清华大学金融学LessonBlack-Scholes

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1、1.股票价格运动规律a.BrownianMotionBtb.Adaptivefunction:measurablew.r.tIto’sIntegralStratonovichIntegralWithdrift:c.股票运动规律Ito’sprocessIto’sLemma:伊藤引理如果f=f(St)是St的二阶可导函数,且二阶导数连续,则f(St)是个伊藤过程,且有Informalproof:对f(s,t)作Taylor展开把dS的表达式代入上式,得2.DerivationofBS6/61.Frictionl

2、essmarket1)Notax,notransactioncost2)assetsareinfinitelydivisible3)Noshortsellrestriction2.Constantriskfreerateof3.Nodividend4.log-normaldistributionconstant,2.DerivationofBS:f:thepriceofacalloptionf(s,t):afunctionofs,t.FromIto’slemma:HedgePortfolio:-1:opti

3、on:sharesofunderlyingFeynman-Kac:R-Nderivative:3.风险中性6/6(1)Feynman-KacFormula如果u(x,t)是偏微分方程的解:边界条件那么其中(2)由于Feynman-Kac可导出Q:风险中性概率,在风险中性世界里,所有资产的预期收益率为,f与无关,最未来现金流预期用无风险利率r的折现｡(3)解STProof:由Ito’slemma:6/6同理可得卖权价值:(或由Putcallparity可知)4.BS公式的推广(1)标的物股票具有已知红利率He

4、dgeportfolio:-1:option(2)Currencyoptions(3)Optionsonfutures(采用盯市交易规则Mark-to-market)6/65.Greeks(1)S(2)SOutofomneyAtthemoneyInthemoneyT-t(3)TimeDecayForEuropeanCallw/odividendTheta(4)6.Volatility(1)Impliedvol(2)Stochasticvolatility6/67.Portforlioinsurance(Ca

5、se:Leland,O’BrienandRubinsteinAssociatesIncorporated:Portfolioinsurance.HBS294-061)(1)WhoarethepotentialbuyersofPI?(2)WhataretheinitialdifficultieswhensellingPI?(3)Whydidn’tLORsimplysellaputanddynamicallyhedgetheshortputposition?(4)IndexFutures(5)Whatisthe

6、destabilizingeffectbehindPI?6/6