Arbitrage Theory in Continuous Time Solutions

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1、SuggestedSolutionsforFinanceIIFall2004IrinaSlinko(FromtheSolutionsManualofRaquelM.Gaspar)1Contents4StochasticIntegrals35DifferentialEquations77ArbitragePricing98CompletenessandHedging159ParityRelationsandDeltaHedging1713SeveralUnderlyingAssets2116Incom

2、pleteMarkets2417Dividends2518CurrencyDerivatives2720BondsandInterestRates3021ShortRateModels3322MartingaleModelsfortheShortRate3523ForwardRateModels3924ChangeofNumeraire4224StochasticIntegralsExercise4.1(a)SinceZ(t)isdeterminist,wehavedZ(t)=αeαtdt=αZ(

3、t)dt.(b)BydefinitionofastochasticdifferentialdZ(t)=g(t)dW(t)(c)UsingItˆo’sformulaα2dZ(t)=eαW(t)dt+αeαW(t)dW(t)2α2=Z(t)dt+αZ(t)dW2(d)UsingItˆo’sformulaandconsideringthedynamicsofX(t)wehaveα2dZ(t)=αeαxdX(t)+eαx(dX(t))22122=Z(t)αµ+ασdt+ασZ(t)dW(t).2(e)Us

4、ingItˆo’sformulaandconsideringthedynamicsofX(t)wehavedZ(t)=2X(t)dX(t)+(d(X(t))2hi=Z(t)2α+σ2dt+2ZσdW(t).Exercise4.3BydefinitionwehavethatthedynamicsofX(t)aregivenbydX(t)=σ(t)dW(t).3ConsiderZ(t)=eiuX(t).ThenusingtheItˆo’sformulawehavethatthedynamicofZ(t)

5、canbedescribedby"#u2dZ(t)=−σ2(t)Z(t)dt+[iuσ(t)]Z(t)dW(t)2FromZ(0)=1weget,u2ZtZtZ(t)=1−σ2(s)Z(s)ds+iuσ(s)Z(s)dW(s).200Takingexpectationswehave,u2ZtZtE[Z(t)]=1−Eσ2(s)Z(s)ds+iuEσ(s)Z(s)dW(s)2002Ztu2=1−σ(s)E[Z(s)]ds+020BysettingE[Z(t)]=m(t)anddiffere

6、ntiatingwithrespecttotwefindanordinarydifferentialequation,∂m(t)u22=−m(t)σ(t)∂t2withtheinitialconditionm(0)=1andwhosesolutionis(Z)u2tm(t)=exp−σ2(s)ds20=E[Z(t)]hi=EeiuX(t)So,X(t)isnormallydistributed.Bythepropertiesofthenormaldistribu-tionthefollowingrel

7、ationhi2iuX(t)iuE[X(t)]−uV[X(t)]Ee=e2whereV[X(t)]isthevarianceofX(t),soitmustbethatE[X(t)]=0andRt2V[X(t)]=σ(s)ds.04Exercise4.5WehaveasubmartingaleifE[X(t)

8、Fs]≥X(s)∀,t≥s.FromthedynamicsofXwecanwriteZZttX(t)=X(s)+µ(z)dz+σ(z)dW(z).ssBytakingexpectation,c

9、onditionedattimes,frombothsideswegetZtE[X(t)

10、Fs]=E[X(s)

11、Fs]+Eµ(z)dzFssZts=X(s)+Eµ(z)dzFss

12、{z}≥0≥X(s)soXisasubmartingale.Exercise4.6SetX(t)=h(W1(t),···,Wn(t)).WehavebyItˆothatXn∂h1Xn∂2hdX(t)=dWi(t)+dWi(t)dWj(t)∂xi2∂xi∂xji=1i,j=1wher