投资学第21章期权定价.ppt

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1、Investments,8th editionBodie,Kane and MarcusSlides by Susan HineSlides by Susan HineMcGraw-Hill/IrwinCopyright 2009 by The McGraw-Hill Companies,Inc.All rights reserved.CHAPTER 21Option ValuationOption Valuation21-2Intrinsic value-profit that could be made if the option was immediately exercisedCall

2、:stock price-exercise pricePut:exercise price-stock price Time value-the difference between the option price and the intrinsic valueOption Values21-3Figure 21.1 Call Option Value before Expiration 21-4Table 21.1 Determinants of Call Option Values21-5Restrictions on Option Value:CallValue cannot be n

3、egativeValue cannot exceed the stock valueValue of the call must be greater than the value of levered equityC S0-(X+D)/(1+rf)TC S0-PV(X)-PV(D)21-6Figure 21.2 Range of Possible Call Option Values21-7Figure 21.3 Call Option Value as a Function of the Current Stock Price 21-8Figure 21.4 Put Option Valu

4、es as a Function of the Current Stock Price 21-910012090Stock PriceC100Call Option Value X=110Binomial Option Pricing:Text Example21-10Alternative PortfolioBuy 1 share of stock at$100Borrow$81.82(10%Rate)Net outlay$18.18PayoffValue of Stock 90 120Repay loan -90-90Net Payoff 0 3018.18300Payoff Struct

5、ureis exactly 3 timesthe CallBinomial Option Pricing:Text Example Continued21-1118.18300C3003C=$18.18C =$6.06Binomial Option Pricing:Text Example Continued21-12Alternative Portfolio-one share of stock and 3 calls written(X=110)Portfolio is perfectly hedgedStock Value90120Call Obligation 0 -30Net pay

6、off90 90Hence 100-3C=81.82 or C=6.06Replication of Payoffs and Option Values21-13Generalizing the Two-State ApproachAssume that we can break the year into two six-month segmentsIn each six-month segment the stock could increase by 10%or decrease by 5%Assume the stock is initially selling at 100Possi

7、ble outcomes:Increase by 10%twiceDecrease by 5%twiceIncrease once and decrease once(2 paths)21-14Generalizing the Two-State Approach Continued1001101219590.25104.5021-15Assume that we can break the year into three intervalsFor each interval the stock could increase by 5%or decrease by 3%Assume the s

8、tock is initially selling at 100Expanding to Consider Three Intervals21-16SS+S+S-S-S+-S+S+-S+-S-Expanding to Consider Three Intervals Continued21-17Possible Outcomes with Three IntervalsEventProbability Final Stock Price3 up 1/8100(1.05)3=115.762 up 1 down 3/8100(1.05)2(.97)=106.941 up 2 down 3/8100

9、(1.05)(.97)2=98.793 down 1/8100(.97)3=91.2721-18Figure 21.5 Probability Distributions21-19Co=SoN(d1)-Xe-rTN(d2)d1=ln(So/X)+(r+2/2)T/(T1/2)d2=d1+(T1/2)whereCo=Current call option valueSo=Current stock priceN(d)=probability that a random draw from a normal distribution will be less than dBlack-Scholes

10、 Option Valuation21-20X=Exercise pricee=2.71828,the base of the natural logr=Risk-free interest rate(annualizes continuously compounded with the same maturity as the option)T=time to maturity of the option in yearsln=Natural log functionStandard deviation of annualized pounded rate of return on the

11、stockBlack-Scholes Option Valuation Continued21-21Figure 21.6 A Standard Normal Curve21-22So=100X =95r =.10T =.25(quarter)=.50d1=ln(100/95)+(.10+(5 2/2)/(5.251/2)=.43 d2=.43+(5.251/2)=.18Call Option Example21-23N(.43)=.6664Table 21.2d N(d).42 .6628 .43.6664 Interpolation .44.6700Probabilities from N

12、ormal Distribution21-24N(.18)=.5714Table 21.2d N(d).16 .5636 .18.5714 .20.5793Probabilities from Normal Distribution Continued21-25Table 21.2 Cumulative Normal Distribution21-26Co=SoN(d1)-Xe-rTN(d2)Co=100 X.6664-95 e-.10 X.25 X.5714 Co=13.70Implied VolatilityUsing Black-Scholes and the actual price

13、of the option,solve for volatility.Is the implied volatility consistent with the stock?Call Option Value21-27Spreadsheet 21.1 Spreadsheet to Calculate Black-Scholes Option Values 21-28Figure 21.7 Using Goal Seek to Find Implied Volatility21-29Figure 21.8 Implied Volatility of the S&P 500(VIX Index)2

14、1-30Black-Scholes Model with DividendsThe call option formula applies to stocks that pay dividendsOne approach is to replace the stock price with a dividend adjusted stock priceReplace S0 with S0-PV(Dividends)21-31Put Value Using Black-ScholesP=Xe-rT 1-N(d2)-S0 1-N(d1)Using the sample call dataS=100

15、 r=.10 X=95 g=.5 T=.2595e-10 x.25(1-.5714)-100(1-.6664)=6.3521-32P =C+PV(X)-So =C +Xe-rT -SoUsing the example dataC =13.70X =95S =100r =.10T =.25P =13.70+95 e-.10 X.25-100P =6.35Put Option Valuation:Using Put-Call Parity21-33Hedging:Hedge ratio or delta The number of stocks required to hedge against

16、 the price risk of holding one optionCall=N(d1)Put=N(d1)-1Option ElasticityPercentage change in the options value given a 1%change in the value of the underlying stockUsing the Black-Scholes Formula21-34Figure 21.9 Call Option Value and Hedge Ratio21-35Buying Puts-results in downside protection with

17、 unlimited upside potentialLimitations Tracking errors if indexes are used for the putsMaturity of puts may be too shortHedge ratios or deltas change as stock values changePortfolio Insurance 21-36Figure 21.10 Profit on a Protective Put Strategy21-37Figure 21.11 Hedge Ratios Change as the Stock Pric

18、e Fluctuates21-38Figure 21.12 S&P 500 Cash-to-Futures Spread in Points at 15 Minute Intervals21-39Hedging On Mispriced OptionsOption value is positively related to volatility:If an investor believes that the volatility that is implied in an options price is too low,a profitable trade is possibleProf

19、it must be hedged against a decline in the value of the stockPerformance depends on option price relative to the implied volatility21-40Hedging and DeltaThe appropriate hedge will depend on the deltaRecall the delta is the change in the value of the option relative to the change in the value of the

20、stockDelta=Change in the value of the optionChange of the value of the stock21-41Mispriced Option:Text ExampleImplied volatility =33%Investor believes volatility should=35%Option maturity =60 daysPut price P =$4.495Exercise price and stock price =$90Risk-free rate r =4%Delta =-.45321-42Table 21.3 Profit on a Hedged Put Portfolio21-43Table 21.4 Profits on Delta-Neutral Options Portfolio21-44Figure 21.13 Implied Volatility of the S&P 500 Index as a Function of Exercise Price