...............................................................................1...............................................................................................5.....................................................15.........................................................25...............................................................................................35......................................................................43....................................................................52........................64DEA....................................................73.....................................................85FirmValueModelwithGaussianInterestRates..............................................ChanKaLeong93........................................................................................................107
1412Aacem@macau.ctm.net
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73DEA2004(DataEnvelopmentAnalysis,DEA)DEA(Decision-makingUnits,DMU)DMUDMUDMUDEA(DMU)DMUWorthington,A.C.,“CostEfficiencyinAustralianLocalGovernment:AComparativeAnalysisofMathematicalProgrammingandEconometricApproaches,”inFinancialAccountabilityandManagement,2000,Vol.16,No.3,pp.201-2242008CharnesA.,CopperW.W.&RhodesE.,“MeasuringtheEfficiencyofDecision-makingUnits,”inEuropeanJournalofOpera-tionalResearch,1978,Vol.2,No.6,pp.429-444;CharnesA.&CopperW.W.,“PrefacetoTopicsinDataEnvelopmentAnalysis,”inAnnalsofOperationsResearch,1985,Vol.2,No.1,pp.59-94.
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8320086DEAFried(1999)DEA1991-20030.9663.4%DEA2004
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872008410
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93FirmValueModelwithGaussianInterestRatesChanKaLeongMemberofMacaoAssociation&EconomicSciencesInthispaper,weapplythefirmvaluemodeltostudypricingproblemsforEuro-peanoptionsforderivativeswithcounterpartydefaultrisk.Wederiveaclosed-formformulaforBlack-ScholestypeoptionswithGaussianinterestrates.TheresultsarebaseonAmmann’stheory(referto[1]),butgeneralizethedriftìandvolatilityófromconstantstofunctionsoftimet.Weprovidecompleteanddetailedproofsinthepaper.Bygettingclosed-formformulaforBlack-Scholestypeoptions,wefirstgivethepriceofderivativesandthefirm’sassets.Thenwewillbuildaforwardmartingalemeasuretogetapricemodelforcredit-riskyoption.Finally,wederiveaclosed-formformulaforBlack-Scholestypeoptionsunderthispricemodel.Inproofs,weusethefollowingmathematicstools:Ito’sformula,martingale,Brownianmotion,GirsanovTheorem(GirsanovTheoremforcorrelatedBrownianMotions),etc.1.Thepriceofderivativesandthefirm’sassetsInthischapter,weconsiderthepricingproblemforEuropeantypeoptionswithcounterpartydefaultrisk,inwhichthepriceofderivativesandthefirm’sassetsarestochasticandcorrelated,givenby(1.1)ThenweassumethattheliabilityisaconstantDwithv0>D.However,inthis
94paper,weassumethatthemarkethasazero-couponbondwithmaturingtimeT,thepriceprocessP(·,T)ofwhichisgivenby(1.2)Whereã(·)istheinstantaneousforwardrate,whichfollowsafamilyofIto’sprocesses:(1.3)Weassumethatthecoefficientìandóareall{Ft}-adapted,andsatisfythatandforalli=S,Vandì(s)=ìP(s,t)andó(s)=óP(s,t).WefurtherassumethatB(·)=(BS(·),BV,(·),BP,(·))isa3-dimensionalcorrelatedstandard{Ft}-Brownianmotionsandforanyi,j=S,V,thereexistsaconstantñijwithñij=ñjiand.SinceeachBi(·)isastandard{Ft}-Brownianmotion,wehavethatñij=1foreachi=S,V;andBP(·)isindependentofBS(·)andBV(·).SothereexistsacorrelationmatrixP3ofB(·)givenby.BecauseoftheGaussianinterestrates,themodelinthispaperismorecomplexthanthemodelwithconstantinterestrates.First,weneedtofindaforwardmartingalemeasureasanequivalentmartingalemeasuretoPsothatwecanderivethepriceofacredit-riskyoptionunder.2.TheForwardMartingaleMeasureWedenoteandforany.Thenweassumethatthe3-dimensional{Ft}-adaptedprocessë=(ëS,ëV,ëP)satisfiesthe
95followingsystemlinearequation:(2.1)LetË=(ëS,ëV,ëP,)bea3-dimensional{Ft}-adaptedprocesssuchthatforeachî.Wedenote,,whichsatisfiestheNovikov’scondition,whereË*isthetransposeofË.Then,wedefineasan{Ft}-martingale.Wefurtherassumethat,foreachî,ëî(t)0foreach,andthereisaconstantñëîsuchthat,.Wedefineanewprobabilitymeasureon(Ω,FT)byforall,thenaccordingtotheGirsanovTheoremonthecorrelatedtheBrownianmotion,underthenewmeasurethe,inwhich(2.2)forî=S,V,P,isa3-dimensionalcorrelatedstandard({FT},)-Brownianmotion,thecorrelationmatrixofwhichisstillP3.Theorem2.1Undertheprobability,thepriceofsecurityS(·),thefirmvalueV(·),andthepriceofbondP(·,s)satisfythefollowingSDEs:(2.3)ProofFromtheSDES(1.1)-(1.2),(2.2),andthesystemofequations(2.1),wecaneasilygetthefirsttwoSDESin(2.3).WeonlyshowthethirdSDE.Fromthethird
96equationin(2.1)wegetthatandsothat,bysubstituting(2.2)forî=Pinto(1.3),wehavethat.ByapplyingtheFubini’stheoremwegettheidentity:andagainapplyingtheFubini’stheoreminitsstandardandstochasticversionswehavethat.Thus,.ItimpliesthatthebondpriceprocessP(t,T)isgivenbythelastSDEin(2.3)under.Werespectivelydefinetheforwardpriceandtheforwardfirmvalueî(t,T)=î(t)P-1(t,T)forallandeachî=S,V.Foreach,wedefine.ThenZPisacontinuous({FT,})-martingale,andbytheGirsanovTheorem,wehavethat,underanewequivalentprobabilitymeasure,whichisdefinedbyforall,(2.4)isastandard{FT}-Brownianmotion.Sinceisindependentofand,(2.5)areallstandard({Ft,})-Brownianmotionssuchthatforall,
97i.e.isa3-dimensionalcorrelatedstandard({Ft,})-BrownianmotionwhosecorrelationmatrixisP3.Theprobabilitymeasure,whichisequivalentto,iscalledtheforwardmartin-galemeasure.3.ThePriceofCredit-RiskyOptionInthissection,weobtainthepriceofcredit-riskyEuropeanoptiondefinedby(2.4)and(2.5)undertheforwardmartingalemeasure.Accordingtotheexpression(2.3),wecaneasilyobtainthefollowingresultbyusingthetranslationin(2.4)and(2.5).Theorem3.1Theforwardmartingalemeasureistheequivalentmartin-galemeasureforthenumeraireP(·,T),i,e.under,theforwardpriceS(t,T)andtheforwardvalueV(t,T),whicharerespectivelygivenbythefollowingSDEs(3.1)areall({Ft},)-martingales.Wedefineforall,anddenote.(3.2)Then,sinceBrownianmotionsandareindependent,isasquareintegrable({Ft,})-martingalesuchthatforeach.Bythecontinuousmartingalerepresentationtheorem,thereexistsaBrownianmotionsuchthat.(3.3)Hence,wehavethat
98(3.4)foreach.Accordingtooptiontheory,wegetthat,atanytime,thepriceX(t)ofacredit-riskycalloptionisgivenby.(3.5)Similarly,thepriceY(t)ofacredit-riskypulloptionisgivenby.(3.6)4.TheBlack-ScholesTypeFormulaNow,wedefinetherecoveryrateprocessbyä(t)=V(t)D–1(whereDisaconstant)forall.ThenwedenoteóSV(t)=ñSVóS(t)óV(t)foreach,suchthat.(4.1)Welet(4.2)and.(4.3)Wealsodefine,whichisajointdistributionfunctionofabivariatestandardnormalrandomvari-ableswiththecorrelationcoefficientñ.WedefinethejointdensityfunctionbyThen,wehavethefollowingexplicitformulaforthepriceprocessX(t),whichisdefinedby(3.5).Theorem4.1Ifóî,î=S,V,P,arealldeterministic,then,undertheassump-
99tionsintheprecedingsections,thepriceprocessX(t)isgivenby(4.4)foreach,withparametersñ=t,T,andProofFrom(3.5)thepriceX(t)canbeexpressedas
100Eachofthefourtermscanbeevaluatedseparately.EvaluationoftermE1WedefineanewequivalentprobabilitymeasureQbyforall,suchthat(4.5)aretwostandard({Ft},Q})-Brwonianmotionswithforeach.Taking(4.5)into(3.4)and(4.1),wegetthat,underQ,(4.6)Thus,letand,from(4.6),wehavethatThus,wehavethatand.
101BytheGirsanovTheorem,andaretwostandardnormalrandomvariables,whichareindependentofFt,andtheircorrelationcoefficientisgivenby.Therefore,withtheBayesruleforconditionalexpectationswegetthat.EvaluationoftermE2.Under,Weletand.ThenwecaneasytogetthatçSandçVarestandardnor-malrandomvariables,whichareallindependentofFtandtheircorrelationcoefficientisñt,T.From(3.4),wehavethosetwoSDEs:(4.7)and.(4.8)Hence,wehavethatandThen,wehaveand.Therefore,wegetthat
102.EvaluationoftermE3.Weletforall,whereisgivenin(3.2.3)andforeach.Thenwelet.(4.9)Thus,isan({Ft},)-martingalesuchthat.Then,accordingtothecontinuousmartingalerepresentationtheorem,thereexistsastandard({Ft},)-Brownianmotionsuchthatforeach,i.e.wegetthat.(4.10)Letñ0Sandñ0Vbetwoconstantssuchthatand.Then,from(4.10)wehavethatAndforall.Wedefineforall,then,accordingtotheGirsanovTheorem,underanewprobabilitymeasuredefinedbyforall,(4.11)arethreestandard{Ft}-Brownianmotionssuchthat;and.From(3.4)and(4.1)wehavethat.
103Hence,bytheBayesruleforconditionalexpectationwegetthat.Using(4.11),wegetthat(4.12)Now,weletand.Then,under,andarestandardnormalvariables,whichareallindependentofFtandtheircorrelationcoefficientisñt,T.From(4.12)wehavethatThen,wehaveand.Therefore,E3canbeexpressedby
104EvaluationoftermE4.Wefirstdefine.Then,accordingtotheGirsnovTheorem,wecandefineanequivalentprobabilitymeasurebyforall.,wehavethatand;(4.13)aretwostandard({Ft},)-Brownianmotionswithforall.Thus,bytheBayesruleforconditionalexpectation,wecangetthat.Ontheotherhand,from(3.2.4)and(3.3.13),wegetthat(4.14)Weletand.Then,under,andarestandardnormalvariables,whichareallindependentofFtandtheircorrelationcoefficientisñt,T.From(4.14),wehavethatSincewehavethefollowingidentitiesandwecangetthat
105CombiningtheevaluationsoftermsE1toE4,wecompletetheproofofTheorem4.1.Similarly,wecanalsoobtainthepriceY(t)onthecredit-riskyclaimoftheEuro-peanputoption.Theorem4.2ThepriceprocessY(t)whichisdefinedby(3.2.6)isgivenbyforeach,wheretheparametersaregiveninTheorem4.1.References:1.M.Ammann,CreditRiskValuation:Methods,Models,andApplications,2ndEdition,Springer-Verlag,2001.2.N.H.Bingham&R.Kiesel,Risk-NeutralValuation:PricingandHedgingofFinancialDerivatives,2ndEdition,Springer-Verlag,2004.3.F.Black&M.Scholes,“ThePricingofOptionsandCorporateLiabilities”,inPoliticalEconomy,1973,Vol.81,pp.637-659.4.K.L.Chan,“AnIntroductiontotheFirmValueModel,”inJournalofMacaoEconomics,2008,Vol.26.5.D.Ding,“StochasticDifferentialEquations,”LectureNotes,UniversityofMacau,2002.6.D.Ding,“StochasticCalculusinFinancialMarkets,”LectureNotes,UniversityofMacau,2002.7.D.Ding,“GirsanovTheoremandNovilov’sCondition,”LectureNotes,UniversityofMacau,2003.8.D.Ding&K.L.Chan,“TheMartingaleApproachforCredit-riskyOptionPricing,”inChineseJournalofAppliedProbabilityandStatistics,2005.9.R.J.Elliott&P.E.Kopp,MathematicsofFinancialMarkets,Springer-Verlag,2000.10.I.I.Gikhman&A.V.Skorokhod,TheTheoryofStochasticProcessI&II,Springer-Verlag,2004.
10611.H.Johnson&R.Stulz,“ThePricingofOptionswithDefaultRisk,”inJournalofFinance,1987,Vol.42,pp.267-280.12.P.Klein,“PricingBlack-ScholesOptionswithCorrelatedCreditRisk,”inJournalofBackingandFinance,1996,Vol.20,pp.1211-1129.13.H.Levy,StochasticDominance:InvestmentDecisionMakingUnderUncertainty,KluwerAca-demicPublishers,199814.R.C.Merton,“OnthePricingofCorporateDebt:theRiskStructureofInterestRates,”inJournalofFinance,1974,Vol.2,pp.449-470.15.P.J.Schonbucher,CreditDerivativesPricingModels:Models,PricingandImplementation,JohnWiler&SonsLtd,2003.16.J.Stampfli&V.Goodman,TheMatiematicsofFinance:ModelingandHeging,ChinaMachinePress,2003.17.E.S.Steven,StochasticCalculusforFinanceII:Continuous-TimeModels,Springer-Verlag,2004.
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