1、CHAPTER 21Option Valuation Intrinsic 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 ValuesFigure 21.1 Call Option Value before Expira
2、tion Table 21.1 Determinants of Call Option ValuesRestrictions on Option Value:Call Call 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=adj
3、usted intrinsic value:C S0-PV(X)-PV(D)(D=dividend)Figure 21.2 Range of Possible Call Option ValuesFigure 21.3 Call Option Value as a Function of the Current Stock Price Early Exercise:Calls The right to exercise an American call early is valueless as long as the stock pays no dividends until the opt
4、ion 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.”Early Exercise:Puts American puts are worth more than European puts,all else equal.The possibility of
5、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.Figure 21.4 Put Option Values as a Function of the Current Stock Price 10012090Stock PriceC100Call Option
6、 Value X=110Binomial Option Pricing:Text ExampleAlternative 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 Example18.183003C3003C=$18.18
7、C =$6.06Binomial Option Pricing:Text Example Alternative 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 ValuesHedge Ratio In the example,the hedge
8、ratio=1 share to 3 calls or 1/3.Generally,the hedge ratio is:00esstock valu of range valuescall of rangedSuSCCHdu Assume 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 Consi
9、der Three IntervalsSS+S+S-S-S+-S+S+-S+-S-Expanding to Consider Three IntervalsPossible 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=$72.90Co=SoN(d1)-Xe-rTN(d2)d1=ln(S
10、o/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 ValuationX=Exercise pricee=2.71828,the base of the natural logr=Risk-free interest rate(annualized,continuousl
11、y 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 ValuationFigure 21.6 A Standard Normal CurveSo=100X =95r =.10T =.25(quarter)=.50(50%per year)Thus:Example 21.1 Black-Scholes Valuation
12、18.25.05.43.43.25.05.25.02510.95100ln221ddUsing 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 DistributionImplied Volatility Implied volatility is volatility fo
13、r 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 ValueBlack-Scholes Model with Dividends The Black Scholes call option formula applies to stocks that do not pay dividend
14、s.What if dividends ARE paid?One approach is to replace the stock price with a dividend adjusted stock priceReplace S0 with S0-PV(Dividends)Example 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-.666
15、4)=$6.35P =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 ParityHedging: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
16、in the options value given a 1%change in the value of the underlying stockUsing the Black-Scholes FormulaFigure 21.9 Call Option Value and Hedge Ratio Buying Puts-results in downside protection with unlimited upside potential Limitations Tracking errors if indexes are used for the putsMaturity of pu
17、ts may be too shortHedge ratios or deltas change as stock values changePortfolio Insurance Figure 21.10 Profit on a Protective Put StrategyFigure 21.11 Hedge Ratios Change as the Stock Price FluctuatesHedging On Mispriced OptionsOption value is positively related to volatility.If an investor believe
18、s 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.Hedging and Delta The appropriate hedge will depend on the delt
19、a.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 stockExample 21.6 Speculating on Mispriced OptionsImplied volatility =33%Investors estimate of
20、true volatility=35%Option maturity =60 daysPut price P =$4.495Exercise price and stock price =$90Risk-free rate =4%Delta =-.453Table 21.3 Profit on a Hedged Put PortfolioExample 21.6 Conclusions As the stock price changes,so do the deltas used to calculate the hedge ratio.Gamma=sensitivity of the de
21、lta to the stock price.Gamma is similar to bond convexity.The hedge ratio will change with market conditions.Rebalancing is necessary.Delta Neutral When 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 s
22、aid to be delta neutral.The portfolio does not change value when the stock price fluctuates.Table 21.4 Profits on Delta-Neutral Options PortfolioEmpirical Evidence on Option Pricing The 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.
