【精品】investments 投资学 （博迪bodie, kane, marcuschap021 option valuation精品ppt课件.ppt

INVESTMENTS 投资学（博迪BODIE,KANE,MARCUS）Chap021 Option ValuationINVESTMENTS|BODIE,KANE,MARCUSIntrinsic value-profit that could be made if the option was immediately exercisedCall:stock price-exercise pricePut:exercise price-stock price Time value-the difference between the option price and the intrinsic valueOption ValuesINVESTMENTS|BODIE,KANE,MARCUSFigure 21.1 Call Option Value before Expiration INVESTMENTS|BODIE,KANE,MARCUSTable 21.1 Determinants of Call Option ValuesINVESTMENTS|BODIE,KANE,MARCUSRestrictions on Option Value:CallCall value cannot be negative.The option payoff is zero at worst,and highly positive at best.Call value cannot exceed the stock value.Value of the call must be greater than the value of levered equity.Lower bound=adjusted intrinsic value:C S0-PV(X)-PV(D)(D=dividend)INVESTMENTS|BODIE,KANE,MARCUSINVESTMENTS|BODIE,KANE,MARCUSINVESTMENTS|BODIE,KANE,MARCUSEarly Exercise:CallsThe right to exercise an American call early is valueless as long as the stock pays no dividends until the option expires.The value of American and European calls is therefore identical.The call gains value as the stock price rises.Since the price can rise infinitely,the call is“worth more alive than dead.”INVESTMENTS|BODIE,KANE,MARCUSEarly Exercise:PutsAmerican puts are worth more than European puts,all else equal.The possibility of early exercise has value because:The value of the stock cannot fall below zero.Once the firm is bankrupt,it is optimal to exercise the American put immediately because of the time value of money.INVESTMENTS|BODIE,KANE,MARCUSFigure 21.4 Put Option Values as a Function of the Current Stock Price INVESTMENTS|BODIE,KANE,MARCUS10012090Stock PriceC100Call Option Value X=110Binomial Option Pricing:Text ExampleINVESTMENTS|BODIE,KANE,MARCUSAlternative 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 Structureis exactly 3 timesthe CallBinomial Option Pricing:Text ExampleINVESTMENTS|BODIE,KANE,MARCUS18.183003C3003C=$18.18C =$6.06Binomial Option Pricing:Text Example INVESTMENTS|BODIE,KANE,MARCUSAlternative Portfolio-one share of stock and 3 calls written(X=110)Portfolio is perfectly hedged:Stock Value90120Call Obligation0 -30Net payoff90 90Hence 100-3C=$81.82 or C=$6.06Replication of Payoffs and Option ValuesINVESTMENTS|BODIE,KANE,MARCUSHedge RatioIn the example,the hedge ratio=1 share to 3 calls or 1/3.Generally,the hedge ratio is:INVESTMENTS|BODIE,KANE,MARCUSAssume that we can break the year into three intervals.For each interval the stock could increase by 20%or decrease by 10%.Assume the stock is initially selling at$100.Expanding to Consider Three IntervalsINVESTMENTS|BODIE,KANE,MARCUSSS+S+S-S-S+-S+S+-S+-S-Expanding to Consider Three IntervalsINVESTMENTS|BODIE,KANE,MARCUSPossible Outcomes with Three IntervalsEventProbabilityFinal Stock Price3 up1/8100(1.20)3=$172.802 up 1 down3/8100(1.20)2(.90)=$129.601 up 2 down3/8100(1.20)(.90)2=$97.203 down1/8100(.90)3=$72.90INVESTMENTS|BODIE,KANE,MARCUSCo=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 Option ValuationINVESTMENTS|BODIE,KANE,MARCUSX=Exercise pricee=2.71828,the base of the natural logr=Risk-free interest rate(annualized,continuously compounded with the same maturity as the option)T=time to maturity of the option in yearsln=Natural log functionStandard deviation of the stockBlack-Scholes Option ValuationINVESTMENTS|BODIE,KANE,MARCUSFigure 21.6 A Standard Normal CurveINVESTMENTS|BODIE,KANE,MARCUSSo=100X =95r =.10T =.25(quarter)=.50(50%per year)Thus:Example 21.1 Black-Scholes ValuationINVESTMENTS|BODIE,KANE,MARCUSUsing a table or the NORMDIST function in Excel,we find that N(.43)=.6664 and N(.18)=.5714.Therefore:Co=SoN(d1)-Xe-rTN(d2)Co=100 X.6664-95 e-.10 X.25 X.5714 Co=$13.70Probabilities from Normal DistributionINVESTMENTS|BODIE,KANE,MARCUSImplied VolatilityImplied volatility is volatility for the stock implied by the option price.Using Black-Scholes and the actual price of the option,solve for volatility.Is the implied volatility consistent with the stock?Call Option ValueINVESTMENTS|BODIE,KANE,MARCUSBlack-Scholes Model with DividendsThe Black Scholes call option formula applies to stocks that do not pay dividends.What if dividends ARE paid?One approach is to replace the stock price with a dividend adjusted stock priceReplace S0 with S0-PV(Dividends)INVESTMENTS|BODIE,KANE,MARCUSExample 21.3 Black-Scholes Put ValuationP=Xe-rT 1-N(d2)-S0 1-N(d1)Using Example 21.2 data:S=100,r=.10,X=95,=.5,T=.25We compute:$95e-10 x.25(1-.5714)-$100(1-.6664)=$6.35INVESTMENTS|BODIE,KANE,MARCUSP =C+PV(X)-So =C +Xe-rT -SoUsing the example dataP =13.70+95 e-.10 X.25-100P =$6.35Put Option Valuation:Using Put-Call ParityINVESTMENTS|BODIE,KANE,MARCUSHedging:Hedge ratio or deltaThe number of stocks required to hedge against 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 FormulaINVESTMENTS|BODIE,KANE,MARCUSFigure 21.9 Call Option Value and Hedge RatioINVESTMENTS|BODIE,KANE,MARCUSBuying Puts-results in downside protection with 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 INVESTMENTS|BODIE,KANE,MARCUSFigure 21.10 Profit on a Protective Put StrategyINVESTMENTS|BODIE,KANE,MARCUSFigure 21.11 Hedge Ratios Change as the Stock Price FluctuatesINVESTMENTS|BODIE,KANE,MARCUSHedging 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 possible.Profit must be hedged against a decline in the value of the stock.Performance depends on option price relative to the implied volatility.INVESTMENTS|BODIE,KANE,MARCUSHedging and DeltaThe appropriate hedge will depend on the delta.Delta is the change in the value of the option relative to the change in the value of the stock,or the slope of the option pricing curve.Delta=Change in the value of the optionChange of the value of the stockINVESTMENTS|BODIE,KANE,MARCUSExample 21.6 Speculating on Mispriced OptionsImplied volatility =33%Investors estimate of true volatility=35%Option maturity =60 daysPut price P =$4.495Exercise price and stock price =$90Risk-free rate =4%Delta =-.453INVESTMENTS|BODIE,KANE,MARCUSTable 21.3 Profit on a Hedged Put PortfolioINVESTMENTS|BODIE,KANE,MARCUSExample 21.6 ConclusionsAs the stock price changes,so do the deltas used to calculate the hedge ratio.Gamma=sensitivity of the delta to the stock price.Gamma is similar to bond convexity.The hedge ratio will change with market conditions.Rebalancing is necessary.INVESTMENTS|BODIE,KANE,MARCUSDelta NeutralWhen you establish a position in stocks and options that is hedged with respect to fluctuations in the price of the underlying asset,your portfolio is said to be delta neutral.The portfolio does not change value when the stock price fluctuates.INVESTMENTS|BODIE,KANE,MARCUSTable 21.4 Profits on Delta-Neutral Options PortfolioINVESTMENTS|BODIE,KANE,MARCUSEmpirical Evidence on Option PricingThe Black-Scholes formula performs worst for options on stocks with high dividend payouts.The implied volatility of all options on a given stock with the same expiration date should be equal.Empirical test show that implied volatility actually falls as exercise price increases.This may be due to fears of a market crash.