1、1Copyright Prentice Hall Inc. 2000. Author: Nick Bagley, bdellaSoft, Inc.ObjectiveTo show how the law of one price maybe used to derive prices of optionsTo show how to infer impliedvolatility from optionprices234Table 15.1 List of IBM Option Prices (Source: Wall Street Journal Interactive Edition, M

2、ay 29, 1998)IBM (IBM)Underlying stock price 120 1/16 Call . Put .StrikeExpiration VolumeLastOpenVolumeLastOpenInterestInterest115Jun1372744837561 3/169692115Oct2584105967115Jan15536 3/440120Jun23773 1/280498732 7/89849120Oct1219 5/162561457 1/81993120Jan9112 1/28842 5259125Jun15641 1/29764175 3/4590

3、0125Oct917 1/22360 731125Jan8710 1/2124 705Table 15.2 List of Index Option Prices (Source: Wall Street Journal Interactive Edition, June 6, 1998)S & P 500 INDEX -AMChicago ExchangeUnderlyingHighLowCloseNetFrom%Change31-DecChangeS&P500 1113.881084.281113.8619.03143.4314.8(SPX)NetOpen Strike V

4、olumeLastChangeInterestJun1110 call 2,08117 1/48 1/215,754Jun1110 put 1,07710-1117,104Jul1110 call 1,27833 1/29 1/23,712Jul1110 put 15223 3/8-12 1/81,040Jun1120 call 8012716,585Jun1120 put 21117-119,947Jul1120 call 6727 1/48 1/45,546Jul1120 put 1027 1/2-114,0336Terninal or Boundary Conditions for Ca

5、ll and Put Options-20020406080100120020406080100120140160180200Underlying PriceDollarsCallPut7Terminal Conditions of a Call and a Put Option with Strike = 100Strike100ShareCallPutShare_PutBondCall_Bond001001001001001009010010010020080100100100300701001001004006010010010050050100100100600401001001007

6、003010010010080020100100100900101001001001000010010010011010011010011012020012010012013030013010013014040014010014015050015010015016060016010016017070017010017018080018010018019090019010019020010002001002008Stock, Call, Put, Bond020406080100120140160180200020406080100120140160180200Stock PriceStock,

7、 Call, Put, Bond, Put+Stock, Call+BondCallPutShare_PutBondCall_BondShare9ShareMaturityStrikePutrfStrikeMaturityStrikeCallMaturity),(1),(1011MaturityMaturityrfForwardMaturityStrikePutrfStrikeMaturityStrikeCall1),(1),(1213Call and Put as a Function of Forward02468101214169092949698100102104106108110Fo

8、rwardPut, Call Valuescallputasy_call_1asy_put_1Strike = ForwardCall = Put14Put and Call as Function of Share Price-1001020304050605060708090100110120130140150Share PricePut and Call Pricescallputasy_call_1asy_call_2asy_put_1asy_put_215Put and Call as Function of Share Price05101520808590951001051101

9、15120Share PricePut and Call Pricescallputasy_call_1asy_call_2asy_put_1asy_put_2PV StrikeStrike16Volatility and Option Prices, P0 = $100, Strike = $100Stock Price Call PayoffPut PayoffLow Volatility CaseRise120200Fall80020Expectation1001010High Volatility CaseRise140400Fall60040Expectation100202017y

10、xSC18yxyx8001202019yxSC10$4010021C20yxSP21yxyx8001202022yxSP10$6010021P23 12 MonthsStockPricexyCallPricexyCallPrice$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) = $202425 21211222121ln21lndNEedNSePdNEedNSeCTdTTdrESdTTdrESdrTdT

11、rTdT26EdNSedNePEdNSedNeCTTESedTTESedTdrrTTdrrTTdrTdr2121222121ln21ln27TSTSPCdNdNSPCdPdNdNSeCTdTdSeEdTTdr39886. 020 If21;21 If21212128Determinants of Option PricesIncreases in:CallPut Stock Price, SIncreaseDecrease Exercise Price, EDecreaseIncrease Volatility, sigmaIncreaseIncrease Time to Expiration

12、, TAmbiguousAmbiguous Interest Rate, rIncreaseDecrease Cash Dividends, dDecreaseIncrease29Value of a Call and Put Options with Strike = Current Stock Price012345678910110.00.10.20.30.40.50.60.70.80.91.0Time-to-MaturityCall and Put Pricecallput30Call and Put Prices as a Function of Volatility01234560

13、.000.020.040.060.080.100.120.140.160.180.20VolatilityCall and Put Pricescallput31Computing Implied Volatilityvolatility0.3154call10.0000strike100.0000share105.0000rate_dom0.0500rate_for0.0000maturity0.2500factor0.0249d_10.4675d_20.3098n_d_10.6799n_d_20.6217call_part_171.3934call_part_2-61.3934error0

14、.0000Insert any number to startFormula for option value minus the actual call value32Computing Implied Volatilityvolatility0.315378127101852call10strike100share105rate_dom0.05rate_for0maturity0.25factor=(rate_dom - rate_for + (volatility2)/2)*maturityd_1=(LN(share/strike)+factor)/(volatility*SQRT(ma

15、turity)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*maturity)call_part_2=- n_d_2*strike*EXP(-rate_dom*maturity)error=call_part_1+call_part_2-call33Construction of Pats Get Rich Portfolio-120-100-80-60-40-200204060805060708090100110120

16、130140150Share PricePortfolio ValuescallP_ShareP_bondPortfolioTangent3435Strategy 1-Week Later-0.50.00.51.01.52.09095100105110115120Share PriceStrategy ValuePortfolioPortfolioLater36Payoffs for Bond and Stock Issues020406080100120020406080100120140160180200Value of Firm (Millions)Value of Bond and S

17、tock (Millions)BondValueStockValue37Probalility Density of a Firms Value0.000.010.020.030.040.050.060.070.080.09020406080100120140160180200Value of a FirmProbability Density38SecurityPayoff State a Payoff State bFirm14070Bond8070Stock60039308,692,57$;7604. 1700;04. 114060YxYxYx40PositionImmediateCas

18、e aCase b6/7 assets-85,714120,00060,000Bond (rf)57,692-60,000-60,000Total28,02260,00004142SecurityTotalmarketValueEquivalentAmountof FirmEquivalentAmountof Rf DebtBonds71,97814,28657,692Stock28,02285,714-57,692Bonds +Stock100,000100,000043026,641,90$76308,692,57000,000,20;xYEVYxVE44PositionImmediate

19、Cash FlowScenario aV1 = 70Scenario bV1 = 140Purchase xof firm- x* V70 x140 xPurchaseY RF Bond- YY (1.04)Y (1.04)TotalPortfolio- x * V - Y708045308,692,57$;7104. 11408004. 17070YxYxYx46753,244,105$71308,692,57273,727,72;xYEVYxVD47Convertible Bond020406080100120140020406080100120140160180200Value of the FirmValue of Stock and Bond IssueC