shreve solution manual

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1、StochasticCalculusforFinance,VolumeIandIIbyYanZengLastupdated:August20,2007Thisisasolutionmanualforthetwo-volumetextbookStochasticcalculusfornance,byStevenShreve.Ifyouhaveanycommentsorndanytypos/errors,pleaseemailmeatyz44@cornell.edu.Thecurrentversionomitsthefollowingproblems.Volu

2、meI:1.5,3.3,3.4,5.7;VolumeII:3.9,7.1,7.2,7.5{7.9,10.8,10.9,10.10.AcknowledgmentIthankHuaLi(agraduatestudentatBrownUniversity)forreadingthroughthissolutionmanualandcommunicatingtomeseveralmistakes/typos.1StochasticCalculusforFinanceI:TheBinomialAssetPricingModel1.TheBinomialNo-Arbitr

3、agePricingModel1.1.Proof.Ifwegettheupsate,thenX1=X1(H)=
0uS0+(1+r)(X0
0S0);ifwegetthedownstate,thenX1=X1(T)=
0dS0+(1+r)(X0
0S0).IfX1hasapositiveprobabilityofbeingstrictlypositive,thenwemusteitherhaveX1(H)>0orX1(T)>0.(i)IfX1(H)>0,then
0uS0+(1+r)(X0
0S0)>0.PluginX0=0,wegetu
0>(1+r)

4、
0.Byconditiond<1+r0.Inthiscase,X1(T)=
0dS0+(1+r)(X0
0S0)=
0S0[d(1+r)]<0.(ii)IfX1(T)>0,thenwecansimilarlydeduce
0<0andhenceX1(H)<0.SowecannothaveX1strictlypositivewithpositiveprobabilityunlessX1isstrictlynegativewithpositiveprobabilityaswell,regardlessthechoiceofth

5、enumber
0.Remark:HeretheconditionX0=0isnotessential,asfarasapropertydenitionofarbitrageforarbitraryX0canbegiven.Indeed,fortheone-periodbinomialmodel,wecandenearbitrageasatradingstrategysuchthatP(X1X0(1+r))=1andP(X1>X0(1+r))>0.First,thisisageneralizationofthecaseX0=0;second,itisp

6、roper"becauseitiscomparingtheresultofanarbitraryinvestmentinvolvingmoneyandstockmarketswiththatofasafeinvestmentinvolvingonlymoneymarket.ThiscanalsobeseenbyregardingX0asborrowedfrommoneymarketaccount.Thenattime1,wehavetopaybackX0(1+r)tothemoneymarketaccount.Insummary,arbitrageisatra

7、dingstrategythatbeatssafe"investment.Accordingly,werevisetheproofofExercise1.1.asfollows.IfX1hasapositiveprobabilityofbeingstrictlylargerthanX0(1+r),theeitherX1(H)>X0(1+r)orX1(T)>X0(1+r).Therstcaseyields
0S0(u1r)>0,i.e.
0>0.SoX1(T)=(1+r)X0+
0S0(d1r)<(1+r)X0.Thesecondcasecanbes

8、imilarlyanalyzed.He