博迪投资学Chap021.ppt

CHAPTER 21,Option Valuation,21-2,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,21-3,Figure 21.1 Call Option Value before Expiration,21-4,Table 21.1 Determinants of Call Option Values,21-5,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 value: C S0 - PV (X) - PV (D) (D=dividend),21-6,Figure 21.2 Range of Possible Call Option Values,21-7,Figure 21.3 Call Option Value as a Function of the Current Stock Price,21-8,Early Exercise: Calls,The 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.”,21-9,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 immediately because of the time value of money.,21-10,Figure 21.4 Put Option Values as a Function of the Current Stock Price,21-11,100,120,90,Stock Price,C,10,0,Call Option Value X = 110,Binomial Option Pricing: Text Example,21-12,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,21-13,18.18,30,0,3C,30,0,3C = $18.18 C = $6.06,Binomial Option Pricing: Text Example,21-14,Alternative Portfolio - one share of stock and 3 calls written (X = 110) Portfolio is perfectly hedged: Stock Value 90 120 Call Obligation 0 -30 Net payoff 90 90 Hence 100 - 3C = $81.82 or C = $6.06,Replication of Payoffs and Option Values,21-15,Hedge Ratio,In the example, the hedge ratio = 1 share to 3 calls or 1/3. Generally, the hedge ratio is:,21-16,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 Consider Three Intervals,21-17,S,S +,S + +,S -,S - -,S + -,S + + +,S + + -,S + - -,S - - -,Expanding to Consider Three Intervals,21-18,Possible Outcomes with Three Intervals,21-19,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,21-20,X = Exercise 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,21-21,Figure 21.6 A Standard Normal Curve,21-22,So = 100 X = 95 r = .10 T = .25 (quarter) = .50 (50% per year) Thus:,Example 21.1 Black-Scholes Valuation,21-23,Using 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 = $13.70,Probabilities from Normal Distribution,21-24,Implied Volatility Implied 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 Value,21-25,Black-Scholes Model with Dividends,The 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 price Replace S0 with S0 - PV (Dividends),21-26,Example 21.3 Black-Scholes Put Valuation,P = Xe-rT 1-N(d2) - S0 1-N(d1) Using Example 21.2 data: S = 100, r = .10, X = 95, = .5, T = .25 We compute: $95e-10x.25(1-.5714)-$100(1-.6664) = $6.35,21-27,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,21-28,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 the Black-Scholes Formula,21-29,Figure 21.9 Call Option Value and Hedge Ratio,21-30,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,21-31,Figure 21.10 Profit on a Protective Put Strategy,21-32,Figure 21.11 Hedge Ratios Change as the Stock Price Fluctuates,21-33,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.,21-34,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 value of the option Change of the value of the stock,21-35,Example 21.6 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 = 4% Delta = -.453,21-36,Table 21.3 Profit on a Hedged Put Portfolio,21-37,Example 21.6 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 similar to bond convexity. The hedge ratio will change with market conditions. Reba