1、INVESTMENTS | BODIE, KANE, MARCUS Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin CHAPTER 18 Option Valuation INVESTMENTS | BODIE, KANE, MARCUS Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise pr
2、ice Put: exercise price - stock price Time value - the difference between the option price and the intrinsic value Option Values INVESTMENTS | BODIE, KANE, MARCUS Figure 18.1 Call Option Value before Expiration INVESTMENTS | BODIE, KANE, MARCUS Table 18.1 Determinants of Call Option Values INVESTMEN
3、TS | BODIE, KANE, MARCUS Restrictions 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 = adjusted intrinsic v
4、alue: C S0 - PV (X) - PV (D) (D=dividend) INVESTMENTS | BODIE, KANE, MARCUS Figure 18.2 Range of Possible Call Option Values INVESTMENTS | BODIE, KANE, MARCUS Figure 18.3 Call Option Value as a Function of the Current Stock Price INVESTMENTS | BODIE, KANE, MARCUS Early Exercise: Calls The right to e
5、xercise 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.”
6、INVESTMENTS | BODIE, KANE, MARCUS Early Exercise: Puts American 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 immediatel
7、y because of the time value of money. INVESTMENTS | BODIE, KANE, MARCUS Figure 18.4 Put Option Values as a Function of the Current Stock Price INVESTMENTS | BODIE, KANE, MARCUS 100 120 90 Stock Price C 10 0 Call Option Value X = 110 Binomial Option Pricing: Text Example INVESTMENTS | BODIE, KANE, MA
8、RCUS Alternative Portfolio Buy 1 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 INVESTMENTS | BODIE, KANE, MARCUS 18.18 3
9、0 0 3C 30 0 3C = $18.18 C = $6.06 Binomial Option Pricing: Text Example INVESTMENTS | BODIE, KANE, MARCUS Alternative Portfolio - one share of stock and 3 calls written (X = 110) Portfolio is perfectly hedged: Stock Value90120 Call Obligation0 -30 Net payoff90 90 Hence 100 - 3C = $81.82 or C = $6.06
10、 Replication of Payoffs and Option Values INVESTMENTS | BODIE, KANE, MARCUS Hedge Ratio In the example, the hedge ratio = 1 share to 3 calls or 1/3. Generally, the hedge ratio is: 00 esstock valu of range valuescall of range dSuS CC H du INVESTMENTS | BODIE, KANE, MARCUS Assume that we can break the
11、 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 Intervals INVESTMENTS | BODIE, KANE, MARCUS S S + S + + S - S - - S + - S + + + S + + - S + - - S - - - Expanding to Consider T
12、hree Intervals INVESTMENTS | BODIE, KANE, MARCUS Possible Outcomes with Three Intervals EventProbabilityFinal Stock Price 3 up1/8100 (1.20)3 = $172.80 2 up 1 down3/8100 (1.20)2 (.90) = $129.60 1 up 2 down3/8100 (1.20) (.90)2 = $97.20 3 down1/8100 (.90)3 = $72.90 INVESTMENTS | BODIE, KANE, MARCUS Co
13、= 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 INVESTMENTS | BODIE, KANE, MARCUS X = Exercise
14、 price e = 2.71828, the base of the natural log r = Risk-free interest rate (annualized, 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 the stock Black-Scholes Option Valuation INVESTMENTS | BO
15、DIE, KANE, MARCUS Figure 18.6 A Standard Normal Curve INVESTMENTS | BODIE, KANE, MARCUS So = 100X = 95 r = .10T = .25 (quarter) = .50 (50% per year) Thus: Example 18.1 Black-Scholes Valuation 18.25. 05 .43. 43. 25. 05 . 25. 0 2 5 10. 95 100 ln 2 2 1 d d INVESTMENTS | BODIE, KANE, MARCUS Using a tabl
16、e 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 = $13.70 Probabilities from Normal Distribution INVESTMENTS | BODIE, KANE, MARCUS Implied Volatility Implied volatility is volatil
17、ity 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 Value INVESTMENTS | BODIE, KANE, MARCUS Black-Scholes Model with Dividends The Black Scholes call option formu
18、la 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 price Replace S0 with S0 - PV (Dividends) INVESTMENTS | BODIE, KANE, MARCUS Example 18.3 Black-Scholes Put Valuation P = Xe-rT 1-N(d2) - S0 1-N(d1) Us
19、ing Example 18.2 data: S = 100, r = .10, X = 95, = .5, T = .25 We compute: $95e-10 x.25(1-.5714)-$100(1-.6664) = $6.35 INVESTMENTS | BODIE, KANE, MARCUS P = C + PV (X) - So = C + Xe-rT - So Using the example data P = 13.70 + 95 e -.10 X .25 - 100 P = $6.35 Put Option Valuation: Using Put-Call Parity
20、 INVESTMENTS | BODIE, KANE, MARCUS Hedging: Hedge ratio or delta The number 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 th
21、e Black-Scholes Formula INVESTMENTS | BODIE, KANE, MARCUS Figure 18.9 Call Option Value and Hedge Ratio INVESTMENTS | BODIE, KANE, MARCUS 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
22、too short Hedge ratios or deltas change as stock values change Portfolio Insurance INVESTMENTS | BODIE, KANE, MARCUS Figure 18.10 Profit on a Protective Put Strategy INVESTMENTS | BODIE, KANE, MARCUS Figure 18.11 Hedge Ratios Change as the Stock Price Fluctuates INVESTMENTS | BODIE, KANE, MARCUS Hed
23、ging 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 pri
24、ce relative to the implied volatility. INVESTMENTS | BODIE, KANE, MARCUS Hedging and Delta The 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
25、value of the option Change of the value of the stock INVESTMENTS | BODIE, KANE, MARCUS Example 18.6 Speculating on Mispriced Options Implied volatility = 33% Investors estimate of true volatility = 35% Option maturity = 60 days Put price P = $4.495 Exercise price and stock price = $90 Risk-free rate
26、 = 4% Delta = -.453 INVESTMENTS | BODIE, KANE, MARCUS Table 18.3 Profit on a Hedged Put Portfolio INVESTMENTS | BODIE, KANE, MARCUS Example 18.6 Conclusions As 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 s
27、imilar to bond convexity. The hedge ratio will change with market conditions. Rebalancing is necessary. INVESTMENTS | BODIE, KANE, MARCUS 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
28、 is said to be delta neutral. The portfolio does not change value when the stock price fluctuates. INVESTMENTS | BODIE, KANE, MARCUS Table 18.4 Profits on Delta-Neutral Options Portfolio INVESTMENTS | BODIE, KANE, MARCUS Empirical 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.
