波动率互换与方差互换定价问题研究

上海交通大学硕士学位论文波动率互换与方差互换定价问题研究姓名：曹智申请学位级别：硕士专业：概率论与数理统计指导教师：林建忠20081201硕士学位论文波动率互换与方差互换定价问题研究PricingofVolatilityandVarianceSwaps曹智上海交通大学2008年12月硕士学位论文波动率互换与方差互换定价问题研究PricingofVolatilityandVarianceSwaps学科专业：概率论与数理统计作者姓名：曹智作者学号：1060719044指导教师：林建忠培养单位：上海交通大学2008年12月Ornstein-Uhlenbeck–I–AbstractThemodernfinancialtheoryisgettingmaturewiththedevelopmentoffinancialmarkets.Oneofthemostoutstandingfeaturesshowedduringtheprogressoffinancialtheoriesisthatmoreandmorequantitativemethodologieshavebeenusedinthefieldsofcapitalmarket,assetpricing,riskmanagement,etc.Reviewingthequantificationprocessofmodernfinance,volatilityisoneoftheessentialissuesallalong.Inthefieldsofderivativepricing,portfoliomanagement,investingandhedgingstrategies,volatilitytakesaveryimportantposition.Previouslynomatterinvestinginvolatil-ityorhedgingvolatilityrisk,wecanonlyachievethisindirectlybyusingoptions.Theemergenceofvolatilityderivativessuchasvarianceswaps,volatilityswapsandvolatilityoptionshavechangedthissituation.Theyprovideapureexposuretovolatil-itywhichmeanstheyaremoreefficientanddirectfortheinvestorswhowanttoinvestinvolatilityorhedgevolatilityrisk.Thispaperinvestigatedthepricingofvarianceswapsandvolatilityswaps,providedanumericalexampletoillustratethepricingprocess,andalsoderivedthepricingformulaforvarianceswapsandvolatilityswapsunderstochasticvolatilitymodel.Keywordsvolatilityswaps;varianceswaps;stochasticvolatility–II–.......................................................................IAbstract.....................................................................II1.................................................................11.1......................................................11.2..............................................................22.....................................32.1Brown...........................................................32.2...........................................................42.2.1Ito..........................................................42.2.2Ito..........................................................52.3.....................................................................62.4..............................................................72.5...........................................................82.6.........................................................93....................................................133.1...............................................133.2..........................................143.2.1SteinandStein...............................................143.2.2Heston......................................................154.............................................174.1..................................................................174.1.1......................................................174.1.2.........................................................184.2.................................................194.3............................................235............................................................256.................326.1...........................................................326.2..............................................................34–III–7.................................................................37....................................................................38.......................................................................40–IV–111.1DeltaDemeterfiCarrWuDemeterfi–1–Ornstein-Uhlenbeck1.223456Ornstein-Uhlenbeck7–2–22Bachelier1900BrownItoBlack-ScholesBrown2.1BrownBrownBrown2.1:(­FP)Wtt¸0Brown(1)WtW0=0(2)PWtN(0;t)(3)PWs+t¡WsN(0;t)FsBrown(1)Wt1(2)Brown10(3)Brown–3–(4)BrownBrownBrownWienerBrownBrown0St=Wt+¹tBrown¹BrownSt=¾Wt+¹t¾1015%5015%Xt=e(¹t+¾Wt)Brown2.22.2.1Ito¢t[0;T]0=t0·t1·¢¢¢·tn=T¢t[tj;tj+1)tk·t·tk+1It=k¡1Xj=0¢tj(Wtj+1¡Wtj)+¢tk(Wt¡Wtk)–4–2It¢tItoIt=Zt0¢udWuIto¢t0·t·TFtEZT0¢2tdt1ItoZt0¢udWu=limn!1Zt0¢u;ndWu¢u;nn!1limn!1EZT0j¢t;n¡¢tj2dt=02.1:It=Rt0¢udWuIto(1)It(2)ItFt(3)It=Rt0¢udWuJt=Rt0¡udWuIt§Jt=Rt0(¢u§¡u)dWu(4)It(5)EI2t=ERt0¢2udu(6)[It;It]=Rt0¢2udu2.2.2ItoItoIto2.2:Wtt¸0Brown¢t£tFtXt=X0+Zt0¢udWu+Zt0£udu–5–ItoItodXt=¢tdWt+£tdtItoIto2.2:Xtt¸0Itof(t;x)ft(t;x)fx(t;x)fxx(t;x)df(t;Xt)=ft(t;Xt)dt+fx(t;Xt)dXt+12fxx(t;Xt)dXtdXtItodXt=¢tdWt+£tdtdXtdXt=¢2tdtdf(t;Xt)=ft(t;Xt)dt+fx(t;Xt)¢tdWt+fx(t;Xt)£tdt+12fxx(t;Xt)¢2tdt(2-1)2.32.3:(­FP)TFt0·t·TF¾Mt0·s·t·T(1)E[MtjFs]=Ms(2)E[MtjFs]¸Ms–6–2(3)E[MtjFs]·MsBrown0·s·tE[WtjFs]=E[(Wt¡Ws)+WsjFs]=E[Wt¡WsjFs]+E[WsjFs]=E[Wt¡Ws]+Ws=Ws(2-2)Brown2.42.3:(­FP)Z1E[Z]=1A2F»P(A)=ZAZ(!)dP(!)(2-3)»PX»E[X]=E[XZ]Z1YE[Y]=»E[YZ]»E»P–7–2.4:P»P(­F)A2FP(A)=0,»P(A)=02.4:P»P(­F)1ZE[Z]=1A2F»P(A)=ZAZ(!)dP(!)Z=d»PdP»PPRadon-Nikodym2.5:Wt0·t·T(­FP)Brown£t0·t·TFtZt=exp(¡Zt0£udWu¡12Zt0£2udu)»Wt=Wt+Zt0£uduEZT0£2uZ2udu1Z=ZTE[Z]=1(2-3)»P»WtBrownGirsanov2.52.6:Wt0·t·T(­FP)BrownFtBrownMt¡Mt=M0+Zt0¡udWu–8–22.6Wt0·t·T(­FP)BrownFtBrowndSt=®tStdt+¾tStdWt(2-4)®t¾tFt¾t1BrownSt=S0exp[Zt0¾sdWs+Zt0(®s¡12¾2s)ds]RtDt=e¡Rt0RsdsdDt=¡RtDtdtDtSt=S0exp[Zt0¾sdWs+Zt0(®s¡Rs¡12¾2s)ds]Itod(DtSt)=(®t¡Rt)DtStdt+¾tDtStdWt®t¡Rt®tRt£t=®t¡Rt¾t–9–d(D