投资学第21章期权定价.ppt

CHAPTER21 OptionValuation Intrinsicvalue profitthatcouldbemadeiftheoptionwasimmediatelyexercisedCall stockprice exercisepricePut exerciseprice stockpriceTimevalue thedifferencebetweentheoptionpriceandtheintrinsicvalue OptionValues Figure21 1CallOptionValuebeforeExpiration Table21 1DeterminantsofCallOptionValues RestrictionsonOptionValue Call ValuecannotbenegativeValuecannotexceedthestockvalueValueofthecallmustbegreaterthanthevalueofleveredequityC S0 X D 1 rf TC S0 PV X PV D Figure21 2RangeofPossibleCallOptionValues Figure21 3CallOptionValueasaFunctionoftheCurrentStockPrice Figure21 4PutOptionValuesasaFunctionoftheCurrentStockPrice 100 120 90 StockPrice C 10 0 CallOptionValueX 110 BinomialOptionPricing TextExample AlternativePortfolioBuy1shareofstockat 100Borrow 81 82 10 Rate Netoutlay 18 18PayoffValueofStock90120Repayloan 90 90NetPayoff030 18 18 30 0 PayoffStructureisexactly3timestheCall BinomialOptionPricing TextExampleContinued 18 18 30 0 C 30 0 3C 18 18C 6 06 BinomialOptionPricing TextExampleContinued AlternativePortfolio oneshareofstockand3callswritten X 110 PortfolioisperfectlyhedgedStockValue90120CallObligation0 30Netpayoff9090Hence100 3C 81 82orC 6 06 ReplicationofPayoffsandOptionValues GeneralizingtheTwo StateApproach Assumethatwecanbreaktheyearintotwosix monthsegmentsIneachsix monthsegmentthestockcouldincreaseby10 ordecreaseby5 Assumethestockisinitiallysellingat100Possibleoutcomes Increaseby10 twiceDecreaseby5 twiceIncreaseonceanddecreaseonce 2paths GeneralizingtheTwo StateApproachContinued 100 110 121 95 90 25 104 50 AssumethatwecanbreaktheyearintothreeintervalsForeachintervalthestockcouldincreaseby5 ordecreaseby3 Assumethestockisinitiallysellingat100 ExpandingtoConsiderThreeIntervals S S S S S S S S S S ExpandingtoConsiderThreeIntervalsContinued PossibleOutcomeswithThreeIntervals EventProbabilityFinalStockPrice3up1 8100 1 05 3 115 762up1down3 8100 1 05 2 97 106 941up2down3 8100 1 05 97 2 98 793down1 8100 97 3 91 27 Figure21 5ProbabilityDistributions Co SoN d1 Xe rTN d2 d1 ln So X r 2 2 T T1 2 d2 d1 T1 2 whereCo CurrentcalloptionvalueSo CurrentstockpriceN d probabilitythatarandomdrawfromanormaldistributionwillbelessthand Black ScholesOptionValuation X Exercisepricee 2 71828 thebaseofthenaturallogr Risk freeinterestrate annualizescontinuouslycompoundedwiththesamematurityastheoption T timetomaturityoftheoptioninyearsln Naturallogfunction Spoundedrateofreturnonthestock Black ScholesOptionValuationContinued Figure21 6AStandardNormalCurve So 100X 95r 10T 25 quarter 50d1 ln 100 95 10 52 2 5 251 2 43d2 43 5 251 2 18 CallOptionExample N 43 6664Table21 2dN d 42 6628 43 6664Interpolation 44 6700 ProbabilitiesfromNormalDistribution N 18 5714Table21 2dN d 16 5636 18 5714 20 5793 ProbabilitiesfromNormalDistributionContinued Table21 2CumulativeNormalDistribution Co SoN d1 Xe rTN d2 Co 100X 6664 95e 10X 25X 5714Co 13 70ImpliedVolatilityUsingBlack Scholesandtheactualpriceoftheoption solveforvolatility Istheimpliedvolatilityconsistentwiththestock CallOptionValue Spreadsheet21 1SpreadsheettoCalculateBlack ScholesOptionValues Figure21 7UsingGoalSeektoFindImpliedVolatility Figure21 8ImpliedVolatilityoftheS P500 VIXIndex Black ScholesModelwithDividends ThecalloptionformulaappliestostocksthatpaydividendsOneapproachistoreplacethestockpricewithadividendadjustedstockpriceReplaceS0withS0 PV Dividends PutValueUsingBlack Scholes P Xe rT 1 N d2 S0 1 N d1 UsingthesamplecalldataS 100r 10X 95g 5T 2595e 10 x 25 1 5714 100 1 6664 6 35 P C PV X So C Xe rT SoUsingtheexampledataC 13 70X 95S 100r 10T 25P 13 70 95e 10X 25 100P 6 35 PutOptionValuation UsingPut CallParity Hedging HedgeratioordeltaThenumberofstocksrequiredtohedgeagainstthepriceriskofholdingoneoptionCall N d1 Put N d1 1OptionElasticityPercentagechangeintheoption svaluegivena1 changeinthevalueoftheunderlyingstock UsingtheBlack ScholesFormula Figure21 9CallOptionValueandHedgeRatio BuyingPuts resultsindownsideprotectionwithunlimitedupsidepotentialLimitationsTrackingerrorsifindexesareusedfortheputsMaturityofputsmaybetooshortHedgeratiosordeltaschangeasstockvalueschange PortfolioInsurance Figure21 10ProfitonaProtectivePutStrategy Figure21 11HedgeRatiosChangeastheStockPriceFluctuates Figure21 12S P500Cash to FuturesSpreadinPointsat15MinuteIntervals HedgingOnMispricedOptions Optionvalueispositivelyrelatedtovolatility Ifaninvestorbelievesthatthevolatilitythatisimpliedinanoption spriceistoolow aprofitabletradeispossibleProfitmustbehedgedagainstadeclineinthevalueofthestockPerformancedependsonoptionpricerelativetotheimpliedvolatility HedgingandDelta TheappropriatehedgewilldependonthedeltaRecallthedeltaisthechangeinthevalueoftheoptionrelativetothechangeinthevalueofthestock Delta ChangeinthevalueoftheoptionChangeofthevalueofthestock MispricedOption TextExample Impliedvolatility 33 Investorbelievesvolatilityshou