定价竞争策略英文.ppt

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1、Lecture#9:Black-Scholesoptionpricingformula·BrownianMotionThefirstformalmathematicalmodeloffinancialassetprices,developedbyBachelier(1900),wasthecontinuous-timerandomwalk,orBrownianmotion.Thiscontinuous-timeprocessiscloselyrelatedtothediscrete-timeversionsoftherandomwal

2、k.·Thediscrete-timerandomwalkPk=Pk-1+k,k=(-)withprobability(1-),P0isfixed.ConsiderthefollowingcontinuoustimeprocessPn(t),t[0,T],whichisconstructedfromthediscretetimeprocessPk,k=1,..nasfollows:Leth=T/nanddefinetheprocessPn(t)=P[t/h]=P[nt/T],t[0,T],where[x]denotes

3、thegreatestintegerlessthanorequaltox.Pn(t)isaleftcontinuousstepfunction.Weneedtoadjust,suchthatPn(t)willconvergewhenngoestoinfinity.ConsiderthemeanandvarianceofPn(T):E(Pn(T))=n(2-1)Var(Pn(T))=4n(-1)2Wewishtoobtainacontinuoustimeversionoftherandomwalk,weshouldexpe

4、ctthemeanandvarianceofthelimitingprocessP(T)tobelinearinT.Therefore,wemusthaven(2-1)T4n(-1)2TThiscanbeaccomplishedbysetting=½*(1+h/),=h·ThecontinuoustimelimitItcabbeshownthattheprocessP(t)hasthefollowingthreeproperties:1.Foranyt1andt2suchthat0t1

5、t1)-P(t2)((t2-t1),2(t2-t1))2.Foranyt1,t2,t3,andt4suchthat0t1

6、andardBrownianMotionwhichisdenotedasB(t).Accordingly,P(t)=t+B(t)Considerthefollowingmoments:E[P(t)

7、P(t0)]=P(t0)+(t-t0)Var[P(t)

8、P(t0)]=2(t-t0)Cov(P(t1),P(t2)=2min(t1,t2)SinceVar[(B(t+h)-B(t))/h]=2/h,therefore,thederivativeofBrownianmotion,B’(t)doesnotexistintheordi

9、narysense,theyarenowheredifferentiable.·StochasticdifferentialequationsDespitethefact,theinfinitesimalincrementofBrownianmotion,thelimitofB(t+h)=B(t)ashapproachestoaninfinitesimaloftime(dt)hasearnedthenotationdB(t)andithasbecomeafundamentalbuildingblockforconstructingot

10、hercontinuoustimeprocess.Itiscalledwhitenoise.ForP(t)defineearlierwehavedP(t)=dt+dB(t).Thisiscalledstochasti