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1、Lecture 9 Black Scholesoptionpricingformula BrownianMotion Thefirstformalmathematicalmodeloffinancialassetprices developedbyBachelier 1900 wasthecontinuous timerandomwalk orBrownianmotion Thiscontinuous timeprocessiscloselyrelatedtothediscrete timeversionsoftherandomwalk Thediscrete timerandomwalk P
2、k 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 Weneedtoa
3、djust suchthatPn t willconvergewhenngoestoinfinity ConsiderthemeanandvarianceofPn T E Pn T n 2 1 Var Pn T 4n 1 2 Wewishtoobtainacontinuoustimeversionoftherandomwalk weshouldexpectthemeanandvarianceofthelimitingprocessP T tobelinearinT Therefore wemusthaven 2 1 T4n 1 2 TThiscanbeaccomplishedbysetting
4、 1 h h Thecontinuoustimelimit ItcabbeshownthattheprocessP t hasthefollowingthreeproperties 1 Foranyt1andt2suchthat0 t1 t2 T P t1 P t2 t2 t1 2 t2 t1 2 Foranyt1 t2 t3 andt4suchthat0 t1 t2t1 t2 t3 t4 T theincrementP t2 P t1 isstatisticallyindependentoftheincrementP t4 P t3 3 ThesamplepathsofP t arecont
5、inuous 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 thederivativeofBrownianmotio
6、n B t doesnotexistintheordinarysense theyarenowheredifferentiable Stochasticdifferentialequations Despitethefact theinfinitesimalincrementofBrownianmotion thelimitofB t h B t ashapproachestoaninfinitesimaloftime dt hasearnedthenotationdB t andithasbecomeafundamentalbuildingblockforconstructingotherc
7、ontinuoustimeprocess 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 d
8、Bdt o dt Ifwetreattermsoforderofo dt asessentiallyzero the dB 2anddBdtarebothnon stochasticvariables dBdtdB dt0dt 00Usingthaboverulewecancalculate dP 2 2dt Itisnotarandomvariable GeometricBrownianmotion IfthearithmeticBrownianmotionP t istakentobethepriceofsomeasset thepricemaybenegative Thepricepro
9、cessp t exp P t whereP t isthearithmeticBrownianmotion iscalledgeometricBrownianmotionorlognormaldiffusion Ito sLemma AlthoughthefirstcompletemathematicaltheoryofBrownianmotionisduetoWiener 1923 itistheseminalcontributionofIto 1951 thatislargelyresponsiblefortheenormousnumberofapplicationsofBrownian
10、motiontoproblemsinmathematics statistics physics chemistry biology engineering andofcourse financialeconomics Inparticular ItoconstructsabroadclassofcontinuoustimestochasticprocessbasedonBrownianmotion nowknownasItoprocessorItostochasticdifferentialequations whichisclosedundergeneralnon lineartransf
11、ormation Ito 1951 providesaformula Ito slemmaforcalculatingexplicitlythestochasticdifferentialequationthatgovernsthedynamicsoff P t df P t f PdP f tdt 2f P2 dP 2 ApplicationsinFinance Alognormaldistributionforstockpricereturnsisthestandardmodelusedinfinancialeconomics Givensomereasonableassumptionsa
12、bouttherandombehaviorofstockreturns alognormaldistributionisimplied Theseassumptionswillcharacterizelognomaldistributioninaveryintuitivemanner LetS t bethestock spriceatdatet Wesubdividedthetimehorizon 0T intonequallyspacedsubintervalsoflengthh WewriteS ih asS i i 0 1 n Letz i bethecontinuouscompoun
13、dedrateofreturnover 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 istheexpectedcontinuousl
14、ycompoundedreturnperunittime AssumptionA3 var z t 2h Technically theseassumptionsensurethatasthetimedecreaseproportionally thebehaviorofthedistributionforS t dosenotexplodenordegeneratetoafixedpoint Assumption1 3impliesthatforanyinfinitesimaltimesubintervals thedistributionforthecontinuouslycompound
15、edreturnz t hasanormaldistributionwithmean h andvariance 2h ThisimpliesthatS t islognormallydistributed LognormaldistributionAttimet t hlnSt h lnSt 2 2 h h0 5 where m s denotesanormaldistributionwithmeanmandstandarddeviations Continuouslycompoundedreturnln St h St 2 2 h h0 5 ExpectedreturnsEt ln St
16、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 2isanestimatorfor h1 2 Example Dailyreturns Fundamentalequationforderivativesecurities StockpricefollowsItoprocess dS S t dt S t dzAtthispoint weassume S t S and S t SLetC S t beaderivativesecurity accordingtoIto slemma th
