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1、單擊此處編輯母版標(biāo)題樣式,*,單擊此處編輯母版文本樣式,第二級(jí),第三級(jí),第四級(jí),第五級(jí),Lecture#9:Black-Scholesoption pricingformula,Brownian Motion,The firstformal mathematicalmodelof financial assetprices,developed byBachelier(1900),was the continuous-time randomwalk,or Brownian motion.Thiscontinuous-time processis closelyrelated to thedisc
2、rete-timeversions of therandom walk.,The discrete-time randomwalk,P,k,=P,k-1,+,k,k,=,(-,)with probability,(1-,),P,0,is fixed.Consider the following continuous time process P,n,(t),t,0,T,which is constructed from the discrete time processP,k,k=1,.nas follows:Leth=T/n anddefine the process,P,n,(t)=P,t
3、/h,=P,nt/T,t,0,T,where x denotesthegreatest integer less than orequalto x.P,n,(t)is a left continuousstepfunction.,We need toadjust,suchthatP,n,(t)will converge when ngoesto infinity.Consider themeanandvariance of P,n,(T):,E(P,n,(T)=n(2,-1),Var(P,n,(T)=4n,(,-1),2,We wish toobtain acontinuoustimevers
4、ion of therandom walk,we should expectthemeanand variance ofthelimiting process P(T)tobe linearin T.Therefore,wemusthave,n(2,-1),T,4n,(,-1),2,T,Thiscan be accomplishedby setting,=*(1+,h/,),=,h,The continuoustimelimit,It cab beshownthattheprocess P(t)has thefollowingthree properties:,1.For any t,1,an
5、d t,2,suchthat0,t,1,t,2,T:,P(t,1,)-P(t,2,),(,(t,2,-t,1,),2,(t,2,-t,1,),2.For any t,1,t,2,t,3,andt,4,suchthat0,t,1,t,2,t,1,t,2,t,3,t,4,T,the increment,P(t,2,)-P(t,1,)isstatistically independentof the increment P(t,4,)-P(t,3,).,3.The samplepathsof P(t)are continuous.,P(t)is calledarithmeticBrownianmot
6、ion orWinner process.,If weset,=0,=1,we obtain standard Brownian Motion whichis denotedas B(t).Accordingly,P(t)=,t+,B(t),Considerthefollowingmoments:,EP(t)|P(t,0,)=P(t,0,)+,(t-t,0,),VarP(t)|P(t,0,)=,2,(t-t,0,),Cov(P(t,1,),P(t,2,)=,2,min(t,1,t,2,),SinceVar(B(t+h)-B(t)/h=,2,/h,therefore,thederivativeo
7、fBrownianmotion,B,(t)doesnotexistintheordinarysense,theyarenowheredifferentiable.,Stochasticdifferentialequations,Despitethefact,theinfinitesimalincrementofBrownianmotion,thelimitofB(t+h)=B(t)ashapproachestoaninfinitesimaloftime(dt)hasearnedthenotationdB(t)andithasbecomeafundamentalbuildingblockforc
8、onstructingothercontinuoustimeprocess.Itiscalledwhitenoise.ForP(t)defineearlierwehavedP(t)=,dt+,dB(t).Thisiscalledstochasticdifferentialequation.ThenaturaltransformationdP(t)/dt=,+,dB(t)/dtdoesn,tmalesensebecausedB(t)/dtisanotwelldefined,(althroughdB(t)is).,ThemomentsofdB(t):,EdB(t)=0,VardB(t)=dt,Ed
9、BdB=dt,VardBdB=o(dt),EdBdt=0,VardBdt=o(dt),Ifwetreattermsoforderofo(dt)asessentiallyzero,the(dB),2,anddBdtarebothnon-stochasticvariables.,|dBdt,dB|dt0,dt|00,Us,ingthaboverulewecancalculate(dP),2,=,2,dt.Itisnotarandomvariable!,GeometricBrownianmotion,IfthearithmeticBrownianmotionP(t)istakentobethepri
10、ceofsomeasset,thepricemaybenegative.Thepriceprocessp(t)=exp(P(t),whereP(t)isthearithmeticBrownianmotion,iscalledgeometricBrownianmotionorlognormaldiffusion.,ItosLemma,AlthoughthefirstcompletemathematicaltheoryofBrownianmotionisduetoWiener(1923),itistheseminalcontributionofIto(1951)thatislargelyrespo
11、nsiblefortheenormousnumberofapplicationsofBrownianmotiontoproblemsinmathematics,statistics,physics,chemistry,biology,engineering,andofcourse,financialeconomics.Inparticular,ItoconstructsabroadclassofcontinuoustimestochasticprocessbasedonBrownianmotion,nowknownasItoprocessorItostochasticdifferentiale
12、quations,whichisclosedundergeneralnon-lineartransformation.,Ito(1951)provides aformula,Itos lemma forcalculatingexplicitly thestochastic differentialequationthat governsthe dynamics of f(P,t):,df(P,t)=,f/,P dP+,f/,t dt+,2,f/,P,2,(dP),2,ApplicationsinFinance,A lognormaldistributionfor stock price ret
13、urnsisthe standard model usedinfinancial economics.Givensomereasonable assumptions about therandombehaviorofstockreturns,alognormaldistribution is implied.These assumptions willcharacterize lognomal distributionina veryintuitive manner.,Let S(t)be thestocks priceat date t.We subdividedthe time horiz
14、on 0T into nequally spacedsubintervals oflength h.We writeS(ih)as S(i),i=0,1,n.Letz(i)be the continuous compounded rate of return over(i-1)h ih,ieS(i)=S(i-1)exp(z(i),i=1,2,.,n.It isclear that S(i)=S(0)expz(1)+z(2)+,+z(i).,The continuouscompoundedreturn onthestockovertheperiod 0T isthesum of theconti
15、nuously compoundedreturns over the n subintervals.,AssumptionA1.Thereturns z(j)are i.i.d.,AssumptionA2.Ez(t)=,h,where,is the expectedcontinuously compoundedreturn perunittime.,AssumptionA3.varz(t)=,2,h.,Technically,these assumptionsensure that asthetimedecrease proportionally,the behavior ofthedistr
16、ibution for S(t)dose notexplode nor degenerate to a fixedpoint.,Assumption1-3implies that for anyinfinitesimaltimesubintervals,the distributionforthe continuously compounded returnz(t)has anormal distributionwithmean,h,and variance,2,h.This impliesthatS(t)is lognormallydistributed.,Lognormaldistribution,At time t t+h,lnS,t+h,lnS,t,+(,-,2,/2)h,h,0.5,where,(m,s)denotes anormal distributionwithmeanm and standarddeviations.,Continuously compounded return,ln(S,t+h,/S,t,),(,-,2,/2)h,h,0.5,Expected ret
定價(jià)競(jìng)爭(zhēng)策略英文
