Springer.Mathematics.for.Finance.An.Introduction.to.Financial.Engineering.(Capinski,2004).Springer

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MathematicsforFinance:AnIntroductiontoFinancialEngineeringMarekCapinskiTomaszZastawniakSpringer SpringerUndergraduateMathematicsSeriesSpringerLondonBerlinHeidelbergNewYorkHongKongMilanParisTokyo AdvisoryBoardP.J.CameronQueenMaryandWestfieldCollegeM.A.J.ChaplainUniversityofDundeeK.ErdmannOxfordUniversityL.C.G.RogersUniversityofCambridgeE.SüliOxfordUniversityJ.F.TolandUniversityofBathOtherbooksinthisseriesAFirstCourseinDiscreteMathematicsI.AndersonAnalyticMethodsforPartialDifferentialEquationsG.Evans,J.Blackledge,P.YardleyAppliedGeometryforComputerGraphicsandCADD.MarshBasicLinearAlgebra,SecondEditionT.S.BlythandE.F.RobertsonBasicStochasticProcessesZ.Brze´zniakandT.ZastawniakElementaryDifferentialGeometryA.PressleyElementaryNumberTheoryG.A.JonesandJ.M.JonesElementsofAbstractAnalysisM.ÓSearcóidElementsofLogicviaNumbersandSetsD.L.JohnsonEssentialMathematicalBiologyN.F.BrittonFields,FlowsandWaves:AnIntroductiontoContinuumModelsD.F.ParkerFurtherLinearAlgebraT.S.BlythandE.F.RobertsonGeometryR.FennGroups,RingsandFieldsD.A.R.WallaceHyperbolicGeometryJ.W.AndersonInformationandCodingTheoryG.A.JonesandJ.M.JonesIntroductiontoLaplaceTransformsandFourierSeriesP.P.G.DykeIntroductiontoRingTheoryP.M.CohnIntroductoryMathematics:AlgebraandAnalysisG.SmithLinearFunctionalAnalysisB.P.RynneandM.A.YoungsonMatrixGroups:AnIntroductiontoLieGroupTheoryA.BakerMeasure,IntegralandProbabilityM.Capi´nskiandE.KoppMultivariateCalculusandGeometryS.DineenNumericalMethodsforPartialDifferentialEquationsG.Evans,J.Blackledge,P.YardleyProbabilityModelsJ.HaighRealAnalysisJ.M.HowieSets,LogicandCategoriesP.CameronSpecialRelativityN.M.J.WoodhouseSymmetriesD.L.JohnsonTopicsinGroupTheoryG.SmithandO.TabachnikovaTopologiesandUniformitiesI.M.JamesVectorCalculusP.C.Matthews MarekCapi´nskiandTomaszZastawniakMathematicsforFinanceAnIntroductiontoFinancialEngineeringWith75Figures1Springer MarekCapi´nskiNowySa
czSchoolofBusiness–NationalLouisUniversity,33-300NowySa
cz,ul.Zielona27,PolandTomaszZastawniakDepartmentofMathematics,UniversityofHull,CottinghamRoad,KingstonuponHull,HU67RX,UKCoverillustrationelementsreproducedbykindpermissionof:AptechSystems,Inc.,PublishersoftheGAUSSMathematicalandStatisticalSystem,23804S.E.Kent-KangleyRoad,MapleValley,WA98038,USA.Tel:(206)432-7855Fax(206)432-7832email:info@aptech.comURL:www.aptech.com.AmericanStatisticalAssociation:ChanceVol8No1,1995articlebyKSandKWHeiner‘TreeRingsoftheNorthernShawangunks’page32fig2.Springer-Verlag:MathematicainEducationandResearchVol4Issue31995articlebyRomanEMaeder,BeatriceAmrheinandOliverGloor‘IllustratedMathematics:VisualizationofMathematicalObjects’page9fig11,originallypublishedasaCDROM‘IllustratedMathematics’byTELOS:ISBN0-387-14222-3,GermaneditionbyBirkhauser:ISBN3-7643-5100-4.MathematicainEducationandResearchVol4Issue31995articlebyRichardJGaylordandKazumeNishidate‘TrafficEngineeringwithCellularAutomata’page35fig2.MathematicainEducationandResearchVol5Issue21996articlebyMichaelTrott‘TheImplicitizationofaTrefoilKnot’page14.MathematicainEducationandResearchVol5Issue21996articlebyLeedeCola‘Coins,Trees,BarsandBells:SimulationoftheBinomialProcess’page19fig3.MathematicainEducationandResearchVol5Issue21996articlebyRichardGaylordandKazumeNishidate‘ContagiousSpreading’page33fig1.MathematicainEducationandResearchVol5Issue21996articlebyJoeBuhlerandStanWagon‘SecretsoftheMadelungConstant’page50fig1.BritishLibraryCataloguinginPublicationDataCapi´nski,Marek,1951-Mathematicsforfinance:anintroductiontofinancialengineering.-(Springerundergraduatemathematicsseries)1.Businessmathematics2.Finance–MathematicalmodelsI.TitleII.Zastawniak,Tomasz,1959-332’.0151ISBN1852333308LibraryofCongressCataloging-in-PublicationDataCapi´nski,Marek,1951-Mathematicsforfinance:anintroductiontofinancialengineering/MarekCapi´nskiandTomaszZastawniak.p.cm.—(Springerundergraduatemathematicsseries)Includesbibliographicalreferencesandindex.ISBN1-85233-330-8(alk.paper)1.Finance–Mathematicalmodels.2.Investments–Mathematics.3.Businessmathematics.I.Zastawniak,Tomasz,1959-II.Title.III.Series.HG106.C362003332.6’01’51—dc212003045431Apartfromanyfairdealingforthepurposesofresearchorprivatestudy,orcriticismorreview,aspermittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced,storedortransmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublishers,orinthecaseofreprographicreproductioninaccordancewiththetermsoflicencesissuedbytheCopyrightLicensingAgency.Enquiriesconcerningreproductionoutsidethosetermsshouldbesenttothepublishers.SpringerUndergraduateMathematicsSeriesISSN1615-2085ISBN1-85233-330-8Springer-VerlagLondonBerlinHeidelbergamemberofBertelsmannSpringerScience+BusinessMediaGmbHhttp://www.springer.co.uk©Springer-VerlagLondonLimited2003PrintedintheUnitedStatesofAmericaTheuseofregisterednames,trademarksetc.inthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantlawsandregulationsandthereforefreeforgeneraluse.Thepublishermakesnorepresentation,expressorimplied,withregardtotheaccuracyoftheinformationcontainedinthisbookandcannotacceptanylegalresponsibilityorliabilityforanyerrorsoromissionsthatmaybemade.Typesetting:Camerareadybytheauthors12/3830-543210Printedonacid-freepaperSPIN10769004 PrefaceTruetoitstitle,thisbookitselfisanexcellentfinancialinvestment.ForthepriceofonevolumeitteachestwoNobelPrizewinningtheories,withplentymoreincludedforgoodmeasure.Howmanyundergraduatemathematicstextbookscanboastsuchaclaim?Buildingonmathematicalmodelsofbondandstockprices,thesetwotheo-riesleadindifferentdirections:Black–Scholesarbitragepricingofoptionsandotherderivativesecuritiesontheonehand,andMarkowitzportfoliooptimisa-tionandtheCapitalAssetPricingModelontheotherhand.Modelsbasedontheprincipleofnoarbitragecanalsobedevelopedtostudyinterestratesandtheirtermstructure.Thesearethreemajorareasofmathematicalfinance,allhavinganenormousimpactonthewaymodernfinancialmarketsoperate.Thistextbookpresentsthematalevelaimedatsecondorthirdyearundergraduatestudents,notonlyofmathematicsbutalso,forexample,businessmanagement,financeoreconomics.Thecontentscanbecoveredinaone-yearcourseofabout100classhours.Smallercoursesonselectedtopicscanreadilybedesignedbychoosingtheappropriatechapters.Thetextisinterspersedwithamultitudeofworkedex-amplesandexercises,completewithsolutions,providingamplematerialfortutorialsaswellasmakingthebookidealforself-study.Prerequisitesincludeelementarycalculus,probabilityandsomelinearalge-bra.Incalculusweassumeexperiencewithderivativesandpartialderivatives,findingmaximaorminimaofdifferentiablefunctionsofoneormorevariables,Lagrangemultipliers,theTaylorformulaandintegrals.Topicsinprobabilityincluderandomvariablesandprobabilitydistributions,inparticularthebi-nomialandnormaldistributions,expectation,varianceandcovariance,condi-tionalprobabilityandindependence.FamiliaritywiththeCentralLimitThe-oremwouldbeabonus.Inlinearalgebrathereadershouldbeabletosolvev viMathematicsforFinancesystemsoflinearequations,add,multiply,transposeandinvertmatrices,andcomputedeterminants.Inparticular,asareferenceinprobabilitytheorywerecommendourbook:M.Capi´nskiandT.Zastawniak,ProbabilityThroughProblems,Springer-Verlag,NewYork,2001.Inmanynumericalexamplesandexercisesitmaybehelpfultouseacom-puterwithaspreadsheetapplication,thoughthisisnotabsolutelyessential.MicrosoftExcelfileswithsolutionstoselectedexamplesandexercisesareavail-ableonourwebpageattheaddressesbelow.WeareindebtedtoNigelCutlandforpromptingustosteerclearofaninaccuracyfrequentlyencounteredinothertexts,ofwhichmorewillbesaidinRemark4.1.Itisalsoagreatpleasuretothankourstudentsandcolleaguesfortheirfeedbackonpreliminaryversionsofvariouschapters.Readersofthisbookarecordiallyinvitedtovisitthewebpagebelowtocheckforthelatestdownloadsandcorrections,ortocontacttheauthors.Yourcommentswillbegreatlyappreciated.MarekCapi´nskiandTomaszZastawniakJanuary2003www.springer.co.uk/M4F Contents1.Introduction:ASimpleMarketModel......................11.1BasicNotionsandAssumptions............................11.2No-ArbitragePrinciple....................................51.3One-StepBinomialModel.................................71.4RiskandReturn.........................................91.5ForwardContracts........................................111.6CallandPutOptions.....................................131.7ManagingRiskwithOptions...............................192.Risk-FreeAssets............................................212.1TimeValueofMoney.....................................212.1.1SimpleInterest.....................................222.1.2PeriodicCompounding..............................242.1.3StreamsofPayments...............................292.1.4ContinuousCompounding...........................322.1.5HowtoCompareCompoundingMethods..............352.2MoneyMarket...........................................392.2.1Zero-CouponBonds................................392.2.2CouponBonds.....................................412.2.3MoneyMarketAccount.............................433.RiskyAssets................................................473.1DynamicsofStockPrices..................................473.1.1Return............................................493.1.2ExpectedReturn...................................533.2BinomialTreeModel......................................55vii viiiContents3.2.1Risk-NeutralProbability............................583.2.2MartingaleProperty................................613.3OtherModels............................................633.3.1TrinomialTreeModel...............................643.3.2Continuous-TimeLimit.............................664.DiscreteTimeMarketModels..............................734.1StockandMoneyMarketModels...........................734.1.1InvestmentStrategies...............................754.1.2ThePrincipleofNoArbitrage.......................794.1.3ApplicationtotheBinomialTreeModel...............814.1.4FundamentalTheoremofAssetPricing...............834.2ExtendedModels.........................................855.PortfolioManagement......................................915.1Risk....................................................915.2TwoSecurities...........................................945.2.1RiskandExpectedReturnonaPortfolio..............975.3SeveralSecurities.........................................1075.3.1RiskandExpectedReturnonaPortfolio..............1075.3.2EfficientFrontier...................................1145.4CapitalAssetPricingModel...............................1185.4.1CapitalMarketLine................................1185.4.2BetaFactor........................................1205.4.3SecurityMarketLine...............................1226.ForwardandFuturesContracts.............................1256.1ForwardContracts........................................1256.1.1ForwardPrice......................................1266.1.2ValueofaForwardContract.........................1326.2Futures.................................................1346.2.1Pricing............................................1366.2.2HedgingwithFutures...............................1387.Options:GeneralProperties................................1477.1Definitions...............................................1477.2Put-CallParity..........................................1507.3BoundsonOptionPrices..................................1547.3.1EuropeanOptions..................................1557.3.2EuropeanandAmericanCallsonNon-DividendPayingStock.............................................1577.3.3AmericanOptions..................................158 Contentsix7.4VariablesDeterminingOptionPrices........................1597.4.1EuropeanOptions..................................1607.4.2AmericanOptions..................................1657.5TimeValueofOptions....................................1698.OptionPricing..............................................1738.1EuropeanOptionsintheBinomialTreeModel...............1748.1.1OneStep..........................................1748.1.2TwoSteps.........................................1768.1.3GeneralN-StepModel..............................1788.1.4Cox–Ross–RubinsteinFormula.......................1808.2AmericanOptionsintheBinomialTreeModel...............1818.3Black–ScholesFormula....................................1859.FinancialEngineering.......................................1919.1HedgingOptionPositions..................................1929.1.1DeltaHedging.....................................1929.1.2GreekParameters..................................1979.1.3Applications.......................................1999.2HedgingBusinessRisk....................................2019.2.1ValueatRisk......................................2029.2.2CaseStudy........................................2039.3SpeculatingwithDerivatives...............................2089.3.1Tools.............................................2089.3.2CaseStudy........................................20910.VariableInterestRates.....................................21510.1Maturity-IndependentYields...............................21610.1.1InvestmentinSingleBonds..........................21710.1.2Duration..........................................22210.1.3PortfoliosofBonds.................................22410.1.4DynamicHedging..................................22610.2GeneralTermStructure...................................22910.2.1ForwardRates.....................................23110.2.2MoneyMarketAccount.............................23511.StochasticInterestRates...................................23711.1BinomialTreeModel......................................23811.2ArbitragePricingofBonds................................24511.2.1Risk-NeutralProbabilities...........................24911.3InterestRateDerivativeSecurities..........................25311.3.1Options...........................................254 xContents11.3.2Swaps............................................25511.3.3CapsandFloors....................................25811.4FinalRemarks...........................................259Solutions.......................................................263Bibliography....................................................303GlossaryofSymbols............................................305Index...........................................................307 1Introduction:ASimpleMarketModel1.1BasicNotionsandAssumptionsSupposethattwoassetsaretraded:onerisk-freeandoneriskysecurity.Theformercanbethoughtofasabankdepositorabondissuedbyagovernment,afinancialinstitution,oracompany.Theriskysecuritywilltypicallybesomestock.Itmayalsobeaforeigncurrency,gold,acommodityorvirtuallyanyassetwhosefuturepriceisunknowntoday.Throughouttheintroductionwerestrictthetimescaletotwoinstantsonly:today,t=0,andsomefuturetime,sayoneyearfromnow,t=1.Morerefinedandrealisticsituationswillbestudiedinlaterchapters.Thepositioninriskysecuritiescanbespecifiedasthenumberofsharesofstockheldbyaninvestor.ThepriceofoneshareattimetwillbedenotedbyS(t).ThecurrentstockpriceS(0)isknowntoallinvestors,butthefuturepriceS(1)remainsuncertain:itmaygoupaswellasdown.ThedifferenceS(1)−S(0)asafractionoftheinitialvaluerepresentstheso-calledrateofreturn,orbrieflyreturn:S(1)−S(0)KS=,S(0)whichisalsouncertain.ThedynamicsofstockpriceswillbediscussedinChap-ter3.Therisk-freepositioncanbedescribedastheamountheldinabankac-count.Asanalternativetokeepingmoneyinabank,investorsmaychoosetoinvestinbonds.ThepriceofonebondattimetwillbedenotedbyA(t).The1 2MathematicsforFinancecurrentbondpriceA(0)isknowntoallinvestors,justlikethecurrentstockprice.However,incontrasttostock,thepriceA(1)thebondwillfetchattime1isalsoknownwithcertainty.Forexample,A(1)maybeapaymentguaranteedbytheinstitutionissuingbonds,inwhichcasethebondissaidtomatureattime1withfacevalueA(1).Thereturnonbondsisdefinedinasimilarwayasthatonstock,A(1)−A(0)KA=.A(0)Chapters2,10and11giveadetailedexpositionofrisk-freeassets.Ourtaskistobuildamathematicalmodelofamarketoffinancialsecuri-ties.Acrucialfirststageisconcernedwiththepropertiesofthemathematicalobjectsinvolved.Thisisdonebelowbyspecifyinganumberofassumptions,thepurposeofwhichistofindacompromisebetweenthecomplexityoftherealworldandthelimitationsandsimplificationsofamathematicalmodel,imposedinordertomakeittractable.Theassumptionsreflectourcurrentpositiononthiscompromiseandwillbemodifiedinthefuture.Assumption1.1(Randomness)ThefuturestockpriceS(1)isarandomvariablewithatleasttwodifferentvalues.ThefuturepriceA(1)oftherisk-freesecurityisaknownnumber.Assumption1.2(PositivityofPrices)Allstockandbondpricesarestrictlypositive,A(t)>0andS(t)>0fort=0,1.Thetotalwealthofaninvestorholdingxstocksharesandybondsatatimeinstantt=0,1isV(t)=xS(t)+yA(t).Thepair(x,y)iscalledaportfolio,V(t)beingthevalueofthisportfolioor,inotherwords,thewealthoftheinvestorattimet.Thejumpsofassetpricesbetweentimes0and1giverisetoachangeoftheportfoliovalue:V(1)−V(0)=x(S(1)−S(0))+y(A(1)−A(0)).Thisdifference(whichmaybepositive,zero,ornegative)asafractionoftheinitialvaluerepresentsthereturnontheportfolio,V(1)−V(0)KV=.V(0) 1.Introduction:ASimpleMarketModel3Thereturnsonbondsorstockareparticularcasesofthereturnonaportfolio(withx=0ory=0,respectively).NotethatbecauseS(1)isarandomvariable,soisV(1)aswellasthecorrespondingreturnsKSandKV.ThereturnKAonarisk-freeinvestmentisdeterministic.Example1.1LetA(0)=100andA(1)=110dollars.ThenthereturnonaninvestmentinbondswillbeKA=0.10,thatis,10%.Also,letS(0)=50dollarsandsupposethattherandomvariableS(1)cantaketwovalues,
52withprobabilityp,S(1)=48withprobability1−p,foracertain00withnon-zeroprobability.Inotherwords,iftheinitialvalueofanadmissibleportfolioiszero,V(0)=0,thenV(1)=0withprobability1.Thismeansthatnoinvestorcanlockinaprofitwithoutriskandwithnoinitialendowment.Ifaportfolioviolatingthisprincipledidexist,wewouldsaythatanarbitrageopportunitywasavailable.Arbitrageopportunitiesrarelyexistinpractice.Ifandwhentheydo,thegainsaretypicallyextremelysmallascomparedtothevolumeoftransactions,makingthembeyondthereachofsmallinvestors.Inaddition,theycanbemoresubtlethantheexamplesabove.SituationswhentheNo-ArbitragePrincipleisviolatedaretypicallyshort-livedanddifficulttospot.Theactivitiesofinvestors(calledarbitrageurs)pursuingarbitrageprofitseffectivelymakethemarketfreeofarbitrageopportunities.Theexclusionofarbitrageinthemathematicalmodeliscloseenoughtorealityandturnsouttobethemostimportantandfruitfulassumption.Ar-gumentsbasedontheNo-arbitragePrinciplearethemaintoolsoffinancialmathematics.1.3One-StepBinomialModelInthissectionwerestrictourselvestoaverysimpleexample,inwhichthestockpriceS(1)takesonlytwovalues.Despiteitssimplicity,thissituationissufficientlyinterestingtoconveytheflavourofthetheorytobedevelopedlateron. 8MathematicsforFinanceExample1.4SupposethatS(0)=100dollarsandS(1)cantaketwovalues,
125withprobabilityp,S(1)=105withprobability1−p,where00withprobabilityp>0.Theportfolioprovidesanarbitrageopportunity,violatingtheNo-ArbitragePrinciple.NowsupposethatA(1)≥Su.Ifthisisthecase,thenattime0:•Sellshortonesharefor$100.•Invest$100risk-free.Asaresult,youwillbeholdingaportfolio(x,y)withx=−1andy=1,againofzeroinitialvalue,V(0)=0.Thefinalvalueofthisportfoliowillbe
−Su+A(1)ifstockgoesup,V(1)=−Sd+A(1)ifstockgoesdown,whichisnon-negative,withthesecondvaluebeingstrictlypositive,sinceA(1)≥Su.Thus,V(1)isanon-negativerandomvariablesuchthatV(1)>0withprobability1−p>0.Onceagain,thisindicatesanarbitrageopportunity,violatingtheNo-ArbitragePrinciple.Thecommonsensereasoningbehindtheaboveargumentisstraightforward:Buycheapassetsandsell(orsellshort)expensiveones,pocketingthedifference.1.4RiskandReturnLetA(0)=100andA(1)=110dollars,asbefore,butS(0)=80dollarsand
100withprobability0.8,S(1)=60withprobability0.2. 10MathematicsforFinanceSupposethatyouhave$10,000toinvestinaportfolio.Youdecidetobuyx=50shares,whichfixestherisk-freeinvestmentaty=60.Then
11,600ifstockgoesup,V(1)=9,600ifstockgoesdown,
0.16ifstockgoesup,KV=−0.04ifstockgoesdown.Theexpectedreturn,thatis,themathematicalexpectationofthereturnontheportfolioisE(KV)=0.16×0.8−0.04×0.2=0.12,thatis,12%.TheriskofthisinvestmentisdefinedtobethestandarddeviationoftherandomvariableKV:22σV=(0.16−0.12)×0.8+(−0.04−0.12)×0.2=0.08,thatis8%.Letuscomparethiswithinvestmentsinjustonetypeofsecurity.Ifx=0,theny=100,thatis,thewholeamountisinvestedrisk-free.InthiscasethereturnisknownwithcertaintytobeKA=0.1,thatis,10%andtheriskasmeasuredbythestandarddeviationiszero,σA=0.Ontheotherhand,ifx=125andy=0,theentireamountbeinginvestedinstock,then
12,500ifstockgoesup,V(1)=7,500ifstockgoesdown,andE(KS)=0.15withσS=0.20,thatis,15%and20%,respectively.Giventhechoicebetweentwoportfolioswiththesameexpectedreturn,anyinvestorwouldobviouslypreferthatinvolvinglowerrisk.Similarly,iftherisklevelswerethesame,anyinvestorwouldoptforhigherreturn.However,inthecaseinhandhigherreturnisassociatedwithhigherrisk.Insuchcircumstancesthechoicedependsonindividualpreferences.TheseissueswillbediscussedinChapter5,whereweshallalsoconsiderportfoliosconsistingofseveralriskysecurities.Theemergingpicturewillshowthepowerofportfolioselectionandportfoliodiversificationastoolsforreducingriskwhilemaintainingtheex-pectedreturn.Exercise1.4Fortheabovestockandbondprices,designaportfoliowithinitialwealthof$10,000splitfifty-fiftybetweenstockandbonds.Computetheex-pectedreturnandriskasmeasuredbystandarddeviation. 1.Introduction:ASimpleMarketModel111.5ForwardContractsAforwardcontractisanagreementtobuyorsellariskyassetataspecifiedfuturetime,knownasthedeliverydate,forapriceFfixedatthepresentmoment,calledtheforwardprice.Aninvestorwhoagreestobuytheassetissaidtoenterintoalongforwardcontractortotakealongforwardposition.Ifaninvestoragreestoselltheasset,wespeakofashortforwardcontractorashortforwardposition.Nomoneyispaidatthetimewhenaforwardcontractisexchanged.Example1.5Supposethattheforwardpriceis$80.Ifthemarketpriceoftheassetturnsouttobe$84onthedeliverydate,thentheholderofalongforwardcontractwillbuytheassetfor$80andcansellitimmediatelyfor$84,cashingthedifferenceof$4.Ontheotherhand,thepartyholdingashortforwardpositionwillhavetoselltheassetfor$80,sufferingalossof$4.However,ifthemarketpriceoftheassetturnsouttobe$75onthedeliverydate,thenthepartyholdingalongforwardpositionwillhavetobuytheassetfor$80,sufferingalossof$5.Meanwhile,thepartyholdingashortpositionwillgain$5bysellingtheassetaboveitsmarketprice.Ineithercasethelossofonepartyisthegainoftheother.Ingeneral,thepartyholdingalongforwardcontractwithdeliverydate1willbenefitifthefutureassetpriceS(1)risesabovetheforwardpriceF.IftheassetpriceS(1)fallsbelowtheforwardpriceF,thentheholderofalongforwardcontractwillsufferaloss.Ingeneral,thepayoffforalongforwardpositionisS(1)−F(whichcanbepositive,negativeorzero).ForashortforwardpositionthepayoffisF−S(1).Apartfromstockandbonds,aportfolioheldbyaninvestormaycontainforwardcontracts,inwhichcaseitwillbedescribedbyatriple(x,y,z).Herexandyarethenumbersofstocksharesandbonds,asbefore,andzisthenumberofforwardcontracts(positiveforalongforwardpositionandnegativeforashortposition).Becausenopaymentisduewhenaforwardcontractisexchanged,theinitialvalueofsuchaportfolioissimplyV(0)=xS(0)+yA(0).AtthedeliverydatethevalueoftheportfoliowillbecomeV(1)=xS(1)+yA(1)+z(S(1)−F). 12MathematicsforFinanceAssumptions1.1to1.5aswellastheNo-ArbitragePrincipleextendreadilytothiscase.TheforwardpriceFisdeterminedbytheNo-ArbitragePrinciple.Inpar-ticular,itcaneasilybefoundforanassetwithnocarryingcosts.Atypicalexampleofsuchanassetisastockpayingnodividend.(Bycontrast,acom-moditywillusuallyinvolvestoragecosts,whileaforeigncurrencywillearninterest,whichcanberegardedasanegativecarryingcost.)AforwardpositionguaranteesthattheassetwillbeboughtfortheforwardpriceFatdelivery.Alternatively,theassetcanbeboughtnowandhelduntildelivery.However,iftheinitialcashoutlayistobezero,thepurchasemustbefinancedbyaloan.Theloanwithinterest,whichwillneedtoberepaidatthedeliverydate,isacandidatefortheforwardprice.Thefollowingpropositionshowsthatthisisindeedthecase.Proposition1.2SupposethatA(0)=100,A(1)=110,andS(0)=50dollars,wheretheriskysecurityinvolvesnocarryingcosts.ThentheforwardpricemustbeF=55dollars,oranarbitrageopportunitywouldexistotherwise.ProofSupposethatF>55.Then,attime0:•Borrow$50.•BuytheassetforS(0)=50dollars.•EnterintoashortforwardcontractwithforwardpriceFdollarsanddeliverydate1.Theresultingportfolio(1,−1,−1)consistingofstock,arisk-freeposition,and2ashortforwardcontracthasinitialvalueV(0)=0.Then,attime1:•ClosetheshortforwardpositionbysellingtheassetforFdollars.•Closetherisk-freepositionbypaying1×110=55dollars.2Thefinalvalueoftheportfolio,V(1)=F−55>0,willbeyourarbitrageprofit,violatingtheNo-ArbitragePrinciple.Ontheotherhand,ifF<55,thenattime0:•Sellshorttheassetfor$50.•Investthisamountrisk-free.•TakealongforwardpositioninstockwithforwardpriceFdollarsanddeliverydate1.Theinitialvalueofthisportfolio(−1,1,1)isalsoV(0)=0.Subsequently,at2time1: 1.Introduction:ASimpleMarketModel13•Cash$55fromtherisk-freeinvestment.•BuytheassetforFdollars,closingthelongforwardposition,andreturntheassettotheowner.YourarbitrageprofitwillbeV(1)=55−F>0,whichonceagainviolatestheNo-ArbitragePrinciple.ItfollowsthattheforwardpricemustbeF=55dollars.Exercise1.5LetA(0)=100,A(1)=112andS(0)=34dollars.IsitpossibletofindanarbitrageopportunityiftheforwardpriceofstockisF=38.60dollarswithdeliverydate1?Exercise1.6SupposethatA(0)=100andA(1)=105dollars,thepresentpriceofpoundsterlingisS(0)=1.6dollars,andtheforwardpriceisF=1.50dollarstoapoundwithdeliverydate1.Howmuchshouldasterlingbondcosttodayifitpromisestopay£100attime1?Hint:Thefor-wardcontractisbasedonanassetinvolvingnegativecarryingcosts(theinterestearnedbyinvestinginsterlingbonds).1.6CallandPutOptionsLetA(0)=100,A(1)=110,S(0)=100dollarsand
120withprobabilityp,S(1)=80withprobability1−p,where01S(0).Ifthisisthecase,thenattime0:112•Issueandsell1optionforC(0)dollars.•Borrow4×100=400dollarsincash(ortakeashortpositiony=−4in111111bondsbysellingthem).•Purchasex=1sharesofstockforxS(0)=1×100=50dollars.22Thecashbalanceofthesetransactionsispositive,C(0)+4A(0)−1S(0)>0.112Investthisamountrisk-free.Theresultingportfolioconsistingofshares,risk-freeinvestmentsandacalloptionhasinitialvalueV(0)=0.Subsequently,attime1:•Ifstockgoesup,thensettletheoptionbypayingthedifferenceof$20betweenthemarketpriceofoneshareandthestrikeprice.Youwillpaynothingifstockgoesdown.ThecosttoyouwillbeC(1),whichcoversbothpossibilities.•Repaytheloanwithinterest(orcloseyourshortpositiony=−4inbonds).11Thiswillcostyou4×110=40dollars.11•Sellthestockfor1S(1)obtainingeither1×120=60dollarsiftheprice22goesup,or1×80=40dollarsifitgoesdown.2Thecashbalanceofthesetransactionswillbezero,−C(1)+1S(1)−4A(1)=0,211regardlessofwhetherstockgoesupordown.Butyouwillbeleftwiththeinitialrisk-freeinvestmentofC(0)+4A(0)−1S(0)plusinterest,thusrealisingan112arbitrageopportunity.Ontheotherhand,ifC(0)+4A(0)<1S(0),then,attime0:112•Buy1optionforC(0)dollars.•Buy4bondsfor4×100=400dollars.111111•Sellshortx=1sharesofstockfor1×100=50dollars.22Thecashbalanceofthesetransactionsispositive,−C(0)−4A(0)+1S(0)>0,112andcanbeinvestedrisk-free.InthiswayyouwillhaveconstructedaportfoliowithinitialvalueV(0)=0.Subsequently,attime1:•Ifstockgoesup,thenexercisetheoption,receivingthedifferenceof$20betweenthemarketpriceofoneshareandthestrikeprice.Youwillreceivenothingifstockgoesdown.YourincomewillbeC(1),whichcoversbothpossibilities.•Sellthebondsfor4A(1)=4×110=40dollars.1111•Closetheshortpositioninstock,paying1S(1),thatis,1×120=60dollars22ifthepricegoesup,or1×80=40dollarsifitgoesdown.2Thecashbalanceofthesetransactionswillbezero,C(1)+4A(1)−1S(1)=0,112regardlessofwhetherstockgoesupordown.Butyouwillbeleftwithan 1.Introduction:ASimpleMarketModel17arbitrageprofitresultingfromtherisk-freeinvestmentof−C(0)−4A(0)+111S(0)plusinterest,againacontradictionwiththeNo-ArbitragePrinciple.2Hereweseeoncemorethatthearbitragestrategyfollowsacommonsensepattern:Sell(orsellshortifnecessary)expensivesecuritiesandbuycheapones,aslongasallyourfinancialobligationsarisingintheprocesscanbedischarged,regardlessofwhathappensinthefuture.Proposition1.3impliesthattoday’spriceoftheoptionmustbe14C(0)=S(0)−A(0)∼=13.6364211dollars.Anyonewhowouldselltheoptionforlessorbuyitformorethanthispricewouldbecreatinganarbitrageopportunity,whichamountstohandingoutfreemoney.Thiscompletesthesecondstepofoursolution.Remark1.2Notethattheprobabilitiespand1−pofstockgoingupordownareirrelevantinpricingandreplicatingtheoption.Thisisaremarkablefeatureofthetheoryandbynomeansacoincidence.Remark1.3Optionsmayappeartobesuperfluousinamarketinwhichtheycanberepli-catedbystockandbonds.Inthesimplifiedone-stepmodelthisisinfactavalidobjection.However,inasituationinvolvingmultipletimesteps(orcontinuoustime)replicationbecomesamuchmoreoneroustask.Itrequiresadjustmentstothepositionsinstockandbondsateverytimeinstantatwhichthereisachangeinprices,resultinginconsiderablemanagementandtransactioncosts.Insomecasesitmaynotevenbepossibletoreplicateanoptionprecisely.Thisiswhythemajorityofinvestorsprefertobuyorselloptions,replicationbeingnormallyundertakenonlybyspecialiseddealersandinstitutions.Exercise1.7LetthebondandstockpricesA(0),A(1),S(0),S(1)beasabove.Com-putethepriceC(0)ofacalloptionwithexercisetime1anda)strikeprice$90,b)strikeprice$110.Exercise1.8LetthepricesA(0),S(0),S(1)beasabove.ComputethepriceC(0)of 18MathematicsforFinanceacalloptionwithstrikeprice$100andexercisetime1ifa)A(1)=105dollars,b)A(1)=115dollars.Aputoptionwithstrikeprice$100andexercisetime1givestherighttoselloneshareofstockfor$100attime1.Thiskindofoptionisworthlessifthestockgoesup,butitbringsaprofitotherwise,thepayoffbeing
0ifstockgoesup,P(1)=20ifstockgoesdown,giventhatthepricesA(0),A(1),S(0),S(1)arethesameasabove.Thenotionofaportfoliomaybeextendedtoallowpositionsinputoptions,denotedbyz,asbefore.Thereplicatingandpricingprocedureforputsfollowsthesamepatternasforcalloptions.Inparticular,thepriceP(0)oftheputoptionisequaltothetime0valueofareplicatinginvestmentinstockandbonds.Remark1.4Thereissomesimilaritybetweenaputoptionandashortforwardposition:bothinvolvesellinganassetforafixedpriceatacertaintimeinthefuture.However,anessentialdifferenceisthattheholderofashortforwardcontractiscommittedtosellingtheassetforthefixedprice,whereastheownerofaputoptionhastherightbutnoobligationtosell.Moreover,aninvestorwhowantstobuyaputoptionwillhavetopayforit,whereasnopaymentisinvolvedwhenaforwardcontractisexchanged.Exercise1.9Onceagain,letthebondandstockpricesA(0),A(1),S(0),S(1)beasabove.ComputethepriceP(0)ofaputoptionwithstrikeprice$100.Aninvestormaywishtotradesimultaneouslyinbothkindsofoptionsand,inaddition,totakeaforwardposition.Insuchcasesnewsymbolsz1,z2,z3,...willneedtobereservedforalladditionalsecuritiestodescribethepositionsinaportfolio.Acommonfeatureofthesenewsecuritiesisthattheirpayoffsdependonthestockprices.Becauseofthistheyarecalledderivativesecuritiesorcontingentclaims.ThegeneralpropertiesofderivativesecuritieswillbediscussedinChapter7.InChapter8thepricingandreplicatingschemeswillbeextendedtomorecomplicated(andmorerealistic)marketmodels,aswellastootherfinancialinstruments. 1.Introduction:ASimpleMarketModel191.7ManagingRiskwithOptionsTheavailabilityofoptionsandotherderivativesecuritiesextendsthepossibleinvestmentscenarios.Supposethatyourinitialwealthis$1,000andcomparethefollowingtwoinvestmentsinthesetupoftheprevioussection:•buy10shares;attime1theywillbeworth
1,200ifstockgoesup,10×S(1)=800ifstockgoesdown;or•buy1,000/13.6364∼=73.3333options;inthiscaseyourfinalwealthwillbe
1,466.67ifstockgoesup,73.3333×C(1)∼=0.00ifstockgoesdown.Ifstockgoesup,theinvestmentinoptionswillproduceamuchhigherreturnthanshares,namelyabout46.67%.However,itwillbedisastrousotherwise:youwillloseallyourmoney.Meanwhile,wheninvestinginshares,youwouldgainjust20%orlose20%.Withoutspecifyingtheprobabilitieswecannotcomputetheexpectedreturnsorstandarddeviations.Nevertheless,onewouldreadilyagreethatinvestinginoptionsismoreriskythaninstock.Thiscanbeexploitedbyadventurousinvestors.Exercise1.10Intheabovesetting,findthefinalwealthofaninvestorwhoseinitialcapitalof$1,000issplitfifty-fiftybetweenstockandoptions.Optionscanalsobeemployedtoreducerisk.Consideraninvestorplanningtopurchasestockinthefuture.ThesharepricetodayisS(0)=100dollars,buttheinvestorwillonlyhavefundsavailableatafuturetimet=1,whenthesharepricewillbecome
160withprobabilityp,S(1)=40withprobability1−p,forsome00istheinterestrate.ThevalueoftheinvestmentwillthusbecomeV(1)=P+rP=(1+r)P.AftertwoyearstheinvestmentwillgrowtoV(2)=(1+2r)P.Considerafractionofayear.Interestistypicallycalculatedonadailybasis:theinterestearnedinonedaywillbe1rP.AfterndaystheinterestwillbenrPand365365thetotalvalueoftheinvestmentwillbecomeV(n)=(1+nr)P.This365365motivatesthefollowingruleofsimpleinterest:Thevalueoftheinvestmentattimet,denotedbyV(t),isgivenbyV(t)=(1+tr)P,(2.1)wheretimet,expressedinyears,canbeanarbitrarynon-negativerealnumber;seeFigure2.1.Inparticular,wehavetheobviousequalityV(0)=P.Thenumber1+rtiscalledthegrowthfactor.Hereweassumethattheinterestraterisconstant.IftheprincipalPisinvestedattimes,ratherthanattime0,thenthevalueattimet≥swillbeV(t)=(1+(t−s)r)P.(2.2)Figure2.1Principalattractingsimpleinterestat10%(r=0.1,P=1) 2.Risk-FreeAssets23Throughoutthisbooktheunitoftimewillbeoneyear.Weshalltransformanyperiodexpressedinotherunits(days,weeks,months)intoafractionofayear.Example2.1Consideradepositof$150heldfor20daysandattractingsimpleinterestatarateof8%.Thisgivest=20andr=0.08.After20daysthedepositwill365growtoV(20)=(1+20×0.08)×150∼=150.66.365365ThereturnonaninvestmentcommencingattimesandterminatingattimetwillbedenotedbyK(s,t).ItisgivenbyV(t)−V(s)K(s,t)=.(2.3)V(s)InthecaseofsimpleinterestK(s,t)=(t−s)r,whichclearlyfollowsfrom(2.2).Inparticular,theinterestrateisequaltothereturnoveroneyear,K(0,1)=r.Asageneralrule,interestrateswillalwaysrefertoaperiodofoneyear,fa-cilitatingthecomparisonbetweendifferentinvestments,independentlyoftheiractualduration.Bycontrast,thereturnreflectsboththeinterestrateandthelengthoftimetheinvestmentisheld.Exercise2.1Asumof$9,000paidintoabankaccountfortwomonths(61days)toattractsimpleinterestwillproduce$9,020attheandoftheterm.Findtheinterestraterandthereturnonthisinvestment.Exercise2.2Howmuchwouldyoupaytodaytoreceive$1,000atacertainfuturedateifyourequireareturnof2%?Exercise2.3Howlongwillittakeforasumof$800attractingsimpleinteresttobecome$830iftherateis9%?Computethereturnonthisinvestment. 24MathematicsforFinanceExercise2.4Findtheprincipaltobedepositedinitiallyinanaccountattractingsim-pleinterestatarateof8%if$1,000isneededafterthreemonths(91days).Thelastexerciseisconcernedwithanimportantgeneralproblem:Findtheinitialsumwhosevalueattimetisgiven.Inthecaseofsimpleinteresttheansweriseasilyfoundbysolving(2.1)fortheprincipal,obtainingV(0)=V(t)(1+rt)−1.(2.4)ThisnumberiscalledthepresentordiscountedvalueofV(t)and(1+rt)−1isthediscountfactor.Example2.2Aperpetuityisasequenceofpaymentsofafixedamounttobemadeatequaltimeintervalsandcontinuingindefinitelyintothefuture.Forexample,supposethatpaymentsofanamountCaretobemadeonceayear,thefirstpaymentdueayearhence.ThiscanbeachievedbydepositingCP=rinabankaccounttoearnsimpleinterestataconstantrater.SuchadepositwillindeedproduceasequenceofinterestpaymentsamountingtoC=rPpayableeveryyear.Inpracticesimpleinterestisusedonlyforshort-terminvestmentsandforcertaintypesofloansanddeposits.Itisnotarealisticdescriptionofthevalueofmoneyinthelongerterm.Inthemajorityofcasestheinterestalreadyearnedcanbereinvestedtoattractevenmoreinterest,producingahigherreturnthanthatimpliedby(2.1).Thiswillbeanalysedindetailinwhatfollows.2.1.2PeriodicCompoundingOnceagain,supposethatanamountPisdepositedinabankaccount,at-tractinginterestataconstantrater>0.However,incontrasttothecaseofsimpleinterest,weassumethattheinterestearnedwillnowbeaddedtotheprincipalperiodically,forexample,annually,semi-annually,quarterly,monthly,orperhapsevenonadailybasis.Subsequently,interestwillbeattractednot 2.Risk-FreeAssets25justbytheoriginaldeposit,butalsobyalltheinterestearnedsofar.Inthesecircumstancesweshalltalkofdiscreteorperiodiccompounding.Example2.3InthecaseofmonthlycompoundingthefirstinterestpaymentofrPwillbe12dueafteronemonth,increasingtheprincipalto(1+r)P,allofwhichwill12attractinterestinthefuture.Thenextinterestpayment,dueaftertwomonths,willthusber(1+r)P,andthecapitalwillbecome(1+r)2P.Afterone121212yearitwillbecome(1+r)12P,afternmonthsitwillbe(1+r)nP,andafter1212tyears(1+r)12tP.Thelastformulaadmitstequaltoawholenumberof12months,thatis,amultipleof1.12Ingeneral,ifminterestpaymentsaremadeperannum,thetimebetweentwoconsecutivepaymentsmeasuredinyearswillbe1,thefirstinterestpay-mmentbeingdueattime1.Eachinterestpaymentwillincreasetheprincipalmbyafactorof1+r.Giventhattheinterestraterremainsunchanged,aftertmyearsthefuturevalueofaninitialprincipalPwillbecomertmV(t)=1+P,(2.5)mbecausetherewillbetminterestpaymentsduringthisperiod.Inthisformulatmtmustbeawholemultipleoftheperiod1.Thenumber1+risthemmgrowthfactor.Theexactvalueoftheinvestmentmaysometimesneedtobeknownattimeinstantsbetweeninterestpayments.Inparticular,thismaybesoiftheaccountisclosedonadaywhennointerestpaymentisdue.Forexample,whatisthevalueafter10daysofadepositof$100subjecttomonthlycompoundingat12%?Onepossibleansweris$100,sincethefirstinterestpaymentwouldbedueonlyafteronewholemonth.Thissuggeststhat(2.5)shouldbeextendedtoarbitraryvaluesoftbymeansofastepfunctionwithstepsofduration1,masshowninFigure2.2.Lateron,inRemark2.6weshallseethattheextensionconsistentwiththeNo-ArbitragePrincipleshouldusetheright-handsideof(2.5)forallt≥0.Exercise2.5Howlongwillittaketodoubleacapitalattractinginterestat6%com-poundeddaily? 26MathematicsforFinanceFigure2.2Annualcompoundingat10%(m=1,r=0.1,P=1)Exercise2.6Whatistheinterestrateifadepositsubjecttoannualcompoundingisdoubledafter10years?Exercise2.7Findandcomparethefuturevalueaftertwoyearsofadepositof$100attractinginterestatarateof10%compoundeda)annuallyandb)semi-annually.Proposition2.1ThefuturevalueV(t)increasesifanyoneoftheparametersm,t,rorPincreases,theothersremainingunchanged.ProofItisimmediatelyobviousfrom(2.5)thatV(t)increasesift,rorPincreases.ToshowthatV(t)increasesasthecompoundingfrequencymincreases,weneedtoverifythatifm0compoundedmtimesayearcanbewrittenasmtrrrV(t)=1+P.mInthelimitasm→∞,weobtainV(t)=etrP,(2.10)wherex1e=lim1+x→∞xisthebaseofnaturallogarithms.Thisisknownascontinuouscompounding.Thecorrespondinggrowthfactorisetr.AtypicalgraphofV(t)isshowninFigure2.3.Figure2.3Continuouscompoundingat10%(r=0.1,P=1)ThederivativeofV(t)=etrPisV(t)=retrP=rV(t).Inthecaseofcontinuouscompoundingtherateofthegrowthisproportionaltothecurrentwealth.Formula(2.10)isagoodapproximationofthecaseofperiodiccompoundingwhenthefrequencymislarge.Itissimplerandlendsitselfmorereadilytotransformationsthantheformulaforperiodiccompounding. 2.Risk-FreeAssets33Exercise2.18Howlongwillittaketoearn$1ininterestif$1,000,000isdepositedat10%compoundedcontinuously?Exercise2.19In1626PeterMinuit,governorofthecolonyofNewNetherland,boughttheislandofManhattanfromIndianspayingwithbeads,cloth,andtrinketsworth$24.Findthevalueofthissuminyear2000at5%com-poundeda)continuouslyandb)annually.Proposition2.2Continuouscompoundingproduceshigherfuturevaluethanperiodiccom-poundingwithanyfrequencym,giventhesameinitialprincipalPandinterestrater.ProofItsufficestoverifythatrttrrtmrme>(1+)=(1+)r.mmrmTheinequalityholdsbecausethesequence(1+)risincreasingandconvergesmtoeasm∞.Exercise2.20Whatwillbethedifferencebetweenthevalueafteroneyearof$100depositedat10%compoundedmonthlyandcompoundedcontinuously?Howfrequentshouldtheperiodiccompoundingbeforthedifferencetobelessthan$0.01?ThepresentvalueundercontinuouscompoundingisobviouslygivenbyV(0)=V(t)e−tr.Inthiscasethediscountfactorise−tr.GiventheterminalvalueV(T),weclearlyhaveV(t)=e−r(T−t)V(T).(2.11) 34MathematicsforFinanceExercise2.21Findthepresentvalueof$1,000,000tobereceivedafter20yearsas-sumingcontinuouscompoundingat6%.Exercise2.22Giventhatthefuturevalueof$950subjecttocontinuouscompoundingwillbe$1,000afterhalfayear,findtheinterestrate.ThereturnK(s,t)definedby(2.3)onaninvestmentsubjecttocontinuouscompoundingfailstobeadditive,justlikeinthecaseofperiodiccompounding.ItprovesconvenienttointroducethelogarithmicreturnV(t)k(s,t)=ln.(2.12)V(s)Proposition2.3Thelogarithmicreturnisadditive,k(s,t)+k(t,u)=k(s,u).ProofThisisaneasyconsequenceof(2.12):V(t)V(u)k(s,t)+k(t,u)=ln+lnV(s)V(t)V(t)V(u)V(u)=ln=ln=k(s,u).V(s)V(t)V(s)IfV(t)isgivenby(2.10),thenk(s,t)=r(t−s),whichenablesustorecovertheinterestratek(s,t)r=.t−sExercise2.23Supposethatthelogarithmicreturnover2monthsonaninvestmentsubjecttocontinuouscompoundingis3%.Findtheinterestrate. 2.Risk-FreeAssets352.1.5HowtoCompareCompoundingMethodsAswehavealreadynoticed,frequentcompoundingwillproduceahigherfu-turevaluethanlessfrequentcompoundingiftheinterestratesandtheinitialprincipalarethesame.Weshallconsiderthegeneralcircumstancesinwhichonecompoundingmethodwillproduceeitherthesameorhigherfuturevaluethananothermethod,giventhesameinitialprincipal.Example2.5Supposethatcertificatespromisingtopay$120afteroneyearcanbepurchasedorsoldnow,oratanytimeduringthisyear,for$100.Thisisconsistentwithaconstantinterestrateof20%underannualcompounding.Ifaninvestordecidedtosellsuchacertificatehalfayearafterthepurchase,whatpricewoulditfetch?Supposeitis$110,afrequentfirstguessbasedonhalvingtheannualprofitof$20.However,thisturnsouttobetoohighaprice,leadingtothefollowingarbitragestrategy:•Borrow$1,000tobuy10certificatesfor$100each.•Aftersixmonthssellthe10certificatesfor$110eachandbuy11newcertificatesfor$100each.Thebalanceofthesetransactionsisnil.•Afteranothersixmonthssellthe11certificatesfor$110each,cashing$1,210intotal,andpay$1,200tocleartheloanwithinterest.Thebalanceof$10wouldbethearbitrageprofit.Asimilarargumentshowsthatthecertificatepriceaftersixmonthscannotbetoolow,say,$109.Thepriceofacertificateaftersixmonthsisrelatedtotheinterestrateundersemi-annualcompounding:Ifthisrateisr,thenthepriceis1001+r2dollarsandviceversa.Arbitragewilldisappearifthecorrespondinggrowth2factor1+roveroneyearisequaltothegrowthfactor1.2underannual2compounding,r21+=1.2,2whichgivesr∼=0.1909,or19.09%.Ifso,thenthecertificatepriceaftersixmonthsshouldbe1001+0.1909∼=109.54dollars.2Theideabasedonconsideringthegrowthfactorsoverafixedperiod,typi-callyoneyear,canbeusedtocompareanytwocompoundingmethods. 36MathematicsforFinanceDefinition2.1Wesaythattwocompoundingmethodsareequivalentifthecorrespondinggrowthfactorsoveraperiodofoneyeararethesame.Ifoneofthegrowthfactorsexceedstheother,thenthecorrespondingcompoundingmethodissaidtobepreferable.Example2.6Semi-annualcompoundingat10%isequivalenttoannualcompoundingat10.25%.Indeed,intheformercasethegrowthfactoroveraperiodofoneyearis20.11+=1.1025,2whichisthesameasthegrowthfactorinthelattercase.Botharepreferabletomonthlycompoundingat9%,forwhichthegrowthfactoroveroneyearisonly120.091+∼=1.0938.12Wecanfreelyswitchfromonecompoundingmethodtoanotherequivalentmethodbyrecalculatingtheinterestrate.Inthechapterstofollowweshallnormallyuseeitherannualorcontinuouscompounding.Exercise2.24Findtherateforcontinuouscompoundingequivalenttomonthlycom-poundingat12%.Exercise2.25Findthefrequencyofperiodiccompoundingat20%tobeequivalenttoannualcompoundingat21%.Insteadofcomparingthegrowthfactors,itisoftenconvenienttocomparetheso-calledeffectiveratesasdefinedbelow.Definition2.2Foragivencompoundingmethodwithinterestratertheeffectiveratereisonethatgivesthesamegrowthfactoroveraoneyearperiodunderannualcompounding. 2.Risk-FreeAssets37Inparticular,inthecaseofperiodiccompoundingwithfrequencymandratertheeffectiverateresatisfiesrm1+=1+re.mInthecaseofcontinuouscompoundingwithraterer=1+r.eExample2.7Inthecaseofsemi-annualcompoundingat10%theeffectiverateis10.25%,seeExample2.6.Proposition2.4Twocompoundingmethodsareequivalentifandonlyifthecorrespondingeffectiveratesrandrareequal,r=r.Thecompoundingmethodwitheeeeeffectiveraterispreferabletotheothermethodifandonlyifr>r.eeeProofThisisbecausethegrowthfactorsoveroneyearare1+rand1+r,respec-eetively.Example2.8InExercise2.8wehaveseenthatdailycompoundingat15%ispreferabletosemi-annualcompoundingat15.5%.Thecorrespondingeffectiveratesreandrcanbefoundfrome3650.151+re=1+∼=1.1618,36521+r=1+0.155∼=1.1610.e2Thismeansthatrisabout16.18%andrabout16.10%.eeRemark2.6Recallthatformula(2.5)forperiodiccompounding,thatis,rtmV(t)=1+P,m 38MathematicsforFinanceadmitsonlytimeinstantstbeingwholemultiplesofthecompoundingperiod1.AnargumentsimilartothatinExample2.5showsthattheappropriateno-mrtmarbitragevalueofaninitialsumPatanytimet≥0shouldbe1+P.mAreasonableextensionof(2.5)isthereforetousetheright-handsideforallt≥0ratherthanjustforwholemultiplesof1.Fromnowonweshallalwaysmusethisextension.IntermsoftheeffectiveraterethefuturevaluecanbewrittenastV(t)=(1+re)P.forallt≥0.ThisappliesbothtocontinuouscompoundingandtoperiodiccompoundingextendedtoarbitrarytimesasinRemark2.6.Proposition2.4impliesthat,giventhesameinitialprincipal,equivalentcompoundingmethodswillproducethesamefuturevalueforalltimest≥0.Similarly,acompoundingmethodpreferabletoanotheronewillproduceahigherfuturevalueforallt>0.Remark2.7Simpleinterestdoesnotfitintotheschemeforcomparingcompoundingmeth-ods.InthiscasethefuturevalueV(t)isalinearfunctionoftimet,whereasitisanexponentialfunctionifeithercontinuousorperiodiccompoundingapplies.Thegraphsofsuchfunctionshaveatmosttwointersectionpoints,sotheycanneverbeequaltooneanotherforalltimest≥0(exceptforthetrivialcaseofzeroprincipal).Exercise2.26WhatisthepresentvalueofanannuityconsistingofmonthlypaymentsofanamountCcontinuingfornyears?Expresstheanswerintermsoftheeffectiveratere.Exercise2.27Whatisthepresentvalueofaperpetuityconsistingofbimonthlypay-mentsofanamountC?Expresstheanswerintermsoftheeffectiveratere. 2.Risk-FreeAssets392.2MoneyMarketThemoneymarketconsistsofrisk-free(moreprecisely,default-free)securi-ties.Anexampleisabond,whichisafinancialsecuritypromisingtheholderasequenceofguaranteedfuturepayments.Risk-freemeansherethatthesepaymentswillbedeliveredwithcertainty.(Nevertheless,eveninthiscaseriskcannotbecompletelyavoided,sincethemarketpricesofsuchsecuritiesmayfluctuateunpredictably;seeChapters10and11.)Therearemanykindsofbondsliketreasurybillsandnotes,treasury,mortgageanddebenturebonds,commercialpapers,andotherswithvariousparticulararrangementsconcern-ingtheissuinginstitution,duration,numberofpayments,embeddedrightsandguarantees.2.2.1Zero-CouponBondsThesimplestcaseofabondisazero-couponbond,whichinvolvesjustasinglepayment.Theissuinginstitution(forexample,agovernment,abankoracom-pany)promisestoexchangethebondforacertainamountofmoneyF,calledthefacevalue,onagivendayT,calledthematuritydate.Typically,thelifespanofazero-couponbondisuptooneyear,thefacevaluebeingsomeroundfigure,forexample100.Ineffect,thepersonorinstitutionwhobuysthebondislendingmoneytothebondwriter.Giventheinterestrate,thepresentvalueofsuchabondcaneasilybecomputed.SupposethatabondwithfacevalueF=100dollarsismaturinginoneyear,andtheannualcompoundingrateris12%.ThenthepresentvalueofthebondshouldbeV(0)=F(1+r)−1∼=89.29dollars.Inreality,theoppositehappens:Bondsarefreelytradedandtheirpricesaredeterminedbymarketforces,whereastheinterestrateisimpliedbythebondprices,Fr=−1.(2.13)V(0)Thisformulagivestheimpliedannualcompoundingrate.Forinstance,ifaone-yearbondwithfacevalue$100isbeingtradedat$91,thentheimpliedrateis9.89%.Forsimplicity,weshallconsiderunitbondswithfacevalueequaltooneunitofthehomecurrency,F=1. 40MathematicsforFinanceTypically,abondcanbesoldatanytimepriortomaturityatthemarketprice.ThispriceattimetisdenotedB(t,T).Inparticular,B(0,T)isthecurrent,time0priceofthebond,andB(T,T)=1isequaltothefacevalue.Again,thesepricesdeterminetheinterestratesbyapplyingformulae(2.6)and(2.11)withV(t)=B(t,T),V(T)=1.Forexample,theimpliedannualcompoundingratesatisfiestheequationB(t,T)=(1+r)−(T−t).Thelastformulahastobesuitablymodifiedifadifferentcompoundingmethodisused.Usingperiodiccompoundingwithfrequencym,weneedtosolvetheequationr−m(T−t)B(t,T)=1+.mInthecaseofcontinuouscompoundingtheequationfortheimpliedratesatisfiesB(t,T)=e−r(T−t).Ofcourseallthesedifferentimpliedratesareequivalenttooneanother,sincethebondpricedoesnotdependonthecompoundingmethodused.Remark2.8Ingeneral,theimpliedinterestratemaydependonthetradingtimetaswellasonthematuritytimeT.Thisisanimportantissue,whichwillbediscussedinChapters10and11.Forthetimebeing,weadoptthesimplifyingassumptionthattheinterestrateremainsconstantthroughouttheperioduptomaturity.Exercise2.28Aninvestorpaid$95forabondwithfacevalue$100maturinginsixmonths.Whenwillthebondvaluereach$99iftheinterestrateremainsconstant?Exercise2.29Findtheinterestratesforannual,semi-annualandcontinuouscom-poundingimpliedbyaunitbondwithB(0.5,1)=0.9455.NotethatB(0,T)isthediscountfactorandB(0,T)−1isthegrowthfactorforeachcompoundingmethod.Theseuniversalfactorsareallthatisneededtocomputethetimevalueofmoney,withoutresortingtothecorrespondinginterestrates.However,interestratesareusefulbecausetheyaremoreintuitive. 2.Risk-FreeAssets41Foranaveragebankcustomertheinformationthataone-year$100bondcanbepurchasedfor$92.59maynotbeasclearastheequivalentstatementthatadepositwillearn8%interestifkeptforoneyear.2.2.2CouponBondsBondspromisingasequenceofpaymentsarecalledcouponbonds.Thesepay-mentsconsistofthefacevaluedueatmaturity,andcouponspaidregularly,typicallyannually,semi-annually,orquarterly,thelastcoupondueatmaturity.Theassumptionofconstantinterestratesallowsustocomputethepriceofacouponbondbydiscountingallthefuturepayments.Example2.9ConsiderabondwithfacevalueF=100dollarsmaturinginfiveyears,T=5,withcouponsofC=10dollarspaidannually,thelastoneatmaturity.Thismeansastreamofpaymentsof10,10,10,10,110dollarsattheendofeachconsecutiveyear.Giventhecontinuouscompoundingrater,say12%,wecanfindthepriceofthebond:V(0)=10e−r+10e−2r+10e−3r+10e−4r+110e−5r∼=90.27dollars.Exercise2.30Findthepriceofabondwithfacevalue$100and$5annualcouponsthatmaturesinfouryears,giventhatthecontinuouscompoundingrateisa)8%orb)5%.Exercise2.31SketchthegraphofthepriceofthebondinExercise2.30asafunctionofthecontinuouscompoundingrater.Whatisthevalueofthisfunctionforr=0?Whatisthelimitasr→∞?Example2.10WecontinueExample2.9.Afteroneyear,oncethefirstcouponiscashed,thebondbecomesafour-yearbondworthV(1)=10e−r+10e−2r+10e−3r+110e−4r∼=91.78 42MathematicsforFinancedollars.Observethatthetotalwealthattime1isV(1)+C=V(0)er.SixmonthslaterthebondwillbeworthV(1.5)=10e−0.5r+10e−1.5r+10e−2.5r+110e−3.5r∼=97.45dollars.Afterfouryearsthebondwillbecomeazero-couponbondwithfacevalue$110andpriceV(4)=110e−r∼=97.56dollars.Aninvestormaychoosetosellthebondatanytimepriortomaturity.Thepriceatthattimecanonceagainbefoundbydiscountingallthepaymentsdueatlatertimes.Exercise2.32SketchthegraphofthepriceofthecouponbondinExamples2.9and2.10asafunctionoftime.Exercise2.33HowlongwillittakeforthepriceofthecouponbondinExamples2.9and2.10toreach$95forthefirsttime?Thecouponcanbeexpressedasafractionofthefacevalue.Assumingthatcouponsarepaidannually,weshallwriteC=iF,whereiiscalledthecouponrate.Proposition2.5Whenevercouponsarepaidannually,thecouponrateisequaltotheinterestrateforannualcompoundingifandonlyifthepriceofthebondisequaltoitsfacevalue.Inthiscasewesaythatthebondsellsatpar.ProofToavoidcumbersomenotationwerestrictourselvestoanexample.Supposethatannualcompoundingwithr=iapplies,andconsiderabondwithfacevalueF=100maturinginthreeyears,T=3.ThenthepriceofthebondisCCF+CrFrFF(1+r)++=++1+r(1+r)2(1+r)31+r(1+r)2(1+r)3 2.Risk-FreeAssets43rFrFFrFF(1+r)=++=+=F.1+r(1+r)2(1+r)21+r(1+r)2Conversely,notethatCCF+C++1+r(1+r)2(1+r)3isone-to-oneasafunctionofr(infact,astrictlydecreasingfunction),soitassumesthevalueFexactlyonce,andweknowthishappensforr=i.Remark2.9Ifabondsellsbelowthefacevalue,itmeansthattheimpliedinterestrateishigherthanthecouponrate(sincethepriceofabonddecreaseswhentheinterestrategoesup).Ifthebondpriceishigherthanthefacevalue,itmeansthattheinterestrateislowerthanthecouponrate.Thismaybeimportantinformationinrealcircumstances,wherethebondpriceisdeterminedbythemarketandgivesanindicationofthelevelofinterestrates.Exercise2.34AbondwithfacevalueF=100andannualcouponsC=8maturingafterthreeyears,atT=3,istradingatpar.Findtheimpliedcontinuouscompoundingrate.2.2.3MoneyMarketAccountAninvestmentinthemoneymarketcanberealisedbymeansofafinancialintermediary,typicallyaninvestmentbank,whobuysandsellsbondsonbehalfofitscustomers(thusreducingtransactioncosts).Therisk-freepositionofaninvestorisgivenbythelevelofhisorheraccountwiththebank.Itisconvenienttothinkofthisaccountasatradableasset,whichisindeedthecase,sincethebondsthemselvesaretradable.Alongpositioninthemoneymarketinvolvesbuyingtheasset,thatis,investingmoney.Ashortpositionamountstoborrowingmoney.First,consideraninvestmentinazero-couponbondclosedpriortomaturity.AninitialamountA(0)investedinthemoneymarketmakesitpossibletopurchaseA(0)/B(0,T)bonds.ThevalueofeachbondwillfetchB(t,T)=e−(T−t)r=erte−rT=ertB(0,T) 44MathematicsforFinanceattimet.Asaresult,theinvestmentwillreachA(0)rtA(t)=B(t,T)=A(0)eB(0,T)attimet≤T.Exercise2.35Findthereturnona75-dayinvestmentinzero-couponbondsifB(0,1)=0.89.Exercise2.36Thereturnonabondoversixmonthsis7%.Findtheimpliedcontinuouscompoundingrate.Exercise2.37AfterhowmanydayswillabondpurchasedforB(0,1)=0.92producea5%return?Theinvestmentinabondhasafinitetimehorizon.ItwillbeterminatedwithA(T)=A(0)erTatthetimeTofmaturityofthebond.ToextendthepositioninthemoneymarketbeyondTonecanreinvesttheamountA(T)intoabondnewlyissuedattimeT,maturingatT>T.TakingA(T)astheinitialinvestmentwithTplayingtheroleofthestartingtime,wehave

A(t)=A(T)er(t−T)=A(0)ertforT≤t≤T.Byrepeatingthisargument,wereadilyarriveattheconclu-sionthataninvestmentinthemoneymarketcanbeprolongedforaslongasrequired,theformulaA(t)=A(0)ert(2.14)beingvalidforallt≥0.Exercise2.38Supposethatonedollarisinvestedinzero-couponbondsmaturingafteroneyear.Attheendofeachyeartheproceedsarereinvestedinnewbondsofthesamekind.Howmanybondswillbepurchasedattheendofyear9?Expresstheanswerintermsoftheimpliedcontinuouscompoundingrate. 2.Risk-FreeAssets45AnalternativewaytoprolonganinvestmentinthemoneymarketforaslongasrequiredistoreinvestthefacevalueofanybondsmaturingattimeTinotherbondsissuedattime0,butmaturingatalatertimet>T.HavinginvestedA(0)initiallytobuyunitbondsmaturingattimeT,wewillhavethesumofA(0)/B(0,T)atourdisposalattimeT.Atthistimewechoseabondmaturingattimet,itspriceatTbeingB(T,t).AttimetthisinvestmentwillbeworthA(0)A(0)rt==A(0)e,B(0,T)B(T,t)B(0,t)thesameasin(2.14).Finally,considercouponbondsasatooltomanufactureaninvestmentinthemoneymarket.SupposeforsimplicitythatthefirstcouponCisdueafteroneyear.Attime0webuyA(0)/V(0)couponbonds.AfteroneyearwecashthecouponandsellthebondforV(1),receivingthetotalsumC+V(1)=V(0)er(seeExample2.10).Becausetheinterestrateisconstant,thissumofmoneyiscertain.Inthiswaywehaveeffectivelycreatedazero-couponbondwithfacevalueV(0)ermaturingattime1.Itmeansthattheschemeworkedoutaboveforzero-couponbondsappliestocouponbondsaswell,resultinginthesameformula(2.14)forA(t).Exercise2.39Thesumof$1,000isinvestedinfive-yearbondswithfacevalue$100and$8couponspaidannually.Allcouponsarereinvestedinbondsofthesamekind.Assumingthatthebondsaretradingatparandtheinterestrateremainsconstantthroughouttheperiodtomaturity,computethenumberofbondsheldduringeachconsecutiveyearoftheinvestment.Aswehaveseen,undertheassumptionthattheinterestrateisconstant,thefunctionA(t)doesnotdependonthewaythemoneymarketaccountisrun,thatis,itneitherdependsonthetypesofbondsselectedforinvestmentnoronthemethodofextendingtheinvestmentbeyondthematurityofthebonds.ThroughoutmostofthisbookweshallassumeA(t)tobedeterministicandknown.Indeed,weassumethatA(t)=ert,whererisaconstantinterestrate.VariableinterestrateswillbeconsideredinChapter10andarandommoneymarketaccountwillbestudiedinChapter11. Thispageintentionallyleftblank 3RiskyAssetsThefuturepricesofanyassetareunpredictabletoacertainextent.Inthischapterweshalltypicallybeconcernedwithcommonstock,thoughanysecuritysuchasforeigncurrency,acommodity,orevenapartiallyunpredictablefuturecashflowcanbeconsidered.Marketpricesdependonthechoicesanddecisionsmadebyagreatnumberofagentsactingunderconditionsofuncertainty.Itisthereforereasonabletotreatthepricesofassetsasrandom.However,littlemorecanbesaidinafullygeneralsituation.Weshallthereforeimposespecificconditionsonassetprices,motivatedbyaneedforthemathematicalmodeltoberealisticandrelevantontheonehand,andtractableontheotherhand.3.1DynamicsofStockPricesThepriceofstockattimetwillbedenotedbyS(t).Itisassumedtobestrictlypositiveforallt.Wetaket=0tobethepresenttime,S(0)beingthecurrentstockprice,knowntoallinvestors.ThefuturepricesS(t)fort>0remainunknown,ingeneral.Mathematically,S(t)canberepresentedasapositiverandomvariableonaprobabilityspaceΩ,thatis,S(t):Ω→(0,∞).TheprobabilityspaceΩconsistsofallfeasiblepricemovement‘scenarios’ω∈Ω.WeshallwriteS(t,ω)todenotethepriceattimetifthemarketfollowsscenarioω∈Ω.47 48MathematicsforFinanceThecurrentstockpriceS(0)knowntoallinvestorsissimplyapositivenumber,butitcanbethoughtofasaconstantrandomvariable.TheunknownfuturepricesS(t)fort>0arenon-constantrandomvariables.Thismeansthatforeacht>0thereareatleasttwoscenariosω,ω∈ΩsuchthatS(t,ω)=S(t,ω).Weassumethattimerunsinadiscretemanner,t=nτ,wheren=0,1,2,3,...andτisafixedtimestep,typicallyayear,amonth,aweek,aday,orevenaminuteorasecondtodescribesomehectictrading.Becausewetakeoneyearastheunitmeasureoftime,amonthcorrespondstoτ=1/12,aweekcorrespondstoτ=1/52,adaytoτ=1/365,andsoon.TosimplifyournotationweshallwriteS(0),S(1),S(2),...,S(n),...insteadofS(0),S(τ),S(2τ),...,S(nτ),...,identifyingnwithnτ.Thisconventionwillinfactbeadoptedformanyothertime-dependentquantities.Example3.1Consideramarketthatcanfollowjusttwoscenarios,boomorrecession,de-notedbyω1andω2,respectively.Thecurrentsharepriceofacertainstockis$10,whichmayriseto$12afteroneyearifthereisaboomorcomedownto$7inthecaseofrecession.InthesecircumstancesΩ={ω1,ω2}and,puttingτ=1,wehaveScenarioS(0)S(1)ω1(boom)1012ω2(recession)107Example3.2Supposethattherearethreepossiblemarketscenarios,Ω={ω1,ω2,ω3},thestockpricestakingthefollowingvaluesovertwotimesteps:ScenarioS(0)S(1)S(2)ω1555860ω2555852ω3555253Thesepricemovementscanberepresentedasatree,seeFigure3.1.Itiscon-venienttoidentifythescenarioswithpathsthroughthetreeleadingfromthesinglenodeontheleft(the‘root’ofthetree)totherightmostbranchtips.Suchatreestructureofpricemovements,iffoundrealisticanddesirable,canreadilybeimplementedinamathematicalmodel. 3.RiskyAssets49Figure3.1TreeofpricemovementsinExample3.2Exercise3.1SketchatreerepresentingthescenariosandpricemovementsinExam-ple3.1.Exercise3.2Supposethatthestockpriceonanygivendaycaneitherbe5%higheror4%lowerthanonthepreviousday.Sketchatreerepresentingpossiblestockpricemovementsoverthenextthreedays,giventhatthepricetodayis$20.Howmanydifferentscenarioscanbedistinguished?3.1.1ReturnItprovesconvenienttodescribethedynamicsofstockpricesS(n)intermsofreturns.Weassumethatthestockpaysnodividends.Definition3.1Therateofreturn,orbrieflythereturnK(n,m)overatimeinterval[n,m](infact[mτ,nτ]),isdefinedtobetherandomvariableS(m)−S(n)K(n,m)=.S(n)Thereturnoverasingletimestep[n−1,n]willbedenotedbyK(n),thatisS(n)−S(n−1)K(n)=K(n−1,n)=,S(n−1)whichimpliesthatS(n)=S(n−1)(1+K(n)).(3.1) 50MathematicsforFinanceExample3.3InthesituationconsideredinExample3.2thereturnsarerandomvariablestakingthefollowingvalues:ScenarioK(1)K(2)ω15.45%3.45%ω25.45%−10.34%ω3−5.45%1.92%Exercise3.3GiventhefollowingreturnsandassumingthatS(0)=45dollars,findthepossiblestockpricesinathree-stepeconomyandsketchatreeofpricemovements:ScenarioK(1)K(2)K(3)ω110%5%−10%ω25%10%10%ω35%−10%10%Remark3.1Ifthestockpaysadividendofdiv(n)attimen,thenthedefinitionofreturnhastobemodified.Typically,whenadividendispaid,thestockpricedropsbythatamount.Sincetherighttoadividendisdecidedpriortothepaymentday,thedropofstockpriceisalreadyreflectedinS(n).Asaresult,aninvestorwhobuysstockattimen−1payingS(n−1)andwishestosellthestockattimenwillreceiveS(n)+div(n)andthereturnmustreflectthis:S(n)−S(n−1)+div(n)K(n)=.S(n−1)Exercise3.4IntroducethenecessarymodificationsinExercise3.3ifadividendof$1ispaidattheendofeachtimestep.Itisimportanttounderstandtherelationshipbetweenone-stepreturnsandthereturnoveralongertimeinterval. 3.RiskyAssets51Example3.4SupposethatS(0)=100dollars.1.ConsiderascenarioinwhichS(1)=110andS(2)=100dollars.InthiscaseK(0,2)=0%,whileK(1)=10%andK(2)∼=−9.09%,thesumoftheone-stepreturnsK(1)andK(2)beingpositiveandgreaterthanK(0,2).2.ConsideranotherscenariowithlowerpriceS(1)=90dollarsandwithS(2)=100dollarsasbefore.ThenK(1)=−10%andK(2)∼=11.11%,theirsumbeingonceagaingreaterthanK(0,2)=0%.3.InascenariosuchthatS(1)=110andS(2)=121dollarswehaveK(0,2)=21%,whichisgreaterthanK(1)+K(2)=10%+10%=20%.Exercise3.5FindK(0,2)andK(0,3)forthedatainExercise3.3andcomparetheresultswiththesumsofone-stepreturnsK(1)+K(2)andK(1)+K(2)+K(3),respectively.Remark3.2Thenon-additivityofreturns,alreadyobservedinChapter2fordeterministicreturns,isworthpointingout,sinceitiscommonpracticetocomputetheav-erageofrecordedpastreturnsasapredictionforthefuture.Thismayresultinmisrepresentingtheinformation,forexample,overestimatingthefuturereturnifthehistoricalpricestendtofluctuate,orunderestimatingiftheydonot.Proposition3.1Thepreciserelationshipbetweenconsecutiveone-stepreturnsandthereturnovertheaggregateperiodis1+K(n,m)=(1+K(n+1))(1+K(n+2))···(1+K(m)).ProofComparethefollowingtwoformulaeforS(m):S(m)=S(n)(1+K(n,m))andS(m)=S(n)(1+K(n+1))(1+K(n+2))···(1+K(m)).BothofthemfollowfromDefinition3.1. 52MathematicsforFinanceExercise3.6Ineachofthefollowingthreescenariosfindtheone-stepreturns,assum-ingthatK(1)=K(2):ScenarioS(0)S(2)ω13541ω23532ω33528Exercise3.7GiventhatK(1)=10%or−10%,andK(0,2)=21%,10%or−1%,findapossiblestructureofscenariossuchthatK(2)takesatmosttwodifferentvalues.Thelackofadditivityisoftenaninconvenience.Thiscanberectifiedbyintroducingthelogarithmicreturnonariskysecurity,motivatedbysimilarconsiderationsforrisk-freeassetsinChapter2.Definition3.2Thelogarithmicreturnoveratimeinterval[n,m](moreprecisely,[τn,τm])isarandomvariablek(n,m)definedbyS(m)k(n,m)=ln.S(n)Theone-steplogarithmicreturnwillbedenotedsimplybyk(n),thatis,S(n)k(n)=k(n−1,n)=ln,S(n−1)sothatS(n)=S(n−1)ek(n).(3.2)TherelationshipbetweenthereturnK(m,n)andthelogarithmicreturnk(m,n)isobviousbycomparingtheirdefinitions,namely1+K(m,n)=ek(m,n).Becauseofthiswecanreadilyswitchfromonereturntotheother. 3.RiskyAssets53Remark3.3Ifthestockpaysadividendofdiv(n)attimenandthisisreflectedinthepriceS(n),thenthefollowingversionofthelogarithmicreturnshouldbeused:S(n)+div(n)k(n)=ln.S(n−1)Consecutiveone-steplogarithmicreturnscanbecombinedinanadditivemannertofindthereturnduringtheoveralltimeperiod.Exercise3.8ForthedatainExample3.2findtherandomvariablesk(1),k(2)andk(0,2).Comparek(0,2)withk(1)+k(2).Proposition3.2Ifnodividendsarepaid,thenk(n,m)=k(n+1)+k(n+2)+···+k(m).ProofOntheonehand,S(m)=S(n)ek(n,m)bythedefinitionofthelogarithmicreturn.Ontheotherhand,usingone-steplogarithmicreturnsrepeatedly,weobtain,S(m)=S(n)ek(n+1)ek(n+2)···ek(m)=S(n)ek(n+1)+k(n+2)+···+k(m).Theresultfollowsbycomparingthesetwoexpressions.3.1.2ExpectedReturnSupposethattheprobabilitydistributionofthereturnKoveracertaintimeperiodisknown.ThenwecancomputethemathematicalexpectationE(K),calledtheexpectedreturn.Example3.5Weestimatetheprobabilitiesofrecession,stagnationandboomtobe1/4,1/2,1/4,respectively.Ifthepredictedannualreturnsonsomestockinthese 54MathematicsforFinancescenariosare−6%,4%,30%,respectively,thentheexpectedannualreturnis111−6%×+4%×+30%×=8%.424Exercise3.9Withtheprobabilitiesofrecession,stagnationandboomequalto1/2,1/4,1/4andthepredictedannualreturnsinthefirsttwoofthesescenar-iosat−5%and6%,respectively,findtheannualreturnintheremainingscenarioiftheexpectedannualreturnisknowntobe6%.Exercise3.10SupposethatthestockpricesinthefollowingthreescenariosareScenarioS(0)S(1)S(2)ω1100110120ω2100105100ω310090100withprobabilities1/4,1/4,1/2,respectively.FindtheexpectedreturnsE(K(1)),E(K(2))andE(K(0,2)).Compare1+E(K(0,2))with(1+E(K(1)))(1+E(K(2))).ThelastexerciseshowsthattherelationestablishedinProposition3.1doesnotextendtoexpectedreturns.Forthatweneedanadditionalassumption.Proposition3.3Iftheone-stepreturnsK(n+1),...,K(m)areindependent,then1+E(K(n,m))=(1+E(K(n+1)))(1+E(K(n+2)))···(1+E(K(m))).ProofThisisanimmediateconsequenceofProposition3.1andthefactthattheexpectationofaproductofindependentrandomvariablesistheproductofexpectations.(NotethatiftheK(i)areindependent,thensoaretherandomvariables1+K(i)fori=n+1,...,m.) 3.RiskyAssets55Exercise3.11Supposethatthetimestepistakentobethreemonths,τ=1/4,andthequarterlyreturnsK(1),K(2),K(3),K(4)areindependentandiden-ticallydistributed.FindtheexpectedquarterlyreturnE(K(1))andtheexpectedannualreturnE(K(0,4))iftheexpectedreturnE(K(0,3))overthreequartersis12%.Remark3.4Inthecaseoflogarithmicreturnsadditivityextendstoexpectedreturns,eveniftheone-stepreturnsarenotindependent.NamelyE(k(n,m))=E(k(n+1))+E(k(n+2))+···+E(k(m)).Thisisbecausetheexpectationofasumofrandomvariablesisthesumofexpectations.Remark3.5Inpracticeitisdifficulttoestimatetheprobabilitiesandreturnsineachsce-nario,neededtocomputetheexpectedreturn.Whatcanreadilybecomputedistheaveragereturnoverapastperiod.Theresultcanbeusedasanestimatefortheexpectedfuturereturn.Forexample,ifthestockpricesonthelast10consecutivedayswere$98,$100,$99,$95,$88,$82,$89,$98,$101,$105,thentheaverageoftheresultingninedailyreturnswouldbeabout0.77%.However,theaverageofthelastfourdailyreturnswouldbeabout6.18%.(Weuselog-arithmicreturnsbecauseoftheiradditivity.)Thisshowsthattheresultmaydependheavilyonthechoiceofdata.UsinghistoricalpricesforpredictionisacomplexstatisticalissuebelongingtoEconometrics,whichisbeyondthescopeofthisbook.3.2BinomialTreeModelWeshalldiscussanextremelyimportantmodelofstockprices.Ontheonehand,themodeliseasilytractablemathematicallybecauseitinvolvesasmallnumberofparametersandassumesanidenticalsimplestructureateachnodeofthetreeofstockprices.Ontheotherhand,itcapturessurprisinglymanyfeaturesofreal-worldmarkets.Themodelisdefinedbythefollowingconditions. 56MathematicsforFinanceCondition3.1Theone-stepreturnsK(n)onstockareidenticallydistributedindependentrandomvariablessuchthat
uwithprobabilityp,K(n)=dwithprobability1−p,ateachtimestepn,where−1r,arguingthatheorsheshouldberewardedwithahigherexpectedreturnasacompensationforrisk.ThereversesituationwhenE(K(1))0.BecausetherandomwalkwNisonlydefinedatdiscretetimesbeingwholemultiplesofthestepτ=1,weconsiderw(t),wheretNNNNisthewholemultipleof1nearesttot.Then,clearly,NtisawholenumberNNforeachN,andwecanwrite√x(1)+x(2)+···+x(NtN)wN(tN)=tN√.NtNAsN→∞,wehavetN→tandNtN→∞,sothatwN(tN)→W(t)√indistribution,whereW(t)=tX.ThelastequalitymeansthatW(t)isnormallydistributedwithmean0andvariancet.Thisargument,basedontheCentralLimitTheorem,worksforanysinglefixedtimet>0.Itispossibletoextendtheresulttoobtainalimitforalltimest≥0simultaneously,butthisisbeyondthescopeofthisbook.ThelimitW(t)iscalledtheWienerprocess(orBrownianmotion).Itinheritsmanyofthepropertiesoftherandomwalk,forexample:1.W(0)=0,whichcorrespondstowN(0)=0.2.E(W(t))=0,correspondingtoE(wN(t))=0(seethesolutionofExer-cise3.25).3.Var(W(t))=t,withthediscretecounterpartVar(wN(t))=t(seethesolu-tionofExercise3.25).4.TheincrementsW(t3)−W(t2)andW(t2)−W(t1)areindependentfor0≤t1≤t2≤t3;soaretheincrementswN(t3)−wN(t2)andwN(t2)−wN(t1).2See,forexample,Capi´nskiandZastawniak(2001). 70MathematicsforFinance5.W(t)hasanormaldistributionwithmean0andvariancet,thatis,with21−xdensity√e2t.ThisisrelatedtothedistributionofwN(t).Thelatteris2πtnotnormal,butapproachesthenormaldistributioninthelimitaccordingtotheCentralLimitTheorem.AnimportantdifferencebetweenW(t)andwN(t)isthatW(t)isdefinedforallt≥0,whereasthetimeinwN(t)isdiscrete,t=n/Nforn=0,1,2,....ThepriceprocessobtainedinthelimitfromSN(t)asN→∞willbedenotedbyS(t).WhileSN(t)satisfiestheapproximateequation(3.8)withtheappropriatesubstitutions,namely11211SN(t+)−SN(t)≈m+σSN(t)+σSN(t)(wN(t+)−wN(t)),N2NNthecontinuous-timestockpricesS(t)satisfyanequationoftheform12dS(t)=m+σS(t)dt+σS(t)dW(t).(3.9)2HeredS(t)=S(t+dt)−S(t)anddW(t)=W(t+dt)−W(t)aretheincrementsofS(t)andW(t)overaninfinitesimaltimeintervaldt.Theexplicitformulaeforthesolutionsarealsosimilar,SN(t)=SN(0)exp(mt+σwN(t))inthediscretecase,whereasS(t)=S(0)exp(mt+σW(t))inthecontinuouscase.Figure3.11DensityofthedistributionofS(10)SinceW(t)hasanormaldistributionwithmean0andvariancet,itfollowsthatlnS(t)hasanormaldistributionwithmeanlnS(0)+mtandvarianceσ2t.Becauseofthisitissaidthatthecontinuous-timepriceprocessS(t)hasthelog 3.RiskyAssets71normaldistribution.ThenumberσiscalledthevolatilityofthepriceS(t).ThedensityofthedistributionofS(t)isshowninFigure3.11fort=10,S(0)=1,m=0andσ=0.1.ThiscanbecomparedwiththediscretedistributioninFigure3.2.Remark3.6Equation(3.9)andtheincrementsdS(t),dW(t)anddtareintroducedaboveonlyinformallybyanalogywiththediscretecase.Theycanbegivenapre-cisestatusinStochasticCalculus,atheorywithfundamentalapplicationsinadvancedmathematicalfinance.Inparticular,(3.9)isanexampleofwhatisknownasastochasticdifferentialequation. Thispageintentionallyleftblank 4DiscreteTimeMarketModelsHavingdiscussedanumberofdifferentmodelsofstockpricedynamics,weshallnowgeneraliseandpursuealittlefurthersomeoftheideasintroducedinChapter1.Inparticular,weshallreformulateandextendthegeneralnotionsandassumptionsunderlyingmathematicalfinancealreadymentionedinthatchapter.AsinChapter3,weassumethattimerunsinstepsoffixedlengthτ.Formanytime-dependentquantitiesweshallsimplifythenotationbywritingninplaceofthetimet=nτofthenthstep.4.1StockandMoneyMarketModelsSupposethatmriskyassetsaretraded.Thesewillbereferredtoasstocks.Theirpricesattimen=0,1,2,...aredenotedbyS1(n),...,Sm(n).Inaddition,investorshaveattheirdisposalarisk-freeasset,thatis,aninvestmentinthemoneymarket.Unlessstatedotherwise,wetaketheinitialleveloftherisk-freeinvestmenttobeoneunitofthehomecurrency,A(0)=1.However,insomenumericalexamplesandexercisesweshalloftentakeA(0)=100forconvenience.Becausethemoneymarketaccountcanbemanufacturedusingbonds(seeChapter2),weshallfrequentlyrefertoarisk-freeinvestmentasapositioninbonds,findingitconvenienttothinkofA(n)asthebondpriceattimen.Theriskypositionsinassetsnumber1,...,mwillbedenotedbyx1,...,xm,73 74MathematicsforFinancerespectively,andtherisk-freepositionbyy.ThewealthofaninvestorholdingsuchpositionsattimenwillbemV(n)=xjSj(n)+yA(n).(4.1)j=1Assumptions1.1to1.5ofChapter1canreadilybeadaptedtothisgeneralsetting.ThemotivationandinterpretationoftheseassumptionsarethesameasinChapter1,withthenaturalchangesfromonetoseveraltimestepsandfromonetoseveralriskyassets.Assumption4.1(Randomness)ThefuturestockpricesS1(n),...,Sm(n)arerandomvariablesforanyn=1,2,....ThefuturepricesA(n)oftherisk-freesecurityforanyn=1,2,...areknownnumbers.Assumption4.2(PositivityofPrices)Allstockandbondpricesarestrictlypositive,S(n)>0andA(n)>0forn=0,1,2,....Assumption4.3(Divisibility,LiquidityandShortSelling)Aninvestormaybuy,sellandholdanynumberxkofstocksharesofeachkindk=1,...,mandtakeanyrisk-freepositiony,whetherintegerorfractional,negative,positiveorzero.Ingeneral,x1,...,xm,y∈R.Assumption4.4(Solvency)Thewealthofaninvestormustbenon-negativeatalltimes,V(n)≥0forn=0,1,2,....Assumption4.5(DiscreteUnitPrices)Foreachn=0,1,2,...thesharepricesS1(n),...,Sm(n)arerandomvariablestakingonlyfinitelymanyvalues. 4.DiscreteTimeMarketModels754.1.1InvestmentStrategiesThepositionsheldbyaninvestorintheriskyandrisk-freeassetscanbealteredatanytimestepbysellingsomeassetsandinvestingtheproceedsinotherassets.Inreallifecashcanbetakenoutoftheportfolioforconsumptionorinjectedfromothersources.Nevertheless,weshallassumethatnoconsumptionorinjectionoffundstakesplaceinourmodelstokeepthingsassimpleaspossible.Decisionsmadebyanyinvestorofwhentoalterhisorherportfolioandhowmanyassetstobuyorsellarebasedontheinformationcurrentlyavailable.Wearegoingtoexcludetheunlikelypossibilitythatinvestorscouldforeseethefuture,aswellasthesomewhatmorelikely(butillegal)onethattheywillactoninsiderinformation.However,allthehistoricalinformationaboutthemarketuptoandincludingthetimeinstantwhenaparticulartradingdecisionisexecutedwillbefreelyavailable.Example4.1Letm=2andsupposethatS1(0)=60,S1(1)=65,S1(2)=75,S2(0)=20,S2(1)=15,S2(2)=25,A(0)=100,A(1)=110,A(2)=121,inacertainmarketscenario.Attime0initialwealthV(0)=3,000dollarsisinvestedinaportfolioconsistingofx1(1)=20sharesofstocknumberone,x2(1)=65sharesofstocknumbertwo,andy(1)=5bonds.Ournotationalconventionistouse1ratherthan0astheargumentinx1(1),x2(1)andy(1)toreflectthefactthatthisportfoliowillbeheldoverthefirsttimestep.Attime1thisportfoliowillbeworthV(1)=20×65+65×15+5×110=2,825dollars.Atthattimethenumberofassetscanbealteredbybuyingorsellingsomeofthem,aslongasthetotalvalueremains$2,825.Forexample,wecouldformanewportfolioconsistingofx1(2)=15sharesofstockone,x2(2)=94sharesofstocktwo,andy(2)=4bonds,whichwillbeheldduringthesecondtimestep.ThevalueofthisportfoliowillbeV(2)=15×75+94×25+4×121=3,959dollarsattime2,whenthepositionsinstocksandbondscanbeadjustedonceagain,aslongasthetotalvalueremains$3,959,andsoon.However,ifnoadjustmentsaremadetotheoriginalportfolio,thenitwillbeworth$2,825attime1and$3,730attime2. 76MathematicsforFinanceDefinition4.1Aportfolioisavector(x1(n),...,xm(n),y(n))indicatingthenumberofsharesandbondsheldbyaninvestorbetweentimesn−1andn.Asequenceofportfoliosindexedbyn=1,2,...iscalledaninvestmentstrategy.Thewealthofaninvestororthevalueofthestrategyattimen≥1ismV(n)=xj(n)Sj(n)+y(n)A(n).j=1Attimen=0theinitialwealthisgivenbymV(0)=xj(1)Sj(0)+y(1)A(0).j=1WehaveseeninExample4.1thatthecontentsofaportfoliocanbeadjustedbybuyingorsellingsomeassetsatanytimestep,aslongasthecurrentvalueoftheportfolioremainsunaltered.Definition4.2Aninvestmentstrategyiscalledself-financingiftheportfolioconstructedattimen≥1tobeheldoverthenexttimestepn+1isfinancedentirelybythecurrentwealthV(n),thatis,mxj(n+1)Sj(n)+y(n+1)A(n)=V(n).(4.2)j=1Example4.2LetthestockandbondpricesbeasinExample4.1.SupposethataninitialwealthofV(0)=3,000dollarsisinvestedbypurchasingx1(1)=18.22sharesofthefirststock,shortsellingx2(1)=−16.81sharesofthesecondstock,andbuyingy(1)=22.43bonds.Thetime1valueofthisportfoliowillbeV(1)=18.22×65−16.81×15+22.43×110=3,399.45dollars.Theinvestorwillbenefitfromthedropofthepriceoftheshortedstock.Thisexampleillustratesthefactthatportfolioscontainingfractionalornegativenumbersofassetsareallowed.Wedonotimposeanyrestrictionsonthenumbersx1(n),...,xm(n),y(n).Thefactthattheycantakenon-integervaluesisreferredtoasdivisibility.Negativexj(n)meansthatstocknumberjissoldshort(inotherwords,a 4.DiscreteTimeMarketModels77shortpositionistakeninstockj),negativey(n)correspondstoborrowingcash(takingashortpositioninthemoneymarket,forexample,byissuingandsellingabond).Theabsenceofanyboundsonthesizeofthesenumbersmeansthatthemarketisliquid,thatis,anynumberofassetsofeachtypecanbepurchasedorsoldatanytime.Inpracticesomesecuritymeasurestocontrolshortsellingmaybeimple-mentedbystockexchanges.Typically,investorsarerequiredtopayacertainpercentageoftheshortsaleasasecuritydeposittocoverpossiblelosses.Iftheirlossesexceedthedeposit,thepositionmustbeclosed.Thedepositcreatesaburdenontheportfolio,particularlyifitearnsnointerestfortheinvestor.However,restrictionsofthiskindmaynotconcerndealerswhoworkforma-jorfinancialinstitutionsholdinglargenumbersofsharesdepositedbysmallerinvestors.Thesesharesmaybeborrowedinternallyinlieuofshortselling.Example4.3WecontinueassumingthatstockpricesfollowthescenarioinExample4.1.Supposethat20sharesofthefirststockaresoldshort,x1(1)=−20.Theinvestorwillreceive20×60=1,200dollarsincash,buthastopayasecuritydepositof,say50%,thatis,$600.Onetimesteplatershewillsufferalossof20×65−1,200=100dollars.Thisissubtractedfromthedepositandthepositioncanbeclosedbywithdrawingthebalanceof600−100=400dollars.Ontheotherhand,if60sharesofthesecondstockareshorted,thatis,x2(1)=−60,thentheinvestorwillmakeaprofitof1,200−60×15=300dollarsafteronetimestep.Thepositioncanbeclosedwithfinalwealth600+300=900dollars.Inbothcasesthefinalbalanceshouldbereducedby600×0.1=60dollars,theinterestthatwouldhavebeenearnedontheamountdeposited,haditbeeninvestedinthemoneymarket.Aninvestorconstructingaportfolioattimenhasnoknowledgeoffuturestockprices.Inparticular,noinsiderdealingisallowed.Investmentdecisionscanbebasedonlyontheperformanceofthemarkettodate.Thisisreflectedinthefollowingdefinition.Definition4.3Aninvestmentstrategyiscalledpredictableifforeachn=0,1,2,...theport-folio(x1(n+1),...,xm(n+1),y(n+1))constructedattimendependsonlyonthenodesofthetreeofmarketscenariosreacheduptoandincludingtimen.Thenextpropositionshowsthatthepositiontakenintherisk-freeassetis 78MathematicsforFinancealwaysdeterminedbythecurrentwealthandthepositionsinriskyassets.Proposition4.1GiventheinitialwealthV(0)andapredictablesequence(x1(n),...,xm(n)),n=1,2,...ofpositionsinriskyassets,itisalwayspossibletofindasequencey(n)ofrisk-freepositionssuchthat(x1(n),...,xm(n),y(n))isapredictableself-financinginvestmentstrategy.ProofPutV(0)−x1(1)S1(0)−···−xm(1)Sm(0)y(1)=A(0)andthencomputeV(1)=x1(1)S1(1)+···+xm(1)Sm(1)+y(1)A(1).Next,V(1)−x1(2)S1(1)−···−xm(2)Sm(1)y(2)=,A(1)V(2)=x1(2)S1(2)+···+xm(2)Sm(2)+y(2)A(2),andsoon.Thisclearlydefinesaself-financingstrategy.Thestrategyispre-dictablebecausey(n+1)canbeexpressedintermsofstockandbondpricesuptotimen.Exercise4.1Findthenumberofbondsy(1)andy(2)heldbyaninvestorduringthefirstandsecondstepsofapredictableself-financinginvestmentstrategywithinitialvalueV(0)=200dollarsandriskyassetpositionsx1(1)=35.24,x1(2)=−40.50,x2(1)=24.18,x2(2)=10.13,ifthepricesofassetsfollowthescenarioinExample4.1.Alsofindthetime1valueV(1)andtime2valueV(2)ofthisstrategy.Example4.4Onceagain,supposethatthestockandbondpricesfollowthescenarioinExample4.1.IfanamountV(0)=100dollarswereinvestedinaportfoliowith 4.DiscreteTimeMarketModels79x1(1)=−12,x2(1)=31andy(1)=2,thenitwouldleadtoinsolvency,sincethetime1valueofthisportfolioisnegative,V(1)=−12×65+31×15+2×110=−95dollars.Suchaportfolio,whichisexcludedbyAssumption4.4,wouldbeimpossibletoconstructinpractice.Noshortpositionwillbeallowedunlessitcanbeclosedatanytimeandinanyscenario(ifnecessary,bysellingotherassetsintheportfoliotoraisecash).Thismeansthatthewealthofaninvestormustbenon-negativeatalltimes.Definition4.4Astrategyiscalledadmissibleifitisself-financing,predictable,andforeachn=0,1,2,...V(n)≥0withprobability1.Exercise4.2Consideramarketconsistingofonerisk-freeassetwithA(0)=10andA(1)=11dollars,andoneriskyassetsuchthatS(0)=10andS(1)=13or9dollars.Onthex,yplanedrawthesetofallportfolios(x,y)suchthattheone-stepstrategyinvolvingriskypositionxandrisk-freepositionyisadmissible.4.1.2ThePrincipleofNoArbitrageWearereadytoformulatethefundamentalprincipleunderlyingallmathe-maticalmodelsinfinance.Itgeneralisesthesimplifiedone-stepversionoftheNo-ArbitragePrincipleinChapter1tomodelswithseveraltimestepsandseveralriskyassets.Whereasthenotionofaportfolioissufficienttostatetheone-stepversion,inthegeneralsettingweneedtouseasequenceofportfoliosforminganadmissibleinvestmentstrategy.Thisisbecauseinvestorscanadjusttheirpositionsateachtimestep.Assumption4.6(No-ArbitragePrinciple)ThereisnoadmissiblestrategysuchthatV(0)=0andV(n)>0withpositiveprobabilityforsomen=1,2,.... 80MathematicsforFinanceExercise4.3ShowthattheNo-ArbitragePrinciplewouldbeviolatediftherewasaself-financingpredictablestrategywithinitialvalueV(0)=0andfinalvalue0=V(2)≥0,suchthatV(1)<0withpositiveprobability.ThestrategyinExercise4.3clearlyviolatesthesolvencyassumption(As-sumption4.4),sinceV(1)maybenegative.Infact,thisassumptionisnotessen-tialfortheformulationoftheNo-ArbitragePrinciple.Anadmissiblestrategyrealisinganarbitrageopportunitycanbefoundwheneverthereisapredictableself-financingstrategy(possiblyviolatingAssumption4.4)suchthatV(0)=0and0=V(n)≥0forsomen>0.Exercise4.4Consideramarketwithonerisk-freeassetandoneriskyassetthatfollowsthebinomialtreemodel.Supposethatwheneverstockgoesup,youcanpredictthatitwillgodownatthenextstep.Findaself-financing(butnotnecessarilypredictable)strategywithV(0)=0,V(1)≥0and0=V(2)≥0.ThisexerciseindicatesthatpredictabilityisanessentialassumptionintheNo-ArbitragePrinciple.Aninvestorwhocouldforeseethefuturebehaviourofstockprices(here,ifstockgoesdownatonestep,youcanpredictwhatitwilldoatthenextstep)wouldalwaysbeabletofindasuitableinvestmentstrategytoensurearisk-freeprofit.Exercise4.5Consideramarketwitharisk-freeassetsuchthatA(0)=100,A(1)=110,A(2)=121dollarsandariskyasset,thepriceofwhichcanfollowthreepossiblescenarios,ScenarioS(0)S(1)S(2)ω1100120144ω210012096ω31009096Isthereanarbitrageopportunityifa)therearenorestrictionsonshortselling,andb)noshortsellingoftheriskyassetisallowed? 4.DiscreteTimeMarketModels81Exercise4.6GiventhebondandstockpricesinExercise4.5,isthereanarbitragestrategyifshortsellingofstockisallowed,butthenumberofunitsofeachassetinaportfoliomustbeaninteger?Exercise4.7GiventhebondandstockpricesinExercise4.5,isthereanarbitragestrategyifshortsellingofstockisallowed,buttransactioncostsof5%ofthetransactionvolumeapplywheneverstockistraded.4.1.3ApplicationtotheBinomialTreeModelWeshallseethatinthebinomialtreemodelwithseveraltimestepsCondi-tion3.2isequivalenttothelackofarbitrage.Proposition4.2Thebinomialtreemodeladmitsnoarbitrageifandonlyifd0,leadingtoarbitrage.Supposethatu≤r.Inthiscase:•Buyonebond.•Sellshort1/S(0)shares.Theresultingportfoliowithx=−1/S(0)andy=1willonceagainhaveinitialvalueV(0)=0.AfteronestepthisportfoliowillbeworthV(1)=r−u≥0ifthestockpricegoesup,orV(1)=r−d>0ifitgoesdown,alsorealisinganarbitrageopportunity.Finally,supposethatd0(acashloaninvestedinstock).ThenV(1)=a(d−r)<0ifthepriceofstockgoesdown.3)a<0(alongpositioninbondsfinancedbyshortingstock).InthiscaseV(1)=a(u−r)<0ifstockgoesup.Arbitrageisclearlyimpossiblewhend0atoneormoreofthesenodes.Bytheone-stepcasethisisimpossibleifd0foreachscenarioω∈ΩandthediscountedstockpricesSj(n)=Sj(n)/A(n)satisfyE∗(Sj(n+1)|S(n))=Sj(n)(4.3)foranyj=1,...,mandn=0,1,2,...,whereE∗(·|S(n))denotestheconditionalexpectationwithrespecttoprobabilityP∗computedoncethestockpriceS(n)becomesknownattimen.TheproofoftheFundamentalTheoremofAssetPricingisquitetechnicalandwillbeomitted.Definition4.5AsequenceofrandomvariablesX(0),X(1),X(2),...suchthatE∗(X(n+1)|S(n))=X(n)foreachn=0,1,2,...issaidtobeamartingalewithrespecttoP∗.Condition(4.3)canbeexpressedbysayingthatthediscountedstockpricesSj(0),Sj(1),Sj(2),...formamartingalewithrespecttoP∗.Thelatteriscalledarisk-neutralormartingaleprobabilityonthesetofscenariosΩ.Moreover,E∗iscalledarisk-neutralormartingaleexpectation. 84MathematicsforFinanceExample4.5LetA(0)=100,A(1)=110,A(2)=121andsupposethatstockpricescanfollowfourpossiblescenarios:ScenarioS(0)S(1)S(2)ω190100112ω290100106ω3908090ω4908080ThetreeofstockpricesisshowninFigure4.2.Therisk-neutralprobabilityP∗isrepresentedbythebranchingprobabilitiesp∗,q∗,r∗ateachnode.ConditionFigure4.2TreeofstockpricesinExample4.5(4.3)forS(n)=S(n)/A(n)canbewrittenintheformofthreeequations,oneforeachnodeofthetree,1008090p∗+(1−p∗)=,110110100112106100q∗+(1−q∗)=,121121110908080r∗+(1−r∗)=.121121110Thesecanbesolvedtofind1924p∗=,q∗=,r∗=.2035Foreachscenario(eachpaththroughthetree)thecorrespondingrisk-neutralprobabilitycanbecomputedasfollows:19219P∗(ω1)=p∗q∗=×=,2033019219P∗(ω2)=p∗(1−q∗)=×1−=,203601941P∗(ω3)=(1−p∗)r∗=1−×=,205251941P∗(ω4)=(1−p∗)(1−r∗)=1−×1−=.205100 4.DiscreteTimeMarketModels85ByTheorem4.4theexistenceofarisk-neutralprobabilityimpliesthatthereisnoarbitrage.4.2ExtendedModelsSecuritiessuchasstock,whicharetradedindependentlyofotherassets,arecalledprimarysecurities.Bycontrast,derivativesecuritiessuchas,forexam-ple,optionsorforwards(inChapter1wehaveseensomesimpleexamplesofthese)arelegalcontractsconferringcertainfinancialrightsorobligationsupontheholder,contingentonthepricesofothersecurities,referredtoastheun-derlyingsecurities.Anunderlyingsecuritymaybeaprimarysecurity,asforaforwardcontractonstock,butitmayalsobeaderivativesecurity,asinthecaseofanoptiononfutures.Aderivativesecuritycannotexistinitsownright,unlesstheunderlyingsecurityorsecuritiesaretraded.Derivativesecuritiesarealsoreferredtoascontingentclaimsbecausetheirvalueiscontingentontheunderlyingsecurities.Forexample,theholderofalongforwardcontractonastockiscommittedtobuyingthestockfortheforwardpriceataspecifiedtimeofdelivery,nomatterhowmuchtheactualstockpriceturnsouttobeatthattime.Thevalueoftheforwardpositioniscontingentonthestock.Itwillbecomepositiveifthemarketpriceofstockturnsouttobehigherthantheforwardpriceondelivery.Ifthestockpriceturnsouttobelowerthantheforwardprice,thenthevalueoftheforwardpositionwillbenegative.Remark4.1TheassumptionsinSection4.1,includingtheNo-ArbitragePrinciple,arestatedforstrategiesconsistingofprimarysecuritiesonly,suchasstocksandbonds(orthemoneymarketaccount).Nevertheless,inmanytextstheyareinvokedinarbitrageproofsinvolvingstrategiesconstructedoutofderivativesecuritiesinadditiontostocksandbonds.Toavoidthisinaccuracytheas-sumptionsneedtobeextendedtostrategiesconsistingofbothprimaryandderivativesecurities.ThesettingofSection4.1,involvingportfoliosofriskystocksandthemoneymarketaccount,willbeextendedtoincluderiskysecuritiesofvariousotherkindsinadditionto(andsometimesinplaceof)stock.Inparticular,tocoverreal-lifesituationsweneedtoincludederivativesecuritiessuchasforwardsoroptions,butalsoprimarysecuritiessuchasbondsofvariousmaturities,the 86MathematicsforFinancefuturepricesofwhichmayberandom(except,ofcourse,atmaturity).Weshallalsorelaxtheassumptionthataninvestmentinamoneymarketaccountshouldberisk-free,withaviewtowardsmodellingrandominterestrates.InthiswaywepreparethestageforadetailedstudyofderivativesecuritiesinChapters6,7and8,andrandombondpricesandthetermstructureofinterestratesinChapters10and11.SecuritiesofvariouskindswillbetreatedonasimilarfootingasstockinSection4.1.WeshalldenotebyS1(n),...,Sm(n)thetimenpricesofmdif-ferentprimarysecurities,typicallymdifferentstocks,thoughtheymayalsoincludeotherassetssuchasforeigncurrency,commoditiesorbondsofvari-ousmaturities.Moreover,thepriceofonedistinguishedprimarysecurity,themoneymarketaccount,willbedenotedbyA(n).Inaddition,weintroducekdifferentderivativesecuritiessuchasforwards,callandputoptions,orindeedanyothercontingentclaims,whosetimenmarketpriceswillbedenotedbyD1(n),...,Dk(n).Asopposedtostocksandbonds,wecannolongerinsistthatthepricesofallderivativesecuritiesshouldbepositive.Forexample,atthetimeofexchangingaforwardcontractitsvalueiszero,whichmayandoftendoesbecomenegativelateronbecausetheholderofalongforwardpositionmayhavetobuythestockaboveitsmarketpriceatdelivery.ThefuturepricesS1(n),...,Sm(n)andA(n)ofprimarysecuritiesandthefuturepricesD1(n),...,Dk(n)ofderivativesecuritiesmayberandomforn=1,2,...,butwedonotruleoutthepossibilitythatsomeofthem,suchasthepricesofbondsatmaturity,mayinfactbeknowninadvance,beingrepresentedbyconstantrandomvariablesorsimplyrealnumbers.AllthecurrentpricesS1(0),...,Sm(0),A(0),D1(0),...,Dk(0)areofcourseknownattime0,thatis,arealsojustrealnumbers.Thepositionsinprimarysecurities,includingthemoneymarketaccount,willbedenotedbyx1,...,xmandy,andthoseinderivativesecuritiesbyz1,...,zk,respectively.ThewealthofaninvestorholdingsuchpositionsattimenwillbemkV(n)=xjSj(n)+yA(n)+ziDi(n),j=1i=1whichextendsformula(4.1).TheassumptionsinSection4.1needtobereplacedbythefollowing.Assumption4.1a(Randomness)TheassetpricesS1(n),...,Sm(n),A(n),D1(n),...,Dk(n)arerandomvari-ablesforanyn=1,2,.... 4.DiscreteTimeMarketModels87Assumption4.2a(PositivityofPrices)Thepricesofprimarysecurities,includingthemoneymarketaccount,arepos-itive,S1(n),...,Sm(n),A(n)>0forn=0,1,2,....Assumption4.3a(Divisibility,LiquidityandShortSelling)Aninvestormaybuy,sellandholdanynumberofassets,whetherintegerorfractional,negative,positiveorzero.Ingeneral,x1,...,xm,y,z1,...,zk∈R.Assumption4.4a(Solvency)Thewealthofaninvestormustbenon-negativeatalltimes,V(n)≥0forn=0,1,2,....Assumption4.5a(DiscreteUnitPrices)Foreachn=0,1,2,...thepricesS1(n),...,Sm(n),A(n),D1(n),...,Dk(n)arerandomvariablestakingonlyfinitelymanyvalues.Definitions4.1to4.4alsoextendimmediatelytothecaseinhand.Definition4.1aAportfolioisavector(x1(n),...,xm(n),y(n),z1(n),...,zk(n))indicatingthenumberofprimaryandderivativesecuritiesheldbyaninvestorbetweentimesn−1andn.Asequenceofportfoliosindexedbyn=1,2,...iscalledaninvestmentstrategy.Thewealthofaninvestororthevalueofthestrategyattimen≥1ismkV(n)=xj(n)Sj(n)+y(n)A(n)+zi(n)Di(n).j=1i=1Attimen=0theinitialwealthisgivenbymkV(0)=xj(1)Sj(0)+y(1)A(0)+zi(1)Di(0).j=1i=1 88MathematicsforFinanceDefinition4.2aAninvestmentstrategyiscalledself-financingiftheportfolioconstructedattimen≥1tobeheldoverthenexttimestepn+1isfinancedentirelybythecurrentwealthV(n),thatis,mkxj(n+1)Sj(n)+y(n+1)A(n)+zi(n+1)Di(n)=V(n).j=1i=1Definition4.3aAninvestmentstrategyiscalledpredictableifforeachn=0,1,2,...theport-folio(x1(n+1),...,xm(n+1),y(n+1),z1(n+1),...,zk(n+1))constructedattimendependsonlyonthenodesofthetreeofmarketscenariosreacheduptoandincludingtimen.Definition4.4aAstrategyiscalledadmissibleifitisself-financing,predictable,andforeachn=0,1,2,...V(n)≥0withprobability1.TheNo-ArbitragePrincipleextendswithoutanymodifications.Assumption4.6a(No-ArbitragePrinciple)ThereisnoadmissiblestrategysuchthatV(0)=0andV(n)>0withpositiveprobabilityforsomen=1,2,....Finally,theFundamentalTheoremofAssetPricingtakesthefollowingform.Theorem4.4a(FundamentalTheoremofAssetPricing)TheNo-ArbitragePrincipleisequivalenttotheexistenceofaprobabilityP∗onthesetofscenariosΩsuchthatP∗(ω)>0foreachscenarioω∈ΩandthediscountedpricesofprimaryandderivativesecuritiesSj(n)=Sj(n)/A(n)andDi(n)=Di(n)/A(n)formmartingaleswithrespecttoP∗,thatis,satisfyE∗(Sj(n+1)|S(n))=Sj(n),E∗(Di(n+1)|S(n))=Di(n) 4.DiscreteTimeMarketModels89foranyj=1,...,m,anyi=1,...,kandanyn=0,1,2,...,whereE∗(·|S(n))denotestheconditionalexpectationwithrespecttoprobabilityP∗computedoncethestockpriceS(n)becomesknownattimen.Example4.6Weshallusethesamescenariosω1,ω2,ω3,ω4,stockpricesS(0),S(1),S(2)andmoneymarketpricesA(0),A(1),A(2)asinExample4.5.Inaddition,weconsideraEuropeancalloptiongivingtheholdertheright(butnoobligation)tobuythestockforthestrikepriceofX=85dollarsattime2.Inthissituationweneedtoconsideranextendedmodelwiththreeassets,thestock,themoneymarket,andtheoption,withunitpricesS(n),A(n),CE(n),respectively,whereCE(n)isthemarketpriceoftheoptionattimen=0,1,2.Thetime2optionpriceisdeterminedbythestrikepriceandthestockprice,CE(2)=max{S(2)−X,0}.ThepricesCE(0)andCE(1)canbefoundusingtheFundamentalTheoremofAssetPricing.(Whichexplainsthenameofthetheorem!)Accordingtothetheorem,thereisaprobabilityP∗suchthatthediscountedstockandoptionpricesS(n)=S(n)/A(n)andCE(n)=CE(n)/A(n)aremartingales,orelseanarbitrageopportunitywouldexist.However,thereisonlyoneprobabilityP∗turningS(n)intoamartingale,namelythatfoundinExample4.5.Asaresult,CE(n)mustbeamartingalewithrespecttothesameprobabilityP.∗ThisgivesEA(1)EEA(0)EC(1)=E∗(C(2)|S(1))andC(0)=E∗(C(1)).A(2)A(1)ThevaluesofP∗foreachscenariofoundinExample4.5cannowbeusedtocomputeCE(1)andthenCE(0).Forexample,A(1)P(ω)CE(2,ω)+P(ω)CE(2,ω)CE(1,ω)=CE(1,ω)=∗11∗2212A(2)P∗(ω1)+P∗(ω2)19×27+19×21110=3060∼=22.7312119+193060dollars.Proceedinginasimilarway,weobtainScenarioCE(0)CE(1)CE(2)ω119.7922.7327.00ω219.7922.7321.00ω319.793.645.00ω419.793.640.00 90MathematicsforFinanceExercise4.8ApplytheFundamentalTheoremofAssetPricingtofindthetime0and1pricesofaputoptionwithstrikeprice$110maturingaftertwosteps,giventhesamescenariosω1,ω2,ω3,ω4,stockpricesS(0),S(1),S(2)andmoneymarketpricesA(0),A(1),A(2)asinExample4.5. 5PortfolioManagementAninvestmentinariskysecurityalwayscarriestheburdenofpossiblelossesorpoorperformance.Inthischapterweanalysetheadvantagesofspreadingtheinvestmentamongseveralsecurities.Eventhoughthemathematicaltoolsinvolvedarequitesimple,theyleadtoformidableresults.5.1RiskFirstofall,weneedtoidentifyasuitablequantitytomeasurerisk.Aninvest-mentinbondsreturning,forexample,8%atmaturityisfreeofrisk,inwhichcasethemeasureofriskshouldbeequaltozero.Ifthereturnonaninvestmentis,say11%or13%,dependingonthemarketscenario,thentheriskisclearlysmallerascomparedwithaninvestmentreturning2%or22%,respectively.However,thespreadofreturnvaluescanhardlybeusedtomeasureriskbe-causeitignorestheprobabilities.Ifthereturnrateis22%withprobability0.99and2%withprobability0.01,theriskcanbeconsideredquitesmall,whereasthesameratesofreturnoccurringwithprobability0.5eachwouldindicatearathermoreriskyinvestment.Thedesiredquantityneedstocapturethefollow-ingtwoaspectsofrisk:1)thedistancesbetweenacertainreferencevalueandtheratesofreturnineachmarketscenarioand2)theprobabilitiesofdifferentscenarios.ThereturnKonariskyinvestmentisarandomvariable.ItisnaturaltotaketheexpectationE(K)asthereferencevalue.ThevarianceVar(K)turns91 92MathematicsforFinanceouttobeaconvenientmeasureofrisk.Exercise5.1ComputetheriskVar(K1),Var(K2)andVar(K3)ineachofthefollowingthreeinvestmentprojects,wherethereturnsK1,K2andK3dependonthemarketscenario:ScenarioProbabilityReturnK1ReturnK2ReturnK3ω10.2512%11%2%ω20.7512%13%22%Whichoftheseisthemostriskyandtheleastriskyproject?Exercise5.2Considertwoscenarios,ωwithprobability1andωwithprobability3.1424SupposethatthereturnonacertainsecurityisK1(ω1)=−2%inthefirstscenarioandK1(ω2)=8%inthesecondscenario.IfthereturnonanothersecurityisK2(ω1)=−4%inthefirstscenario,findthereturnK2(ω2)intheotherscenariosuchthatthetwosecuritieshavethesamerisk.InsomecircumstancesthestandarddeviationσK=Var(K)ofthereturnisamoreconvenientmeasureofrisk.Ifaquantityismeasuredincertainunits,thenthestandarddeviationwillbeexpressedinthesameunits,soitcanberelateddirectlytotheoriginalquantity,incontrasttovariance,whichwillbeexpressedinsquaredunits.Example5.1LetthereturnonaninvestmentbeK=3%or−1%,bothwithprobability0.5.ThentheriskisVar(K)=0.0004orσK=0.02,dependingonwhetherwechoosethevarianceorstandarddeviation.Nowsup-posethatthereturnonanotherinvestmentisdoublethatonthefirstinvest-ment,beingequalto2K=6%or−2%,alsowithprobability0.5each.ThentheriskofthesecondinvestmentwillbeVar(2K)=0.0016orσ2K=0.04.Theriskasmeasuredbythevarianceisquadrupled,whilethestandarddevi-ationissimplydoubled. 5.PortfolioManagement93Thisillustratesthefollowinggeneralrule:Var(aK)=a2Var(K),σaK=|a|σKforanyrealnumbera.Remark5.1AnothernaturalwaytoquantifyriskwouldbetousethevarianceVar(k)(orthestandarddeviationσk)ofthelogarithmicreturnk.ThechoicebetweenKandkisdictatedtoalargeextentbythepropertiesneededtohandlethetaskinhand.Forexample,ifoneisinterestedinasequenceofinvestmentsfollowingoneanotherintime,thenthevarianceofthelogarithmicreturnmaybemoreusefulasameasureofrisk.Thisisbecauseoftheadditivityofrisksbasedonlogarithmicreturns:Var(k(0,n))=Var(k(1))+···+Var(k(n)),wherek(i)isthelogarithmicreturnintimestepi=1,...,nandk(0,n)isthelogarithmicreturnoverthewholetimeintervalfrom0ton,providedthatthek(i)areindependent.Theaboveformulaholdsbecausek(0,n)=k(1)+···+k(n)byProposition3.2,andthevarianceofasumofindependentrandomvariablesisthesumoftheirvariances.(Thisisnotnecessarilysowithoutindependence.)However,inthepresentchapterweshallbeconcernedwithaportfolioofseveralsecuritiesheldsimultaneouslyoverasingletimestep.ThepropertiesofE(K)andVar(K),whereKistheordinaryreturnontheportfolio(seeformulae(5.4)and(5.5)below),aremuchmoreconvenientforthispurposethanthoseforthelogarithmicreturn.Exercise5.3ConsidertworiskysecuritieswithreturnsK1andK2givenbyScenarioProbabilityReturnK1ReturnK2ω10.510.53%7.23%ω20.513.87%10.57%Computethecorrespondinglogarithmicreturnsk1andk2andcompareVar(k1)withVar(k2)andVar(K1)withVar(K2). 94MathematicsforFinance5.2TwoSecuritiesWebeginadetaileddiscussionoftherelationshipbetweenriskandexpectedreturninthesimplesituationofaportfoliowithjusttworiskysecurities.Example5.2Supposethatthepricesoftwostocksbehaveasfollows:ScenarioProbabilityReturnK1ReturnK2ω10.510%−5%ω20.5−5%10%Ifwesplitourmoneyequallybetweenthesetwostocks,thenweshallearn5%ineachscenario(losing5%ononestock,butgaining10%ontheother).Eventhoughaninvestmentineitherstockseparatelyinvolvesrisk,wehavereducedtheoverallrisktonilbysplittingtheinvestmentbetweenthetwostocks.Thisisasimpleexampleofdiversification,whichisparticularlyeffectiveherebecausethereturnsarenegativelycorrelated.Inadditiontothedescriptionofaportfoliointermsofthenumberofsharesofeachsecurityheld(developedinSection4.1),weshallintroduceanotherveryconvenientnotationtodescribetheallocationoffundsbetweenthesecurities.Example5.3SupposethatthepricesoftwokindsofstockareS1(0)=30andS2(0)=40dollars.WeprepareaportfolioworthV(0)=1,000dollarsbypurchasingx1=20sharesofstocknumber1andx2=10sharesofstocknumber2.Theallocationoffundsbetweenthetwosecuritiesis30×2010×40w1==60%,w2==40%.1,0001,000Thenumbersw1andw2arecalledtheweights.IfthestockpriceschangetoS1(1)=35andS2(1)=39dollars,thentheportfoliowillbeworthV(1)=20×35+10×39=1,090dollars.Observethatthisamountisnolongersplitbetweenthetwosecuritiesas60%to40%,butasfollows:20×3510×39∼=64.22%,∼=35.78%,1,0901,090eventhoughtheactualnumberofsharesofeachstockintheportfolioremainsunchanged. 5.PortfolioManagement95Theweightsaredefinedbyx1S1(0)x2S2(0)w1=,w2=,V(0)V(0)wherex1andx2aresharenumbersofstock1and2intheportfolio.Thismeansthatwkisthepercentageoftheinitialvalueoftheportfolioinvestedinsecuritynumberk.Observethattheweightsalwaysaddupto100%,x1S1(0)+x2S2(0)V(0)w1+w2===1.(5.1)V(0)V(0)Ifshortsellingisallowed,thenoneoftheweightsmaybenegativeandtheotheronegreaterthan100%.Example5.4SupposethataportfolioworthV(0)=1,000dollarsisconstructedbytakingalongpositioninstocknumber1andashortpositioninstocknumber2inExample5.3withweightsw1=120%andw2=−20%.TheportfoliowillconsistofV(0)1,000x1=w1=120%×=40,S1(0)30V(0)1,000x2=w2=−20%×=−5S2(0)40sharesoftype1and2.IfthestockpriceschangeasinExample5.3,thenthisportfoliowillbeworthS1(1)S2(1)V(1)=x1S1(1)+x2S2(1)=V(0)w1+w2S1(0)S2(0)3539=1,000120%×−20%×=1,2053040dollars,benefitingfromboththeriseofthepriceofstock1andthefallofstock2.However,asmallinvestormayhavetofacesomerestrictionsonshortselling.Forexample,itmaybenecessarytopayasecuritydepositequalto50%ofthesumraisedbyshortingstocknumber2.Thedeposit,whichwouldamountto50%×200=100dollars,canbeborrowedattherisk-freerateandtheinterestpaidonthisloanwillneedtobesubtractedfromthefinalvalueV(1)oftheportfolio. 96MathematicsforFinanceExercise5.4ComputethevalueV(1)ofaportfolioworthinitiallyV(0)=100dollarsthatconsistsoftwosecuritieswithweightsw1=25%andw2=75%,giventhatthesecuritypricesareS1(0)=45andS2(0)=33dollarsinitially,changingtoS1(1)=48andS2(1)=32dollars.WecanseeinExample5.4andExercise5.4thatV(1)/V(0)dependsonthepricesofsecuritiesonlythroughtheratiosS1(1)/S1(0)=1+K1andS2(1)/S2(0)=1+K2.Thisindicatesthatthereturnontheportfolioshoulddependonlyontheweightsw1,w2andthereturnsK1,K2oneachofthetwosecurities.Proposition5.1ThereturnKVonaportfolioconsistingoftwosecuritiesistheweightedaverageKV=w1K1+w2K2,(5.2)wherew1andw2aretheweightsandK1andK2thereturnsonthetwocomponents.ProofSupposethattheportfolioconsistsofx1sharesofsecurity1andx2sharesofsecurity2.ThentheinitialandfinalvaluesoftheportfolioareV(0)=x1S1(0)+x2S2(0),V(1)=x1S1(0)(1+K1)+x2S2(0)(1+K2)=V(0)(w1(1+K1)+w2(1+K2)).Asaresult,thereturnontheportfolioisV(1)−V(0)KV==w1K1+w2K2.V(0)Exercise5.5Findthereturnonaportfolioconsistingoftwokindsofstockwithweightsw1=30%andw2=70%ifthereturnsonthecomponentsare 5.PortfolioManagement97asfollows:ScenarioReturnK1ReturnK2ω112%−4%ω210%7%Remark5.2Asimilarformulato(5.2)holdsforlogarithmicreturns,ekV=wek1+wek2.(5.3)12However,thisisnotparticularlyusefuliftheexpectationsandvariancesorstandarddeviationsofreturnsneedtoberelatedtotheweights.Ontheotherhand,aswillbeseenbelow,formula(5.2)lendsitselfwelltothistask.Exercise5.6Verifyformula(5.3).5.2.1RiskandExpectedReturnonaPortfolioTheexpectedreturnonaportfolioconsistingoftwosecuritiescaneasilybeexpressedintermsoftheweightsandtheexpectedreturnsonthecomponents,E(KV)=w1E(K1)+w2E(K2).(5.4)Thisfollowsatoncefrom(5.2)bytheadditivityofmathematicalexpectation.Example5.5Considerthreescenarioswiththeprobabilitiesgivenbelow(atrinomialmodel).Letthereturnsontwodifferentstocksinthesescenariosbeasfollows:ScenarioProbabilityReturnK1ReturnK2ω1(recession)0.2−10%−30%ω2(stagnation)0.50%20%ω3(boom)0.310%50%TheexpectedreturnsonstockareE(K1)=−0.2×10%+0.5×0%+0.3×10%=1%,E(K2)=−0.2×30%+0.5×20%+0.3×50%=19%. 98MathematicsforFinanceSupposethatw1=60%ofavailablefundsisinvestedinstock1and40%instock2.TheexpectedreturnonsuchaportfolioisE(KV)=w1E(K1)+w2E(K2)=0.6×1%+0.4×19%=8.2%.Exercise5.7ComputetheweightsinaportfolioconsistingoftwokindsofstockiftheexpectedreturnontheportfolioistobeE(KV)=20%,giventhefollowinginformationonthereturnsonstock1and2:ScenarioProbabilityReturnK1ReturnK2ω1(recession)0.1−10%10%ω2(stagnation)0.50%20%ω3(boom)0.420%30%TocomputethevarianceofKVweneedtoknownotonlythevariancesofthereturnsK1andK2onthecomponentsintheportfolio,butalsothecovariancebetweenthetworeturns.Theorem5.2ThevarianceofthereturnonaportfolioisgivenbyVar(K)=w2Var(K)+w2Var(K)+2wwCov(K,K).(5.5)V11221212ProofSubstitutingK=wK+wKandcollectingthetermswithw2,w2andV112212w1w2,wecomputeVar(K)=E(K2)−E(K)2VVV=w2[E(K2)−E(K)2]+w2[E(K2)−E(K)2]111222+2w1w2[E(K1K2)−E(K1)E(K2)]=w2Var(K)+w2Var(K)+2wwCov(K,K).11221212 5.PortfolioManagement99Toavoidclutter,weintroducethefollowingnotationfortheexpectationandvarianceofaportfolioanditscomponents:µV=E(KV),σV=Var(KV),µ1=E(K1),σ1=Var(K1),µ2=E(K2),σ2=Var(K2).WeshallalsousethecorrelationcoefficientCov(K1,K2)ρ12=.(5.6)σ1σ2Formulae(5.4)and(5.5)canbewrittenasµV=w1µ1+w2µ2,(5.7)σ2=w2σ2+w2σ2+2wwρσσ.(5.8)V1122121212Remark5.3ForriskysecuritiesthereturnsK1andK2arealwaysassumedtobenon-constantrandomvariables.Becauseofthisσ1,σ2>0andρ12iswelldefined,sincethedenominatorσ1σ2in(5.6)isnon-zero.Example5.6Weusethefollowingdata:ScenarioProbabilityReturnK1ReturnK2ω1(recession)0.4−10%20%ω2(stagnation)0.20%20%ω3(boom)0.420%10%Wewanttocomparetheriskofaportfoliosuchthatw1=40%andw2=60%withtherisksofitscomponentsasmeasuredbythevariance.Directcomputationsgiveσ2∼=0.0184,σ2∼=0.0024,ρ∼=−0.96309.1212By(5.8)σ2∼=(0.4)2×0.0184+(0.6)2×0.0024V√√+2×0.4×0.6×(−0.96309)×0.0184×0.0024∼=0.000736.Observethatthevarianceσ2issmallerthanσ2andσ2.V12 100MathematicsforFinanceExample5.7Consideranotherportfoliowithweightsw1=80%andw2=20%,allotherthingsbeingthesameasinExample5.6.Thenσ2∼=(0.8)2×0.0184+(0.2)2×0.0024V√√+2×0.8×0.2×(−0.96309)×0.0184×0.0024∼=0.009824,whichisbetweenσ2andσ2.12Proposition5.3Thevarianceσ2ofaportfoliocannotexceedthegreaterofthevariancesσ2V1andσ2ofthecomponents,2σ2≤max{σ2,σ2},V12ifshortsalesarenotallowed.ProofLetusassumethatσ2≤σ2.Ifshortsalesarenotallowed,thenw,w≥0and1212w1σ1+w2σ2≤(w1+w2)σ2=σ2.Sincethecorrelationcoefficientsatisfies−1≤ρ12≤1,itfollowsthatσ2=w2σ2+w2σ2+2wwρσσV1122121212≤w2σ2+w2σ2+2wwσσ1122121222=(w1σ1+w2σ2)≤σ2.Ifσ2≥σ2,theproofisanalogous.12Example5.8Nowconsideraportfoliowithweightsw1=−50%andw2=150%(allowingshortsalesofsecurity1),alltheotherdatabeingthesameasinExample5.6.Thevarianceofthisportfolioisσ2∼=(−0.5)2×0.0184+(1.5)2×0.0024V√√+2×(−0.5)×1.5×(−0.96309)×0.0184×0.0024∼=0.0196,whichisgreaterthanbothσ2andσ2.12 5.PortfolioManagement101Exercise5.8UsingthedatainExample5.6,findtheweightsinaportfoliowithexpectedreturnµ=46%andcomputetheriskσ2ofthisportfolio.VVThecorrelationcoefficientalwayssatisfies−1≤ρ12≤1.Thenextpropo-sitionisconcernedwiththetwospecialcaseswhenρ12assumesoneoftheextremevalues1or−1,whichmeansperfectpositiveornegativecorrelationbetweenthesecuritiesintheportfolio.Proposition5.4Ifρ12=1,thenσV=0whenσ1=σ2andσ2σ1w1=−,w2=.(5.9)σ1−σ2σ1−σ2(Shortsalesarenecessary,sinceeitherw1orw2isnegative.)Ifρ12=−1,thenσV=0forσ2σ1w1=,w2=.(5.10)σ1+σ2σ1+σ2(Noshortsalesarenecessary,sincebothw1andw2arepositive.)ProofLetρ12=1.Then(5.8)takestheformσ2=w2σ2+w2σ2+2wwσσ=(wσ+wσ)2V112212121122andσ2=0ifandonlyifwσ+wσ=0.Thisisequivalenttoσ=σandV112212(5.9)becausew1+w2=1.Nowletρ12=−1.Then(5.8)becomesσ2=w2σ2+w2σ2−2wwσσ=(wσ−wσ)2V112212121122andσ2=0ifandonlyifwσ−wσ=0.ThelastequalityisequivalenttoV1122(5.10)becausew1+w2=1.EachportfoliocanberepresentedbyapointwithcoordinatesσVandµVontheσ,µplane.Figure5.1showstwotypicallinesrepresentingportfolioswithρ12=−1(left)andρ12=1(right).Theboldsegmentscorrespondtoportfolioswithoutshortselling. 102MathematicsforFinanceFigure5.1Typicalportfoliolineswithρ12=−1and1Supposethatρ12=−1.ItfollowsfromtheproofofProposition5.4thatσV=|w1σ1−w2σ2|.Inaddition,µV=w1µ1+w2µ2by(5.7)andw1+w2=1by(5.1).Wecanchooses=w2asaparameter.Then1−s=w1andσV=|(1−s)σ1−sσ2|,µV=(1−s)µ1+sµ2.TheseparametricequationsdescribethelineinFigure5.1withabrokenseg-mentbetween(σ1,µ1)and(σ2,µ2).Assincreases,thepoint(σV,µV)movesalongthelineinthedirectionfrom(σ1,µ1)to(σ2,µ2).Ifρ12=1,thenσV=|w1σ1+w2σ2|.Wechooses=w2asaparameteronceagain,andobtaintheparametricequationsσV=|(1−s)σ1+sσ2|,µV=(1−s)µ1+sµ2ofthelineinFigure5.1withastraightsegmentbetween(σ1,µ1)and(σ2,µ2).Ifnoshortsellingisallowed,then0≤s≤1inbothcases,whichcorrespondstotheboldlinesegments.Exercise5.9Supposethattherearejusttwoscenariosω1andω2andconsidertworiskysecuritieswithreturnsK1andK2.ShowthatK1=aK2+bforsomenumbersa=0andb,anddeducethatρ12=1or−1.Ournexttaskistofindaportfoliowithminimumriskforanygivenρ12suchthat−1<ρ12<1.Again,wetakes=w2asaparameter.Then(5.7) 5.PortfolioManagement103and(5.8)taketheformµV=(1−s)µ1+sµ2,(5.11)σ2=(1−s)2σ2+s2σ2+2s(1−s)ρσσ.(5.12)V121212Obviously,µasafunctionofsisastraightlineandσ2isaquadraticfunctionVVofswithapositivecoefficientats2(namelyσ2+σ2−2ρσσ>σ2+σ2−121212122σσ=(σ−σ)2≥0).Theproblemofminimisingthevarianceσ2(or,1212Vequivalently,thestandarddeviationσV)ofaportfolioissolvedinthenexttheorem.Firstwefindtheminimumwithoutanyrestrictionsonshortsales.Ifshortsalesarenotallowed,weshallhavetotakeintoaccountthebounds0≤s≤1ontheparameter.Theorem5.5For−1<ρ12<1theportfoliowithminimumvarianceisattainedatσ2−ρσσ11212s0=22.(5.13)σ1+σ2−2ρ12σ1σ2Ifshortsalesarenotallowed,thenthesmallestvarianceisattainedat0ifs0<0,smin=s0if0≤s0≤1,1if12σ2+2σ2−4σσ=2(σ−σ)2≥0,121212121212whichshowsthatthereisaminimumats0.Itisaglobalminimumbecauseσ2isaquadraticfunctionofs.VIfshortsalesarenotallowed,thenweneedtofindtheminimumfor0≤s≤1.If0≤s0≤1,thentheminimumisats0.Ifs0<0,thentheminimumisat0,andifs>1,thenitisat1,sinceσ2isaquadraticfunctionofswith0Vapositivecoefficientats2.ThisisillustratedinFigure5.2.Theboldpartsofthecurvecorrespondtoportfolioswithnoshortselling. 104MathematicsforFinanceFigure5.2Theminimumofσ2asafunctionofsVThelineontheσ,µplanedefinedbytheparametricequations(5.11)and(5.12)representsallpossibleportfolioswithgivenσ1,σ2>0and−1≤ρ12≤1.Theparameterscanbeanyrealnumberwhenevertherearenorestrictionsonshortselling.Ifshortsellingisnotallowed,then0≤s≤1andweonlyobtainasegmentoftheline.Assincreasesfrom0to1,thecorrespondingpoint(σV,µV)travelsalongthelineinthedirectionfrom(σ1,µ1)to(σ2,µ2).Figure5.3showstwotypicalexamplesofsuchlines,withρ12closetobutgreaterthan−1(left)andwithρ12closetobutsmallerthan1(right).Portfolioswithoutshortsellingareindicatedbytheboldlinesegments.Figure5.3Typicalportfoliolineswith−1<ρ12<1Figure5.4illustratesthefollowingcorollary.Corollary5.6Supposethatσ1≤σ2.Thefollowingthreecasesarepossible:1)If−1≤ρ<σ1,thenthereisaportfoliowithoutshortsellingsuchthat12σ2σV<σ1(lines4and5inFigure5.4);2)Ifρ=σ1,thenσ≥σforeachportfolio(line3inFigure5.4);12σ2V1 5.PortfolioManagement1053)Ifσ1<ρ≤1,thenthereisaportfoliowithshortsellingsuchthatσ212σV<σ1,butforeachportfoliowithoutshortsellingσV≥σ1(lines1and2inFigure5.4).Figure5.4Portfoliolinesforvariousvaluesofρ12Proof1)If−1≤ρ<σ1,thenσ1>s>0.Butσ1<1,so0σ1foreveryportfoliowithoutshortselling.Theabovecorollaryisimportantbecauseitshowswhenitispossibletoconstructaportfoliowithrisklowerthanthatofanyofitscomponents.Incase1)thisispossiblewithoutshortselling.Incase3)thisisalsopossible,butonlyifshortsellingisallowed.Incase2)itisimpossibletoconstructsuchaportfolio.Example5.9Supposethatσ2=0.0041,σ2=0.0121,ρ=0.9796.1212Clearly,σ<σandσ1<ρ<1,sothisiscase3)inCorollary5.6.Ourtask12σ212willbetofindtheportfoliowithminimumriskwithandwithoutshortselling. 106MathematicsforFinanceUsingTheorem5.5,wecomputes0∼=−1.1663,smin=0.Itfollowsthatintheportfoliowithminimumrisktheweightsofsecuritiesshouldbew1∼=2.1663andw2∼=−1.1663ifshortsellingisallowed.Withoutshortsellingw1=1andw2=0.Exercise5.10ComputetheweightsintheportfoliowithminimumriskforthedatainExample5.6.Doesthisportfolioinvolveshortselling?Weconcludethissectionwithabriefdiscussionofportfoliosinwhichoneofthesecuritiesisrisk-free.Thevarianceoftheriskysecurity(astock)ispositive,whereasthatoftherisk-freecomponent(abond)iszero.Proposition5.7ThestandarddeviationσVofaportfolioconsistingofariskysecuritywithexpectedreturnµ1andstandarddeviationσ1>0,andarisk-freesecuritywithreturnrFandstandarddeviationzerodependsontheweightw1oftheriskysecurityasfollows:σV=|w1|σ1.ProofLetσ>0andσ=0.Then(5.7)reducestoσ2=w2σ2,andtheformulafor12V11σVfollowsbytakingthesquareroot.Figure5.5Portfoliolineforoneriskyandonerisk-freesecurity 5.PortfolioManagement107Thelineontheσ,µplanerepresentingportfoliosconstructedfromoneriskyandonerisk-freesecurityisshowninFigure5.5.Asusual,theboldlinesegmentcorrespondstoportfolioswithoutshortselling.5.3SeveralSecurities5.3.1RiskandExpectedReturnonaPortfolioAportfolioconstructedfromndifferentsecuritiescanbedescribedintermsoftheirweightsxiSi(0)wi=,i=1,...,n,V(0)wherexiisthenumberofsharesoftypeiintheportfolio,Si(0)istheinitialpriceofsecurityi,andV(0)istheamountinitiallyinvestedintheportfolio.Itwillproveconvenienttoarrangetheweightsintoaone-rowmatrixw=w1w2···wn.Justlikefortwosecurities,theweightsadduptoone,whichcanbewritteninmatrixformas1=uwT,(5.14)whereu=11···1isaone-rowmatrixwithallnentriesequalto1,wTisaone-columnmatrix,thetransposeofw,andtheusualmatrixmultiplicationrulesapply.Theat-tainablesetconsistsofallportfolioswithweightswsatisfying(5.14),calledtheattainableportfolios.SupposethatthereturnsonthesecuritiesareK1,...,Kn.Theexpectedreturnsµi=E(Ki)fori=1,...,nwillalsobearrangedintoaone-rowmatrixm=µ1µ2···µn.Thecovariancesbetweenreturnswillbedenotedbycij=Cov(Ki,Kj).Theyaretheentriesofthen×ncovariancematrixc11c12···c1nc21c22···c2nC=.............cn1cn2···cnn 108MathematicsforFinanceItiswellknownthatthecovariancematrixissymmetricandpositivedefinite.Thediagonalelementsaresimplythevariancesofreturns,cii=Var(Ki).In−1whatfollowsweshallassume,inaddition,thatChasaninverseC.Proposition5.8Theexpectedreturnµ=E(K)andvarianceσ2=Var(K)ofaportfolioVVVVwithweightswaregivenbyµ=mwT,(5.15)Vσ2=wCwT.(5.16)VProofTheformulaforµVfollowsbythelinearityofexpectation,nnµ=E(K)=EwK=wµ=mwT.VViiiii=1i=1Forσ2weusethelinearityofcovariancewithrespecttoeachofitsarguments,Vnσ2=Var(K)=VarwKVViii=1nnn=CovwiKi,wjKj=wiwjciji=1j=1i,j=1=wCwT.Exercise5.11ComputetheexpectedreturnµVandstandarddeviationσVofaport-folioconsistingofthreesecuritieswithweightsw1=40%,w2=−20%,w3=80%,giventhatthesecuritieshaveexpectedreturnsµ1=8%,µ2=10%,µ3=6%,standarddeviationsσ1=1.5,σ2=0.5,σ3=1.2andcorrelationsρ12=0.3,ρ23=0.0,ρ31=−0.2.Weshallsolvethefollowingtwoproblems:1.Tofindaportfoliowiththesmallestvarianceintheattainableset.Itwillbecalledtheminimumvarianceportfolio. 5.PortfolioManagement1092.TofindaportfoliowiththesmallestvarianceamongallportfoliosintheattainablesetwhoseexpectedreturnisequaltoagivennumberµV.Thefamilyofsuchportfolios,parametrisedbyµV,iscalledtheminimumvari-anceline.Sincethevarianceisacontinuousfunctionoftheweights,boundedbelowby0,theminimumclearlyexistsinbothcases.Proposition5.9(MinimumVariancePortfolio)Theportfoliowiththesmallestvarianceintheattainablesethasweights−1uCw=,uC−1uTprovidedthatthedenominatorisnon-zero.ProofWeneedtofindtheminimumof(5.16)subjecttotheconstraint(5.14).TothisendwecanusethemethodofLagrangemultipliers.LetusputF(w,λ)=wCwT−λuwT,whereλisaLagrangemultiplier.EquatingtozerothepartialderivativesofFwithrespecttotheweightswiweobtain2wC−λu=0,thatis,λ−1w=uC,2whichisanecessaryconditionforaminimum.Substitutingthisintocon-straint(5.14)weobtainλ−1T1=uCu,2−1whereweusethefactthatCisasymmetricmatrixbecauseCis.Solvingthisforλandsubstitutingtheresultintotheexpressionforwwillgivetheassertedformula.Proposition5.10(MinimumVarianceLine)Theportfoliowiththesmallestvarianceamongattainableportfolioswithex-pectedreturnµVhasweights!!!!!1uC−1mT!!uC−1uT1!!!uC−1+!!mC−1!µmC−1mT!!mC−1uTµ!VVw=!!,!uC−1uTuC−1mT!!!!mC−1uTmC−1mT! 110MathematicsforFinanceprovidedthatthedeterminantinthedenominatorisnon-zero.TheweightsdependlinearlyonµV.ProofHereweneedtofindtheminimumof(5.16)subjecttotwoconstraints(5.14)and(5.15).WetakeG(w,λ,µ)=wCwT−λuwT−µmwT,whereλandµareLagrangemultipliers.ThepartialderivativesofGwithrespecttotheweightswiequatedtozerogiveanecessaryconditionforaminimum,2wC−λu−µm=0,whichimpliesthatλ−1µ−1w=uC+mC.22Substitutingthisintotheconstraints(5.14)and(5.15),weobtainasystemoflinearequationsλ−1Tµ−1T1=uCu+uCm,22λ−1Tµ−1TµV=mCu+mCm,22tobesolvedforλandµ.Theassertedformulafollowsbysubstitutingthesolutionintotheexpressionforw.Example5.10(3securities)Considerthreesecuritieswithexpectedreturns,standarddevia-tionsofreturnsandcorrelationsbetweenreturnsµ1=0.10,σ1=0.28,ρ12=ρ21=−0.10,µ2=0.15,σ2=0.24,ρ23=ρ32=0.20,µ3=0.20,σ3=0.25,ρ31=ρ13=0.25.Wearrangetheµi’sintoaone-rowmatrixmand1’sintoaone-rowmatrixu,m=0.100.150.20,u=111.Nextwecomputetheentriescij=ρijσiσjofthecovariancematrixC,andfindtheinversematrixtoC,0.0784−0.00670.017513.9542.544−4.396C∼=−0.00670.05760.0120,C−1∼=2.54418.548−4.2740.01750.01200.0625−4.396−4.27418.051 5.PortfolioManagement111FromProposition5.9wecancomputetheweightsintheminimumvarianceportfolio.SinceuC−1∼=12.10216.8189.382,uC−1uT∼=38.302,weobtainuC−1w=∼=0.3160.4390.245.uC−1uTTheexpectedreturnandstandarddeviationofthisportfolioare√µ=mwT∼=0.146,σ=wCwT∼=0.162.VVTheminimumvariancelinecanbecomputedusingProposition5.10.TothisendwecomputeuC−1∼=12.10216.8189.382,mC−1∼=0.8982.1822.530,uC−1uT∼=38.302,mC−1mT∼=0.923,uC−1mT=mC−1uT∼=5.609.SubstitutingtheseintotheformulaforwinProposition5.10,weobtaintheweightsintheportfoliowithminimumvarianceamongallportfolioswithex-pectedreturnµV:w∼=1.578−8.614µV0.845−2.769µV−1.422+11.384µV.Thestandarddeviationofthisportfoliois√σV=wCwT∼=0.237−2.885µV+9.850µ2.VExercise5.12Amongallattainableportfoliosconstructedusingthreesecuritieswithexpectedreturnsµ1=0.20,µ2=0.13,µ3=0.17,standarddeviationsofreturnsσ1=0.25,σ2=0.28,σ3=0.20,andcorrelationsbetweenreturnsρ12=0.30,ρ23=0.00,ρ31=0.15,findtheminimumvarianceportfolio.Whataretheweightsinthisportfolio?Alsocomputetheexpectedreturnandstandarddeviationofthisportfolio.Exercise5.13AmongallattainableportfolioswithexpectedreturnµV=20%con-structedusingthethreesecuritiesinExercise5.12findtheportfoliowiththesmallestvariance.Computetheweightsandthestandarddeviationofthisportfolio. 112MathematicsforFinanceExample5.11(3securitiesvisualised)TherearetwoconvenientwaystovisualiseallportfoliosthatcanbeconstructedfromthethreesecuritiesinExample5.10.OneispresentedinFigure5.6.Heretwoofthethreeweights,namelyw2andw3,Figure5.6Attainableportfoliosonthew2,w3planeareusedasparameters.Theremainingweightisgivenbyw1=1−w2−w3.(Ofcourseanyothertwoweightscanalsobeusedasparameters.)Eachpointonthew2,w3planerepresentsadifferentportfolio.Theverticesofthetrianglerepresenttheportfoliosconsistingofonlyoneofthethreesecurities.Forexample,thevertexwithcoordinates(1,0)correspondstoweightsw1=0,w2=1andw3=0,thatis,representsaportfoliowithallmoneyinvestedinsecuritynumber2.Thelinesthroughtheverticescorrespondtoportfoliosconsistingoftwosecuritiesonly.Forexample,thelinethrough(1,0)and(0,1)correspondstoportfolioscontainingsecurities2and3only.Pointsinsidethetriangle,includingtheboundaries,correspondtoportfolioswithoutshortselling.Forexample,(2,1)representsaportfoliowith10%oftheinitialfundsinvestedinsecurity1,5240%insecurity2,and50%insecurity3.Pointsoutsidethetrianglecorrespondtoportfolioswithoneortwoofthethreesecuritiesshorted.Theminimumvariancelineisastraightlinebecauseofthelineardependenceoftheweightsontheexpectedreturn.ItisrepresentedbytheboldlineinFigure5.6.Figure5.7showsanotherwaytovisualiseattainableportfoliosbyplot-tingtheexpectedreturnofaportfolioagainstthestandarddeviation.Thisissometimescalledtherisk–expectedreturngraph.Thethreepointsindicatedinthispicturecorrespondtoportfoliosconsistingofonlyoneofthethreesecurities.Forinstance,theportfoliowithallfundsinvestedinsecurity2isrepresentedbythepoint(0.24,0.15).Thelinespassingthroughapairofthesethreepointscorrespondtoportfoliosconsistingofjusttwosecurities.Thesearethetwo-securitylinesstudiedindetailinSection5.2.Forexample,allportfo- 5.PortfolioManagement113lioscontainingsecurities2and3onlylieonthelinethrough(0.24,0.15)and(0.25,0.20).ThethreepointsandthelinespassingthroughthemcorrespondtotheverticesofthetriangleandthestraightlinespassingthroughtheminFigure5.6.Theshadedarea(bothdarkandlight),includingtheboundary,representsportfoliosthatcanbeconstructedfromthethreesecurities,thatis,allattainableportfolios.Theboundary,shownasaboldline,istheminimumvarianceline.TheshapeofitisknownastheMarkowitzbullet.ThedarkerpartoftheshadedareacorrespondstotheinteriorofthetriangleinFigure5.6,thatis,itrepresentsportfolioswithoutshortselling.Figure5.7Attainableportfoliosontheσ,µplaneItisinstructivetoimaginehowthewholew2,w3planeinFigure5.6ismappedontotheshadedarearepresentingallattainableportfoliosinFig-ure5.7.Namely,thew2,w3planeisfoldedalongtheminimumvarianceline,be-ingsimultaneouslywarpedandstretchedtoattaintheshapeoftheMarkowitzbullet.Thismeans,inparticular,thatpairsofpointsonoppositesidesoftheminimumvariancelineonthew2,w3planearemappedintosinglepointsontheσ,µplane.Inotherwords,eachpointinsidetheshadedareainFigure5.7correspondstotwodifferentportfolios.However,eachpointontheminimumvariancelinecorrespondstoasingleportfolio.Example5.12(3securitieswithoutshortselling)ForthesamethreesecuritiesasinExam-ples5.10and5.11,Figure5.8showswhathappensifnoshortsellingisallowed.Allportfolioswithoutshortsellingarerepresentedbytheinteriorandbound-aryofthetriangleonthew1,w2planeandbytheshadedareawithboundaryontheσ,µplane.Theminimumvariancelinewithoutshortsellingisshownasaboldlineinbothplots.Forcomparison,theminimumvariancelinewithshortsellingisshownasabrokenline. 114MathematicsforFinanceFigure5.8PortfolioswithoutshortsellingExercise5.14ForportfoliosconstructedwithandwithoutshortsellingfromthethreesecuritiesinExercise5.12computetheminimumvariancelineparame-trisedbytheexpectedreturnandsketchita)onthew2,w3planeandb)ontheσ,µplane.Alsosketchthesetofallattainableportfolioswithandwithoutshortselling.5.3.2EfficientFrontierGiventhechoicebetweentwosecuritiesarationalinvestorwill,ifpossible,choosethatwithhigherexpectedreturnandlowerstandarddeviation,thatis,lowerrisk.Thismotivatesthefollowingdefinition.Definition5.1Wesaythatasecuritywithexpectedreturnµ1andstandarddeviationσ1dominatesanothersecuritywithexpectedreturnµ2andstandarddeviationσ2wheneverµ1≥µ2andσ1≤σ2.Thisdefinitionreadilyextendstoportfolios,whichcanofcoursebeconsideredassecuritiesintheirownright.Remark5.4Giventwosecuritiessuchthatonedominatestheother,thedominatedsecuritymayappearquiteredundantonfirstsight.Nevertheless,itcanalsobeofsome 5.PortfolioManagement115use.EmployingthetechniquesofSection5.2,itmaybepossibletoconstructportfoliosconsistingofthetwosecuritieswithsmallerriskthaneitherofthesecurities,asinFigure5.9,inwhichthesecuritywithσ2,µ2isdominatedbythatwithσ1,µ1.Figure5.9ReductionofriskusingadominatedsecurityDefinition5.2Aportfolioiscalledefficientifthereisnootherportfolio,exceptitself,thatdominatesit.Thesetofefficientportfoliosamongallattainableportfoliosiscalledtheefficientfrontier.Everyrationalinvestorwillchooseanefficientportfolio,alwayspreferringadominatingportfoliotoadominatedone.However,differentinvestorsmayselectdifferentportfoliosontheefficientfrontier,dependingontheirindividualpreferences.Giventwoefficientportfolioswithµ1≤µ2andσ1≤σ2,acautiouspersonmaypreferthatwithlowerriskσ1andlowerexpectedreturnµ1,whileothersmaychooseaportfoliowithhigherriskσ2,regardingthehigherexpectedreturnµ2ascompensationforincreasedrisk.Inparticular,anefficientportfoliohasthehighestexpectedreturnamongallattainableportfolioswiththesamestandarddeviation(thesamerisk),andhastheloweststandarddeviation(thelowestrisk)amongallattainableportfolioswiththesameexpectedreturn.Asaresult,theefficientfrontiermustbeasubsetoftheminimumvarianceline.Tounderstandthestructureoftheefficientfrontierweshallfirststudytheminimumvariancelineinmoredetailandthenselectasuitablesubset.Proposition5.11Takeanytwodifferentportfoliosontheminimumvarianceline,withweightswandw.Thentheminimumvariancelineconsistsofportfolioswithweights 116MathematicsforFinancecw+(1−c)wforanyc∈Randonlyofsuchportfolios.ProofByProposition5.10theminimumvariancelineconsistsofportfolioswhoseweightsaregivenbyacertainlinearfunctionoftheexpectedreturnµVontheportfolio,w=aµ+b.IfwandwaretheweightsoftwodifferentportfoliosVontheminimumvarianceline,thenw=aµV
+bandw=aµV

+bforsomeµV
=µV

.BecausenumbersoftheformcµV
+(1−c)µV

forc∈Rexhaustthewholerealline,itfollowsthatportfolioswithweightscw+(1−c)wforc∈Rexhaustthewholeminimumvarianceline.Thispropositionisimportant.Itmeansthattheminimumvariancelinehasthesameshapeasthesetofportfoliosconstructedfromtwosecurities,studiedingreatdetailinSection5.2.Italsomeansthattheshapeoftheattainablesetontheσ,µplane(theMarkowitzbullet),whichwehaveseensofarforportfoliosconstructedfromtwoorthreesecurities,willinfactbethesameforanynumberofsecurities.Oncetheshapeoftheminimumvariancelineisunderstood,distinguishingtheefficientfrontieriseasy,alsointhecaseofnsecurities.ThisisillustratedinFigure5.10.Theefficientfrontierconsistsofallportfoliosontheminimumvariancelinewhoseexpectedreturnisgreaterthanorequaltotheexpectedreturnontheminimumvarianceportfolio.Figure5.10EfficientfrontierconstructedfromseveralsecuritiesThenextpropositionprovidesapropertyoftheefficientfrontierwhichwillproveusefulintheCapitalAssetPricingModel.Proposition5.12Theweightswofanyportfoliobelongingtotheefficientfrontier(exceptfor 5.PortfolioManagement117theminimumvarianceportfolio)satisfytheconditionγwC=m−µu(5.17)forsomerealnumbersγ>0andµ.ProofLetwbetheweightsofaportfolio,otherthantheminimumvarianceportfolio,belongingtotheefficientfrontier.Theportfoliohasexpectedreturn√µV=mwTandstandarddeviationσ=wCwT.Ontheσ,µplanewedrawtheVtangentlinetotheefficientfrontierthroughthepointrepresentingtheportfolio.Thislinewillintersecttheverticalaxisatsomepointwithcoordinateµ,themwT−µgradientofthelinebeing√.ThisgradientismaximalamongalllineswCwTpassingthroughthepointontheverticalaxiswithcoordinateµandintersectingthesetofattainableportfolios.ThemaximumistobetakenoverallweightswsubjecttotheconstraintuwT=1.WeputmwT−µF(w,λ)=√−λuwT,wCwTwhereλisaLagrangemultiplier.Anecessaryconditionforaconstrainedmax-imumisthatthepartialderivativesofFwithrespecttotheweightsshouldbezero.ThisgivesµV−µm−λσVu=2wC.σVMultiplyingbywTontherightandusingtheconstraint,wefindthatλ=µ.σVForγ=µV−µthisgivestheassertedcondition.Becausethetangentlinehasσ2Vpositiveslope,wehaveµV>µ,thatis,γ>0.Remark5.5Aninterpretationofγandµfollowsclearlyfromtheproof:γσVisthegradientofthetangentlinetotheefficientfrontieratthepointrepresentingthegivenportfolio,µbeingtheinterceptofthistangentlineontheσ,µplane.Exercise5.15InamarketconsistingofthethreesecuritiesinExercise5.12,considertheportfolioontheefficientfrontierwithexpectedreturnµV=21%.ComputethevaluesofγandµsuchthattheweightswinthisportfoliosatisfyγwC=m−µu. 118MathematicsforFinance5.4CapitalAssetPricingModelInthedayswhencomputerswhereslowitwasdifficulttouseportfoliotheory.Foramarketwithn=1,000tradedsecuritiesthecovariancematrixCwillhaven2=1,000,000entries.Tofindtheefficientfrontierwehavetocomputethe−1inversematrixC,whichiscomputationallyintensive.AccurateestimationofCmayposeconsiderableproblemsinpractice.TheCapitalAssetPricingModel(CAPM)providesasolutionthatismuchmoreefficientcomputation-ally,doesnotinvolveanestimateofC,butoffersadeep,evenifsomewhatoversimplified,insightintosomefundamentaleconomicissues.WithintheCAPMitisassumedthateveryinvestorusesthesamevaluesofexpectedreturns,standarddeviationandcorrelationsforallsecurities,makinginvestmentdecisionsbasedonlyonthesevalues.Inparticular,everyinvestorwillcomputethesameefficientfrontieronwhichtoselecthisorherportfo-lio.However,investorsmaydifferintheirattitudetorisk,selectingdifferentportfoliosontheefficientfrontier.5.4.1CapitalMarketLineFormnowonweshallassumethatarisk-freesecurityisavailableinadditiontonriskysecurities.Thereturnontherisk-freesecuritywillbedenotedbyrF.Thestandarddeviationisofcoursezerofortherisk-freesecurity.Consideraportfolioconsistingoftherisk-freesecurityandaspecifiedriskysecurity(possiblyaportfolioofriskysecurities)withexpectedreturnµ1andstandarddeviationσ1>0.ByProposition5.7allsuchportfoliosformabrokenlineontheσ,µplaneconsistingoftworectilinearhalf-lines,seeFigure5.5.Bytakingportfolioscontainingtherisk-freesecurityandasecuritywithσ1,µ1anywhereintheattainablesetrepresentedbytheMarkowitzbulletontheσ,µplane,wecanconstructanyportfoliobetweenthetwohalf-linesshowninFigure5.11.Theefficientfrontierofthisnewsetofportfolios,whichmaycontaintherisk-freesecurity,istheupperhalf-linetangenttotheMarkowitzbulletandpassingthroughthepointwithcoordinates0,rF.AccordingtotheassumptionsoftheCAPM,everyrationalinvestorwillselecthisorherportfolioonthishalf-line,calledthecapitalmarketline.Thisargumentworksaslongastherisk-freereturnrFisnottoohigh,sotheupperhalf-lineistangenttothebullet.(IfrFistoohigh,thentheupperhalf-linewillnolongerbetangenttothebullet.)ThetangencypointwithcoordinatesσM,µMplaysaspecialrole.Everyportfolioonthecapitalmarketlinecanbeconstructedfromtherisk-freese-curityandtheportfoliowithstandarddeviationσMandexpectedreturnµM. 5.PortfolioManagement119Figure5.11Efficientfrontierforportfolioswitharisk-freesecuritySinceeveryinvestorwillselectaportfolioonthecapitalmarketline,everyonewillbeholdingaportfoliowiththesamerelativeproportionsofriskysecurities.ButthismeansthattheportfoliowithstandarddeviationσMandexpectedreturnµMhastocontainallriskysecuritieswithweightsequaltotheirrela-tiveshareinthewholemarket.Becauseofthispropertyitiscalledthemarketportfolio.Inpracticethemarketportfolioisapproximatedbyasuitablestockexchangeindex.Thecapitalmarketlinejoiningtherisk-freesecurityandthemarketport-foliosatisfiestheequationµM−rFµ=rF+σ.(5.18)σMForaportfolioonthecapitalmarketlinewithriskσthetermµM−rFσiscalledσMtheriskpremium.Thisadditionalreturnabovetherisk-freelevelprovidescompensationforexposuretorisk.Example5.13WeshallapplyProposition5.12tocomputethemarketportfolioforatoymarketconsistingofthethreesecuritiesinExample5.10andarisk-freesecuritywithreturnrF=5%.Theweightswinthemarketportfolio,whichbelongstotheefficientfrontier,satisfycondition(5.17),whichimpliesthat−1γw=(m−µu)C.FromtheproofofProposition5.12weknowthatµ=rFbecausethecapitalmarketline,tangenttotheefficientfrontieratthepointrepresentingthemarketportfolio,intersectstheµaxisatrF.SubstitutingthenumericalvaluesfromExample5.10,wefindthatγw∼=0.2931.3412.061. 120MathematicsforFinanceSincewmustsatisfy(5.14),itfollowsthatγ∼=3.694andtheweightsinthemarketportfolioarew∼=0.0790.3630.558.Exercise5.16Supposethattherisk-freereturnisrF=5%.ComputetheweightsinthemarketportfolioconstructedfromthethreesecuritiesinExercise5.11.Alsocomputetheexpectedreturnandstandarddeviationofthemarketportfolio.5.4.2BetaFactorItisimportanttounderstandhowthereturnKVonagivenportfolioorasinglesecuritywillreacttotrendsaffectingthewholemarket.TothisendwecanplotthevaluesofKVforeachmarketscenarioagainstthoseofthereturnKMonthemarketportfolioandcomputethelineofbestfit,alsoknownastheregressionlineorthecharacteristicline.InFigure5.12thevaluesofKMaremarkedalongthexaxisandthevaluesofKValongtheyaxis.Theequationofthelineofbestfitwillbey=βVx+αV.Figure5.12LineofbestfitForanygivenβandαthevaluesoftherandomvariableα+βKMcanberegardedaspredictionsforthereturnonthegivenportfolio.Thedifferenceε=KV−(α+βKM)betweentheactualreturnKVandthepredictedreturn 5.PortfolioManagement121α+βKMiscalledtheresidualrandomvariable.TheconditiondefiningthelineofbestfitisthatE(ε2)=E(K2)−2βE(KK)+β2E(K2)+α2−2αE(K)+2αβE(K)VVMMVMasafunctionofβandαshouldattainitsminimumatβ=βVandα=αV.Inotherwords,thelineofbestfitshouldleadtopredictionsthatareascloseaspossibletothetruevaluesofKV.Anecessaryconditionforaminimumisthatthepartialderivativeswithrespecttoβandαshouldbezeroatβ=βVandα=αV.ThisleadstothesystemoflinearequationsαE(K)+βE(K2)=E(KK),VMVMVMαV+βVE(KM)=E(KV),whichcanbesolvedtofindthegradientβVandinterceptαVofthelineofbestfit,Cov(KV,KM)βV=2,αV=µV−βVµM.σMHereweemploytheusualnotationµ=E(K),µ=E(K)andσ2=VVMMMVar(KM).Exercise5.17SupposethatthereturnsKVonagivenportfolioandKMonthemarketportfoliotakethefollowingvaluesindifferentmarketscenarios:ScenarioProbabilityReturnKVReturnKMω10.1−5%10%ω20.30%14%ω30.42%12%ω30.24%16%ComputethegradientβVandinterceptαVofthelineofbestfit.Definition5.3WecallCov(KV,KM)βV=2σMthebetafactorofthegivenportfolioorindividualsecurity. 122MathematicsforFinanceThebetafactorisanindicatorofexpectedchangesinthereturnonaparticularportfolioorindividualsecurityinresponsetothebehaviourofthemarketasawhole.SinceµV=βVµM+αV,thereturnonasecuritywithapositivebetafactortendstoincreaseasthereturnonthemarketportfolioincreases,whilethereturnonasecuritywithanegativebetafactortendstoincreaseifthereturnonthemarketportfoliogoesdown.Inwhatfollowswediscussanotherinterpretationofthebetafactor.Theriskσ2=Var(K)ofasecurityorportfoliocanbewrittenasVVσ2=Var(ε)+β2σ2.VVVMThisformulaiseasytoverifyuponsubstitutingtheexpressionεV=KV−(αV+βVKM)fortheresidualrandomvariable.ThefirsttermVar(εV)iscalledtheresidualvarianceordiversifiablerisk.Itvanishesforthemarketportfolio,Var(εM)=0.Thispartofriskcan‘diversifiedaway’byinvestinginthemarketportfolio.Thesecondtermβ2σ2iscalledthesystematicorundiversifiableVMrisk.Themarketportfolioinvolvesonlythiskindofrisk.ThebetafactorβVcanberegardedasameasureofsystematicriskassociatedwithasecurityorportfolio.Thisinterpretationofthebetafactorisofcrucialimportance.IntheCAPMsystematicrisk,measuredbyβV,willbelinkedtotheexpectedreturnµVandhencetothepricingofindividualsecuritiesandportfolios:Thehigherthesystematicrisk,thehigherthereturnrequiredbyinvestorsasapremiumforexposuretothiskindofrisk.However,diversifiableriskwillattractnoadditionalpremium,havingnoeffectonµV.Thisisbecausediversifiableriskcanbeeliminatedbyspreadinganinvestmentinaportfolioofmanysecuritiesand,inparticular,byinvestinginthemarketportfolio.ThenextsectionisdevotedtoestablishingthelinkbetweenβVandµV.Exercise5.18ShowthatthebetafactorβVofaportfolioconsistingofnsecuritieswithweightsw1,...,wnisgivenbyβV=w1β1+···+wnβn,whereβ1,...,βnarethebetafactorsofthesecurities.5.4.3SecurityMarketLineConsideranarbitraryportfoliowithweightswV.TheweightsinthemarketportfoliowillbedenotedbywM.Themarketportfoliobelongstotheefficientfrontieroftheattainablesetofportfoliosconsistingofriskysecurities.Thus,byProposition5.12γwMC=m−µu 5.PortfolioManagement123forsomenumbersγ>0andµ.ThebetafactoroftheportfoliowithweightswVcan,therefore,bewrittenasCov(K,K)wCwTγ(m−µu)wTµ−µβ=VM=MV=V=V.Vσ2wCwTγ(m−µu)wTµ−µMMMMMTofindµconsidertherisk-freesecurity,withreturnrFandbetafactorβF=0.SubstitutingβFandrFforβVandµVintheaboveequation,wefindthatµ=rF.Wehaveprovedthefollowingremarkableproperty.Theorem5.13TheexpectedreturnµVonaportfolio(oranindividualsecurity)isalinearfunctionofthebetacoefficientβVoftheportfolio,µV=rF+(µM−rF)βV.(5.19)Theexpectedreturnplottedagainstthebetacoefficientofanyportfolioorindividualsecuritywillformastraightlineontheβ,µplane,calledthesecuritymarketline.ThisisshowninFigure5.13,inwhichthesecuritymarketlineisplottednexttothecapitalmarketlineforcomparison.Anumberofdifferentportfoliosandindividualsecuritiesareindicatedbydotsinbothgraphs.Figure5.13CapitalmarketlineandsecuritymarketlineSimilarlyasinformula(5.18)forthecapitalmarketline,theterm(µM−rF)βVin(5.19)istheriskpremium,interpretedascompensationforexposuretosystematicrisk.However,(5.18)appliesonlytoportfoliosonthecapitalmarketline,whereas(5.19)ismuchmoregeneral:Itappliestoallportfoliosandindividualsecurities.Exercise5.19ShowthatthecharacteristiclinesofallsecuritiesintersectatacommonpointintheCAPM.Whatarethecoordinatesofthispoint? 124MathematicsforFinanceTheCAPMdescribesastateofequilibriuminthemarket.Everyoneisholdingaportfolioofriskysecuritieswiththesameweightsasthemarketportfolio.Anytradesthatmaybeexecutedbyinvestorswillonlyaffecttheirsplitoffundsbetweentherisk-freesecurityandthemarketportfolio.Asaresult,thedemandandsupplyofallsecuritieswillbebalanced.Thiswillremainsoaslongastheestimatesofexpectedreturnsandbetafactorssatisfy(5.19).However,assoonassomenewinformationaboutthemarketbecomesavail-abletoinvestors,itmayaffecttheirestimatesofexpectedreturnsandbetafactors.Thenewestimatedvaluesmaynolongersatisfy(5.19).Suppose,forexample,thatµV>rF+(µM−rF)βVforaparticularsecurity.Inthiscaseinvestorswillwanttoincreasetheirrelativepositioninthissecurity,whichoffersahigherexpectedreturnthanrequiredascompensationforsystematicrisk.Demandwillexceedsupply,thepriceofthesecuritywillbegintoriseandtheexpectedreturnwilldecline.Ontheotherhand,ifthereverseinequalityµVS(T),thenthesituationwillbereversed.Thepayoffsat125 126MathematicsforFinancedeliveryareS(T)−F(0,T)foralongforwardpositionandF(0,T)−S(T)forashortposition;seeFigure6.1.Figure6.1PayoffforlongandshortforwardpositionsatdeliveryIfthecontractisinitiatedattimetS(0)erT.Inthiscase,attime0•borrowtheamountS(0)untiltimeT;•buyoneshareforS(0);•takeashortforwardposition,thatis,agreetoselloneshareforF(0,T)attimeT.Then,attimeT•sellthestockforF(0,T);•payS(0)erTtocleartheloanwithinterest.Thiswillbringarisk-freeprofitofF(0,T)−S(0)erT>0,contrarytotheNo-ArbitragePrinciple.Next,supposethatF(0,T)0,againacontradictionwiththeNo-ArbitragePrinciple.Theproofof(6.2)issimilar.Simplyreplace0byt,observingthatthetimeelapsedbetweenexchangingtheforwardcontractanddeliveryisnowT−t.InamarketwithrestrictionsonshortsalesofstocktheinequalityF(0,T)S(t).ThedifferenceF(t,T)−S(t),whichiscalledthebasis,convergesto0astT.Remark6.2UnderperiodiccompoundingtheforwardpriceisgivenbyrmTF(0,T)=S(0)(1+).mIntermsofzero-couponbondprices,thisformulabecomesF(0,T)=S(0)B(0,T)−1.Thelastformulaisinfactmoregeneral,requiringnoassumptionaboutcon-stantinterestrates.IncludingDividends.Weshallgeneralisetheformulafortheforwardpricetocoverassetsthatgenerateincomeduringthelifetimeoftheforwardcontract.Theincomemaybeintheformofdividendsoraconvenienceyield.Weshallalsocoverthecasewhentheassetinvolvessomecosts(calledthecostofcarry),suchasstorageorinsurance,bytreatingthecostsasnegativeincome.Supposethatthestockistopayadividenddivatanintermediatetimetbetweeninitiatingtheforwardcontractanddelivery.Attimetthestockpricewilldropbytheamountofthedividendpaid.Theformulafortheforwardprice,whichinvolvesthepresentstockprice,canbemodifiedbysubtractingthepresentvalueofthedividend.Theorem6.2Theforwardpriceofastockpayingdividenddivattimet,where0[S(0)−e−rtdiv]erT.Weshallconstructanarbitragestrategy.Attime0•enterintoashortforwardcontractwithforwardpriceF(0,T)anddeliverytimeT;•borrowS(0)dollarsandbuyoneshare.Attimet•cashthedividenddivandinvestitattherisk-freeratefortheremainingtimeT−t.AttimeT•selltheshareforF(0,T);•payS(0)erTtocleartheloanwithinterestandcollecter(T−t)div.Thefinalbalancewillbepositive:F(0,T)−S(0)erT+er(T−t)div>0,acontradictionwiththeNo-ArbitragePrinciple.Ontheotherhand,supposethatF(0,T)<[S(0)−e−rtdiv]erT.Inthiscase,attime0•enterintoalongforwardcontractwithforwardpriceF(0,T)anddeliveryattimeT;•sellshortoneshareandinvesttheproceedsS(0)attherisk-freerate.Attimet•borrowdivdollarsandpayadividendtothestockowner.AttimeT•buyoneshareforF(0,T)andcloseouttheshortpositioninstock;•cashtherisk-freeinvestmentwithinterest,collectingtheamountS(0)erT,andpayer(T−t)divtocleartheloanwithinterest.Thefinalbalancewillagainbepositive,−F(0,T)+S(0)erT−er(T−t)div>0,completingtheproof. 130MathematicsforFinanceTheformulacaneasilybegeneralisedtothecasewhendividendsarepaidmorethanonce:F(0,T)=[S(0)−div]erT,(6.4)0wherediv0isthepresentvalueofalldividendsdueduringthelifetimeoftheforwardcontract.Exercise6.3Considerastockwhosepriceon1Januaryis$120andwhichwillpayadividendof$1on1July2000and$2on1October2000.Theinterestrateis12%.Isthereanarbitrageopportunityifon1January2000theforwardpricefordeliveryofthestockon1November2000is$131?Ifso,computethearbitrageprofit.Exercise6.4Supposethattherisk-freerateis8%.However,asasmallinvestor,youcaninvestmoneyat7%onlyandborrowat10%.DoeseitherofthestrategiesintheproofofProposition6.2giveanarbitrageprofitifF(0,1)=89andS(0)=83dollars,anda$2dividendispaidinthemiddleoftheyear,thatis,attime1/2?DividendYield.Dividendsareoftenpaidcontinuouslyataspecifiedrate,ratherthanatdiscretetimeinstants.Forexample,inacaseofahighlydiversi-fiedportfolioofstocksitisnaturaltoassumethatdividendsarepaidcontinu-ouslyratherthantotakeintoaccountfrequentpaymentsscatteredthroughouttheyear.Anotherexampleisforeigncurrency,attractinginterestatthecorre-spondingrate.Weshallfirstderiveaformulafortheforwardpriceinthecaseofforeigncurrency.LetthepriceofoneBritishpoundinNewYorkbeP(t)dollars,andlettherisk-freeinterestratesforinvestmentsinBritishpoundsandUSdollarsberGBPandrUSD,respectively.Letuscomparethefollowingstrategies:A:InvestP(0)dollarsattheraterUSDfortimeT.B:Buy1poundforP(0)dollars,investitfortimeTattheraterGBP,andtakeashortpositioninerGBPTpoundsterlingforwardcontractswithdeliverytimeTandforwardpriceF(0,T).Bothstrategiesrequirethesameinitialoutlay,sothefinalvaluesshouldbealsothesame:P(0)erUSDT=erGBPTF(0,T).ItfollowsthatF(0,T)=P(0)e(rUSD−rGBP)T.(6.5) 6.ForwardandFuturesContracts131Next,supposethatastockpaysdividendscontinuouslyataraterdiv>0,calledthe(continuous)dividendyield.Ifthedividendsarereinvestedinthestock,thenaninvestmentinoneshareheldattime0willincreasetobecomeerdivTsharesattimeT.(Thesituationissimilartocontinuouscompounding.)Consequently,inordertohaveoneshareattimeTweshouldbeginwithe−rdivTsharesattime0.Thisobservationisusedinthearbitrageproofbelow.Theorem6.3TheforwardpriceforstockpayingdividendscontinuouslyataraterdivisF(0,T)=S(0)e(r−rdiv)T.(6.6)ProofSupposethatF(0,T)>S(0)e(r−rdiv)T.Inthiscase,attime0•enterintoashortforwardcontract;•borrowtheamountS(0)e−rdivTtobuye−rdivTshares.Betweentime0andTcollectthedividendspaidcontinuously,reinvestingtheminthestock.AttimeTyouwillhave1share,asexplainedabove.Atthattime•selltheshareforF(0,T),closingouttheshortforwardposition;•payS(0)e(r−rdiv)Ttocleartheloanwithinterest.ThefinalbalanceF(0,T)−S(0)e(r−rdiv)T>0willbeyourarbitrageprofit.NowsupposethatF(0,T)0,contrarytotheNo-ArbitragePrinciple. 132MathematicsforFinanceIngeneral,ifthecontractisinitiatedattimet0willbeyourarbitrageprofit.WeleavethecasewhenV(t)>[F(t,T)−F(0,T)]e−r(T−t)asanexercise.Exercise6.6ShowthatV(t)>[F(t,T)−F(0,T)]e−r(T−t)leadstoanarbitrageop-portunity.ObservethatV(0)=0,whichistheinitialvalueoftheforwardcontract,andV(T)=S(T)−F(0,T)(sinceF(T,T)=S(T)),whichistheterminalpayoff.Forastockpayingnodividendsformula(6.8)givesV(t)=[S(t)er(T−t)−S(0)erT]e−r(T−t)=S(t)−S(0)ert.(6.9)Themessageis:Ifthestockpricegrowsatthesamerateasarisk-freeinvest-ment,thenthevalueoftheforwardcontractwillbezero.Growthabovetherisk-freerateresultsinagainfortheholderofalongforwardposition.Remark6.3ConsideracontractwithdeliverypriceXratherthanF(0,T).Thevalueofthiscontractattimetwillbegivenby(6.8)withF(0,T)replacedbyX,V(t)=[F(t,T)−X]e−r(T−t).X 134MathematicsforFinanceSuchacontractmayhavenon-zerovalueinitially.InthecaseofastockpayingnodividendsV(0)=[F(0,T)−X]e−rT=S(0)−Xe−rT.(6.10)XForastockpayingonedividendbetweentimes0andTtheinitialvalueofthecontractisV(0)=S(0)−div−Xe−rT,X0div0beingthevalueofthedividenddiscountedtotime0.Forastockpayingdividendscontinuouslyataraterdiv,theinitialvalueofthecontractisV(0)=S(0)e−rdivT−Xe−rT.XExercise6.7Supposethatthepriceofastockis$45atthebeginningoftheyear,therisk-freerateis6%,anda$2dividendistobepaidafterhalfayear.Foralongforwardpositionwithdeliveryinoneyear,finditsvalueafter9monthsifthestockpriceatthattimeturnsouttobea)$49,b)$51.6.2FuturesOneofthetwopartiestoaforwardcontractwillbelosingmoney.Thereisalwaysariskofdefaultbythepartysufferingaloss.Futurescontractsaredesignedtoeliminatesuchrisk.Weassumeforawhilethattimeisdiscretewithstepsoflengthτ,typicallyaday.Justlikeaforwardcontract,afuturescontractinvolvesanunderlyingassetandaspecifiedtimeofdelivery,astockwithpricesS(n)forn=0,1,...andtimeT,say.Inadditiontotheusualstockprices,themarketdictatestheso-calledfuturespricesf(n,T)foreachstepn=0,1,...suchthatnτ≤T.Thesepricesareunknownattime0,exceptforf(0,T),andweshalltreatthemasrandomvariables.Asinthecaseofaforwardcontract,itcostsnothingtoinitiateafuturesposition.Thedifferenceliesinthecashflowduringthelifetimeofthecontract.AlongforwardcontractinvolvesjustasinglepaymentS(T)−F(0,T)atde-livery.Afuturescontractinvolvesarandomcashflow,knownasmarkingtomarket.Namely,ateachtimestepn=1,2,...suchthatnτ≤Ttheholderofalongfuturespositionwillreceivetheamountf(n,T)−f(n−1,T) 6.ForwardandFuturesContracts135ifpositive,orwillhavetopayitifnegative.Theoppositepaymentsapplyforashortfuturesposition.Thefollowingtwoconditionsareimposed:1.Thefuturespriceatdeliveryisf(T,T)=S(T).2.Ateachtimestepn=0,1,...suchthatnτ≤Tthevalueofafuturespositioniszero.(Ateachstepn≥1thisvalueiscomputedaftermarkingtomarket.)Thesecondconditionmeansthat,inparticular,itcostsnothingtoclose,openoralterafuturespositionatanytimestepbetween0andT.Remark6.4Toensurethattheobligationsinvolvedinafuturespositionarefulfilled,certainpracticalregulationsareenforced.Eachinvestorenteringintoafuturescontracthastopayadeposit,calledtheinitialmargin,whichiskeptbytheclearinghouseascollateral.Inthecaseofalongfuturespositiontheamountf(n,T)−f(n−1,T)isaddedtothedepositifpositiveorsubtractedifnegativeateachtimestepn,typicallyonceaday.(Theoppositeamountisaddedorsubtractedforashortfuturesposition.)Anyexcessthatbuildsupabovetheinitialmargincanbewithdrawnbytheinvestor.Ontheotherhand,ifthedepositdropsbelowacertainlevel,calledthemaintenancemargin,theclearinghousewillissueamargincall,requestingtheinvestortomakeapaymentandrestorethedeposittotheleveloftheinitialmargin.Afuturespositioncanbeclosedatanytime,inwhichcasethedepositwillbereturnedtotheinvestor.Inparticular,thefuturespositionwillbeclosedimmediatelybytheclearinghouseiftheinvestorfailstorespondtoamargincall.Asaresult,theriskofdefaultiseliminated.Example6.1Supposethattheinitialmarginissetat10%andthemaintenancemarginat5%ofthefuturesprice.Thetablebelowshowsascenariowithfuturespricesf(n,T).Thecolumnslabelled‘margin1’and‘margin2’showthedepositatthebeginningandattheendofeachday,respectively.The‘payment’columncon-tainstheamountspaidtotopupthedeposit(negativenumbers)orwithdrawn 136MathematicsforFinance(positivenumbers).nf(n,T)cashflowmargin1paymentmargin20140opening:0−14141138−2120122130−84−9133140+1023+9144150+1024+915closing:15+150total:10Onday0afuturespositionisopenedanda10%depositpaid.Onday1thefuturespricedropsby$2,whichissubtractedfromthedeposit.Onday2thefuturespricedropsfurtherby$8,triggeringamargincallbecausethedepositfallsbelow5%.Theinvestorhastopay$9torestorethedeposittothe10%level.Onday3theforwardpriceincreasesand$9iswithdrawn,leavinga10%margin.Onday4theforwardpricegoesupagain,allowingtheinvestortowithdrawanother$9.Attheendofthedaytheinvestordecidestoclosetheposition,collectingthebalanceofthedeposit.Thetotalofallpaymentsis$10,theincreaseinthefuturespricebetweenday0and4.Remark6.5Animportantfeatureofthefuturesmarketisliquidity.Thisispossibleduetostandardisationandthepresenceofaclearinghouse.Onlyfuturescontractswithparticulardeliverydatesaretraded.Moreover,futurescontractsoncom-moditiessuchasgoldortimberspecifystandardiseddeliveryarrangementsaswellasstandardisedphysicalpropertiesoftheassets.Theclearinghouseactsasanintermediary,matchingthetotalofalargenumberofshortandlongfuturespositionsofvarioussizes.Theclearinghousealsomaintainsthemargindepositforeachinvestortoeliminatetheriskofdefault.Thisisincontrasttoforwardcontractsnegotiateddirectlybetweentwoparties.6.2.1PricingWeshallshowthatinsomecircumstancestheforwardandthefuturespricesarethesame.Letrbetherisk-freerateundercontinuouscompounding.Theorem6.5Iftheinterestrateisconstant,thenf(0,T)=F(0,T). 6.ForwardandFuturesContracts137ProofSupposeforsimplicitythatmarkingtomarketisperformedatjusttwointer-mediatetimeinstants0r,thenthebasisispositive:b(t,T)=S(t)(1−e(r−rdiv)(T−t)). 6.ForwardandFuturesContracts141Goingbacktotheproblemofdesigningahedge,supposethatwewishtosellanassetattimetX,0otherwise.Thispayoffisarandomvariable,contingentonthepriceS(T)oftheunderlyingontheexercisedateT.(Thisexplainswhyoptionsareoftenreferredtoascontingentclaims.)Itisconvenienttousethenotation
+xifx>0,x=0otherwise.forthepositivepartofarealnumberx.ThenthepayoffofaEuropeancalloptioncanbewrittenas(S(T)−X)+.Foraputoptionthepayoffis(X−S(T))+.Sincethepayoffsarenon-negative,apremiummustbepaidtobuyanoption.Ifnopremiumhadtobepaid,aninvestorpurchasinganoptioncouldundernocircumstanceslosemoneyandwouldinfactmakeaprofitwheneverthepayoffturnedouttobepositive.ThiswouldbecontrarytotheNo-ArbitragePrinciple.Thepremiumisthemarketpriceoftheoption.Establishingboundsandsomegeneralpropertiesforoptionpricesistheprimarygoalofthepresentchapter.Thenextchapterwillbedevotedtode-tailedtechniquesofcomputingtheseprices.Weassumethatoptionsarefreelytraded,thatis,canreadilybeboughtandsoldatthemarketprice.ThepricesofcallsandputswillbedenotedbyCE,PEforEuropeanoptionsandCA,PAforAmericanoptions,respectively.Thesameconstantinterestraterwillapplyforlendingandborrowingmoneywithoutrisk,andcontinuouscompoundingwillbeused.Example7.1On22March1997EuropeancallsonRolls-Roycestockwithstrikeprice220pencetobeexercisedon22May1997tradedat19.5penceattheLondonInternationalFinancialFuturesExchange(LIFFE).Supposethatthepurchaseofsuchanoptionwasfinancedbyaloanat5.23%compoundedcontinuously,so0.0523×2∼=19.67pencewouldhavetobepaidbackontheexercisethat19.5e12date.Theinvestmentwouldbringaprofitifthestockpriceturnedouttobehigherthan220+19.67=239.67penceontheexercisedate. 7.Options:GeneralProperties149Exercise7.1FindthestockpriceontheexercisedateforaEuropeanputoptionwithstrikeprice$36andexercisedateinthreemonthstoproduceaprofitof$3iftheoptionisboughtfor$4.50,financedbyaloanat12%compoundedcontinuously.Thegainofanoptionbuyer(writer)isthepayoffmodifiedbythepremiumCEorPEpaid(received)fortheoption.AttimeTthegainofthebuyerofaEuropeancallis(S(T)−X)+−CEerT,wherethetimevalueofthepremiumistakenintoaccount.ForthebuyerofaEuropeanputthegainis(X−S(T))+−PEerT.ThesegainsareillustratedinFigure7.1.ForthewriterofanoptionthegainsareCEerT−(S(T)−X)+foracallandPEerT−(X−S(T))+foraputoption.Notethatthepotentiallossforabuyerofacallorputisalwayslimitedtothepremiumpaid.Forawriterofanoptionthelosscanbemuchhigher,evenunboundedinthecaseofacalloption.Figure7.1Payoffs(solidlines)andgains(brokenlines)forabuyerofEuro-peancallsandputsExercise7.2Findtheexpectedgain(orloss)foraholderofaEuropeancalloptionwithstrikeprice$90tobeexercisedin6monthsifthestockpriceontheexercisedatemayturnouttobe$87,$92or$97withprobability13each,giventhattheoptionisboughtfor$8,financedbyaloanat9%compoundedcontinuously. 150MathematicsforFinance7.2Put-CallParityInthissectionweshallmakeanimportantlinkbetweenthepricesofEuropeancallandputoptions.Consideraportfolioconstructedbyandwritingandsellingoneputandbuyingonecalloption,bothwiththesamestrikepriceXandexercisedateT.Addingthepayoffsofthelongpositionincallsandtheshortpositioninputs,weobtainthepayoffofalongforwardcontractwithforwardpriceXanddeliverytimeT.Indeed,ifS(T)≥X,thenthecallwillpayS(T)−Xandtheputwillbeworthless.IfS(T)S(0)−Xe−rT.(7.2)Inthiscaseanarbitragestrategycanbeconstructedasfollows:Attime0•buyoneshareforS(0); 7.Options:GeneralProperties151•buyoneputoptionforPE;•writeandsellonecalloptionforCE;•investthesumCE−PE−S(0)(orborrow,ifnegative)onthemoneymarketattheinterestrater.Thebalanceofthesetransactionsis0.Then,attimeT•closeoutthemoneymarketposition,collecting(orpaying,ifnegative)thesum(CE−PE−S(0))erT;•selltheshareforXeitherbyexercisingtheputifS(T)≤XorsettlingtheshortpositionincallsifS(T)>X.Thebalancewillbe(CE−PE−S(0))erT+X,whichispositiveby(7.2),contradictingtheNo-ArbitragePrinciple.NowsupposethatCE−PEXorsettlingtheshortpositioninputsifS(T)≤X,andclosetheshortpositioninstock.Thebalancewillbe(S(0)−CE+PE)erT−X,positiveby(7.3),onceagaincontradictingtheNo-ArbitragePrinciple.Exercise7.3Supposethatastockpayingnodividendsistradingat$15.60ashare.Europeancallsonthestockwithstrikeprice$15andexercisedateinthreemonthsaretradingat$2.83.Theinterestrateisr=6.72%,com-poundedcontinuously.WhatisthepriceofaEuropeanputwiththesamestrikepriceandexercisedate?Exercise7.4Europeancallandputoptionswithstrikeprice$24andexercisedateinsixmonthsaretradingat$5.09and$7.78.Thepriceoftheunder- 152MathematicsforFinancelyingstockis$20.37andtheinterestrateis7.48%.Findanarbitrageopportunity.Remark7.1WecanmakeasimplebutpowerfulobservationbasedonTheorem7.1:ThepricesofEuropeancallsandputsdependinthesamewayonanyvariablesabsentintheput-callparityrelation(7.1).Inotherwords,thedifferenceofthesepricesdoesnotdependonsuchvariables.Asanexample,considertheexpectedreturnonstockIfthepriceofacallshouldgrowalongwiththeexpectedreturn,whichonfirstsightseemsconsistentwithintuitionbecausehigherstockpricesmeanhigherpayoffsoncalls,thenthepriceofaputwouldalsogrow.Thelatter,however,contradictscommonsensebecausehigherstockpricesmeanlowerpayoffsonputs.Becauseofthis,onecouldarguethatputandcallpricesshouldbeindependentoftheexpectedreturnonstock.WeshallseethatthisisindeedthecaseoncetheBlack–ScholesformulaisderivedforcallandputoptionsinChapter8.Followingtheargumentpresentedatthebeginningofthissection,wecanreformulateput-callparityasfollows:CE−PE=V(0),(7.4)XwhereVX(0)isthevalueofalongforwardcontract,see(6.10).NotethatifXisequaltothetheoreticalforwardpriceS(0)erToftheasset,thenthevalueoftheforwardcontractiszero,V(0)=0,andsoCE=PE.Formula(7.4)allowsXustogeneraliseput-callparitybydrawingontherelationshipsestablishedinRemark6.3.Namely,ifthestockpaysadividendbetweentimes0andT,thenV(0)=S(0)−div−Xe−rT,wheredivisthepresentvalueofthedividend.X00ItfollowsthatCE−PE=S(0)−div−Xe−rT.(7.5)0Ifdividendsarepaidcontinuouslyatarater,thenV(0)=S(0)e−rdivT−divXXe−rT,soCE−PE=S(0)e−rdivT−Xe−rT.(7.6)Exercise7.5Outlineanarbitrageproofof(7.5).Exercise7.6Outlineanarbitrageproofof(7.6). 7.Options:GeneralProperties153Exercise7.7ForthedatainExercise6.5,findthestrikepriceforEuropeancallsandputstobeexercisedinsixmonthssuchthatCE=PE.ForAmericanoptionsput-callparitygivesonlyanestimate,ratherthanastrictequalityinvolvingputandcallprices.Theorem7.2(Put-CallParityEstimates)ThepricesofAmericanputandcalloptionswiththesamestrikepriceXandexpirytimeTonastockthatpaysnodividendssatisfyS(0)−Xe−rT≥CA−PA≥S(0)−X.ProofSupposethatthefirstinequalityfailstohold,thatis,CA−PA−S(0)+Xe−rT>0.Thenwecanwriteandsellacall,andbuyaputandashare,financingthetransactionsonthemoneymarket.IftheholderoftheAmericancallchoosestoexerciseitattimet≤T,thenweshallreceiveXfortheshareandsettlethemoneymarketposition,endingupwiththeputandapositiveamountX+(CA−PA−S(0))ert=(Xe−rt+CA−PA−S(0))ert≥(Xe−rT+CA−PA−S(0))ert>0.Ifthecalloptionisnotexercisedatall,wecanselltheshareforXbyexercisingtheputattimeTandclosethemoneymarketposition,alsoendingupwithapositiveamountX+(CA−PA−S(0))erT>0.NowsupposethatCA−PA−S(0)+X<0.Inthiscasewecanwriteandsellaput,buyacallandsellshortoneshare,investingthebalanceonthemoneymarket.IftheAmericanputisexercisedattimet≤T,thenwecanwithdrawXfromthemoneymarkettobuyashareandclosetheshortsale.Weshallbeleftwiththecalloptionandapositiveamount(−CA+PA+S(0))ert−X>Xert−X≥0. 154MathematicsforFinanceIftheputisnotexercisedatall,thenwecanbuyashareforXbyexercisingthecallattimeTandclosetheshortpositioninstock.Onclosingthemoneymarketposition,weshallalsoendupwithapositiveamount(−CA+PA+S(0))erT−X>XerT−X>0.Thetheorem,therefore,holdsbytheNo-ArbitragePrinciple.Exercise7.8ModifytheproofofTheorem7.2toshowthatS(0)−Xe−rT≥CA−PA≥S(0)−div−X0forastockpayingadividendbetweentime0andtheexpirytimeT,wherediv0isthevalueofthedividenddiscountedtotime0.Exercise7.9ModifytheproofofTheorem7.2toshowthatS(0)−Xe−rT≥CA−PA≥S(0)e−rdivT−Xforastockpayingdividendscontinuouslyataraterdiv.7.3BoundsonOptionPricesFirstofall,wenotetheobviousinequalitiesCE≤CA,PE≤PA,(7.7)forEuropeanandAmericanoptionswiththesamestrikepriceXandexpirytimeT.TheyholdbecauseanAmericanoptiongivesatleastthesamerightsasthecorrespondingEuropeanoption.Figure7.3showsascenarioofstockpricesinwhichthepayoffofaEuropeancalliszeroattheexercisetimeT,whereasthatofanAmericancallwillbepositiveiftheoptionisexercisedatanearliertimet0.ThisprovesthatCES(0)−Xifr>0.BecausethepriceoftheAmericanoptionisgreaterthanthepayoff,theoptionwillsoonerbesoldthanexercisedattime0.Thechoiceof0asthestartingtimeisofcoursearbitrary.Replacing0byanygiventCE,thenwriteandsellanAmericancallandbuyaEuropeancall,investingthebalanceCA−CEattheinterestrater.IftheAmericancallisexercisedattimet≤T,thenborrowashareandsellitforXtosettleyourobligationaswriterofthecalloption,investingXatrater.Then,attimeTyoucanusetheEuropeancalltobuyashareforXandcloseyourshortpositioninstock.Yourarbitrageprofitwillbe(CA−CE)erT+Xer(T−t)−X>0.IftheAmericanoptionisnotexercisedatall,youwillendupwiththeEuropeanoptionandanarbitrageprofitof(CA−CE)erT>0.ThisprovesthatCA=CE.Theorem7.4mayseemcounter-intuitiveatfirstsight.Whileitispossible 158MathematicsforFinancetogainS(t)−XbyexercisinganAmericancalloptionifS(t)>Xattimet0.Iftheoptionisnotexercisedat 7.Options:GeneralProperties159all,thefinalbalancewillalsobepositive,PAerT>0,atexpiry.Theseresultscanbesummarisedasfollows.Proposition7.5ThepricesofAmericancallandputoptionsonastockpayingnodividendssatisfytheinequalitiesmax{0,S(0)−Xe−rT}≤CACE(X),PE(X)αCE(X)+(1−α)CE(X). 162MathematicsforFinanceWecanwriteandsellanoptionwithstrikepriceX,andpurchaseαoptionswithstrikepriceXand1−αoptionswithstrikepriceX,investingthebalanceCE(X)−αCE(X)+(1−α)CE(X)>0withoutrisk.Iftheoptionwith+strikepriceXisexercisedatexpiry,thenweshallhavetopay(S(T)−X).Wecanraisetheamountα(S(T)−X)++(1−α)(S(T)−X)+byexercisingαcallswithstrikeXand1−αcallswithstrikeX.Inthiswaywewillrealiseanarbitrageprofitbecauseofthefollowinginequality,whichiseasytoverify(thedetailsarelefttothereader,seeExercise7.15):+++(S(T)−X)≤α(S(T)−X)+(1−α)(S(T)−X).(7.9)Convexityforputoptionsfollowsfromthatforcallsbyput-callparity(7.1).Alternatively,anarbitrageargumentcanbegivenalongsimilarlinesasforcalloptions.Exercise7.15Verifyinequality(7.9).Remark7.3AccordingtoProposition7.8,CE(X)andPE(X)areconvexfunctionsofX.Geometrically,thismeansthatiftwopointsonthegraphofthefunctionarejoinedwithastraightline,thenthegraphofthefunctionbetweenthetwopointswillliebelowtheline.ThisisillustratedinFigure7.6forcallprices.Figure7.6ConvexityofcallpricesCE(X)DependenceontheUnderlyingAssetPrice.ThecurrentpriceS(0)oftheunderlyingassetisgivenbythemarketandcannotbealtered.However,wecanconsideranoptiononaportfolioconsistingofxshares,worthS=xS(0).ThepayoffofaEuropeancallwithstrikepriceXonsuchaportfoliotobeexercised++attimeTwillbe(xS(T)−X).Foraputthepayoffwillbe(X−xS(T)).WeshallstudythedependenceofoptionpricesonS.Assumingthatallremainingvariablesarefixed,weshalldenotethecallandputpricesbyCE(S)andPE(S). 7.Options:GeneralProperties163Remark7.4Eventhoughoptionsonaportfolioofstocksareoflittlepracticalsignificance,thefunctionsCE(S)andPE(S)areimportantbecausetheyalsoreflectthede-pendenceofoptionpricesonverysuddenchangesofthepriceoftheunderlyingsuchthattheremainingvariablesremainalmostunaltered.Proposition7.9IfSPE(S),thatis,CE(S)isastrictlyincreasingfunctionandPE(S)astrictlydecreasingfunctionofS.ProofSupposethatCE(S)≥CE(S)forsomeSαCE(S)+(1−α)CE(S).Wewriteandsellacallonaportfoliowithxshares,andpurchaseαcallsonaportfoliowithxsharesand1−αcallsonaportfoliowithxshares,investingthebalanceCE(S)−αCE(S)−(1−α)CE(S)withoutrisk.Iftheoptionsold+isexercisedattimeT,thenweshallhavetopay(xS(T)−X).Tocoverthisliabilitywecanexercisetheotheroptions.Since+++(xS(T)−X)≤α(xS(T)−X)+(1−α)(xS(T)−X),thisisanarbitragestrategy.Theinequalityforputoptionscanbeprovedbyasimilararbitrageargumentorusingput-callparity.7.4.2AmericanOptionsIngeneral,AmericanoptionshavesimilarpropertiestotheirEuropeancounter-parts.Onedifficultyistheabsenceofput-callparity;weonlyhavetheweakerestimatesinTheorem7.2.Inaddition,wehavetotakeintoaccountthepossi-bilityofearlyexercise.DependenceontheStrikePrice.WeshalldenotethecallandputpricesbyCA(X)andPA(X)toemphasisethedependenceonX,keepinganyothervariablesfixed.ThefollowingpropositionisobviousforthesamereasonsasforEuropeanoptions:Higherstrikepricemakestherighttobuylessvaluableandtherighttosellmorevaluable.Proposition7.12IfXCA(X),PA(X)αCA(X)+(1−α)CA(X).WewriteanoptionwithstrikepriceXandbuyαoptionswithstrikepriceXand(1−α)optionswithstrikepriceX,investingwithoutriskthepositivebalanceofthesetransactions.Ifthewrittenoptionisexercisedattimet≤T,thenweexercisebothoptionsheld.Inthiswayweshallachievearbitragebecause+++(S(t)−X)≤α(S(t)−X)+(1−α)(S(t)−X).Theproofforputoptionsissimilar. 7.Options:GeneralProperties167DependenceontheUnderlyingAssetPrice.Onceagain,weshallcon-sideroptionsonaportfolioofxshares.ThepricesofAmericancallsandputsonsuchaportfoliowillbedenotedbyCA(S)andPA(S),whereS=xS(0)isthevalueoftheportfolio,allremainingvariablesbeingfixed.Thepayoffsat++timetare(xS(t)−X)forcallsand(X−xS(t))forputs.Proposition7.15IfSPA(S).ProofSupposethatCA(S)≥CA(S)forsomeSαCA(S)+(1−α)CA(S).Wecanwriteandsellacallonaportfoliowithxshares,andpurchaseαcallsonaportfoliowithxsharesand1−αcallsonaportfoliowithxshares,allthreeoptionssharingthesamestrikepriceXandexpirytimeT.ThepositivebalanceCA(S)−αCA(S)−(1−α)CA(S)ofthesetransactionscanbeinvestedwithoutrisk.Ifthewrittenoptionisexercisedattimet≤T,thenweshallhavetopay(xS(t)−X)+,wherex=αx+(1−α)x.Wecanexercisetheothertwooptionstocovertheliability.Thisisanarbitragestrategybecause+++(xS(t)−X)≤α(xS(t)−X)+(1−α)(xS(t)−X).Theproofforputoptionsissimilar.DependenceontheExpiryTime.ForAmericanoptionswecanalsofor-mulateageneralresultonthedependenceoftheirpricesontheexpirytimeT.Toemphasisethisdependence,weshallnowwriteCA(T)andPA(T)forthepricesofAmericancallsandputs,assumingthatallothervariablesarefixed.Proposition7.18IfTCA(T).WewriteandselloneoptionexpiringattimeTandbuyonewiththesamestrikepricebutexpiringattimeT,invest-ingthebalancewithoutrisk.Ifthewrittenoptionisexercisedattimet≤T,wecanexercisetheotheroptionimmediatelytocoverourliability.Theposi-tivebalanceCA(T)−CA(T)>0investedwithoutriskwillbeourarbitrageprofit.Theargumentisthesameforputs.7.5TimeValueofOptionsThefollowingconvenientterminologyisoftenused.WesaythatattimetacalloptionwithstrikepriceXis•inthemoneyifS(t)>X,•atthemoneyifS(t)=X,•outofthemoneyifS(t)X.Alsoconvenient,thoughlessprecise,arethetermsdeepinthemoneyanddeepoutofthemoney,whichmeanthatthedifferencebetweenthetwosidesintherespectiveinequalitiesisconsiderable.AnAmericanoptioninthemoneywillbringapositivepayoffifexercisedimmediately,whereasanoptionoutofthemoneywillnot.WeusethesametermsforEuropeanoptions,thoughtheirmeaningisdifferent:Eveniftheoptioniscurrentlyinthemoney,itmaynolongerbesoontheexercisedate,whenthepayoffmaywellturnouttobezero.AEuropeanoptioninthemoneyisnomorethanapromisingasset.Definition7.1Attimet≤TtheintrinsicvalueofacalloptionwithstrikepriceXisequal+to(S(t)−X).Theintrinsicvalueofaputoptionwiththesamestrikeprice+is(X−S(t)).Wecanseethattheintrinsicvalueiszeroforoptionsoutofthemoneyoratthemoney.Optionsinthemoneyhavepositiveintrinsicvalue.Thepriceofan 170MathematicsforFinanceoptionatexpiryTcoincideswiththeintrinsicvalue.ThepriceofanAmericanoptionpriortoexpirymaybegreaterthantheintrinsicvaluebecauseofthepossibilityoffuturegains.ThepriceofaEuropeanoptionpriortotheexercisetimemaybegreaterorsmallerthantheintrinsicvalue.Definition7.2Thetimevalueofanoptionisthedifferencebetweenthepriceoftheoptionanditsintrinsicvalue,thatis,E+C(t)−(S(t)−X)foraEuropeancall,E+P(t)−(X−S(t))foraEuropeanput,A+C(t)−(S(t)−X)foranAmericancall,A+P(t)−(X−S(t))foranAmericanput.Example7.3Letusexaminesometypicaldata.Supposethatthecurrentpriceofstockis$125.23pershare.Considerthefollowing:IntrinsicValueTimeValueOptionPriceSrikePriceCallPutCallPutCallPut11015.230.003.172.8418.402.841205.230.007.046.4612.276.461300.005.236.784.416.789.64AnAmericancalloptionwithstrikeprice$110isinthemoneyandhas$15.23intrinsicvalue.Theoptionpricemustbeatleastequaltotheintrinsicvalue,sincetheoptionmaybeexercisedimmediately.Typically,thepricewillbehigherthantheintrinsicvaluebecauseofthepossibilityoffuturegains.Ontheotherhand,aputoptionwithstrikeprice$110willbeoutofthemoneyanditsintrinsicvaluewillbezero.Thepositivepriceoftheputisentirelyduetothepossibilityoffuturegains.Similarrelationshipsforotherstrikepricescanbeseeninthetable.ThetimevalueofaEuropeancallasafunctionofSisshowninFigure7.8.Itcanneverbenegative,andforlargevaluesofSitexceedsthedifferenceX−Xe−rT.ThisisbecauseoftheinequalityCE(S)≥S−Xe−rT,seePropo-sition7.3.ThemarketvalueofaEuropeanputmaybelowerthanitsintrinsicvalue,thatis,thetimevaluemaybenegative,seeFigure7.9.Thismaybesoonlyiftheputoptionisinthemoney,SV(0),thenwewritethederivativesecurityandtakealongpositioninthestrategy.Ourobligationwillbecoveredbythestrategy,thedifferenceD(0)−V(0)being173 174MathematicsforFinanceourarbitrageprofit.IfD(0)X.HenceNE−NNkN−kkN−kC(0)=(1+r)p∗(1−p∗)S(0)(1+u)(1+d)−X.kk=mThiscanbewrittenasCE(0)=x(1)S(0)+y(1),relatingtheoptionpricetotheinitialreplicatingportfoliox(1),y(1),whereNNx(1)=(1+r)−Npk(1−p)N−k(1+u)k(1+d)N−k,∗∗kk=mNNy(1)=−X(1+r)−Npk(1−p)N−k.∗∗kk=mTheexpressionforx(1)canberewrittenasNN1+uk1+dN−kNNkx(1)=p(1−p)=q(1−q)N−k,∗∗k1+r1+rkk=mk=mwhere1+uq=p∗.1+r(Notethatp1+uand(1−p)1+dadduptoone.)Similarformulaecanbe∗1+r∗1+rderivedforputoptions,eitherdirectlyorusingput-callparity.Theseimportantresultsaresummarisedinthefollowingtheorem,inwhichΦ(m,N,p)denotesthecumulativebinomialdistributionwithNtrialsandprobabilitypofsuccessineachtrial,mNkN−kΦ(m,N,p)=p(1−p).kk=0 8.OptionPricing181Theorem8.5(Cox–Ross–RubinsteinFormula)InthebinomialmodelthepriceofaEuropeancallandputoptionwithstrikepriceXtobeexercisedafterNtimestepsisgivenbyCE(0)=S(0)[1−Φ(m−1,N,q)]−(1+r)−NX[1−Φ(m−1,N,p)],∗PE(0)=−S(0)Φ(m−1,N,q)+(1+r)−NXΦ(m−1,N,p).∗Theinitialreplicatingportfoliox(1),y(1)isgivenbyx(1)y(1)foracall1−Φ(m−1,N,q)−(1+r)−NX[1−Φ(m−1,N,p)]∗foraput−Φ(m−1,N,q)(1+r)−NXΦ(m−1,N,p)∗Exercise8.7LetS(0)=50dollars,r=5%,u=0.3andd=−0.1.FindthepriceofaEuropeancallandputwithstrikepriceX=60dollarstobeexercisedafterN=3timesteps.Exercise8.8LetS(0)=50dollars,r=0.5%,u=0.01andd=−0.01.Findm,x(1),andthepriceCE(0)ofaEuropeancalloptionwithstrikeX=60dollarstobeexercisedafterN=50timesteps.Exercise8.9Considerthescenarioinwhichstockgoesupateachstep.AtwhichstepwillthedeltaofaEuropeancallbecome1?8.2AmericanOptionsintheBinomialTreeModelEventheformulationofaprecisemathematicaldefinitionofanAmericantypecontingentclaimpresentssomedifficulties.Nevertheless,theinformaldescrip-tionissimple:Theoptioncanbeexercisedatanytimestepnsuchthat0≤n≤N,withpayofff(S(n)).Ofcourse,itcanbeexercisedonlyonce.ThepriceofanAmericanoptionattimenwillbedenotedbyDA(n). 182MathematicsforFinanceTobeginwith,weshallanalyseanAmericanoptionexpiringafter2timesteps.Unlesstheoptionhasalreadybeenexercised,atexpiryitwillbeworthDA(2)=f(S(2)),wherewehavethreevaluesdependingonthevaluesofS(2).Attime1theop-tionholderwillhavethechoicetoexerciseimmediately,withpayofff(S(1)),ortowaituntiltime2,whenthevalueoftheAmericanoptionwillbecomef(S(2)).Thevalueofwaitingcanbecomputedbytreatingf(S(2))asaone-stepEuro-peancontingentclaimtobepricedattime1,whichgivesthevalue1[p∗f(S(1)(1+u))+(1−p∗)f(S(1)(1+d))]1+rattime1.Ineffect,theoptionholderhasthechoicebetweenthelattervalueortheimmediatepayofff(S(1)).TheAmericanoptionattime1will,therefore,beworththehigherofthetwo,"#A1D(1)=maxf(S(1)),[p∗f(S(1)(1+u))+(1−p∗)f(S(1)(1+d))]1+r=f1(S(1))(arandomvariablewithtwovalues),where"#1f1(x)=maxf(x),[p∗f(x(1+u))+(1−p∗)f(x(1+d))].1+rAsimilarargumentgivestheAmericanoptionvalueattime0,"#A1D(0)=maxf(S(0)),[p∗f1(S(0)(1+u))+(1−p∗)f1(S(0)(1+d))].1+rExample8.1ToillustratetheaboveprocedureweconsideranAmericanputoptionwithstrikepriceX=80dollarsexpiringattime2onastockwithinitialpriceS(0)=80dollarsinabinomialmodelwithu=0.1,d=−0.05andr=0.05.(Weconsideraput,asweknowthatthereisnodifferencebetweenAmericanandEuropeancalloptions,seeTheorem7.4.)Thestockvaluesaren01296.8088.000Ee−rtS(t)1=Ee−ruS(u)1,(8.8)∗S(u)m+xσ∼=−37.50%,whereN(x)∼=5%,sox∼=−1.645.(HereN(x)isthenormaldistributionfunction(8.10)withmean0andvariance1.)Hencewithprobability95%thefuturepriceS(1)willsatisfyS(1)>S(0)em+xσ∼=68.83dollars,andso,giventhatr=8%,VaR=S(0)er−S(0)em+xσ∼=39.50dollars.Exercise9.8EvaluateVaRat95%confidencelevelforaone-yearinvestmentof$1,000intoeurosiftheinterestrateforrisk-freeinvestmentsineurosisrEUR=4%andtheexchangeratefromeurosintoUSdollarsfollowsthelognormaldistributionwithm=1%andσ=15%.Takeintoaccounttheforegoneopportunityofinvestingdollarswithoutrisk,giventhattherisk-freeinterestratefordollarsisrUSD=5%. 9.FinancialEngineering203Exercise9.9Supposethat$1,000isinvestedinEuropeancalloptionsonastockwithcurrentpriceS(0)=60dollars.Theoptionsexpireafter6monthswithstrikepriceX=40dollars.Assumethatσ=30%,r=8%,andtheexpectedlogarithmicreturnonstockis12%.ComputeVaRafter6monthsat95%confidencelevel.Findthefinalwealthifthestockpricegrowsattheexpectedrate.Findthestockpricelevelthatwillbeexceededwith5%probabilityandcomputethecorrespondingfinalpayoff.9.2.2CaseStudyWeshalldiscussanumberofwaysinwhichVaRcanbemanagedwiththeaidofderivativesecurities.Themethodswillbeillustratedbyasimpleexampleofbusinessactivity.Case9.1AcompanymanufacturesgoodsintheUKforsaleintheUSA.Theinvestmenttostartproductionis5millionpounds.AdditionalfundscanberaisedbyborrowingBritishpoundsat16%tofinanceahedgingstrategy.Therateofreturndemandedbyinvestors,bearinginmindtheriskinvolved,is25%.Thesalesarepredictedtogenerate8milliondollarsattheendoftheyear.Themanufacturingcostsare3millionpoundsperyear.Theinterestrateis8%fordollarsand11%forpounds.Thecurrentrateofexchangeis1.6dollarstoapound.Thevolatilityofthelogarithmicreturnontherateofexchangeisestimatedat15%.Thecompanypays20%taxonearnings.Firstnotethattosatisfytheexpectationsofinvestorsthecompanyshouldbeabletoachieveaprofitof1.25millionpoundsayeartopaythedividend.Alowerprofitwouldmeanaloss.Theprofitdependsontherateofexchangedattheendoftheyear,hencesomeriskemerges.(Weassumethattheothervalueswillbeaspredicted.)Tobeginwith,supposethatnoactionistakentomanagetherisk.1.UnhedgedPosition.Iftheexchangeratedturnsouttobe1.6dollarstoapoundattheendoftheyear,thenthenetearningswillbe1.6millionpounds,asshowninthefollowingprofitandlossstatement(allamountsin 204MathematicsforFinancepounds):sales5,000,000costofsales−3,000,000earningsbeforetax2,000,000tax−400,000earningsaftertax1,600,000dividend−1,250,000result350,000Thesurplusincomewillbe0.35millionpounds.However,iftheexchangeratedbecomes2dollarstoapound,thecom-panywillendupwithalossof0.45millionpounds(andthedividendwillinfacthavetobereduced):sales4,000,000costofsales−3,000,000earningsbeforetax1,000,000tax−200,000earningsaftertax800,000dividend−1,250,000result−450,000LetuscomputeVaR.Weassumethattherateofexchangehaslog-normaldistributionwithmeanreturnequaltothedifferencebetweentheinterestrates,8%−11%=−3%.1Withthevolatilityofthereturnontheexchangerateat15%,thereturnontheinvestmentwillexceed−3%+1.65×15%=21.75%withprobability95%.Thiscorrespondstoanexchangerated=1.6×e21.75%∼=1.9887dollarstoapound,forwhichtheincomestatementwillbeasfollows(allamountsroundedtothenearestpound):sales4,022,728costofsales−3,000,000earningsbeforetax1,022,728tax−204,546earningsaftertax818,182dividend−1,250,000result−431,8181Thisassumptioncanbejustifiedasfollows:Ifapoundisinvestedwithoutriskforoneyearandthenconvertedtodollarsataratedknowninadvance,toavoid11%8%−3%arbitrageweshouldhaved×e=1.6×e,sod=1.6×e.Thisgives−3%logarithmicreturnontheexchangerate.Forarandomexchangerateitisthereforenaturaltoassumethemeanlogarithmicreturntobe−3%. 9.FinancialEngineering205Asaresult,VaR∼=431,818dollars.Thefinalbalanceasafunctionoftheexchangeratedis8,000,000b(d)=80%×(−3,000,000)−1,250,000d6,400,000=−3,650,000.dThebreakevenexchangerate,whichsolvesb(d)=0,isapproximatelyequalto1.7534dollarstoapound.Inanoptimisticscenarioinwhichthepoundweakens,forexample,downto1.5dollars,thefinalbalancewillbeabout£616,666.Thequestionishowtomanagethisriskexposure.2.ForwardContract.Theeasiestsolutionwouldbetofixtheexchangerateinadvancebyenteringintoalongforwardcontract.Theforwardrateis1.6×e−3%∼=1.5527dollarstoapound.Asaresult,thecompanycanobtainthefollowingstatementwithguaranteedsurplus,butnopossibilityoffurthergainsshouldtheexchangeratebecomemorefavourable:sales5,152,315costofsales−3,000,000earningsbeforetax2,152,315tax−430,463netincome1,721,852dividend−1,250,000result471,8523.FullHedgewithOptions.Optionscanbeusedtoensurethattherateofexchangeiscappedatacertainlevel,whilstthebenefitsassociatedwithfavourableexchangeratemovementsareretained.However,thismaybecostlybecauseofthepremiumpaidforoptions.Thecompanycanbuycalloptionsontheexchangerate.AEuropeancalltobuyonepoundwithstrikeprice1.6dollarstoapoundwillcost£0.0669.2Supposethatthecompanybuys5millionofsuchoptions,payinga£334,510premium,whichtheyhavetoborrowat16%.Theinterestistaxdeductible,makingtheloanlesscostly.Nevertheless,thefinalresultis2ForoptionsoncurrenciestheBlack–Scholesformulahastobemodifiedbyreplacingtherisk-freeinterestraterbythedifferencebetweentherisk-freeratesforthecurrencies,inourcase:−3%. 206MathematicsforFinancedisappointing:sales5,000,000costofsales−3,000,000earningsbeforeinterestandtax2,000,000interest−53,522earningsbeforetax1,946,478tax−389,296netincome1,557,182loanrepaid−334,510dividend−1,250,000result−27,328Theoptimal(inthesenseofminimisingtheloss)strikepriceis1.5734dollarstoapound,resultinginalossof£24,283.Iftheexchangeratedropsto1.5dollarstoapound,theoptionswillnotbeexercisedandthesumobtainedfromsaleswillreach£5,333,333,withapositivefinalresultof£239,339.Thisstrategyleadstoabetterresultthanthehedgeinvolvingaforwardcontractonlyiftherateofexchangedropsbelow1.42dollarstoapound.4.PartialHedgewithOptions.Toreducethecostofoptionsthecompanycanhedgepartiallybybuyingcalloptionstocoveronlyafractionofthedollaramountfromsales.Supposethatthecompanybuys2,500,000unitsofthesamecalloptionasabove,payingahalfofthepreviouspremium.Ahalfoftherevenueisthenexposedtorisk.TofindVaRat95%confidencelevelweassumethatthissumisexchangedat1.9887dollarstoapound,asinthecaseofanunhedgedposition,theotherhalfbeingexchangedattheexerciseprice:sales4,511,364costofsales−3,000,000earningsbeforeinterestandtax1,511,364interest−26,761earningsbeforetax1,484,603tax−296,921netincome1,187,682loanrepaid−167,255dividend−1,250,000result−229,573Iftheexchangeratedropsto1.5dollarstoapound,thecompanywillhaveasurplusof£428,003. 9.FinancialEngineering2075.CombinationofOptionsandForwardContracts.Finally,letusinves-tigatewhathappensifthecompanyhedgeswithbothkindsofderivatives.Halfoftheirpositionwillbehedgedwithoptions.Intheworstcasesce-nariotheywillbuypoundsforhalfoftheirdollarrevenueattherateof1.6dollarstoapound,theremaininghalfbeingexchangedattheforwardrateof1.5527dollarstoapound.Theoutcomeisshownbelow,wherewesummarisetheresultingVaRforallstrategiesconsidered(theresultbelowisequaltominusVaR):strategy12345result−431,818471,852−27,328−229,573222,263Thesevaluesarecomputedat95%confidencelevel,correspondingtotheexchangerateof1.9887dollarstoapound.Clearly,VaRprovidesonlypartialinformationaboutpossibleoutcomesofvariousstrategies.Figure9.1showsthegraphsofthefinalresultasafunctionoftheexchangeratedforeachoftheabovestrategies.Thegraphsarelabelledbythestrategynumberasabove.Thestrategyusingaforwardcontract(strat-Figure9.1Comparisonofvariousstrategiesegy2)appearstobethesafestone.Anadventurousinvestorwhostronglybelievesthatthepoundwillweakenconsiderablymayprefertoremainuncov-ered(strategy1).Avarietyofmiddle-of-the-roadstrategiesarealsoavailable.Theprobabilitydistributionoftheexchangeratedshouldalsobetakenintoaccountwhenexaminingthegraphs. 208MathematicsforFinance9.3SpeculatingwithDerivatives9.3.1ToolsOptionscanbeusedasbuildingblockstodesignsophisticatedinvestmentin-struments.Weshallconsideraninvestorwithspecificviewsonthefuturebe-haviourofstockpricesandwillingtotakerisks.Ourtaskwillbetodesignaportfolioofsecuritieswithaprescribedpayoffprofilethatwouldsatisfythiskindofinvestor.Supposethattheinvestorexpectsthestockpricetoriseandwantstogambleonthat.Onesimplewayistobuyacalloption.AnoptionwithstrikepriceXclosetothecurrentstockpriceisconsiderablycheaperthanthestockitself,creatingariskyleverageposition,aswillbeseeninthecasestudytofollow.ThepremiummaybereducedbysellingacalloptionwithstrikepriceX>X.Inthiswaywecanbuildaso-calledbullspreadwithpayoffshowninFigure9.2.Thisstrategywillbringagoodreturnifstockpriceincreasesaremoderate.Figure9.2BullspreadUsingputoptionswithstrikepricesX62.Asaresult,theanalystwillarriveatthefollowingvalues:1.Stock.UnderthemodifiedprobabilityQµ∼=2.6788%,σ∼=3.9257%.SS2.CallOptions.TakeacallwithstrikepriceX=58dollars.HenceCE∼=3.2923dollars(thispriceisfoundinthebinomialmodelwithoutanyrestrictionontherangeofstockpricesafter20days).Foraninvestmentinoptionswefindthatµ∼=8.8816%,σ∼=71.095%.CC3.BullSpread.Constructthespreadbypurchasingacallwithstrike$58andsellingacallwithstrike$60.Thepremiumreceivedforthelatteris$2.10,henceasinglespreadcosts$1.18.Theexpectedreturnandriskareµ∼=38.4094%,σ∼=52.3997%.bullbull4.BullSpreadCombinedwithRisk-FreeAsset.Investing94.58%ofthecapitalintherisk-freeassetandtheremainderinabullspread,wecanconstructaportfolioPwiththesameexpectedreturnasstock,butlowerrisk,µ∼=2.6788%,σ∼=2.8396%.PPFromthepointofviewofVaR,weconsidertheworstcasescenario(amongthoseadmittedbytheinvestor)whenS(20)∼=58.59dollars,whichmayhappenwithconditionalprobability0.2597.Inthisscenarioeachoftheaboveinvest-mentswillbringaloss,whichcanberegardedasVaRat74.03%confidencelevel.ThevaluesofVaRandthemarketpriceofriskarecollectedbelow:BullspreadwithInvestmentStockCalloptionsBullspreadrisk-freeassetMarketprice0.50.10.70.7ofriskVaR$447$12,426$7,602$412at74.03% 214MathematicsforFinanceThebullspreadcombinedwithrisk-freewillclearlybepreferabletotheotherinvestmentsasithasthehighestmarketpriceofriskandlowestVaR.Exercise9.10Checktheabovecomputationsandconsideramodificationsuchthatthebullspreadisconstructedbybuyingacallwithstrikeprice$60andsellingacallwithstrikeprice$62.Computetheexpectedreturn,riskandVaR.Exercise9.11Withintheframeworkofthebinomialmodelusedaboveconsiderananalystwhohasreasonstobelievethatthestockpricewillfall,butnomorethan20%after20days.Forabearspreadwithstrikes$56and$58constructedfromputoptionscomputetheexpectedreturn,risk,andVaRfortheworstpossibleoutcome. 10VariableInterestRatesThischapterbeginswithamodelinwhichtheinterestratesimpliedbybondsdonotdependonmaturity.Iftheratesaredeterministic,thentheymustbeconstantandthemodelturnsouttobetoosimpletodescribeanyreal-lifesituation.Inanextensionallowingrandomchangesofinterestratestheproblemofriskmanagementwillbedealtwithbyintroducingamathematicaltoolcalledthedurationofbondinvestments.Finally,weshallshowthatmaturity-dependentratescannotbedeterministiceither,preparingthemotivationandnotationforthenextchapter,inwhichamodelofstochasticrateswillbeexplored.AsinChapter2,B(t,T)willdenotethepriceattimet(therunningtime)ofazero-couponunitbondmaturingattimeT(thematuritytime).Thede-pendenceontwotimevariablesgivesrisetosomedifficultiesinmathematicalmodelsofbondprices.Thesepricesareexactlywhatisneededtodescribethetimevalueofmoney.InChapter2wesawhowbondpricesimplytheinterestrate,undertheassumptionthattherateisconstant.Here,wewanttorelaxthisrestriction,allowingvariableinterestrates.Inthischapterandthenextonetimewillbediscrete,thoughsomepartsofthetheorycaneasilybeextendedtocontinuoustime.Weshallfixatimestepτ,writingt=τnfortherunningtimeandT=τNforthematuritytime.Inthemajorityofexamplesweshalltakeeitherτ=1orτ=1.Thenotation12B(n,N)willbeemployedinsteadofB(t,T)forthepriceofazero-couponunitbond.Weshallusecontinuouscompounding,bearinginmindthatitsimplifiesnotationandmakesitpossibletohandletimestepsofanylengthconsistently.215 216MathematicsforFinance10.1Maturity-IndependentYieldsThepresentvalueofazero-couponunitbonddeterminesaninterestratecalledtheyieldanddenotedbyy(0)toemphasisethefactthatitiscomputedattime0:B(0,N)=e−Nτy(0).Foradifferentrunningtimeinstantnsuchthat00wereknownattime0,theny(0)=y(n)orelseanarbitragestrategycouldbefound.ProofSupposethaty(0)y(n)canbedealtwithinasimilarmanner.Exercise10.1Letτ=1.Findarbitrageiftheyieldsareindependentofmaturity,12andunitbondsmaturingattime6(halfayear)aretradedatB(0,6)=0.9320dollarsandB(3,6)=0.9665dollars,bothpricesbeingknownattime0. 10.VariableInterestRates217AsaconsequenceofProposition10.1,iftheyieldisindependentofmaturityanddeterministic(thatis,y(n)isknowninadvanceforanyn≥0),thenitmustbeconstant,y(n)=yforalln.ThisisthesituationinChapter2,whereallthebondpricesweredeterminedbyasingleinterestrate.Theyieldy(n)=y,independentofn,isthenequaltotheconstantrisk-freeinterestratedenotedpreviouslybyr.Historicalbondpricesshowadifferentpicture:Theyieldsimpliedbythebondpricesrecordedinthepastclearlyvarywithtime.Inanarbitrage-freemodel,toadmityieldsvaryingwithtimebutindependentofmaturityweshouldallowthemtoberandom,soitisimpossibletopredictinadvancewhethery(n)willbehigherorlowerthany(0).Weassume,therefore,thatateachtimeinstanttheyieldy(n)isapositiverandomnumberindependentofthematurityoftheunderlyingbond.Ourgoalistoanalysethereturnonabondinvestmentandtheimminentriskarisingfromrandomchangesofinterestrates.SupposethatweintendtoinvestacertainsumofmoneyPforafixedperiodofNtimesteps.Iftheyieldyremainsconstant,then,asobservedinChapter2,ourterminalwealthwillbePeNτy.Thiswillbeourbenchmarkfordesigningstrategieshedgedagainstunpredictableinterestratemovements.10.1.1InvestmentinSingleBondsIfweinvestinzero-couponbondsandkeepthemtomaturity,therateofreturnisguaranteed,sincethefinalpaymentisfixedinadvanceandisnotaffectedbyanyfuturechangesofinterestrates.However,ifwechoosetocloseoutourinvestmentpriortomaturitybysellingthebonds,wefacetheriskthattheinterestratesmaychangeinthemeantimewithanadverseeffectonthefinalvalueoftheinvestment.Example10.1Supposeweinvestinbondsforaperiodofsixmonths.Letτ=1.Webuya12numberofunitbondsthatwillmatureafteroneyear,payingB(0,12)=0.9300foreach.Thispriceimpliesaratey(0)∼=7.26%.Sincewearegoingtosellthebondsattimen=6,weareconcernedwiththepriceB(6,12)or,equivalently,withthecorrespondingratey(6).Letusdiscusssomepossiblescenarios:1.Therateisstable,y(6)=7.26%.ThebondpriceisB(6,12)∼=0.9644andthelogarithmicreturnontheinvestmentis3.63%,ahalfoftheinterestrate,inlinewiththeadditivityoflogarithmicreturns.2.Theratedecreasestoy(6)=6.26%,say.(Theconventionisthat0.01%is 218MathematicsforFinanceonebasispoint,soheretheratedropsby100basispoints.)ThenB(6,12)∼=0.9692,whichismorethaninscenario1.Asaresult,wearegoingtoearnmore,achievingalogarithmicreturnof4.13%.3.Therateincreasestoy(6)=8.26%.Inthiscasethelogarithmicreturnonourinvestmentwillbe3.13%,whichislowerthaninscenario1,thebondpricebeingB(6,12)∼=0.9596.Wecanseeapatternhere:Oneisbetteroffiftheratedropsandworseoffiftherateincreases.Ageneralformulaforthereturnonthiskindofinvestmentiseasytofind.Supposethattheinitialyieldy(0)changesrandomlytobecomey(n)=y(0)attimen.HenceB(0,N)=e−y(0)τN,B(n,N)=e−y(n)τ(N−n),andthereturnonaninvestmentclosedattimenwillbek(0,n)=lnB(n,N)=lney(0)τN−y(n)τ(N−n)=y(0)τN−y(n)τ(N−n).B(0,N)Wecanseethatthereturndecreasesastheratey(n)increases.Theimpactofaratechangeonthereturndependsonthetiming.Forexample,ifτ=1,12N=12andn=6,thenarateincreaseof120basispointswillreducethereturnby0.6%ascomparedtothecasewhentherateremainsunchanged.Exercise10.2Letτ=1.Invest$100insix-monthzero-couponbondstradingat12B(0,6)=0.9400dollars.Aftersixmonthsreinvesttheproceedsinbondsofthesamekind,nowtradingatB(6,12)=0.9368dollars.Findtheimpliedinterestratesandcomputethenumberofbondsheldateachtime.Computethelogarithmicreturnontheinvestmentoveroneyear.Exercise10.3SupposethatB(0,12)=0.8700dollars.Whatistheinterestrateafter6monthsifaninvestmentfor6monthsinzero-couponbondsgivesalogarithmicreturnof14%?Exercise10.4Inthisexercisewetakeafinertimescalewithτ=1.(Ayearis360assumedtohave360dayshere.)SupposethatB(0,360)=0.9200dollars,therateremainsunchangedforthefirstsixmonths,goesupby200basis 10.VariableInterestRates219pointsonday180,andremainsatthisleveluntiltheendoftheyear.Ifabondisboughtatthebeginningoftheyear,onwhichdayshoulditbesoldtoproducealogarithmicreturnof4.88%ormore?Aninvestmentincouponbondsismorecomplicated.Evenifthebondiskepttomaturity,thecouponsarepaidinthemeantimeandcanbereinvested.Thereturnonsuchaninvestmentdependsontheinterestratesprevailingatthetimeswhenthecouponsaredue.Firstconsidertherelativelysimplecaseofaninvestmentterminatedassoonasthefirstcouponispaid.Example10.2Letusinvestthesumof$1,000in4-yearbondswithfacevalue$100and$10annualcoupons.Acouponbondofthiskindcanberegardedasacollectionoffourzero-couponbondsmaturingafter1,2,3and4yearswithfacevalue$10,$10,$10and$110,respectively.Supposethatsuchcouponbondstradeat$91.78,whichcanbeexpressedasthesumofthepricesofthefourzero-couponbonds,91.78=10e−y(0)+10e−2y(0)+10e−3y(0)+110e−4y(0).(Thelengthofatimestepisτ=1.)Thisequationcanbesolvedtofindtheyield,y(0)∼=12%.Wecanaffordtobuy10.896couponbonds.Afteroneyearwecashthecoupons,collecting$108.96,andsellthebonds,whicharenow3-yearcouponbonds.Considerthreescenarios:1.Afteroneyeartheinterestrateremainsunchanged,y(1)=12%,thecouponbondsbeingvaluedat10e−0.12+10e−2×0.12+110e−3×0.12∼=93.48dollars,andweshallreceive108.96+1,018.52∼=1,127.48dollarsintotal.2.Theratedropsto10%.Asaresult,thecouponbondswillbeworth10e−0.1+10e−2×0.1+110e−3×0.1∼=98.73dollarseach.Weshallendupwith$1,184.63.3.Therategoesupto14%,thecouponbondstradingat$88.53.Thefinalvalueofourinvestmentwillbe$1,073.51.Exercise10.5Findtheratey(1)suchthatthelogarithmicreturnontheinvestmentinExample10.2willbea)12%,b)10%,c)14%. 220MathematicsforFinanceIfthelifetimeofourinvestmentexceedsoneyear,wewillbefacingtheproblemofreinvestingcoupons.Inthefollowingexampleweassumethatthecouponsareusedtopurchasethesamebond.Example10.3WebeginasinExample10.2,butourintentionistoterminatetheinvestmentafter3years.Afteroneyearwereinvestthecouponsobtainedinthesame,nowa3-year,couponbond.Considerthefollowingscenariosafteroneyear:1.Therateremainsthesamefortheperiodofourinvestment,y(0)=y(1)=y(2)=y(3)=12%.Thebondpriceis$93.48,soforthe$108.96receivedfromcouponswecanbuy1.17additionalbonds,increasingthenumberofbondsheldto12.06.Wecanmonitorthevalueofourinvestmentbysimplymultiplyingthenumberofbondsheldbythecurrentbondprice.Werepeatthisinthefollowingyear.Afterthreeyearswecashthecouponsandsellthebonds,thefinalvalueoftheinvestmentbeing$1,433.33.Thisnumberwillbeusedasabenchmarkforotherscenarios.Observethat1,433.33∼=1,000e3×12%,thesameasthevalueafter3yearsof$1,000investedonzero-couponbonds.Thebuildingblocksofourinvestmentaresummarisedinthetablebelow.Year0123Rate12%12%12%12%PVofcoupon1$8.87$10.00PVofcoupon2$7.87$8.87$10.00PVofcoupon3$6.98$7.87$8.87$10.00PVofcoupon4$6.19$6.98$7.87$8.87PVoffacevalue$61.88$69.77$78.66$88.69Bondprice$91.78$93.48$95.40$97.56Cashedcoupons$108.96$120.60$133.26Additionalbonds1.171.26Numberofbonds10.9012.0613.33Valueofinvestment$1,000.00$1,127.50$1,271.25$1,433.332.Supposethattherategoesdownby2%afteroneyearandthenremainsatthenewlevel.Thedropoftherateresultsinanincreaseofallbondprices.Thenumberofadditionalbondsthatcanbeboughtforthecouponsislowerthaninscenario1.Nevertheless,thefinalvalueoftheinvestmentis 10.VariableInterestRates221higherbecausesoisthepriceatwhichwesellthebondsafterthreeyears.Year0123Rate12%10%10%10%PVofcoupon1$8.87$10.00PVofcoupon2$7.87$9.05$10.00PVofcoupon3$6.98$8.19$9.05$10.00PVofcoupon4$6.19$7.41$8.19$9.05PVoffacevalue$61.88$74.08$81.87$90.48Bondprice$91.78$98.73$99.11$99.53Cashedcoupons$108.96$119.99$132.10Additionalbonds1.101.21Numberofbonds10.9012.0013.21Valueofinvestment$1,000.00$1,184.65$1,309.25$1,446.943.Iftherateincreasesto14%andstaysthere,thebondswillbecheaperthaninScenario1.Thefinalvalueoftheinvestmentwillbedisappointing.Year0123Rate12%14%14%14%PVofcoupon1$8.87$10.00PVofcoupon2$7.87$8.69$10.00PVofcoupon3$6.98$7.56$8.69$10.00PVofcoupon4$6.19$6.57$7.56$8.69PVoffacevalue$61.88$65.70$75.58$86.94Bondprice$91.78$88.53$91.83$95.63Cashedcoupons$108.96$121.26$134.46Additionalbonds1.231.32Numberofbonds10.9012.1313.45Valueofinvestment$1,000.00$1,073.53$1,234.85$1,420.41Asamotivationforcertaintheoreticalnotions,considertheaboveinvest-ment,withthesamepossiblescenarios,butinvolvingaspeciallydesignedse-curity,acouponbondwithannualcouponspaying$32,allotherparametersremainingunchanged.Theresultsareasfollows:ScenarioValueafter3years12%,12%,12%,12%$1,433.3312%,10%,10%,10%$1,433.6812%,14%,14%,14%$1,433.78Itisremarkablethatanychangeininterestratesimprovestheresultofourinvestment.Wedonotloseiftherateschangeunfavourably.Ontheother 222MathematicsforFinancehand,wedonotgaininothercircumstances.Thisisexplainedbythefact,thatacertainparameterofthebond,calleddurationanddefinedbelow,isexactlyequaltothelifetimeofourinvestment.Insomesense,thebondbehavesapproximatelylikeazero-couponbondwithprescribedmaturity.Exercise10.6Checkthenumbersgivenintheabovetables.Exercise10.7Computethevalueafterthreeyearsof$1,000investedina4-yearbondwith$32annualcouponsand$100facevalueiftheratesinconsecutiveyearsareasfollows:Scenario1:12%,11%,12%,12%;Scenario2:12%,13%,12%,12%.Designaspreadsheetandexperimentwithvariousinterestrates.10.1.2DurationWehaveseenthatvariableinterestleadstouncertaintyastothefuturevalueofaninvestmentinbonds.Thismaybeundesirable,orevenunacceptable,forexampleforapensionfundmanager.Weshallintroduceatoolwhichmakesitpossibletoimmunisesuchaninvestment,atleastinthespecialsituationofmaturity-independentratesconsideredinthissection.Fornotationalsimplicitywedenotethecurrentyieldy(0)byy.ConsideracouponbondwithcouponsC1,C2,...,CNpayableattimes0<τn1<τn2<...<τnNandfacevalueF,maturingattimeτnN.ItscurrentpriceisgivenbyP(y)=Ce−τn1y+Ce−τn2y+···+(C+F)e−τnNy.(10.1)12NThedurationofthecouponbondisdefinedtobeτnCe−τn1y+τnCe−τn2y+···+τn(C+F)e−τnNy1122NND(y)=.(10.2)P(y)ThenumbersCe−τn1y/P(y),Ce−τn2y/P(y),...,(C+F)e−τnNy/P(y)are12Nnon-negativeandadduptoone,sotheymayberegardedasweightsorproba-bilities.Itcanbesaidthatthedurationisaweightedaverageoffuturepaymenttimes.Thedurationofanyfuturecashflowcanbedefinedinasimilarmanner. 10.VariableInterestRates223Durationmeasuresthesensitivityofthebondpricetochangesintheinterestrate.Toseethiswecomputethederivativeofthebondpricewithrespecttoy,dP(y)=−τnCe−τn1y−τnCe−τn2y−···−τn(C+F)e−τnNy,1122NNdywhichgivesdP(y)=−D(y)P(y).dyThelastformulaissometimestakenasthedefinitionofduration.Example10.4A6-yearbondwith$10annualcoupons,$100facevalueandyieldof6%hasadurationof4.898years.A6-yearbondwiththesamecouponsandyield,butwith$500facevalue,willhaveadurationof5.671years.Thedurationofanyzero-couponbondisequaltoitslifetime.Exercise10.8A2-yearbondwith$100facevaluepaysa$6couponeachquarterandhas11%yield.Computetheduration.Exercise10.9Whatshouldbethefacevalueofa5-yearbondwith10%yield,paying$10annualcouponstohaveduration4?Findtherangeofdurationsthatcanbeobtainedbyalteringthefacevalue,aslongasacouponcannotexceedthefacevalue.Ifthefacevalueisfixed,say$100,findthelevelofcouponsforthedurationtobe4.Whatdurationscanbemanufacturedinthisway?Exercise10.10ShowthatPisaconvexfunctionofy.Ifweinvestinabondwiththeintentiontoclosetheinvestmentattimet,thenthefuturevalueofthemoneyinvestedinasinglebondwillbeP(y)ety,providedthattheinterestrateremainsunchanged(beingequaltotheinitialyieldy(0)).Toseehowsensitivethisamountistointerestratechangescomputethederivativewithrespecttoy,dtydtytyty(P(y)e)=P(y)e+tP(y)e=(t−D(y))P(y)e.dydy 224MathematicsforFinanceIfthedurationofthebondisexactlyt,thendty(P(y)e)=0.dyIfthederivativeiszeroatsomepoint,thenthegraphofthefunctionis‘flat’nearthispoint.Thismeansthatsmallchangesoftheratewillhavelittleeffectonthefuturevalueoftheinvestment.10.1.3PortfoliosofBondsIfabondofdesirabledurationisnotavailable,itmaybepossibletocreateasyntheticonebyinvestinginasuitableportfolioofbondsofdifferentdurations.Example10.5Iftheinitialinterestrateis14%,thena4-yearbondwithannualcouponsC=10andfacevalueF=100hasduration3.44years.Azero-couponbondwithF=100andN=1hasduration1.Aportfolioconsistingoftwobonds,oneofeachkind,canberegardedasasinglebondwithcouponsC1=110,C2=C3=C4=10,F=100.Itsdurationcanbecomputedusingthegeneralformula(10.2),whichgives2.21years.Weshallderiveaformulaforthedurationofaportfoliointermsofthedurationsofitscomponents.DenotebyPA(y)andPB(y)thevaluesoftwobondsAandBwithdurationsDA(y)andDB(y).TakeaportfolioconsistingofabondsAandbbondsB,itsvaluebeingaPA(y)+bPB(y).Thetaskoffindingthedurationoftheportfoliowillbedividedintotwosteps:1.FindthedurationofaportfolioconsistingofabondsoftypeA.WeshallwriteaAtodenotesuchaportfolio.ItspriceisobviouslyaPA(y).Sinced(aPA(y))=−DA(y)(aPA(y)),dyitfollowsthatDaA(y)=DA(y).ThisisclearifweexaminethecashflowofaA.Eachcouponandthefacevaluearemultipliedbya,whichcancelsoutinthecomputationofdurationdirectlyfrom(10.2).2.FindthedurationofaportfolioconsistingofonebondAandonebondB,whichwillbedenotedbyA+B.ThepriceofthisportfolioisPA(y)+PB(y). 10.VariableInterestRates225Differentiatingthelastexpression,weobtainddd(PA(y)+PB(y))=PA(y)+PB(y)dydydy=−DA(y)PA(y)−DB(y)PB(y).Thelasttermcanbewrittenas−DA+B(y)(PA(y)+PB(y))ifweputPA(y)PB(y)DA+B(y)=DA(y)+DB(y).PA(y)+PB(y)PA(y)+PB(y)ThismeansthatDA+B(y)isalinearcombinationofDA(y)andDB(y),thecoefficientsbeingthepercentageweightsofeachbondintheportfolio.FromtheaboveconsiderationsweobtainthegeneralformulaDaA+bB(y)=DA(y)wA+DB(y)wB,whereaPA(y)bPA(y)wA=,wB=,aPA(y)+bPB(y)aPA(y)+bPB(y)arethepercentageweightsofindividualbonds.Ifweallownegativevaluesofaorb(whichcorrespondstowritingabondinsteadofpurchasingit,inotherwords,toborrowingmoneyinsteadofinvest-ing),then,giventwodurationsDA=DB,thedurationDoftheportfoliocantakeanyvaluebecausewB=1−wAandD=DAwA+DB(1−wA)=DB+wA(DA−DB).ThevalueofDcanevenbenegative,whichcorrespondstoanegativecashflow,thatis,sumsofmoneytobepaidratherthanreceived.Example10.6LetDA=1andDB=3.Wewishtoinvest$1,000for6months.Forthedurationtomatchthelifetimeoftheinvestmentweneed0.5=wA+3wB.SincewA+wB=1,itfollowsthatwB=−0.25andwA=1.25.WithPA=0.92dollarsandP=1.01dollars,weinvest$1,250in1250∼=1,358.70bondsAB0.92andweissue250∼=247.52bondsB.1.01Exercise10.11FindthenumberofbondsoftypeAandBtobeboughtifDA=2,DB=3.4,PA=0.98,PB=1.02andyouneedaportfolioworth$5,000withduration6. 226MathematicsforFinanceExercise10.12Invest$1,000inaportfolioofbondswithduration2using1-yearzero-couponbondswith$100facevalueand4-yearbondswith$15annualcouponsand$100facevaluethattradeat$102.Aportfoliowithdurationmatchingtheinvestmentlifetimeisinsensitivetosmallchangesofinterestrates.Howeverinpracticeweshallhavetomodifytheportfolioif,forexample,theinvestmentisfor3yearsandoneofthebondsisazero-couponbondexpiringafteroneyear.Inaddition,thedurationmay,asweshallseebelow,gooffthetarget.Asaresult,itwillbecomenecessarytoupdatetheportfolioduringthelifetimeoftheinvestment.Thisisthesubjectofthenextsubsection.10.1.4DynamicHedgingEvenifaportfolioisselectedwithdurationmatchingthedesiredinvestmentlifetime,thiswillonlybevalidattheinitialinstant,sincedurationchangeswithtimeaswellaswiththeinterestrate.Example10.7Takea5-yearbondwith$10annualcouponsand$100facevalue.Ify=10%,thenthedurationwillbeabout4.16years.Beforethefirstcouponispaidthedurationdecreasesinlinewithtime:After6monthsitwillbe3.66,andafter9months4.16−0.75=3.31.Ifthedurationmatchesourinvestment’slifetimeandtheinterestratesdonotchange,noactionwillbenecessaryuntilacouponbecomespayable.Assoonasthefirstcouponispaidafteroneyear,thebondwillbecomea4-yearonewithduration3.48,nolongerconsistentwiththeinvestmentlifetime.Exercise10.13Assumingthattheinterestratedoesnotchange,showthatbeforethefirstcouponispaidthedurationaftertimetwillD−t,whereDisthedurationcomputedattime0.Thenextexampleshowstheimpactoftheinterestrateonduration. 10.VariableInterestRates227Example10.8ThebondinExample10.7willhaveduration4.23ify=6%,and4.08ify=14%.Exercise10.14Showthatthedurationofa2-yearbondwithannualcouponsdecreasesastheyieldincreases.Durationwillnowbeappliedtodesignaninvestmentstrategyimmunetointerestratechanges.Thiswillbedonebymonitoringthepositionattheendofeachyear,ormorefrequentlyifneeded.Forclarityofexpositionwerestrictourselvestoanexample.Setthelifetimeoftheinvestmenttobe3yearsandthetargetvaluetobe$100,000.Supposethattheinterestrateis12%initially.Weinvest$69,767.63,whichwouldbethepresentvalueof$100,000iftheinterestrateremainedconstant.Werestrictourattentiontotwoinstruments,a5-yearbondAwith$10annualcouponsand$100facevalue,anda1-yearzero-couponbondBwiththesamefacevalue.WeassumethatanewbondoftypeBisalwaysavailable.InsubsequentyearsweshallcombineitwithbondA.Attime0thebondpricesare$90.27and$88.69,respectively.WefindD∼=4.12andtheweightsw∼=0.6405,w∼=0.3595whichgiveaportfolioAABwithduration3.Wesplittheinitialsumaccordingtotheweights,spending$44,687.93tobuya∼=495.05bondsAand$25,079.70tobuyb∼=282.77bondsB.Considersomepossiblescenariosoffutureinterestratechanges.1.Afteroneyeartherateincreasesto14%.Thevalueofourportfolioisthesumof:•thefirstcouponsofbondsA:$4,950.51,•thefacevalueofcashedbondsB:$28,277.29,•themarketvalueofbondsAheld,whicharenow4-yearbondssellingat$85.65:$42,403.53.Thisgives$75,631.32altogether.ThedurationofbondsAisnow3.44.Thedesireddurationis2,sowefindwA∼=0.4094andwB∼=0.5906andarriveatthenumberofbondstobeheldintheportfolio:361.53bondsAand513.76bondsB.(Thismeansthatwehavetosell133.52bondsAandbuy513.76newbondsB.)a)Aftertwoyearstheratedropsto9%.Tocomputeourwealthweadd:•thecouponsofA:$3,615.30, 228MathematicsforFinance•thefacevaluesofB:$51,376.39,•themarketvalueofA,sellingat$101.46:$36,682.22.Theresultis$91,673.92.WeinvestallthemoneyinbondsB,sincetherequireddurationisnow1.(Thepayoffofthesebondsisguaranteednextyear.)Wecanaffordtobuy1,003.07bondsBsellingat$91.39.Theterminalvalueoftheinvestmentwillbeabout$100,307.b)Aftertwoyearstherategoesupto16%.Wecashthesameamountasaboveforcouponsandzero-couponbonds,butbondsAarenowcheaper,sellingat$83.85,sowehavelessmoneyintotal:$85,305.68.However,thezero-couponbondsarenowcheapaswell,sellingat$85.21,andwecanaffordtobuy1,001.07ofthem,endingupwith$100,107.2.Afteroneyeartheratedropsto9%.Inasimilarwayasbefore,wearriveatthecurrentvalueoftheinvestmentbyaddingthecouponsofA,thefacevalueofBandthemarketvalueofbondsAheld,obtaining$83,658.73.ThenwefindtheweightswA∼=0.4013,wB∼=0.5987,determiningournewportfolioof329.56bondsAand548.04bondsB.(Wehavetosell165.50bondsAandbuy548.04newbondsB.)a)Aftertwoyearstherategoesupto14%.Wecash$3,295.55fromthecouponsofA,whichtogetherwiththe$54,803.77obtainedfromBandthemarketvalueof$29,174.39ofbondsAgives$87,273.72intotal.Webuy1003.89newzero-couponbondsB,endingupwith$100,389after3years.b)Aftertwoyearstheratedropsto6%.Ourwealthwillthenbe$94,405.29,wecanaffordtobuy1,002.43bondsB,andthefinalvalueofourin-vestmentwillbe$100,243.Aswecansee,weendupwithmorethan$100,000ineachscenario.1Exercise10.15Designaninvestmentof$20,000inaportfolioofduration2yearscon-sistingoftwokindsofcouponbondsmaturingafter2years,withannualcoupons,bondAwith$20couponsand$100facevalue,andbondBwith$5couponsand$500facevalue,giventhattheinitialrateis8%.Howmuchwillthisinvestmentbeworthafter2years?1Itcanbeshownthatthefuturevalueattimetofabondinvestmentwithdurationequaltothasaminimumiftherateyremainsunchanged.Thismeansthatratejumpsinamodelwithyieldsindependentofmaturityleadtoarbitrage.Inanarbitrage-freemodelwithratejumps,theyieldsmustthereforedependonmaturity. 10.VariableInterestRates22910.2GeneralTermStructureHereweshalldiscussamodelofbondpriceswithouttheconditionthattheyieldshouldbeindependentofmaturity.ThepricesB(n,N)ofzero-couponunitbondswithvariousmaturitiesde-termineafamilyofyieldsy(n,N)byB(n,N)=e−(N−n)τy(n,N).Notethattheyieldshavetobepositive,sinceB(n,N)hastobelessthan1fornB(0,n)B(n,N)canbedealtwithinasimilarmanner,byadoptingtheoppositestrategy.Employingtherepresentationofbondpricesintermsofyields,wehaveB(0,N)τny(0,n)−τNy(0,N)B(n,N)==e.B(0,n)Thiswouldmeanthatallbondsprices(andsothewholetermstructure)aredeterminedbytheinitialtermstructure.However,itisclearthatonecannotexpectthistoholdinrealbondmarkets.Inparticular,thisrelationisnotsupportedbyhistoricaldata.Thisshowsthatassumingdeterministicbondpriceswouldgotoofarinreducingthecomplexityofthemodel.Wehavenochoicebuttoallowthefuturetermstructuretoberandom,onlytheinitialtermstructurebeingknownwithcertainty.Inwhatfollows,futurebondpriceswillberandom,aswillbethequantitiesdeterminedbythem.10.2.1ForwardRatesWebeginwithanexampleshowinghowtosecureinadvancetheinterestrateforadeposittobemadeoraloantobetakenatsomefuturetime.Example10.10Supposethatthebusinessplanofyourcompanywillrequiretakingaloanof$100,000oneyearfromnowinordertopurchasenewequipment.Youexpecttohavethemeanstorepaytheloanafteranotheryear.Youwouldliketoarrangetheloantodayatafixedinterestrate,ratherthantogambleonfuturerates.Supposethatthespotratesarey(0,1)=8%andy(0,2)=9%(withτ=1).Youbuy1,000one-yearbondswith$100facevalue,paying100,000e−8%∼=92,311.63dollars.Thissumisborrowedfor2yearsat9%.Afteroneyearyouwillreceivethe$100,000fromthebonds,andaftertwoyearsyoucansettletheloanwithinterest,thetotalamounttopaybeing92,311.63e2×9%∼=110,517.09dollars.Thus,theinterestrateontheconstructedfutureloanwill 232MathematicsforFinancebeln(110,517.09/100,000)∼=10%.Financialintermediariesmaysimplifyyourtaskbyofferingaso-calledForwardRateAgreementandperformtheaboveconstructionoftheloanonyourbehalf.Exercise10.17Explainhowadepositof$50,000forsixmonthscanbearrangedtostartinsixmonthsandfindtherateify(0,6)=6%andy(0,12)=7%,whereτ=1.12Ingeneral,theinitialforwardratef(0,M,N)isaninterestratesuchthatB(0,N)=B(0,M)e−(N−M)τf(0,M,N),so1B(0,N)lnB(0,N)−lnB(0,M)f(0,M,N)=−ln=−.τ(N−M)B(0,M)τ(N−M)Notethatthisrateisdeterministic,sinceitisworkedoutusingthepresentbondprices.Itcanbeconvenientlyexpressedintermsoftheinitialtermstructure.Insertintotheaboveexpressionthebondpricesasdeterminedbytheyields,B(0,N)=e−τNy(0,N)andB(0,M)=e−τMy(0,M),togetNy(0,N)−My(0,M)f(0,M,N)=.(10.5)N−MExercise10.18SupposethatthefollowingspotratesareprovidedbycentralLondonbanks(LIBOR,theLondonInterbankOfferRate,istherateatwhichmoneycanbedeposited;LIBID,theLondonInterbankBidRate,istherateatwhichmoneycanbeborrowed):RateLIBORLIBID1month8.41%8.59%2months8.44%8.64%3months9.01%9.23%6months9.35%9.54%Asabankmanageractingforacustomerwhowishestoarrangealoanof$100,000inamonth’stimeforaperiodof5months,whatratecouldyouofferandhowwouldyouconstructtheloan?Supposethatanotherinsti-tutionoffersthepossibilityofmakingadepositfor4months,starting2 10.VariableInterestRates233monthsfromnow,atarateof10.23%.Doesthispresentanarbitrageop-portunity?Allratesstatedinthisexercisearecontinuouscompoundingrates.Astimepasses,thebondpriceswillchangeand,consequently,sowilltheforwardrates.Theforwardrateovertheinterval[M,N]determinedattimenr>dofChapter3is 248MathematicsforFinancereplacedbyk(n,N;sn−1u)>τr(n−1;sn−1)>k(n,N;sn−1d).(11.2)Anyfuturecashflowcanbereplicatedinasimilarfashion.Consider,forexample,acouponbondwithfixedcoupons.Example11.6Takeacouponbondmaturingattime2withfacevalueF=100,payingcouponsC=10attimes1and2.Wepricethefuturecashflowbyusingthezero-couponbondmaturingattime3astheunderlyingsecurity.ThecouponbondpricePataparticulartimewillnotincludethecoupondue(theso-calledex-couponprice).AssumethatthestructureofthebondpricesisasinFigure11.10.Considertime1.InstateutheshortrateisdeterminedbythepriceB(1,2;u)=0.9947,sowehaver(1;u)∼=6.38%.HenceP(1;u)∼=109.4170.InstatedweuseB(1,2;d)=0.9913tofindr(1;d)∼=10.49%andP(1;d)∼=109.0485.Considertime0.Thecashflowattime1whichwearetoreplicateincludesthecoupondue,soitisgivenbyP(1;u)+10∼=119.417andP(1;d)+10∼=119.0485.Theshortrater(0)∼=11.94%determinesthemoneymarketaccountasinExample11.5,A(1)=1.01,andwefindx∼=92.1337,y∼=28.3998.HenceP(0)∼=118.009isthepresentpriceofthecouponbond.Analternativeistousethespotyields:y(0,1)∼=11.94%andy(0,2)∼=10.41%todiscountthefuturepaymentswiththesameresult:118.009∼=10×exp(−1×11.94%)+110×exp(−2×10.41%).1212Ingeneral,P(0)=C1exp{−τy(0,1)}+C2exp{−2τy(0,2)}+···+(CN+F)exp{−Nτy(0,N)}.(11.3)(Forsimplicityweincludealltimesteps,soCk=0atthetimestepskwhennocouponispaid.)Ateachtimekwhenacouponispaid,thecashflowisthesumofthe(deterministic)couponandthe(stochastic)priceoftheremainingbond:Ck+P(k;sk)=Ck+Ck+1exp{−τy(k,k+1;sk)}+···+(Cn+F)exp{−τ(n−k)y(k,n;sk)}.Quiteoftenthecouponsdependonotherquantities.Inthiswayacouponbondmaybecomeaderivativesecurity.Animportantbenchmarkcaseisde-scribedbelow,wherethecouponsarecomputedasfractionsofthefacevalue. 11.StochasticInterestRates249Thesefractions,definingthecouponrate,areobtainedbyconvertingtheshortratetoanequivalentdiscretecompoundingrate.Inpractice,whenτisoneday,thecouponratewillbetheovernightLIBORrate.Proposition11.1AcouponbondmaturingattimeNwithrandomcouponsCk(sk−1)=(exp{τr(k−1;sk−1)}−1)F(11.4)for0r2!m!ifmisanintegergreaterthan1.2.13Denotetheinterestratebyr,theamountborrowedbyPandtheamountofeachinstalmentbyC,PPrC==,PA(r,5)1−(1+r)−5seeExample2.4.Letn=1,2,3,4or5.Thepresentvalueoftheoutstandingbalanceaftern−1instalmentsarepaidisequaltotheamountborrowedreducedbythepresentvalueofthefirstn−1instalments:6−nCC(1+r)−1P−−···−=P.1+r(1+r)n−1(1+r)5−1Theactualoutstandingbalanceremainingaftern−1instalmentsarepaidcan−(n−1)befoundbydividingtheabovebythediscountfactor(1+r),whichgives5n−1(1+r)−(1+r)P.(S.1)5(1+r)−1Theinterestincludedinthenthinstalmentis,therefore,5n−1(1+r)−(1+r)Pr.(S.2)5(1+r)−1 Solutions269Thecapitalrepaidaspartofthenthinstalmentisthedifferencebetweentheoutstandingbalanceoftheloanafterthe(n−1)standafterthenthinstal-ment.By(S.1)thisdifferenceisequaltonn−1n−1(1+r)−(1+r)r(1+r)P=P.(S.3)55(1+r)−1(1+r)−1PuttingPtobe$1,000andrtobe15%in(S.1),(S.2)and(S.3),wecancomputethefigurescollectedinthetable:t(years)interestpaid($)capitalrepaid($)outstandingbalance($)0——1,000.001150.00148.32851.682127.75170.56681.123102.17196.15484.97472.75225.57259.40538.91259.400.002.14Theamountyoucanaffordtoborrowis−101−(1+0.18)PA(18%,10)×10,000=×10,000∼=44,9410.18dollars.2.15Thepresentvalueofthebalanceafter40yearsis−401−(1+0.05)PA(5%,40)×1,200=×1,200∼=20,5910.05−40dollars.Dividingbythediscountfactor(1+0.05),wefindthattheactualbalanceafter40yearswillbe20,591∼=144,960(1+0.05)−40dollars.2.16TheannualpaymentswillamounttoC=100,000∼=13,586.80PA(6%,10)dollarseach.Theoutstandingbalancetobepaidtoclearthemortgageafter8years(oncethe8thannualpaymentismade)willbePA(6%,2)×C∼=24,909.93dollars.22.17SupposethatpaymentsC,C(1+g),C(1+g),...aremadeafter1year,2years,3years,andsoon.Iftheinterestrateisconstantandequaltor,thenthe−1−2presentvaluesofthesepaymentsareC(1+r),C(1+g)(1+r),C(1+2−3g)(1+r),....Thepresentvalueoftheinfinitestreamofpaymentsis,therefore,2CC(1+g)C(1+g)C+++...=.1+r23r−g(1+r)(1+r) 270MathematicsforFinanceTheconditiong110.51.mThismeansthatmshouldbegreaterthanabout55.19.2.21Thepresentvalueis1,000,000e−20×0.06∼=301,194dollars.2.22Theratersatisfies0.5r950e=1,000.Itfollowsthatr=1ln1,000∼=0.1026,0.5950thatis,about10.26%.2.23Theinterestrateis0.03r==0.18,2/12thatis,18%. Solutions2712.24Theratersatisfies12r0.12e=1+.12Thesolutionisr∼=0.1194,about11.94%.2.25Thefrequencymsatisfiesm0.21+=1+0.21.mWhencem=2.0.12r2.26Ifmonthlycompoundingataraterapplies,then1+=1+reandthe12presentvalueoftheannuityisCCCV(0)=1+r+r2+···+r12n121+1+1212CCC=++···+n(1+re)1/12(1+re)2/12(1+re)−n1−(1+re)=C.1/12(1+re)−16r2.27Ifbimonthlycompoundingataraterapplies,then1+=1+reandthe6presentvalueoftheperpetuityisCCCV(0)=1+r+r2+r3+···61+1+66CCC=+++···1/62/63/6(1+re)(1+re)(1+re)1=C.1/6(1+re)−12.28Wesolvetheequation1100=95(1+r)2forrtofindtheimpliedeffectiveratetobeabout10.80%.Ifthisrateremainsconstant,thenthebondpricewillreach$99atatimetsuchthat(1−t)100=99(1+r)2.Thesolutionist∼=0.402years,thatis,about0.402×365∼=146.73days.Thebondpricewillreach$99onday147.2.29Theinterestrateforannualcompoundingimpliedbythebondcanbefoundbysolvingtheequation−(1−0.5)(1+r)=0.9455forr.Thesolutionisabout11.86%.Bysolvingtheequationr−2(1−0.5)1+=0.9455,2weobtainthesemi-annualrateofabout11.53%,andsolving−r(1−0.5)e=0.9455,wefindthecontinuousratetobeabout11.21%. 272MathematicsforFinance2.30a)Ifthecontinuouscompoundingrateis8%,thenthepriceofthebondwillbe5e−0.08+5e−2×0.08+5e−3×0.08+105e−4×0.08∼=89.06dollars.b)Iftherateis5%,thenthepriceofthebondwillbe5e−0.05+5e−2×0.05+5e−3×0.05+105e−4×0.05∼=99.55dollars.2.31Thepriceofthebondasafunctionofthecontinuouscompoundingratercan−r−2r−3r−4rbeexpressedas5e+5e+5e+105e.ThegraphofthisfunctionisshowninFigureS.1.Whenr0,thepriceapproaches5+5+5+105=120dollars.Inthelimitasr∞thepricetendstozero.FigureS.1BondpriceversusinterestrateinExercise2.312.32ThetimetpriceofthecouponbondinExamples2.9and2.10isr(t−1)r(t−2)r(t−3)r(t−4)r(t−5)10e+10e+10e+10e+110eif0≤t<1,r(t−2)r(t−3)r(t−4)r(t−5)10e+10e+10e+110eif1≤t<2,r(t−3)r(t−4)r(t−5)10e+10e+110eif2≤t<3,r(t−4)r(t−5)10e+110eif3≤t<4,r(t−5)110eif4≤t<5.ThegraphisshowninFigureS.2.2.33InFigureS.2wecanseethatthebondpricewillreach$95forthefirsttimedur-r(t−1)r(t−2)r(t−3)ingyearone,whenthebondpriceisgivenby10e+10e+10e+r(t−4)r(t−5)10e+110e.Puttingr=0.12,wecanfindthedesiredtimetwhenthepricewillreach$95bysolvingtheequationr(t−1)r(t−2)r(t−3)r(t−4)r(t−5)10e+10e+10e+10e+110e=95.Thisgivest∼=0.4257yearsor155.4days.2.34Sincethebondistradingatpar,itsinitialpriceisthesameasthefacevalueF=100.Theimpliedcontinuouscompoundingratercanbefoundbysolvingtheequation−r−2r−3r8e+8e+108e=100.Thisgivesr∼=0.0770or7.70%. Solutions273FigureS.2CouponbondpriceversustimeinExercise2.322.35Bysolvingtheequation−1(1+r)=0.89wefindthatr∼=0.1236,thatis,theeffectiverateimpliedbythebondisabout12.36%.Thepriceofthebondafter75dayswillbe7575B(75/365,1)=B(0,1)(1+r)365=0.89(1+0.1236)365∼=0.9115andthereturnwillbeK(0,75/365)=B(75/365,1)−B(0,1)∼=0.9115−0.89∼=0.0242,B(0,1)0.89about2.42%.−0.5r2.36Theinitialpriceofasix-monthunitbondise,whererdenotestheimpliedcontinuousrate.Ifthebondistoproducea7%returnoversixmonths,then−0.5r1−e=0.07,e−0.5rwhichgivesr∼=0.1353,or13.53%.2.37Thecontinuousrateimpliedbythebondsatisfies−re=0.92.rtThesolutionisr∼=0.0834.Attimetthebondwillbeworth0.92e.Itwillproducea5%returnatatimetsuchthatrt0.92e−0.92=0.05,0.92whichgivest∼=0.5851yearsor213.6days.r2.38Attime0webuy1/B(0,1)=ebonds,attime1weincreaseourholdingstor2r(n+1)re/B(1,2)=ebonds,andgenerallyattimenwepurchaseeone-yearbonds. 274MathematicsforFinance2.39Becausethebondistradingatparandtheinterestratesremainconstant,thepriceofthebondatthebeginningofeachyearwillbe$100.Thesumof$1,000willbuy10bondsatthebeginningofyearone.Attheendoftheyearthecouponswillpay10×8=80dollars,enoughtobuy0.8bondsat$100each.Asaresult,theinvestorwillbeholding10+0.8=10.8bondsinyeartwo.Attheendofthatyearthecouponswillpay10.8×8=86.4dollarsand0.864additionalbondswillbepurchasedat$100each.Thenumberofbondsheldinyearthreewillbe10.8+0.864=11.664,andsoon.Ingeneral,thenumberofbondsheldinyearnwillben−18101+,100whichgives10.0000,10.8000,11.6640,12.5971,13.6049bondsheldinyearsonetofive.Chapter33.1ThetreerepresentingthescenariosandpricemovementsinExample3.1isshowninFigureS.3.FigureS.3TreeofpricemovementsinExample3.13.2ThetreerepresentingthescenariosandpricemovementsisshowninFig-ureS.4.Therearealtogethereightscenariosrepresentedbythepathsthroughthetreeleadingfromthe‘root’onthelefttowardstherightmostbranchtips.3.3Wecanuse(3.1)tofindScenarioS(0)S(1)S(2)S(3)ω145.0049.5051.9846.78ω245.0047.2551.9857.17ω345.0047.2542.5346.78ThetreeisshowninFigureS.5.3.4Whendividendsarepayable,formula(3.1)becomesS(n)=S(n−1)(1+K(n))−div(n),whichgivesScenarioS(0)S(1)S(2)S(3)ω145.0048.5049.9343.93ω245.0046.2549.8853.86ω345.0046.2540.6343.69 Solutions275FigureS.4TreeofpricemovementsinExercise3.2FigureS.5TreeofpricemovementsinExercise3.33.5Thereturnovertwotimestepsascomparedtothesumofone-stepreturnscantakethefollowingvalues:ScenarioK(0,2)K(1)+K(2)ω115.50%15.00%ω215.50%15.00%ω3−5.50%−5.00%OverthreetimestepswehaveScenarioK(0,3)K(1)+K(2)+K(3)ω13.95%5.00%ω227.05%25.00%ω33.95%5.00%Thesumofone-stepreturnstendstobelargerthenthereturnoverthetotalintervalifthesignofone-stepreturnsalternates.3.6FirstfindK(0,2)andthentheone-stepreturnsK(1)=K(2)=Kfromthe 276MathematicsforFinance2relation1+K(0,2)=(1+K)(assumingthat1+K>0):ScenarioK(0,2)K(1)=K(2)ω117.14%8.23%ω2−8.57%−4.38%ω3−20.00%−10.56%3.7Theformula1+K(0,2)=(1+K(1))(1+K(2))canbeusedtofindK(2).Forexample,thefollowingscenariosandvaluesofK(2)areconsistentwiththeconditionsofExercise3.7:ScenarioK(0,2)K(1)K(2)ω121.00%10.00%10.00%ω210.00%10.00%0.00%ω3−1.00%−10.00%10.00%Thisisnottheonlypossiblesolution.Anotheronecanbeobtainedfromtheabovebychangingscenarioω2toScenarioK(0,2)K(1)K(2)ω210.00%−10.00%22.22%withtheothertwoscenariosunaltered.3.8ForthethreescenariosinExample3.2wefindScenariok(1)k(2)k(0,2)ω15.31%3.39%8.70%ω25.31%−10.92%−5.61%ω3−5.61%1.90%−3.70%Inallthreecasesk(0,2)=k(1)+k(2).3.9LetKdenotethereturninthethirdscenario.Iftheexpectedreturnisequalto6%,then111×(−5%)+×6%+×K=6%.244SolvingforK,wefindthatthereturninthethirdscenariomustbe28%.3.10First,computethereturnsK(1),K(2)andK(0,2)ineachscenario:ScenarioK(1)K(2)K(0,2)ω110.00%9.09%20.00%ω25.00%−4.76%0.00%ω3−10.00%11.11%0.00%ItfollowsthatE(K(1))∼=0.25×10.00%+0.25×5.00%−0.5×10.00%∼=−1.25%,E(K(2))∼=0.25×9.09%−0.25×4.76%+0.5×11.11%∼=6.64%,E(K(0,2))∼=0.25×20.00%+0.25×0.00%+0.5×0.00%∼=5.00%.Clearly,(1+E(K(1)))(1+E(K(2)))∼=1.0530
=1.0500∼=1+E(K(0,2)). Solutions2773.11SincethequarterlyreturnsK(1),K(2),K(3),K(4)areindependentandiden-ticallydistributed,E(K(1))=E(K(2))=E(K(3))=E(K(4))and31+E(K(0,3))=(1+E(K(1)),41+E(K(0,4))=(1+E(K(1)).Thus,ifE(K(0,3))=12%,thentheexpectedquarterlyreturnE(K(1))∼=3.85%andtheexpectedannualreturnE(K(0,4))∼=16.31%.3.12ByCondition3.1therandomvariablesS(1)S(2)S(3)=1+K(1),=1+K(2),=1+K(3)(S.4)S(0)S(1)S(2)areindependent,eachtakingtwovalues1+dand1+uwithprobabilitiespand1−p,respectively.ThepriceS(2),whichistheproductofS(0)andthefirsttwooftheserandomvariables,takesuptofourvaluescorrespondingtothefourpricemovementscenarios,thatis,pathsthroughthetwo-steptreeofstockpricesshowninFigure3.3(inwhichS(0)=1forsimplicity).AmongthesefourvaluesofS(2)thereareinfactonlythreedifferentones,22S(0)(1+u)withprobabilityp,S(2)=S(0)(1+u)(1+d)withprobability2p(1−p),22S(0)(1+d)withprobability(1−p).ThepriceS(3),whichistheproductofS(0)andthethreeindependentrandomvariablesin(S.4),takesuptoeightvaluescorrespondingtotheeightpricemovementscenarios,thatis,pathsthroughthethree-steptreeofstockpricesinFigure3.4(withS(0)=1forsimplicity).AmongtheseeightvaluesofS(3)thereareonlyfourdifferentones,33S(0)(1+u)withprobabilityp,22S(0)(1+u)(1+d)withprobability3p(1−p),S(3)=22S(0)(1+u)(1+d)withprobability3p(1−p),33S(0)(1+d)withprobability(1−p).3.13ThetopvaluesofS(1)andS(2)canbeusedtofindu:u=92−87∼=0.0575.87Next,uandthetopvalueofS(1)givethevalueofS(0):S(0)∼=87∼=82.271+0.0575dollars.Finally,disdeterminedbyS(0)andthebottomvalueofS(1):d∼=76−82.27∼=−0.0762.82.27 278MathematicsforFinance3.14Giventhecontinuousrisk-freerateof14%andthetimestepτ=1/12,wefindtheone-stepreturn0.14/12r=e−1∼=0.0117.ByCondition3.2,u>r∼=0.0117.ThismeansthatthemiddlevalueS(0)(1+u)(1+d)ofS(2)mustbenolessthanabout22(1+0.0117)(1−0.01)∼=22.04dollars.3.15Considerthesystemofequations2S(0)(1+u)=32,S(0)(1+u)(1+d)=28,2S(0)(1+d)=x.Itfollowsthat321+u28==281+dx2andx=28/32=24.50dollars.However,thetreecannotbereconstructeduniquely.ForanyvalueS(0)>0onecanfinduanddconsistentwiththedata.3.16Thevaluesofuanddcanbefoundbysolvingtheequations2S(0)(1+u)=121,2S(0)(1+d)=100,andselectingthosesolutionsthatsatisfy1+u>0and1+d>0.IfS(0)=100,thenu=0.1andd=0.IfS(0)=104,thenu∼=0.0786andd∼=−0.0194.3.17Weonlyneedtoconsiderthevaluesofdbetween−1andr,thatis,−10ifV(1)<0.Thisis,therefore,anadmissiblestrategyrealisinganarbitrageopportunity.4.4Ifyouhappentoknowthatanincreaseinstockpricewillbefollowedbyafallatthenextstep,thenadoptthisstrategy:•Attime0donotinvestineitherasset.•Attime1checkwhetherthestockhasgoneupordown.Ifdown,thenagaindonotinvestineitherasset.Butifstockhasgoneup,thensellshortoneshareforS(0)u,investingtheproceedsrisk-free.Clearly,thetime0andtime1valueofthisstrategyisV(0)=V(1)=0.Ifstockgoesdownatthefirststep,thenthevalueattime2willbeV(2)=0.Butifstockgoesupatthefirststep,thenitwillgodownatthesecondstepandV(2)=S(0)u(r−d)willbepositive,asrequired,sinceu>r>d.(ThenotationisasinSection3.2.)Clearly,thisisnotapredictablestrategy,whichmeansthatnoarbitragehasbeenachieved.4.5a)Iftherearenoshort-sellingrestrictions,thenthefollowingstrategywillrealiseanarbitrageopportunity:•Attime0donotinvestatall.•Attime1checkthepriceS(1).IfS(1)=120dollars,thenonceagaindonotinvestatall.ButifS(1)=90dollars,thensellshortoneshareoftheriskyassetandinvesttheproceedsrisk-free.Thetime0andtime1valueofthisadmissiblestrategyis0.Thevalueattime2willbe
0inscenariosω1andω2,3inscenarioω3,whichmeansthatarbitragecanbeachieved.b)Ina)abovearbitragehasbeenachievedbyutilisingthebehaviourofstockpricesatthesecondstepinscenarioω2:Thereturnontheriskyassetislowerthantherisk-freereturn.Thus,shortingtheriskyassetandinvestingtheproceedsrisk-freecreatesarbitrage.However,whenshortsellingofriskyassetsisdisallowed,thenthisarbitrageopportunitywillbebeyondthereachofinvestors.4.6ThearbitragestrategydescribedinSolution4.5involvesbuyingafractionofabond.IfS(1)=90dollars,thenoneshareofstockshouldbeshortedand9ofabondpurchasedattime1.Toobtainanarbitragestrategyinvolving11anintegernumberofunitsofeachasset,multiplythesequantitiesby11,thatis,shortsell11sharesofstockandbuy9bonds.4.7Supposethattransactioncostsof5%applywheneverstockisboughtorsold.AninvestorwhotriedtofollowthestrategyinSolution4.5,shortsellingoneshareofstockattime1ifS(1)=90dollars,wouldhavetopaytransactioncostsof90×5%=4.50dollars.Iftheremainingamountof90−4.50=85.50121dollarswereinvestedrisk-free,itwouldbeworth85.5×=94.05dollars110attime2.Butclosingtheshortpositioninstockwouldcost$96,makingthefinalwealthnegative.Asaresult,thereisnoarbitragestrategy. 282MathematicsforFinance4.8Theputoptiongivestheright(butnoobligation)toselloneshareofstockforthestrikepriceX=110dollarsattime2.Weconsideranextendedmodelwiththreeassets,thestock,themoneymarket,andtheoption.TheunitpricesEEoftheseassetsareS(n),A(n),P(n),whereP(n)isthemarketpriceoftheputoptionattimen=0,1,2.Thetime2priceoftheputoptionisEP(2)=max{X−S(2),0}.AccordingtotheFundamentalTheoremofAssetPricing,thediscountedstockEEandoptionpricesS(n)=S(n)/A(n)andP(n)=P(n)/A(n)shouldformamartingaleundersomeprobabilitymeasureP∗,orelseanarbitrageopportu-nitywouldarise.FromExample4.5weknowthatthereisonlyoneprobabilityEP∗turningS(n)intoamartingale.Becauseofthis,P(n)mustbeamartin-galeunderthesameprobabilityP∗.ItfollowsthatEA(1)EEA(0)EP(1)=E∗(P(2)|S(1))andP(0)=E∗(P(1)).A(2)A(1)UsingthevaluesofP∗foreachscenariofoundinExample4.5,wecancomputeEEP(1)andP(0).ForexampleEEEEA(1)P∗(ω3)P(2,ω3)+P∗(ω4)P(2,ω4)P(1,ω3)=P(1,ω4)=A(2)P∗(ω3)+P∗(ω4)11110×20+×30=25100∼=20.0012111+25100dollars.Inthismanner,weobtainEEEScenarioP(0)P(1)P(2)ω11.961.210.00ω21.961.214.00ω31.9620.0020.00ω41.9620.0030.00Chapter55.1InthefirstinvestmentprojectE(K1)=0.12×0.25+0.12×0.75=0.12,22Var(K1)=(0.12−0.12)×0.25+(0.12−0.12)×0.75=0.InthesecondprojectE(K2)=0.11×0.25+0.13×0.75=0.125,22Var(K2)=(0.11−0.125)×0.25+(0.13−0.125)×0.75=0.000075.Finally,inthethirdprojectE(K3)=0.02×0.25+0.22×0.75=0.17,22Var(K3)=(0.02−0.17)×0.25+(0.22−0.17)×0.75=0.0075.Thefirstprojectistheleastriskyone,infact,itisrisk-free.Thethirdprojectinvolvesthehighestrisk. Solutions2835.2FirstweputK2(ω2)=xandcomputeVar(K1)=0.001875,2Var(K2)=0.1875x+0.015x+0.0003.ThetwosecuritieswillhavethesameriskifVar(K1)=Var(K2).Thisgivesthefollowingequation20.0003+0.1875x+0.015x=0.001875withtwosolutionsx=−0.14or0.06.ThismeansthatK2(ω2)=−14%or6%.ki=1+K5.3Firstweusetheformulaeifori=1,2tocomputethelogarithmicreturnsandthenworkoutthevarianceofeachreturn:ScenarioProbabilityK1K2k1k2ω10.510.53%7.23%10.01%6.98%ω20.513.89%10.55%13.01%10.03%Variance0.0002820.0002760.0002240.000232WefindthatVar(K1)>Var(K2),whereasVar(k1)0willbeyourarbitrageprofit.6.7By(6.3)theinitialforwardpriceisF(0,1)∼=45.72dollars.Thistakesintoaccountthedividendpaidattime1/2.a)IfS(9/12)=49dollars,thenF(9/12,1)∼=49.74dollarsby(6.2).Itfollowsby(6.8)thatV(9/12)∼=3.96dollars.b)IfS(9/12)=51dollars,thenF(9/12)∼=51.77dollarsandV(9/12)∼=5.96dollars.6.8Lett=1/365,T=1/4.Weapplytheformula(6.11)togetr(T−t)rTf(t,T)−f(0,T)=S(t)e+S(0)e=0rtifS(t)=S(0)e,thatis,ifthestockgrowsattherisk-freerate.r(T−t)6.9Sincef(t,T)=S(t)e,therandomvariablesS(t)andf(t,T)areperfectlyr(T−t)correlatedwithρS(t)f(t,T)=1andσf(t,T)=eσS(t).ItfollowsthatN=−r(T−t)e.6.10ObservethatTheorem6.5ontheequalityoffuturesandforwardpricesappliesalsointhecaseofanassetwithdividendspaidcontinuously.Wecan,therefore,use(6.6)toobtainr1ln14,100∼=2.20%.div=8%−0.7513,500 288MathematicsforFinance6.11Thereturnontheindexwillbe3.37%.ForrF=1%thisgivesthefuturespricesf(0,3)∼=916.97andf(1,3)∼=938.49.IfthebetacoefficientforaportfolioisβV=1.5,thentheexpectedreturnonthisportfoliowillbeµV∼=4.56%.ToconstructaportfoliowithβV=0andinitialvalueV(0)=100dollars,supplementtheoriginalportfoliowithN∼=0.1652shortfuturescontracts(observethatNisthesameasinExample6.4).Iftheactualreturnontheoriginalportfolioduringthefirsttimestephappenstobeequaltotheexpectedreturn,thenitsvalueafteronestepwillbeV(1)∼=104.56dollars.MarkingtomarketrequirestheholderofN∼=0.1652shortforwardcontractstopay$3.56.Asaresult,afteronestepthevalueoftheportfoliowithforwardcontractswillbeV(1)∼=104.56−3.56=101.00dollars,onceagainmatchingtherisk-freegrowth.Chapter77.1Theinvestmentwillbringaprofitof+0.12×3(36−S(T))−4.50e12=3,whereS(T)isthestockpriceontheexercisedate.ThisgivesS(T)∼=28.36dollars.+0.09×67.2E((S(T)−90)−8e12)∼=−5.37dollars.E−3×0.0672E∼=1.98dollars.7.3Byput-callparity2.83−P=15.60−15.00e12,soP−0.0748×67.4Put-callparityisviolated,5.09−7.78>20.37−24e12.ArbitragecanberealisedasinthefirstpartoftheproofofTheorem7.1:•Buyasharefor$20.37;•Buyaputoptionfor$7.78;•Writeandsellacalloptionfor$5.09;•Borrow$23.06attheinterestrateof7.48%.Thebalanceofthesetransactionsiszero.Aftersixmonths•Sellonesharefor$24byexercisingtheputoptionorsettlingtheshortpositionincalls,dependingonwhetherthesharepriceturnsouttobebeloworabovethestrikeprice;1×0.0748∼=23.94dollars•Repaytheloanwithinterest,amountingto23.06e2intotal.Thebalanceof24−23.96=0.06dollarswillbeyourarbitrageprofit.EE−rT7.5IfC−P>S(0)−div0−Xe,thenattime0buyashareandaputoption,writeandsellacalloption,andinvest(orborrow,ifnegative)thebalanceonthemoneymarketattheinterestrater.Assoonasyoureceivethedividend,investitattherater.AttheexercisetimeTclosethemoneymarketinvestmentandselltheshareforX,eitherbyexercisingtheputifEES(T)0willbeyourarbitrageprofit.EE−rTOntheotherhand,ifC−PXorsettlingtheputifS(T)≤X,andclosetheshortpositioninstock.YourEErTarbitrageprofitwillbe(−C+P+S(0)−div0)e−X>0. Solutions2897.6IfCE−PE0.Iftheputoptionisnotexercisedatall,thenyoucanusethecalltobuyashareforXattimeT,closingtheshortpositioninstock.Youwillalsobeleftwithapositivefinalbalance(−CE+PE+S(0)e−rdivT−Xe−rT)erT>0.Ontheotherhand,ifCE−PE>S(0)e−rdivT−Xe−rT,thentheoppositestrategywillalsoleadtoarbitrage.7.7Thestrikepriceisequaltotheforwardprice(moreprecisely,theexchangerate)of0.9883eurostoadollarcomputedinSolution6.5.−rTAA7.8IfS(0)−XeXe−Xe≥0.(Iftheputisexercisedbeforethedividendbecomesdue,youcanincreaseyourarbitrageprofitbyclosingouttheshortpositioninstockimmediately,inwhichcaseyouwouldnothavetopaythedividend.)IftheputisnotexercisedbeforeexpiryT,thenthesecondpartofSolution7.5applies.−rTAA7.9IfS(0)−Xe0plusanydividendsaccumulatedsincethesharewasboughtattimet.Iftheputoptionisnotexercisedatall,thenyoucanusethecalltobuyashareforXattimeT,closingtheshortpositioninstock.Youwillalsobeleftwithapositivefinalbalance(−CA+PA+S(0)e−rdivT)erT−X>(−CA+PA+S(0)e−rdivT−X)erT>0.EA7.10IfC>C,thenwriteandsellaEuropeancallandpurchaseanAmericanEAcall.ThedifferenceC−Cwillbeyourarbitrageincome.Tokeepit,donot 290MathematicsforFinanceexercisetheAmericanoptionuntilexpiry,wheneitherbothoptionswillturnoutworthless,orthelossfromsettlingtheEuropeancallwillberecoveredbyexercisingtheAmericancall.Theargumentforputoptionsissimilar.E7.11SupposethatC≥S(0)−div0.Inthiscasewriteandsellacalloptionandbuystock,investingthebalanceonthemoneymarket.Assoonasyoureceivethedividends,alsoinvestthemonthemoneymarket.Ontheexercisedateyoucansellthestockforatleastmin(S(T),X),settlingthecalloption.YourErTfinalwealthwillbepositive,(C−S(0)+div0)e+min(S(T),X)>0.ThisEprovesthatCdandX>S,soC(0)increasesasuincreases.EThederivativeofC(0)withrespecttodisequaltou(u−r)[S(0)(1+u)−X](u−r)[S−X]−=−,(1+r)(u−d)2(1+r)(u−d)2uwhichisnegative,sincerXism=35.EFromtheCox–Ross–RubinsteinformulaweobtainC(0)∼=3.4661dollarsandx(1)=[1−Φ(m−1,N,q)]∼=0.85196shares. Solutions2938.9ThedeltaofaEuropeancallbecomesequalto1atthefirststepnsuchthattheoptionwillbeinthemoneyindependentlyofwhetherthestockgoesupnordownatthenextstep,thatis,S(0)(1+u)(1+d)>X(inthiscasen+1S(0)(1+u)>Xaswell).ThisgiveslnX−lnS(0)−ln(1+d)n>.ln(1+u)8.10Thestockvaluesaren012379.8672.60<66.00<68.97S(n)60.00<62.70<57.00<59.5754.15<51.44TheAmericanputpricesaren01230.000.00<0.50<0.00AP(n)2.52<1.10<5.00<2.437.85<10.56Theoptionwillbeexercisedearlyinthednodeattime1orintheddnodeattime2(boldfigures).8.11ThevaluesoftheEuropeanandAmericancallsarethesame,n01252.8034.91m+xσ∼=−23.68%,wherex∼=−1.645,sothatN(x)∼=5%.The1,000dollarsconvertedintoeuros,investedwithoutriskattheraterEUR,andcon-vertedbackintodollarsafteroneyear,willgive1,000erEURekdollars.Withprobability95%thisamountwillsatisfy1,000erEURek>1,000erEURem+xσ∼=821.40dollars.Ontheotherhand,1,000dollarsinvestedattheraterUSDwouldhavegrownto1,000erUSD∼=1,051.27dollars.Asaresult,VaR=1,000erUSD−1,000erEURem+xσ∼=229.88dollars.9.9Asinglecallcosts$21.634.Wepurchaseapproximately46.22options.Withprobability5%thestockpricewillbelessthan$49.74.Weshallstillbeabletoexercisetheoptions,cashing$450.18intheborderlinecase.Thealternativerisk-freeinvestmentof$1,000at8%wouldgrowto$1,040.81.HenceVaR∼=590.63dollars.Ifthestockgrowsattheexpectedrate,reaching$63.71,thenweshallobtain$1,095.88whentheoptionsareexercised.With5%probabilitythestockpricewillbeabove$81.6andthenouroptionswillbeworthatleast$1,922.75.9.10Thecostofasinglebullspreadis$0.8585,withexpectedreturn29.6523%,standarddeviation99.169%,andVaRequalto$15,000(at74.03%confidencelevel).If92.95%ofthecapitalisinvestedwithoutriskandtheremainderinthebullspread,thentheexpectedreturnwillthesameasonstock,withriskof6.9957%andVaRequalto$650.9.11Aputwithstrikeprice$56costs$0.426.Aputwithstrikeprice$58costs$0.9282.Theexpectedreturnonthebearspreadis111.4635%,theriskreach-ing177.2334%.Theworstcasescenario(amongthoseadmittedbythean-alyst)iswhenthestockpricedropsto$58.59.Inthisscenario,whichwillhappenwithconditionalprobability0.3901,theinvestorwillloseeverything,soVaR=15,000dollarsat60.99%confidencelevel.Chapter1010.1Theyieldsarey(0)∼=14.08%andy(3)∼=13.63%.ThusB(0,3)=e−3τy(0)∼=0.9654dollars.Arbitragecanbeachievedasfollows:•Attime0buya6-monthbondforB(0,6)=0.9320dollars,raisingthemoneybyissuing0.9654ofa3-monthbond,whichsellsatB(0,3)∼=0.9654dollars.•Attime3(after3months)issue0.9989ofa3-monthbond,whichsellsatB(3,6)=0.9665dollars,andusetheproceedsof$0.9654tosettlethefractionofa3-monthbondissuedattime0.•Attime6(afterhalfayear)the6-monthbondboughtattime0willpay$1,outofwhich$0.9989willsettlethefractionofa3-monthbondissuedattime3.Thebalanceof$0.0011willbethearbitrageprofit.10.2Theimpliedratesarey(0)∼=12.38%andy(6)∼=13.06%.Investing$100,wecanbuy106.38bondsnowand113.56aftersixmonths.Thelogarithmicreturnoveroneyearisln(113.56/100)∼=12.72%,thearithmeticmeanofthesemi-annualreturns. 298MathematicsforFinance10.3Toachieveareturnof14%,wewouldhavetosellthebondfor0.8700e14%∼=1.0007dollars,whichisimpossible.(Azero-couponbondcanneverfetchapricehigherthanitsfacevalue.)k−τy(6)Ingeneral,wehavetosolvetheequationB(0,12)e=etofindy(6),wherekistheprescribedlogarithmicreturn.Theleft-handsidemustbesmallerthan1.10.4Duringthefirstsixmonthstherateisy(n)∼=8.34%,forn=0,...,179,andduringtherestoftheyeary(n)∼=10.34%,forn=180,...,360.Thebondshouldbesoldfor0.92e4.88%∼=0.9660dollarsormore.Thiscannotbeachievedduringthefirstsixmonths,sincethehighestpricebeforetheratechangesisB(179,360)∼=0.9589dollars.OnthedayoftheratechangeB(180,360)∼=0.9496dollars,andwehavetowaituntildayn=240,onwhichthebondpricewillexceedtherequired$0.9660forthefirsttime.10.5Theratecanbefoundbyusingaspreadsheetwithgoalseekfacilitytosolvetheequation−y(1)−2y(1)−3y(1)k10.896×10+10e+10e+110e=1,000e.Thisgivesy(1)∼=12.00%fork=12%incasea),y(1)∼=12.81%fork=10%incaseb)andy(1)∼=11.19%fork=14%incasec).10.6ThenumberswerefoundusinganExcelspreadsheetwithaccuracyhigherthanthedisplayed2decimalpoints.10.7Scenario1:$1,427.10;Scenario2:$1,439.69.10.8Formula(10.2)canbeapplieddirectlytofindD∼=1.6846.10.9Thedurationisequalto4ifthefacevalueis$73.97.Thesmallestpossibleduration,whichcorrespondstofacevalueF=$0,isabout2.80years.ForveryhighfacevaluesFthedurationiscloseto5,approachingthisnumberasFgoestoinfinity.WhenF=100,thecouponvalueC∼=13.52givesdurationof4years.Ifthecouponvalueiszero,thenthedurationis5years.ForveryhighcouponvaluesCtendingtoinfinitythedurationapproachesabout2.80years.10.10SincethesecondderivativeofP(y)ispositive,2dP(y)=(τn2−τn1y2−τn2y2−τnNy1)C1e+(τn2)C2e+···+(τnN)(CN+F)edy2>0,Pisaconvexfunctionofy.10.11Solvingthesystem6=2wA+3.4wB,wA+wB=1,wefindwA∼=−1.8571andwB∼=2.8571.Asaresult,weinvest$14,285.71tobuy14,005.60bondsB,raisingtheshortfallof$9,285.71byissuing9,475.22bondsA.10.12TheyieldonthecouponbondAisabout13.37%,sothepriceofthezero-couponbondBis$87.48.Thecouponbondhasduration3.29andwefindtheweightstobewA∼=0.4366andwB∼=0.5634.Thismeansthatweinvest$436.59tobuy4.2802bondsAand$563.41tobuy6.4403bondsB.10.13Directlyfromthedefinition(10.2)ofdurationwecomputethedurationDtat Solutions299yttimet(notethatthebondpricegrowsbyafactorofe),1(τn−y(τn1−t)−y(τnN−t)Dt=1−t)C1e+···+(τnN−t)(CN+F)eeytP(y)1−τn1y+···+(τn−τnNy=(τn1−t)C1eN−t)(CN+F)eP(y)=D0−t,−τn1y/P(y),C−τn2y−τnNysincetheweightsC1e2e/P(y),...,(CN+F)e/P(y)adduptoone.10.14DenotetheannualpaymentsbyC1,C2andthefacevaluebyF,sothat−y−2yP(y)=C1e+(C2+F)e,−y−2yC1e+2(C2+F)eD(y)=.P(y)ComputethederivativeofD(y)toseethatitisnegative:−3yd−C1(C2+F)eD(y)=<0.dyP(y)210.15Wefirstfindthepricesanddurationsofthebonds:PA(y)∼=120.72,PB(y)∼=434.95,DA(y)∼=1.8471,DB(y)∼=1.9894.TheweightswA∼=−7.46%,wB∼=107.46%giveduration2,whichmeansthatwehavetobuy49.41bondsBandissue12.35bondsA.Afteroneyearweshallreceive$247.05fromthecouponsofBandwillhavetopaythesameamountforthecouponsofA.Ourfinalamountwillbe$23,470.22,exactlyequaltothefuturevalueof$20,000at8%,independentlyofanyratechanges.10.16Ifthetermstructureistobeflat,thentheyieldy(0,6)=8.16%appliestoanyothermaturity,whichgivesB(0,3)=0.9798dollarsandB(0,9)=0.9406dollars.10.17Issueandsell500bondsmaturingin6monthswith$100facevalue,obtaining$48,522.28.Usethissumtobuy520.4054one-yearbonds.After6monthssettlethebondsissuedbypaying$50,000.Afteroneyearcashthefacevalueofthebondspurchased.Theresultingrateis8%.10.18Youneedtodeposit100,000e−8.41%/12∼=99,301.62dollarsforonemonth,whichwillgrowtothedesiredlevelof$100,000,andborrowthesameamountfor6monthsat9.54%.Yourcustomerwillreceive$100,000after1monthandwillhavetopay99,301.62e9.54%/2∼=104,153.09dollarsafter6months,whichimpliesaforwardrateof9.77%.(Theratecanbeobtaineddirectlyfrom(10.5).)Theratefora4-monthloanstartingin2monthsisf(0,2,6)=6×9.35%−2×8.64%∼=10.09%,4soadepositat10.23%wouldgiveanarbitrageopportunity.10.19Toseethattheforwardratesf(n,N)maybenegative,letusanalysethecasewithn=0forsimplicity.Thenf(0,N)=(N+1)y(0,N+1)−Ny(0,N) 300MathematicsforFinanceandf(0,N)<0requiresthat(N+1)y(0,N+1)Snbyn(n+1)toget(aftercancellations)nan+1>a1+···+an.Thelatteristrue,sincean+1>aiforalli=1,...,n.10.21ThevaluesofB(0,2),B(0,3),B(1,3)havenoeffectonthevaluesofthemoneymarketaccount.10.22a)Foraninvestmentof$100inzero-couponbonds,dividetheinitialcashbythepriceofthebondB(0,3)togetthenumberofbondsheld,102.82,whichgivesfinalwealthof$102.82.Thelogarithmicreturnis2.78%.b)Foraninvestmentof$100insingle-periodzero-couponbonds,computethenumberofbondsmaturingattime1as100/B(0,1)∼=100.99.Then,attime1findthenumberofbondsmaturingattime2inasimilarway,100.99/B(1,2)∼=101.54.Finally,wearriveat101.54/B(2,3)∼=102.51bonds,eachgivingadollarattime3.Thelogarithmicreturnis2.48%.c)Aninvestmentof$100inthemoneymarketaccount,forwhichwereceive100A(3)∼=102.51attime3,producesthesamelogarithmicreturnof2.48%asinb).Chapter1111.1Webeginfromtheright,thatis,fromthefacevaluesofthebonds,firstcomputingthevaluesofB(2,3)inallstates.ThesenumberstogetherwiththeknownreturnsgiveB(1,3;u)andB(1,3;d).These,inturn,determinethemissingreturnsk(2,3;ud)=0.20%andk(2,3;dd)=0.16%.Thesameisdoneforthefirststep,resultingink(1,3;d)=0.23%.ThebondpricesaregiveninFigureS.10.FigureS.10BondpricesinSolution11.111.2Becauseoftheadditivityofthelogarithmicreturns,k(1,3;u)+k(2,3;uu)+k(3,3;uuu)=0.64%givesthereturnintheperiodofthreeweeks.Toobtaintheyieldwehavetorescaleittothewholeyearbymultiplyingby52/3,hencey(0,3)=11.09%.Notethatwemusthavek(1,3;u)+k(2,3;ud)+k(3,3;udu)= Solutions3010.64%whichallowsustofindk(2,3;ud)=0.20%.Theothermissingreturnscanbecomputedinasimilarmanner,firstk(1,3;d),thenk(2,3;dd).11.3ThebondpricesaregiveninFigureS.11.FigureS.11BondpricesinSolution11.311.4ThemoneymarketaccountisgiveninFigureS.12.Notethatthevaluesforthe‘up’movementsarelowerthanforthe‘down’movements.Thisisrelatedtothefactthattheyielddecreasesasthebondpriceincreases,andourtreesarebasedonbondpricemovements.FigureS.12MoneymarketaccountinSolution11.411.5ThepricesB(1,2;u)=0.9980andB(1,2;d)=0.9975arefoundbydiscountingthefacevalue1tobereceivedattime2,usingtheshortratesr(1;u)andr(1;d).ThepriceB(0,2)=0.9944canbefoundbythereplicationprocedure.11.6Attime2thecouponsare0.5227or0.8776,dependingonwhetherweareintheupordownstateattime1.Attime1thecouponis0.9999.11.7Attime1wefind18.0647=(0.8159×20+0.1841×10)/1.0052intheupstateand1.7951=(0.1811×10+0.8189×0)/1.0088inthedownstate.Next,applyingthesameformulaagain,weobtain7.9188=(0.3813×18.3928+0.6187×1.7951)/1.01.11.8Thereisanarbitrageopportunityattime1intheupstate.ThepriceB(1,2;u)=0.9924impliesthatthegrowthfactorinthemoneymarketis1.00766,whereasthepricesofthebondmaturingattime3implygrowthfac-tors1.01159and1.00783.Torealisearbitrage,bondswithmaturity3shouldbebought,thepurchasefinancedbyaloaninthemoneymarket. 302MathematicsforFinance11.9Usingformula(11.5)andtheshortratesgiven,wefindthefollowingstructureofbondprices:B(2,3;uu)=0.9931B(1,3;u)=0.9859