Black_Scholes期权定价的风险_英文_

201012ChineseJournalofAppliedProbabilityandStatisticsVol.26No.6Dec.2010TheRiskofBlack-ScholesOptionPricing¤XuSong(SchoolofFinanceandStatistics,EastChinaNormalUniversity,Shanghai,200241)(DepartmentofMathematicsandComputerScience,HuainanNormalUniversity,Huainan,232001)AbstractTheBlack-Scholesoptionpricingformulaisderivedwiththeassumptionthatthehedgingiscontinuous.Inpractice,thereisnotradingwhenthestockmarketisclosed.Sotheadjustmentofportfoliosisdiscontinuous,andtheriskofoptionpricingexists.Weconsidertheriskofoptionpricingcausedbythiskindofdiscontinuoushedging,andgivetheratioofriskofseveraloptionsinAmericanstockmarket.Wecanseethattheratioismostlyexceed5%andtheriskoftraditionalpricingmethodofoptioncan'tbeignored.Keywords:Black-Scholesmodel,hedging,risk,optionpricing.AMSSubjectClassi¯cation:91B28,60G46.x1.IntroductionThepricingofoptionisoftenbasedoncontinuoustimemodelsuchasBlack-Scholesmodel.Toobtainthepriceofanoption,oneconstructsrisklessportfoliosandassumesthehedgingiscontinuous.Buttheinvestorcan'thedgecontinuouslyinthecurrentworld.Discretetimehedgingandtheerrorincludingboththeequidistanttime-netandthegeneraltime-netwithagivencardinalityn(n!1)havebeenconsideredbyGobetandTemam(2001),Geiss(1999,2002)andHujo(2006).Butthecasethattheinvestorcan'tadjustportfolioswhenthestockmarketisclosedisnotconsidered.Sincethereisnotradingwhenthestockmarketisclosed,thetime-netwithouttradingistheperiodofthestockmarketclosed,itisevenlongerthanthetradingtimeeveryday.Inthispaper,weconsidertheriskofoptionpricingcausedbythiskindofdiscontinuoushedging.Wecanseefromthesampleoptionstheriskisatleastfrom3.13%to37.08%,itistoolargetoneglect.Furthermore,undercertaincondition,theboundoftheriskshouldbelargerwiththedecreaseoftherisk-freeinterestrater,justasinthecurrently¯nancialcrisis.¤TheprojectsupportedbyKeyProgramsforNaturalScienceFoundationofAnhuiProvince(KJ2010A234),NaturalScienceFoundationinAnhuiUniversities(KJ2010B451),Qualityofteachingandeducationalreformcon-structionprojectinHuainanNormalUniversity(TSZY200902)andKeysubjectconstructionfundsofappliedmathematicsinHuainanNormalUniversity.ReceivedFebruary24,2009.RevisedMarch22,2010.:Black-Scholes663In1973,FischerBlackandMyronScholesgavethefamousBlack-Scholesmodel:dSt=¹Stdt+¾StdfWt;¹;¾areconstants,(1.1)wherefWtisastandardBrownianmotionon(­;F;P).(St)t2Tisde¯nedon(­;F;P),andisadaptedtoa¯ltrationF=(Ft)t2T,whereTisatimeindexsetandTµ[0;T]forsomeT0.Undertheneutral-riskprobabilityQ,themodelisdSt=rStdt+¾StdWt,thatisSt=S0e¾Wt+(r¡¾2=2)t,whereWtisastandardBrownianmotionwithrespecttoQandWt=fWt+[(¹¡r)=¾]t,ristherisk-freeinterestrate,and¾isthevolatilityoftheunderlyingstock.Mathematically,thepriceoftheoptionisgivenby:c=E[e¡rTf(ST)jF0],wherefisthepayo®function,f2L2(ST),Ft=¾(Ws;0·s·t),t2[0;1).EdenotestheexpectationcorrespondingtoQ.TakeV(t;y)=Ey(e¡r(T¡t)f(ST¡t)),thenc=V(0;y)andV(t;y)solvestheCauchyproblem:8:@@tV(t;y)+12¾2y2@2@y2V(t;y)+ry@@yV(t;y)¡rV(t;y)=0;(t;y)2[0;T)£(0;1);V(T;y)=f(y);y2(0;1):LeteV=e¡rtV,x=e¡rty,then8:@@teV(t;x)+12¾2x2@2@x2eV(t;x)=0;(t;x)2[0;T)£(0;1);eV(T;x)=e¡rTf(erTx);x2(0;1):NowapplyIt^o'sformulatoeV(t;x),weconcludee¡rTf(ST)=c+ZT0'tdeSt;(1.2)whereeSt=e¡rtStisthediscountedpriceoftheriskyasset,anddeSt=¾eStdWt,anditiseasilyknowneStisamartingalewithrespecttothe¯ltrationFandtheprobabilitymeasureQ;thedeltahedgingstrategy'tisgivenby't=(@=@y)V(t;St),0·t·T,and'2C1;2([0;T)£R+),whereCp;q([a;b)£B)=ff:[a;b)£B!R,allpossiblepartialderivativesoff,whereonedi®erentiatesatmostp-timeswithrespecttothe¯rstvariableandatmostq-timeswithrespecttothesecondone,existandcontinuouson[a;b)£Bg.Takeªt=ª(t;St)=('t¡Àa)¾eSt,0·a·t·b,whereaissomeclosetime,bisthenextopentimeaftera.Àa2Fa,Àaistheoptimalvariancehedgingstrategy,thatis,664consideringwecan'ttradejustinthetimeinterval(a;b),Àaisthehedgingstrategye'wecantakeattimeawhichminimizeESa¯¯¯Zba('t¡e't)deSt¯¯¯2:ItisknownbyZhang[5]Àa=E³Zba'ueS2udujFa´E³ZbaeS2udujFa´:(1.3)WeconsiderEuropeancalloption.Assumethepriceoftheunderlyingstocksatisfy(1.1)andsupu2[0;T]Ejª(u;Su)j21.ForanEuropeancalloptionwithmaturityTandstrikepriceK,V(t;St)=StN(d1)¡Ke¡r(T¡t)N(d2);(1.4)whered1=logStK+³r+¾22´(T¡t)¾pT¡t;d2=d1¡¾pT¡t:N(¢)isthestandardnormalcumulativeprobabilitydistributionfunction,i.e.N(x)=Zx¡11p2¼e¡t2=2dt:TakeAmericanstockmarketforexample,takingnoaccountoflegalholidaysexceptweekends,thedurationofopenis6.5hours,andoftheclose17.5hoursintradingdays,andthemarketisclosedinweekends.Wechoosetheclosetimeofthestockmarketinthechosendayastheoriginaltimet0.LetN=365T;t0i=1365³i+17:524´;ti=1365i;i=0;1;¢¢¢;N¡1:Thentiistheclosetimeofthei'thday,t0iistheopentimeofthe(i+1)'thday.Be-causewecan'ttradewhenthemarketisclosed,wetaketheexercisableZT0'dtdeStastheapproximationofZT0'tdeSt.Inotherwords,thetraditionaloptionpricingequationisob-tainedbytheassumptionthatthehedgingiscontinuoussothattheconstructedportfolioisriskless,tohaveaperfecthedging,theinvestormusttradeateachtimet2[0;T]andhold'tunitsoftheunderlyingassetasin(1.2).Butinpracticeonecan'ttradewhenthestockmarketisclosed.Sotheadjustmentofportfoliosisdiscontinuou