Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 1 Chapter 17 Option Valuation

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 2 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

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 3 Time Value of Options: Call Option value X Stock Price Value of Call Intrinsic Value Time value

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 4 Factors Influencing Option Values: Calls FactorEffect on value Stock price increases Exercise price decreases Volatility of stock price increases Time to expirationincreases Interest rate increases Dividend Ratedecreases

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 5 Binomial Option Pricing: Text Example Stock Price C 75 0 Call Option Value X = 125

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 6 Binomial Option Pricing: Text Example 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

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 7 Binomial Option Pricing: Text Example C C = $53.70 C = $26.85

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 8 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 Value50200 Call Obligation Net payoff50 50 Hence C = or C = 26.85

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 9 Black-Scholes Option Valuation C o = S o e -  T 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.

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 10 Black-Scholes Option Valuation X = Exercise price.  = Annual dividend yield of underlying stock 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

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 11 Call Option Example S o = 100X = 95 r =.10T =.25 (quarter)  =.50  = 0 d 1 = [ln(100/95)+(.10-0+(  5 2 /2))]/(  5 .25 1/2 ) =.43 d 2 =.43 - ((  5 .25 1/2 ) =.18

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 12 Probabilities from Normal Dist. N (.43) =.6664 Table 17.2 d N(d) Interpolation

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 13 Probabilities from Normal Dist. N (.18) =.5714 Table 17.2 d N(d)

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 14 Call Option Value C o = S o e -  T 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?

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 15 Put Option Value: Black-Scholes P=Xe -rT [1-N(d 2 )] - S 0 e -  T [1-N(d 1 )] Using the sample data P = $95e (-.10X.25) ( ) - $100 ( ) P = $6.35

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 16 Put Option Valuation: Using Put-Call Parity 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

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 17 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 (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

Essentials of Investments © 2001 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition Irwin / McGraw-Hill Bodie Kane Marcus 18 Portfolio Insurance - Protecting Against Declines in Stock Value 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