第15章期权与或有要求权(金融学,厦门大学)

1第15章:期权与或有要求权Copyright©PrenticeHallInc.2000.Author:NickBagley,bdellaSoft,Inc.学习目的•学习如何从一价法则中推导期权定价公式•学习如何从期权价格中获得隐含波动性2第15章主要内容15.1期权概述15.2用期权投资15.3看跌－看涨平价关系15.4波动性与期权价格15.52状态期权定价15.6动态复制与二叉树模型15.7Black-Scholes模型15.8隐含波动性15.9公司债务和股权的或有要求权分析15.10信贷担保15.11期权定价方法的其他应用3学习目的•学习如何利用一价法则推导期权价格•学习如何从期权价格中获得隐含波动性4Table15.1ListofIBMOptionPrices(Source:WallStreetJournalInteractiveEdition,May29,1998)IBM(IBM)Underlyingstockprice1201/16Call.Put.StrikeExpirationVolumeLastOpenVolumeLastOpenInterestInterest115Jun13727448375613/169692115Oct……2584105967115Jan……155363/440120Jun237731/2804987327/89849120Oct12195/1625614571/81993120Jan91121/28842……5259125Jun156411/297641753/45900125Oct9171/22360……731125Jan87101/2124……705Table15.2ListofIndexOptionPrices(Source:WallStreetJournalInteractiveEdition,June6,1998)S&P500INDEX-AMChicagoExchangeUnderlyingHighLowCloseNetFrom%Change31-DecChangeS&P5001113.881084.281113.8619.03143.4314.8(SPX)NetOpenStrikeVolumeLastChangeInterestJun1110call2,081171/481/215,754Jun1110put1,07710-1117,104Jul1110call1,278331/291/23,712Jul1110put152233/8-121/81,040Jun1120call8012716,585Jun1120put21117-119,947Jul1120call67271/481/45,546Jul1120put10271/2-114,0336TerninalorBoundaryConditionsforCallandPutOptions-20020406080100120020406080100120140160180200UnderlyingPriceDollarsCallPut7TerminalConditionsofaCallandaPutOptionwithStrike=100Strike100ShareCallPutShare_PutBondCall_Bond001001001001001009010010010020080100100100300701001001004006010010010050050100100100600401001001007003010010010080020100100100900101001001001000010010010011010011010011012020012010012013030013010013014040014010014015050015010015016060016010016017070017010017018080018010018019090019010019020010002001002008Stock,Call,Put,Bond020406080100120140160180200020406080100120140160180200StockPriceStock,Call,Put,Bond,Put+Stock,Call+BondCallPutShare_PutBondCall_BondShare9看跌-看涨平价公式TE1CPSrf10复合债券•通过看涨-看跌平价关系的四个等式创新复合证券:C=S+P-BS=C-P+BP=C-S+BB=S+P-C11期货和远期•从上一章得，远期的贴现价值等于现货价格•平价关系成为TTE11FCPrfrf12欧式期权的应用•如果(FE)那么(CP)•如果(F=E)那么(C=P)•如果(FE)那么(CP)•E为交割价格•F为潜含资产远期价格•C为看涨期权价格•P看跌期权价格13CallandPutasaFunctionofForward02468101214169092949698100102104106108110ForwardPut,CallValuescallputasy_call_1asy_put_1Strike=ForwardCall=Put14PutandCallasFunctionofSharePrice-1001020304050605060708090100110120130140150SharePricePutandCallPricescallputasy_call_1asy_call_2asy_put_1asy_put_215PutandCallasFunctionofSharePrice0510152080859095100105110115120SharePricePutandCallPricescallputasy_call_1asy_call_2asy_put_1asy_put_2PVStrikeStrike16VolatilityandOptionPrices,P0=$100,Strike=$100StockPriceCallPayoffPutPayoffLowVolatilityCaseRise120200Fall80020Expectation1001010HighVolatilityCaseRise140400Fall60040Expectation100202017二项分布模型:看涨一个看涨期权,C,可以通过下面方式复合而成•购买股票S的x部分,同时卖空y的无风险债券•比例x称为保值比率yxSC18二项分布模型:看涨•具体化:–给定两种状态下期权到期时的股票价格和期权价格:–求解可得x=1/2,y=40yxyx8001202019二项分布模型:看涨•结论:–将参数值x=1/2,y=40代入等式–得:yxSC10$4010021C20二项分布模型:看跌复合看跌期权P,可以通过下面方式复合而成•卖空x比例的股票,同时买进无风险债券y•x称为保值比率yxSP21二项分布模型:看跌•具体化:–给定两种状态下期权到期时的股票价格和期权价格:–求解，可得x=1/2,y=60yxyx8001202022二项分布模型:看跌•结论:–将参数值x=1/2,y=60代入公式–可得:yxSP10$6010021P23动态复制看涨期权的决策树---------0Months----------------------------6Months----------------12MonthsStockPricexyCallPricexyCallPrice$120.00$20.00$110.00$10.00100.00%-$100.00$100.0050.00%-$45.00$0.00$90.00$0.000.00%$0.00$80.00$0.00($120*100%)+(-$100)=$2024Black-Scholes模型:记号•C=看涨期权价格•P=看跌期权价格•S=股票价格•E=执行价格•T=到期期限•ln(.)=自然对数•e=2.71828...•N(.)=累积正态分布•以下为年连续复利•r=国内无风险利率•d=国外无风险利率或常股利收益率•σ=波动性25Black-Scholes模型:公式21211222121ln21lndNEedNSePdNEedNSeCTdTTdrESdTTdrESdrTdTrTdT26Black-Scholes模型:公式(远期型)EdNSedNePEdNSedNeCTTESedTTESedTdrrTTdrrTTdrTdr2121222121ln21ln27Black-Scholes模型:公式(简化)TSTSPCdNdNSPCdPdNdNSeCTdTdSeEdTTdr39886.020If21;21If21212128DeterminantsofOptionPricesIncreasesin:CallPutStockPrice,SIncreaseDecreaseExercisePrice,EDecreaseIncreaseVolatility,sigmaIncreaseIncreaseTimetoExpiration,TAmbiguousAmbiguousInterestRate,rIncreaseDecreaseCashDividends,dDecreaseIncrease29ValueofaCallandPutOptionswithStrike=CurrentStockPrice012345678910110.00.10.20.30.40.50.60.70.80.91.0Time-to-MaturityCallandPutPricecallput30CallandPutPricesasaFunctionofVolatility01234560.000.020.040.060.080.100.120.140.160.180.20VolatilityCallandPutPricescallput31ComputingImpliedVolatilityvolatility0.3154call10.0000strike100.0000share105.0000rate_dom0.0500rate_for0.0000maturity0.2500factor0.0249d_10.4675d_20.3098n_d_10.6799n_d_20.6217call_part_171.3934call_part_2-61.3934error0.0000InsertanynumbertostartFormulaforoptionvalueminustheactualcallvalue32ComputingImpliedVolatilityvolatility0.315378127101852call10strike100share105rate_dom0.05rate_for0maturity0.25factor=(rate_dom-rate_for+(volatility^2)/2)*maturityd_1=(LN(share/strike)+factor)/(volatility*SQRT(maturity))d_2=d_1-volatility*SQRT(maturity)n_d_1=NORMSDIST(d_1)n_d_2=NORMSDIST(d_2)call_part_1=n_d_1*share*EXP(-rate_for*ma