定价策略：Black-Scholes option pricing formula

Lecture#9:Black-ScholesoptionpricingformulaBrownianMotionThefirstformalmathematicalmodeloffinancialassetprices,developedbyBachelier(1900),wasthecontinuous-timerandomwalk,orBrownianmotion.Thiscontinuous-timeprocessiscloselyrelatedtothediscrete-timeversionsoftherandomwalk.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]denotesthegreatestintegerlessthanorequaltox.Pn(t)isaleftcontinuousstepfunction.Weneedtoadjust,suchthatPn(t)willconvergewhenngoestoinfinity.ConsiderthemeanandvarianceofPn(T):E(Pn(T))=n(2-1)Var(Pn(T))=4n(-1)2Wewishtoobtainacontinuoustimeversionoftherandomwalk,weshouldexpectthemeanandvarianceofthelimitingprocessP(T)tobelinearinT.Therefore,wemusthaven(2-1)T4n(-1)2TThiscanbeaccomplishedbysetting=½*(1+h/),=hThecontinuoustimelimitItcabbeshownthattheprocessP(t)hasthefollowingthreeproperties:1.Foranyt1andt2suchthat0t1t2T:P(t1)-P(t2)((t2-t1),2(t2-t1))2.Foranyt1,t2,t3,andt4suchthat0t1t2t1t2t3t4T,theincrementP(t2)-P(t1)isstatisticallyindependentoftheincrementP(t4)-P(t3).3.ThesamplepathsofP(t)arecontinuous.P(t)iscalledarithmeticBrownianmotionorWinnerprocess.Ifweset=0,=1,weobtainstandardBrownianMotionwhichisdenotedasB(t).Accordingly,P(t)=t+B(t)Considerthefollowingmoments:E[P(t)|P(t0)]=P(t0)+(t-t0)Var[P(t)|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)doesnotexistintheordinarysense,theyarenowheredifferentiable.StochasticdifferentialequationsDespitethefact,theinfinitesimalincrementofBrownianmotion,thelimitofB(t+h)=B(t)ashapproachestoaninfinitesimaloftime(dt)hasearnedthenotationdB(t)andithasbecomeafundamentalbuildingblockforconstructingothercontinuoustimeprocess.Itiscalledwhitenoise.ForP(t)defineearlierwehavedP(t)=dt+dB(t).Thisiscalledstochasticdifferentialequation.ThenaturaltransformationdP(t)/dt=+dB(t)/dtdoesn’tmalesensebecausedB(t)/dtisanotwelldefined(althroughdB(t)is).ThemomentsofdB(t):E[dB(t)]=0Var[dB(t)]=dtE[dBdB]=dtVar[dBdB]=o(dt)E[dBdt]=0Var[dBdt]=o(dt)Ifwetreattermsoforderofo(dt)asessentiallyzero,the(dB)2anddBdtarebothnon-stochasticvariables.|dBdtdB|dt0dt|00Usingthaboverulewecancalculate(dP)2=2dt.Itisnotarandomvariable!GeometricBrownianmotionIfthearithmeticBrownianmotionP(t)istakentobethepriceofsomeasset,thepricemaybenegative.Thepriceprocessp(t)=exp(P(t)),whereP(t)isthearithmeticBrownianmotion,iscalledgeometricBrownianmotionorlognormaldiffusion.Ito’sLemmaAlthoughthefirstcompletemathematicaltheoryofBrownianmotionisduetoWiener(1923),itistheseminalcontributionofIto(1951)thatislargelyresponsiblefortheenormousnumberofapplicationsofBrownianmotiontoproblemsinmathematics,statistics,physics,chemistry,biology,engineering,andofcourse,financialeconomics.Inparticular,ItoconstructsabroadclassofcontinuoustimestochasticprocessbasedonBrownianmotion–nowknownasItoprocessorItostochasticdifferentialequations–whichisclosedundergeneralnon-lineartransformation.Ito(1951)providesaformula–Ito’slemmaforcalculatingexplicitlythestochasticdifferentialequationthatgovernsthedynamicsoff(P,t):df(P,t)=f/PdP+f/tdt+½2f/P2(dP)2ApplicationsinFinanceAlognormaldistributionforstockpricereturnsisthestandardmodelusedinfinancialeconomics.Givensomereasonableassumptionsabouttherandombehaviorofstockreturns,alognormaldistributionisimplied.Theseassumptionswillcharacterizelognomaldistributioninaveryintuitivemanner.LetS(t)bethestock'spriceatdatet.Wesubdividedthetimehorizon[0T]intonequallyspacedsubintervalsoflengthh.WewriteS(ih)asS(i),i=0,1,…,n.Letz(i)bethecontinuouscompoundedrateofreturnover[(i-1)hih],ieS(i)=S(i-1)exp(z(i)),i=1,2,..,n.ItisclearthatS(i)=S(0)exp[z(1)+z(2)+…+z(i)].Thecontinuouscompoundedreturnonthestockovertheperiod[0T]isthesumofthecontinuouslycompoundedreturnsoverthensubintervals.AssumptionA1.Thereturns{z(j)}arei..i.d.AssumptionA2.E[z(t)]=h,whereistheexpectedcontinuouslycompoundedreturnperunittime.AssumptionA3.var[z(t)]=2h.Technically,theseassumptionsensurethatasthetimedecreaseproportionally,thebehaviorofthedistributionforS(t)dosenotexplodenordegeneratetoafixedpoint.Assumption1-3impliesthatforanyinfinitesimaltimesubintervals,thedistributionforthecontinuouslycompoundedreturnz(t)hasanormaldistributionwithmeanh,andvariance2h.ThisimpliesthatS(t)islognormallydistributed.LognormaldistributionAttimett+hlnSt+h~[lnSt+(-2/2)h,h0.5]where(m,s)denotesanormaldistributionwithmeanmandstandarddeviations.Continuouslycompoundedreturnln(St+h/St)~[(-2/2)h,h0.5]ExpectedreturnsEt[ln(St+h/St)]=(-2/2)hEt[St+h/St]=exp(h)VarianceofreturnsVart[ln(St+h/St)]=2hVart[St+h/St]=exp(2h)(exp(2h)-1)Estimationofn+1:numberofstockobservationsSj:stockpriceattheendofjthinterval,j=1,…nh:lengthoftimeintervalsinyearsLetuj=ln[Sj+Dj)/Sj-1]u=(u1+…+un)/nisanestimatorfor(-2/2)h,s={[(u1-u)2+…+(un-u)2]/(n-1)}1/2isanestimato