Ch13_TheGreekLetters(金融工程学,华东师大)

Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.1TheGreekLettersChapter13Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.2Example•AFIhasSOLDfor$300,000aEuropeancallon100,000sharesofanon-dividendpayingstock:S0=49X=50r=5%=20%=13%T=20weeks•TheBlack-Scholesvalueoftheoptionis$240,000•HowdoestheFIhedgeitsrisk?Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.3Naked&CoveredPositions•Nakedposition(裸期权头寸策略)TakeNOaction•Coveredposition(抵补期权头寸策略)Buy100,000sharestodayBothstrategiesleavetheFIexposedtosignificantriskOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.4Stop-LossStrategyThisinvolves–Fullycoveringtheoptionassoonasitmovesin-the-money–Stayingnakedtherestofthetime•ThisdeceptivelysimplehedgingstrategydoesNOTworkwell!!!•Transactionscosts,discontinuityofprices,andthebid-askbouncekillsitOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.5Delta•Delta()istherateofchangeoftheoptionpricewithrespecttotheunderlying•Figure13.2(p.311)fSOptionPriceABStockPriceSlope=•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.6DeltaHedging•Thisinvolvesmaintainingadeltaneutralportfolio•ThedeltaofaEuropeancallonastockpayingdividendsatarateqis•ThedeltaofaEuropeanputis•Thehedgepositionmustbefrequentlyrebalanced•Deltahedgingawrittenoptioninvolvesa“BUYhigh,SELLlow”tradingruleqTdNe)(1qTdNe]1)([1•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.7DeltaNeutralPortfolioExample(in-the-money)Cum.CostofCostStockSharesSharesIncl.Int.WeekPriceDeltaPurch.Purch.InterestCost049.0000.52252,2002,557.82,557.82.5148.1200.458(6,400)(308.0)2,252.32.2247.3700.400(5,800)(274.7)1,979.81.91854.6200.9901,20065.55,197.35.01955.8701.0001,00055.95,258.25.12057.2501.00000.05,263.3…………………Table13.2(p.314)•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.8DeltaNeutralPortfolioExample(out-of-the-money)Cum.CostofCostStockSharesSharesIncl.Int.WeekPriceDeltaPurch.Purch.InterestCost049.0000.52252,2002,557.82,557.82.5149.7500.5684,600228.02,789.22.7252.0000.70513,700712.43,504.33.41848.1300.18312,100582.41,109.61.11946.6300.007(17,600)(820.7)290.00.32048.1200.000(700)(33.7)256.6…………………Table13.3(p.315)•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.9DeltaforFutures•FromChapter3,wehavewhereT*isthematurityoffuturescontract•Thus,thedeltaofafuturescontractis•So,ifHAistherequiredpositionintheassetfordeltahedgingandHFistherequiredpositioninfuturesforthesamedeltahedging,*00erTSF**e)e(rTrTSSSFArTArTFHHH**ee1•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.10DeltaforotherFutures•Forastockorstockindexpayingacontinuousdividend,•Foracurrency,SpeculativeMarkets,Finance665Spring2003BrianBalyeatATqrFHH*)(eATrrFHHf*)(e•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.11Gamma•Gamma()istherateofchangeofdelta()withrespecttothepriceoftheunderlying•Figure13.9(p.325)[foracallorput]22SfS•GammaStockPriceXOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.12EquationforGamma•TheGamma()foraEuropeancallorputpayingacontinuousdividendqiswhereTSdNqT01e)('2/1121e21)()('ddndN•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.13GammaAddressesDeltaHedgingErrorsCausedByCurvature•Figure13.7(p.322)CallPriceSCStockPriceS'C''C'•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.14Theta•Theta()ofaderivative(oraportfolioofderivatives)istherateofchangeofthevaluewithrespecttothepassageoftime•Figure13.6(p.321)ft0ThetaTimetoMaturityAt-the-MoneyIn-the-MoneyOut-of-the-Money•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.15EquationsforTheta•TheTheta()ofanEuropeancalloptionpayingadividendatrateqis•TheTheta()ofanEuropeanputoptionpayingadividendatrateqis•)(ee)(2e)('21010dNrKdNqSTdNSrTqTqTc)(ee)(2e)('21010dNrKdNqSTdNSrTqTqTpOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.16RelationshipAmongDelta,Gamma,and,Theta•Foranon-dividendpayingstock•ThisfollowsfromtheBlack-Scholesdifferentialequation•2212(13.7)rSSrf222122(11.15)fffrSSrftSSOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity13.17Vega•Vega()istherateofchangeofaderivativesportfoliowithrespecttovolatility•Figure14.11(p.317)[foraca