清华大学金融学Lesson6BlackScholes

1、文档根源为:从网络采集整理.word版本可编写.支持.1股票价钱运动规律a.BrownianMotionBtb.Adaptivefunction:measurableFttNfw,tdBtlimf(wti,ti)(BtidtiBti)ItosIntegral0Nt0limf(w1)(BtidtBti)StratonovichIntegralti2dtiWithdrift:dXt(wt,t)dtf(wt,t)dBt股票运动规律St0S0dStuwt,tdtwt,tdBtItosprocessdStuSt,tdtSt,tdBtItosLemma:伊藤引理假如f=f(St)是St的二阶可导函数，且二

2、阶导数连续，则f(St)是个伊藤过程，且有Informalproof:对f(s,t)作Taylor睁开把dS的表达式代入上式，得2.DerivationofBS1.FrictionlessmarketNotax,notransactioncostassetsareinfinitelydivisibleNoshortsellrestrictionConstantriskfreerateofrfNodividendlog-normaldistributionconstant,2.DerivationofBS:f:thepriceofacalloptionf(s,t):afunctionofs,t.

3、FromItoslemma:HedgePortfolio:-1:option:sharesofunderlyingsFeynman-Kac:Qrw)R-Nderivative:SE(P3.风险中性1文档根源为:从网络采集整理.word版本可编写.支持.Feynman-KacFormula假如u(x,t)是偏微分方程的解:u(x,t)u12(x,t)2uq(x)u0界限条件tx2x2u(x,T)(x)Tq(xs)dsXtx那么u(x,t)E(XT)et此中dXtx,tdtx,tdBt(2)因为Feynman-Kac可导出风险中性概率,在风险中性世界里,全部财产的预期利润率为rf,f与没关,最将来

4、现金流预期用无风险利率r的折现。3）解STProof:由Itoslemma:同理可得卖权价值：（或由Putcallparity可知）4.BS公式的推行（1）标的物股票拥有已知盈利率Hedgeportfolio:dSSdtSdwt-1:optionCurrencyoptionsOptionsonfutures(采纳盯市交易规则Mark-to-market)5.Greeks(1)CCN(d1)SSPPN(d1)S(2)c2cN(d1)SS2STtCPSXer(Tt)CP1SS2c2P0S2S2CPS2文档根源为:从网络采集整理.word版本可编写.支持.f(3)TimeDecaytOutofomn

5、eyForEuropeanCallw/odividendThetac(4)cSTtN(d1)0CPAtthemoney6.VolatilityInthemoneyT-t(1)ImpliedvolV(2)Stochasticvolatility7.Portforlioinsurance(Case:Leland,OBrienandRubinsteinAssociatesIncorporated:Portfolioinsurance.HBS294-061)WhoarethepotentialbuyersofPI?WhataretheinitialdifficultieswhensellingPI?WhydidntLORsimplysellaputanddynami