canbedeleted.Withthestateprice,condition(2.17)canchangeto中华证券学习网c06!0qz:~(2.20)16中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comWhenc06=0andc16=0orsolutionisintheinterior,thef.c.o.leadsto@1uq~=;(2.21)@0uwhichsaysthatthestatepriceisthemarginalrateofsubstitutionbetweendate-1consumptionanddate-0con-sumption.(2.21)canalsobeobtainedby(1.9)and(2.15)Assofar,wegettwomethodstocalculatetheequi-libriumprice.Oneistosolvetheportfolioandconsump-tionchoiceinequilibriumfrom(1.2)to(1.4)withmarketclearingcondition,whichcanbecalleddirectlypricing.Theotheristosolvetheproblemof(2.16),(2.18),(2.20)withmarketclearingcondition,whichcanbecalledpric-ingwithstateprice.www.1000zq.com中华证券学习网17中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comwww.1000zq.com中华证券学习网18中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comChapter3ArbitrageandPositivePricing3.1arbitrageandequivalentrepresentationDenition3.1AnarbitrageisaportfoliohsuchthatTTThX>0;hX6=0;ph60:(3.1)orTTTwww.1000zq.comhX=0;hX=0;ph<0:(3.2)Denition3.2AstrongarbitrageisaportfoliohsuchthatTThx>0;ph<0:(3.3)Denition3.3Arisk-freearbitrageisaportfoliohsuchthatXSTThX=es;>0;ph60:(3.4)i=1or中华证券学习网TThX=0;ph<0:(3.5)19中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comDenition3.4AfunctionalF(x)ispositiveiifF(x)>0;8x>0:(3.6)It'sstrictlypositiveiifF(x)>0;8x>0andx6=0:(3.7)ForalinearfunctionalF(x),thereexistsavectorfsuchthatF(x)=fTx.Then(3.6)isequivalenttof>0and(3.7)isequivalenttof>0.Corollary3.1Ifthereisnoportfoliowithpositiveandnonzeropayo,thenanyarbitrageisastrongarbitrageandaportfoliowithzeropayoandstrictlynegativepricewww.1000zq.comProof:Whenthereisnoportfoliowithpositiveandnonzeropayo,(3.1)isexcluded.Thenaarbitrageisdenedby(3.2),whichisalsothedenitionofstrongarbitrage.Theorem3.1Thatastrongarbitrageexistsisequiv-alentto(1)thatthelawofonepricedoesnothold.(2)thatthepayopricingfunctional中华证券学习网q(z)islinearandpositive.20中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.com(3)thateverystatepriceispositiveandatleastonestatepriceisstrictlypositive,q~>0;q~6=0.Proof:(1)necessaryconditionWhenthereisastrongarbitrage,zeropayoispricedstrictlynegatively.Byproposition2.2,thelawofonepricedoesnothold.sucientconditionByproposition2.2,thezeropay-ocanbepricedanyifthelawofonepricedosenothold.Withoutlossofgenerality,letitbestrictlyneg-ative.Thenitisastrongarbitrage.(2)Thelinearpartisguarantiedbytheorem2.1.Exclusionofarbitrageisequivalenttothatq(z)>0ifz>0,z2M.Thisisthedenitionofpositivity.(3)Thestatepriceresultisobvious.www.1000zq.comProposition3.1Exclusionofrisk-freearbitragecan-notguarantytheexclusionofarbitrage.PSProof:Considertheportfolioh,hTX=e;>ii=1PS0.Whenei>0andes<0forsomes,risk-freei=1arbitrageisexcludedbutitisanarbitrage.Theorem3.2中华证券学习网Thatthereisnoarbitrageisequivalentto21中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.com(1)thatthepayopricingfunctionalq(z)islinearandstrictlypositive.(2)thateverystatepriceisstrictly,q~0.Theproofissimilartothatintheorem3.1.3.2diagrammaticrepresentationNote1.istheangleof;2RS,thenTcos=;(3.8)kkkkrPwherek:kdenotestheEucliddistance,kk=i.iwww.1000zq.com3.3arbitrageandoptimalportfoliopricingTheorem3.3Ifthereexistsanagent'soptimalport-folioatgivenpriceandtheagentutilityfunctionisstrictlyincreasingatdate0andincreasingatdate1,thenthereexistsnostrongarbitrage.proof:Leth;c0;c1betheoptimalportfolioandcon-sumptionallocation.Suppose,bycontradiction,thatastrongarbitrageexistsandletitbe中华证券学习网h^,h^TX>0,pTh<^0.BecausecpTh^6!pT(h+h^),c+00122中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comh^TX6!+(h+h^)TX,portfolioallocationh+h^and1consumptionallocationcpTh,c+h^TXarealsobud-01getfeasible.Andtheyarestrictlypreferredtoh;c0;c1becausetheutilityfunctionisstrictlyincreasingatdate0andincreasingatdate1.Thisiscontradict.Theorem3.4Ifthereexistsanagent'soptimalport-folioatgivenpriceandtheagentutilityfunctionisstrictlyincreasing,thenthereexistsnoarbitrage.Theproofissimilartotheorem3.3.Notestoexample3.6.2Atthegivenpricep=TT(p1;p2)=(1;0),u6(1h1)+min(1+h1;2+h2)62;3+h2h1).Whenh1=h2=0,u=2isthewww.1000zq.commaximumandtheyareoptimalportfolioallocation.Butthesecurity2is,inthiscase,anarbitrage,becauseitspayois(0;1)>0anditspriceis0.Theorem3.4doesnotholdherebecausetheutilityfunctionisincreasingbutnotstrictlyincreasing.Notestoexample3.6.2Atthegivenpricep=TT(p1;p2)=(1;1),f.c.o.leadstoc1=c2andh1=2h2ifthecorrespondentLagrangemultiplierisnotzero.h=TT(h1;h2)=(2h2;h2)=(2;1)isanoptimalportfolioallocationandastrongarbitrage.Theorem3.3doesnot中华证券学习网holdherebecausetheutilityfunctionisnotstrictlyin-23中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comcreasingatdate0becausedate-0consumptiondoesnotentertheutilityfunction.Iftheutilityfunctionisstrictlyincreasing,thentheabsenceofarbitrageisthenecessaryconditionfortheexistenceofanoptimalportfoliobytheorem3.3and3.4.Whentheconsumptionisrestrictedtobepositive,thenthenecessaryconditionisalsosucientcondition.Theorem3.5Ifthereisnoarbitrageatgivensecu-ritypriceandiftheagent'sconsumptionisrestrictedtobepositive,thenthereexistsanoptimalportfolio.Proof:(1)Thelawofonepriceholdsbecausetheex-clusionofarbitragebytheorem3.1.Let(c0;c1)andh2RJbetheoptimalallocation.Ifthereisredun-www.1000zq.comdantsecurity,thenthereexistsanotherportfolioh^2Rk;(k0:(3.10)Dividingbothsidesofequation(3.9)and(3.10)andletingthengotoinnity,wegetwww.1000zq.compTh^6!;(3.11)0andXTh^>0:(3.12)Thus,h^isanarbitrage,leadingtocontradiction.Notestoexample3.6.6Thef.c.o.sareLc0=1=0;(3.13)中华证券学习网Lc1=1+1=0;(3.14)Lc2=1+2=0;(3.15)25中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comLh=p+1+2=0:(3.16)Whenp6=0,theaboveconditionscannotbemetandtherewillbenooptimalportfolio.Therewillbenoar-bitragebecausethatthereisonlyonesecurityandthatthepricewillbepositiveifthepayoispositive.Dierentfromtotheorem3.5,thereexistnoarbitrageevenwhenthereisnooptimalportfolio.Thereasonisthatthepositiveconsumptionconditionisnotimposed.Whenitisimposed,formulas(4.13)to(4.16)canbeneg-ativeandtheformula(4.17)canbemet.Thef.c.o.sarethusmetandtherewillbeoptimalportfolio.Thearbi-trageisalsoexcludedandthetheorem3.5holds.www.1000zq.com中华证券学习网26中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comChapter4PortfolioRestriction4.1theimplicationandeectofshortsalesrestrictionThetypicalshortsalesrestrictiononsecurityjtakestheformofhj>bj;bj>0www.1000zq.com:(4.1)Thereare3implicationsfor(4.2):++Theagentdoesnotholdsanyinitialholdingsonsecurityjandcannotsellanyofit.++Theagentholdssomeinitialholdingonsecurityjandcansellnomorethanthoseholdings.++Theagentholdssomeinitialholdingsonsecurityjandcansellsomemorethanthoseholding.Wedenotethesetofsecuritieswithshortsalesrestriction中华证券学习网byJ027中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comThereisbesidesanotherpatternofshortrestriction,inwhichtheagentcanonlyshortsellsomesecuritywithsomecollateral.Inthiscase,theendowmentcanbeviewedasthecollateral.Forthesecurityjwithshortsalesrestrictionwithcollateral,therestrictioncanbewrittenhjxj+!j>0:(4.2)Theconditionwithcollateralofendowmentismorestrin-gentthanthepositiveconstraint.Whentheconsumptionispositive,someholdingsofsecuritiesmaybenegative.Seetherstexampleinthetextbook.Inthepresenceofshortsalesrestriction,,theagent'sconsumption-portfoliochoiceturnsintomaxu(c0;c1)(4.3)c0;c1;hs:t:c6!pTwww.1000zq.comh;(4.4)00c6!+XTh;(4.5)11hj>bj;8j2J0:(4.6)TheLagrangefunctionfortheaboveproblemisXTTTL=u+(!0phc0)+(!+Xhc1)+j(hj+bj):(4.7)j2J0Thef.c.o.ofhleadstoXxjsspj=;j=2J0;(4.8)s中华证券学习网Xxjss+jpj=;j2J0:(4.9)s28中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comWhenweneglectthecondition(4.6),thef.c.oofc0;c1leadto@u=;=rcu:(4.10)@c0Becausetheneglectdoesnotaecttheand,wecansubstituteequation(4.10)into(4.8)and(4.9)andgetXxjs@supj=;j=2J0;(4.11)@0usXxjs@supj>;j2J0:(4.12)@0usWeknowfromchapter2thatthelawofonepriceholdsandthepricetakestheformof(4.11)inequilibriumwithoutshortsalesrestriction.Whentherestrictionisimposed,therestrictionmaybebindingandthepricetakestheformof(4.12).Whenitisthestrictlyinequitywww.1000zq.comin(4.12),thelawofonepricemaynothold.Example4.4.1issuchexample.Withtheshortsalerestriction,thearbitrageneedstobemodied.Denition4.1AnunlimitedarbitrageisaportfoliohsuchthatTTTXh>0;Xh6=0;ph60;hj>08j2J0:(4.13)or中华证券学习网TTTXh=0;Xh=0;ph<08j2J0:(4.14)29中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comDenition4.2Anunlimitedstrongarbitrageisaport-foliohsuchthatTTXh>0;ph<0;hj>08j2J0:(4.15)Denition4.3Anlimited(strong)arbitrageisan(strong)arbitragethatisnotaunlimited(strong)ar-bitrage.4.2bid-askpriceInthereal-worldnancialmarket,theagentsbuysecuri-tiesataskprices(p;p2RJ),whicharethepricestheaaspecialistsell,andsellsecuritiesatbidprices(pb;pb2RJ),whicharethepricesthespecialistbuy.Thewww.1000zq.combid-askspreadisdenedbypapb=ps.Withthebid-askspread,theagent'sportfoliocanbeviewedasportfoliosdividedintotowportfolioswithshortsalesrestriction,portfolioofha;(ha>0)inlongpositionwithpricepaandpayomatrixX,portfolioofhb;(hb>0)inshortpositionwithpriceofpbandpayomatrixX.TheunlimitedarbitrageisredenedDenition4.4Anunlimitedarbitrageisaportfoliohsuchthat中华证券学习网TTTX(hahb)>0;X(hahb)6=0;p(hahb)60;ha>0;hb>0:(4.16)30中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comorTTX(hahb)=0;p(hahb)<0;ha>0;hb>0:(4.17)Denition4.5Anunlimitedstrongarbitrageisaport-foliohsuchthatTTX(hahb)>0;p(hahb)<0;ha>0;hb>0:(4.18)Corollary4.1Whenthereisnoarbitrage,thebid-askspreadispositive.Theconsumption-portfolioequilibriumisalsorede-nedasportfolioallocation(hi;hi),(ci;ci)solvingtheab0ifollowingproblemwww.1000zq.commaxu(ci;ci)(4.19)01ci;ci;hi01s:t:ci6!ipT(hihi);(4.20)00abci6!i+XT(hihi);(4.21)11abPPhi=hi;(4.22)baiihi>0;hi>0:(4.23)abNotestoexample4.8.1p40-42.Because中华证券学习网ci;ci>>,agent1willnotsellsecurity2and12agent2willnotsellsecurity1.Sotheimpliedassumption31中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comisthatoneagentwillnotbuy(orsell)asecurityiftheotheragentdoesnotsell(orbuy)thesecurity.Therelationshipbetweenhandtcanalsobeobtainedbyderivinghbytin(4.34),@h3h=<0:@t2p+3twww.1000zq.com中华证券学习网32中华证券学习网www.1000zq.com\n中华证券学习网www.1000zq.comIndexarbitrage,19limitedarbitrage,30limitedstrongarbitrage,30risk-freearbitrage,19strongarbitrage,19unlimitedarbitrage,29,30unlimitedstrongarbitrage,30,31Arrowsecurity,2,16assetspan,2,9bid-askspread,30complete,3directlypricing,17www.1000zq.comgeneralequilibrium,6,7lawofoneprice,9,29payomatrix,1,3,6payopricingfunctional,16pricingwithstateprice,17securitymarketeconomy,4shortsalesrestriction,27stateclaim,2stateprice,16,17中华证券学习网33中华证券学习网www.1000zq.com