Chapter 18: Option Valuation
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Chapter 18 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Option Valuation Slide 18-1 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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 Slide 18-2 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Option Values 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 Slide 18-3 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Time Value of Options: Call Option value Value of Call Intrinsic Value Time value X Slide 18-4 Stock Price Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Factors Influencing Option Values: Calls Factor Stock price Exercise price Volatility of stock price Time to expiration Interest rate Dividend Rate Slide 18-5 Effect on value increases decreases increases increases increases decreases Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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 > S 0 - ( X + D ) / ( 1 + R f )T C > S0 - PV ( X ) - PV ( D ) Slide 18-6 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Allowable Range for Call Call Value Lower Bound = S0 - PV (X) - PV (D) PV (X) + PV (D) Slide 18-7 S0 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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 Slide 18-8 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Black-Scholes Option Valuation 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 Slide 18-9 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Black-Scholes Option Valuation (cont’d) 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 Slide 18-10 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Call Option Example So = 100 r = .10 = .50 d1 X = 95 T = .25 (quarter) 2 ln(100 / 95) (.10 .5 / 2) .25 .5 .25 .43 d2 .43 .5 .25 .18 Slide 18-11 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Probabilities from Normal Distribution N (.43) = .6664 Table 18.2 d .42 .43 .44 Slide 18-12 N(d) .6628 .6664 Interpolation .6700 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Probabilities from Normal Distribution N (.18) = .5714 Table 18.2 d .16 .18 .20 Slide 18-13 N(d) .5636 .5714 .5793 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Call Option Value 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? Slide 18-14 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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 Slide 18-15 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Put Option Valuation: Using Put-Call Parity 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 Slide 18-16 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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 Slide 18-17 Replace S0 with S0 - PV (Dividends) Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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 Slide 18-18 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Using the Black-Scholes Formula 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 Slide 18-19 Percentage change in the option’s value given a 1% change in the value of the underlying stock Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Portfolio Insurance - Protecting Against Declines in Stock Value Buying Puts - results in downside protection with unlimited upside potential Limitations Slide 18-20 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 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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 Slide 18-21 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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. Change in the value of the option Delta change in the value of the stock Slide 18-22 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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 Slide 18-23 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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 Slide 18-24 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Profit Position on Hedged Put Portfolio Value of put as function of stock price: implied volatility = 35% Stock Price 89 90 Put Price $5.254 $4.785 Profit/loss per put .759 .290 91 $4.347 (.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 Slide 18-25 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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 Slide 18-26 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Binomial Option Pricing: Text Example 200 100 75 C 50 Stock Price Slide 18-27 0 Call Option Value X = 125 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Binomial Option Pricing: Text Example Alternative Portfolio Buy 1 share of stock at $100 Borrow $46.30 (8% Rate) 53.70 Net outlay $53.70 Payoff Value of Stock 50 200 Repay loan - 50 -50 Net Payoff 0 150 Slide 18-28 150 0 Payoff Structure is exactly 2 times the Call Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Binomial Option Pricing: Text Example 150 53.70 75 C 0 0 2C = $53.70 C = $26.85 Slide 18-29 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Another View of Replication of Payoffs and Option Values Alternative Portfolio - one share of stock and 2 calls written (X = 125) Portfolio is perfectly hedged Stock Value Call Obligation Net payoff 50 0 50 200 -150 50 Hence 100 - 2C = 46.30 or C = 26.85 Slide 18-30 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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) Slide 18-31 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Generalizing the Two-State Approach 121 110 104.50 100 95 Slide 18-32 90.25 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Expanding to Consider Three Intervals 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 Slide 18-33 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Expanding to Consider Three Intervals S+++ S++ S++- S+ S+- S S- Slide 18-34 S+-S-- S--- Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition Possible Outcomes with Three Intervals Event Probability Stock Price 3 up 1/8 100 (1.05)3 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 Slide 18-35 =115.76 = 91.27 Copyright © McGraw-Hill Ryerson Limited, 2003 Bodie Kane Marcus Perrakis Ryan INVESTMENTS, Fourth Canadian Edition 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 Slide 18-36 Copyright © McGraw-Hill Ryerson Limited, 2003
