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1、StochasticCalculusandProcessesFE5204LectureNotes7ApplicationstoMathematicalFinancex1.Introduction.Take(•;F;IP;fFtgt¸0)tobea¯lteredprobabilityspace,ofwhichthe¯ltrationfFtgt¸0satis¯estheusualconditions.Onthis¯lteredprobabilityspace,wewillde¯neamathematicalmarketinthi
2、ssetofnotes.Tobeginwith,letusconsiderthecelebratedBlack-Scholesmodel,i.e.,amarketwithtwoinvestmentpossibilitiesS(¢)=(S0(¢);S1(¢))suchthat(a)Bond:dS0(t)=rS0(t)dtwithS0(0)=1,and(b)Stock:dS1(t)=¹S1(t)dt+¾S1(t)dBt,(wherefBt;t¸0gisstandardBrownianmotionon(•;F;IP).Wewill
3、alsoassumethatthe¯ltrationisgeneratedbythisBrownianmotion.)(ThisisknownasaBlack-Scholesmodel.)Obviously,S(t)=ert.0IthasbeenshownearlierthatitcanbesolvedforS1(t),whichistheso-calledgeometricBrownianmotiongivenby(¹¡1¾2)t+¾BtS1(t)=S1(0)e2:11.Portfolio.µ(t)=(®(t);¯(t))
4、,ofwhich²®(t)denotesthenumberofbondsheldattimet,while²¯(t)denotesthenumberofstocksheldatt.Givenµ=(®;¯),thecorrespondingwealthWµ(t)attimetisWµ(t)=µ(t)¢S(t)=®(t)S(t)+¯(t)S(t);(1:1)01where¢"inµ(t)¢S(t)denotestheusualinnerproductinEuclidianspaces.Foreasynotation,wewil
5、ldenotebyW(t)thecorrespondingwealthwheneverthereisnoconfusion.Theportfolioissaidtobeself-¯nancingifdWµ(t)=µ(t)¢dS(t)=®(t)dS(t)+¯(t)dS(t);(1:2)01(whichmeansthatnomoneyisbroughtinortakenoutfromthesystem).Equivalently,ZtWµ(t)=Wµ(0)+µ(s)¢dS(s)0ZZ(1:3)tt=Wµ(0)+®(s)dS(s)
6、+¯(s)dS(s):0100Remark.Supposethatµ=(®;¯)isalsoanIt^oprocess.SinceWµ(t)=®(t)S(t)+¯(t)S(t);01accordingtoIt^o'sformula,wehavedWµ(t)=S(t)d®(t)+®(t)dS(t)+dh®;Si000t+S1(t)d¯(t)+¯(t)dS1(t)+dh¯;S1it:So,undertheassumptionthatµ=(®;¯)isanIt^oprocess,theportfolioisself-¯nancin
7、gifandonlyifS0(t)d®(t)+dh®;S0it+S1(t)d¯(t)+dh¯;S1it´0:22.Question.WhatistherelationbetweentheterminalwealthW(T)andtheself-¯nancingportfolioµ=(®;¯)?Toseethismoreclearly,notethatby(1.1)wehaveW(t)¡¯(t)S1(t)®(t)=:(1:4)S0(t)Substitutingitinto(1.2),onegetsW(t)¡¯(t)S1(t)d
8、W(t)=dS0(t)+¯(t)dS1(t):(1:5)S0(t)SincedS0(t)=S0(t)=rdt,(1.5)canbere-writtenasdW(t)=rW(t)dt¡r¯(t)S1(t)dt+¯(t)[¹S1(t)dt+¾S1(t)dBt]:Hence,·¸¹¡rdW(t)
