 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill 21-1 Options Valuation Chapter 21

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill 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

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill 21-3 Time Value of Options: Call Option value X Stock Price Value of Call Intrinsic Value Time value

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill 21-4 FactorEffect on value Stock price increases Exercise price decreases Volatility of stock price increases Time to expirationincreases Interest rate increases Dividend Ratedecreases Factors Influencing Option Values: Calls

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill 21-5 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 > S 0 - PV ( X ) - PV ( D )

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill 21-6 Allowable Range for Call Call Value S0S0 PV (X) + PV (D) Upper bound = S 0 Lower Bound = S 0 - PV (X) - PV (D)

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill Stock Price C 75 0 Call Option Value X = 125 Binomial Option Pricing: Text Example

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill 21-8 Alternative Portfolio Buy 1 share of stock at $100 Borrow $46.30 (8% Rate) Net outlay $53.70 Payoff Value of Stock Repay loan Net Payoff Payoff Structure is exactly 2 times the Call Binomial Option Pricing: Text Example

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill C C = $53.70 C = $26.85 Binomial Option Pricing: Text Example

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill Alternative Portfolio - one share of stock and 2 calls written (X = 125) Portfolio is perfectly hedged Stock Value50200 Call Obligation Net payoff50 50 Hence C = or C = Another View of Replication of Payoffs and Option Values

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill 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)

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill Generalizing the Two-State Approach

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill 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

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill S S + S + + S - S - - S + - S S S S Expanding to Consider Three Intervals

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill Possible Outcomes with Three Intervals EventProbabilityStock Price 3 up 1/8100 (1.05) 3 = up 1 down 3/8100 (1.05) 2 (.97)= up 2 down 3/8100 (1.05) (.97) 2 = down 1/8100 (.97) 3 = 91.27

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill C o = S o N(d 1 ) - Xe -rT N(d 2 ) d 1 = [ln(S o /X) + (r +  2 /2)T] / (  T 1/2 ) d 2 = d 1 + (  T 1/2 ) where C o = Current call option value. S o = Current stock price N(d) = probability that a random draw from a normal dist. will be less than d. Black-Scholes Option Valuation

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill X = Exercise price. e = , the base of the nat. 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 Option Valuation

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill S o = 100X = 95 r =.10T =.25 (quarter)  =.50 d 1 = [ln(100/95) + (.10+(  5 2 /2))] / (  5 .25 1/2 ) =.43 d 2 =.43 + ((  5 .25 1/2 ) =.18 Call Option Example

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill N (.43) =.6664 Table 17.2 d N(d) Interpolation Probabilities from Normal Dist

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill N (.18) =.5714 Table 17.2 d N(d) Probabilities from Normal Dist.

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill C o = S o N(d 1 ) - Xe -rT N(d 2 ) C o = 100 X e -.10 X.25 X.5714 C o = 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

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill P = C + PV (X) - S o = C + Xe -rT - S o Using the example data C = 13.70X = 95S = 100 r =.10T =.25 P = e -.10 X P = 6.35 Put Option Valuation: Using Put-Call Parity

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill 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 S 0 with S 0 - PV (Dividends)

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill Hedging: Hedge ratio or delta The number of stocks required to hedge against the price risk of holding one option Call = N (d 1 ) Put = N (d 1 ) - 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

 The McGraw-Hill Companies, Inc., 1999 INVESTMENTS Fourth Edition Bodie Kane Marcus Irwin/McGraw-Hill 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