Bài giảng Essentials of Investments - Chapter 16 Option Valuation

Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-1 Chapter 16 Option Valuation Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-2 Chapter Summary  Objective: To discuss factors that affect option prices and to present quantitative option pricing models.  Factors influencing option values  Black-Scholes option valuation  Using the Black-Scholes formula  Binomial Option Pricing Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-3  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 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-4 Time Value of Options: Call Option value X Stock Price Value of Call Intrinsic Value Time value Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-5 Factor Effect on value Stock price increases Exercise price decreases Volatility of stock price increases Time to expiration increases Interest rate increases Dividend Rate decreases Factors Influencing Option Values: Calls Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-6 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 ) Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-7 Allowable Range for Call Call Value S0 PV (X) + PV (D) Lower Bound = S0 - PV (X) - PV (D) Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-8 Summary Reminder  Objective: To discuss factors that affect option prices and to present quantitative option pricing models.  Factors influencing option values  Black-Scholes option valuation  Using the Black-Scholes formula  Binomial Option Pricing Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-9 Co = SoN(d1) - Xe -rTN(d2) d1 = [ln(So/X) + (r +  2/2)T] / (T1/2) d2 = d1 + (T 1/2) 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 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-10 X = Exercise price e = 2.71828, 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 continuously compounded rate of return on the stock Black-Scholes Option Valuation (cont’d) Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-11 So = 100 X = 95 r = .10 T = .25 (quarter) = .50 Call Option Example 43. 25.5. 25.)2/5.10(.)95/100ln( d 2 1    18.25.5.43.d2  Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-12 N (.43) = .6664 Table 18.2 d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700 Probabilities from Normal Distribution Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-13 N (.18) = .5714 Table 18.2 d N(d) .16 .5636 .18 .5714 .20 .5793 Probabilities from Normal Distribution Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-14 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 actual price of the option, solve for volatility.  Is the implied volatility consistent with the stock? Call Option Value Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-15 Put Value using Black-Scholes P = Xe-rT [1-N(d2)] - S0 [1-N(d1)] Using the sample call data S = 100 r = .10 X = 95 g = .5 T = .25 P= 95e-10x.25(1-.5714)-100(1-.6664)=6.35 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-16 P = C + PV (X) - So = C + Xe-rT - So Using the example data C = 13.70 X = 95 S = 100 r = .10 T = .25 P = 13.70 + 95 e -.10 x .25 - 100 P = 6.35 Put Option Valuation: Using Put-Call Parity Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-17 Adjusting the Black-Scholes Model for 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) Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-18 Summary Reminder  Objective: To discuss factors that affect option prices and to present quantitative option pricing models.  Factors influencing option values  Black-Scholes option valuation  Using the Black-Scholes formula  Binomial Option Pricing Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-19  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 option’s value given a 1% change in the value of the underlying stock Using the Black-Scholes Formula Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-20  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 - Protecting Against Declines in Stock Value Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-21 Hedging Bets on Mispriced Options  Option value is positively related to volatility  If an investor believes that the volatility that is implied in an option’s 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 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-22 Hedging and Delta  The appropriate hedge will depend on the delta.  Recall the delta is the change in the value of the option relative to the change in the value of the stock. stocktheofvaluetheinchange optiontheofvaluetheinChange Delta  Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-23 Mispriced Option: Text Example Implied volatility = 33% Investor believes volatility should = 35% Option maturity = 60 days Put price P = $4.495 Exercise price and stock price = $90 Risk-free rate r = 4% Delta = -.453 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-24 Hedged Put Portfolio Cost to establish the hedged position 1000 put options at $4.495 / option $ 4,495 453 shares at $90 / share 40,770 Total outlay 45,265 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-25 Profit Position on Hedged Put Portfolio Value of put as function of stock price: implied volatility = 35% Stock Price 89 90 91 Put Price $5.254 $4.785 $4.347 Profit/loss per put .759 .290 (.148) Value of and profit on hedged portfolio Stock Price 89 90 91 Value of 1,000 puts $ 5,254 $ 4,785 $ 4,347 Value of 453 shares 40,317 40,770 41,223 Total 45,571 45,555 45,570 Profit 306 290 305 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-26 Summary Reminder  Objective: To discuss factors that affect option prices and to present quantitative option pricing models.  Factors influencing option values  Black-Scholes option valuation  Using the Black-Scholes formula  Binomial Option Pricing Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-27 100 200 50 Stock Price C 75 0 Call Option Value X = 125 Binomial Option Pricing: Text Example Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-28 Alternative Portfolio Buy 1 share of stock at $100 Borrow $46.30 (8% Rate) Net outlay $53.70 Payoff Value of Stock 50 200 Repay loan - 50 -50 Net Payoff 0 150 53.70 150 0 Payoff Structure is exactly 2 times the Call Binomial Option Pricing: Text Example Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-29 53.70 150 0 C 75 0 2C = $53.70 C = $26.85 Binomial Option Pricing: Text Example Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-30  Alternative Portfolio - one share of stock and 2 calls written (X = 125)  Portfolio is perfectly hedged Stock Value 50 200 Call Obligation 0 -150 Net payoff 50 50 Hence 100 - 2C = 46.30 or C = 26.85 Another View of Replication of Payoffs and Option Values Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-31 Generalizing the Two-State Approach  Assume 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) Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-32 Generalizing the Two-State Approach 100 110 121 95 90.25 104.50 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-33  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 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-34 S S + S + + S - S - - S + - S + + + S + + - S + - - S - - - Expanding to Consider Three Intervals Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-35 Possible Outcomes with Three Intervals Event Probability Stock Price 3 up 1/8 100 (1.05)3 =115.76 2 up 1 down 3/8 100 (1.05)2 (.97) =106.94 1 up 2 down 3/8 100 (1.05) (.97)2 = 98.79 3 down 1/8 100 (.97)3 = 91.27 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 18-36 Multinomial Option Pricing  Incomplete markets  If the stock return has more than two possible outcomes it is not possible to replicate the option with a portfolio containing the stock and the riskless asset  Markets are incomplete when there are fewer assets than there are states of the world (here possible stock outcomes)  No single option price can be then derived by arbitrage methods alone  Only upper and lower bounds exist on option prices, within which the true option price lies  An appropriate pair of such bounds converges to the Black-Scholes price at the limit