1、CHAPTER 21,Option Valuation,Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put: exercise price - stock price Time value - the difference between the option price and the intrinsic value,Option Values,Figure 21.1 Call Option Valu

2、e before Expiration,Table 21.1 Determinants of Call Option Values,Restrictions on Option Value: Call,Value cannot be negative Value cannot exceed the stock value Value of the call must be greater than the value of levered equity C S0 - ( X + D ) / ( 1 + rf )T C S0 - PV ( X ) - PV ( D ),Figure 21.2 R

3、ange of Possible Call Option Values,Figure 21.3 Call Option Value as a Function of the Current Stock Price,Figure 21.4 Put Option Values as a Function of the Current Stock Price,100,120,90,Stock Price,C,10,0,Call Option Value X = 110,Binomial Option Pricing: Text Example,Alternative Portfolio Buy 1

4、share of stock at $100 Borrow $81.82 (10% Rate) Net outlay $18.18 Payoff Value of Stock 90 120 Repay loan - 90 - 90 Net Payoff 0 30,18.18,30,0,Payoff Structure is exactly 3 times the Call,Binomial Option Pricing: Text Example Continued,18.18,30,0,C,30,0,3C = $18.18 C = $6.06,Binomial Option Pricing:

5、 Text Example Continued,Alternative Portfolio - one share of stock and 3 calls written (X = 110) Portfolio is perfectly hedged Stock Value90120 Call Obligation 0 -30 Net payoff90 90 Hence 100 - 3C = 81.82 or C = 6.06,Replication of Payoffs and Option Values,Generalizing the Two-State Approach,Assume

6、 that we can break the year into two six-month segments In each six-month segment the stock could increase by 10% or decrease by 5% Assume the stock is initially selling at 100 Possible outcomes: Increase by 10% twice Decrease by 5% twice Increase once and decrease once (2 paths),Generalizing the Tw

7、o-State Approach Continued,100,110,121,95,90.25,104.50,Assume that we can break the year into three intervals For each interval the stock could increase by 5% or decrease by 3% Assume the stock is initially selling at 100,Expanding to Consider Three Intervals,S,S +,S + +,S -,S - -,S + -,S + + +,S +

8、+ -,S + - -,S - - -,Expanding to Consider Three Intervals Continued,Possible Outcomes with Three Intervals,EventProbability Final Stock Price 3 up 1/8100 (1.05)3 =115.76 2 up 1 down 3/8100 (1.05)2 (.97)=106.94 1 up 2 down 3/8100 (1.05) (.97)2= 98.79 3 down 1/8100 (.97)3= 91.27,Figure 21.5 Probabilit

9、y Distributions,Co = SoN(d1) - Xe-rTN(d2) d1 = ln(So/X) + (r + 2/2)T / (T1/2) d2 = d1 + (T1/2) where Co = Current call option value So = Current stock price N(d) = probability that a random draw from a normal distribution will be less than d,Black-Scholes Option Valuation,X = Exercise price e = 2.71

10、828, the base of the natural log r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option) T = time to maturity of the option in years ln = Natural log function Standard deviation of annualized cont. compounded rate of return on the stock,Black-Scholes Opt

11、ion Valuation Continued,Figure 21.6 A Standard Normal Curve,So = 100X = 95 r = .10T = .25 (quarter) = .50 d1 = ln(100/95) + (.10+(5 2/2) / (5.251/2) = .43 d2 = .43 + (5.251/2) = .18,Call Option Example,N (.43) = .6664 Table 21.2 d N(d) .42 .6628 .43.6664 Interpolation .44.6700,Probabilities from Nor

12、mal Distribution,N (.18) = .5714 Table 21.2 d N(d) .16 .5636 .18.5714 .20.5793,Probabilities from Normal Distribution Continued,Table 21.2 Cumulative Normal Distribution,Co = SoN(d1) - Xe-rTN(d2) Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70 Implied Volatility Using Black-Scholes and the act

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

14、,Black-Scholes Model with Dividends,The call option formula applies to stocks that pay dividends One approach is to replace the stock price with a dividend adjusted stock price Replace S0 with S0 - PV (Dividends),Put Value Using Black-Scholes,P = Xe-rT 1-N(d2) - S0 1-N(d1) Using the sample call data

15、 S = 100 r = .10 X = 95 g = .5 T = .25 95e-10 x.25(1-.5714)-100(1-.6664) = 6.35,P = C + PV (X) - So = C + Xe-rT - So Using the example data C = 13.70X = 95S = 100 r = .10T = .25 P = 13.70 + 95 e -.10 X .25 - 100 P = 6.35,Put Option Valuation: Using Put-Call Parity,Hedging: Hedge ratio or delta The n

16、umber of stocks required to hedge against the price risk of holding one option Call = N (d1) Put = N (d1) - 1 Option Elasticity Percentage change in the options value given a 1% change in the value of the underlying stock,Using the Black-Scholes Formula,Figure 21.9 Call Option Value and Hedge Ratio,

17、Buying Puts - results in downside protection with unlimited upside potential Limitations Tracking errors if indexes are used for the puts Maturity of puts may be too short Hedge ratios or deltas change as stock values change,Portfolio Insurance,Figure 21.10 Profit on a Protective Put Strategy,Figure

18、 21.11 Hedge Ratios Change as the Stock Price Fluctuates,Figure 21.12 S&P 500 Cash-to-Futures Spread in Points at 15 Minute Intervals,Hedging On Mispriced Options,Option 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,Hedging and Delta,The appr