1. Chapter 15 Option Valuation

2. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Option Values Intrinsic value – Time value -

3. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Time Value of Options: Call Option value X Stock Price Value of Call Intrinsic Value Time value

4. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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

5. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Binomial Option Pricing: Text Example 100 200 50 Stock Price C 75 0 Call Option Value X = 125

6. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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 53.70 150 0 Payoff Structure is exactly 2 times the Call

7. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Binomial Option Pricing: Text Example 53.70 150 0 C 75 0 2C = C =

8. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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 Hence 100 - 2C = 46.30 or C = 26.85

9. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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.

10. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Black-Scholes Option Valuation X = Exercise price.  = Annual dividend yield of underlying stock 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 cont. compounded rate of return on the stock

11. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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 ) = d 2 =.43 - ((  5 .25 1/2 ) =

12. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Probabilities from Normal Dist. N (.43) =.6664 Table 17.2 d N(d).42.6628.43.6664 Interpolation.44.6700

13. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Probabilities from Normal Dist. N (.18) =.5714 Table 17.2 d N(d).16.5636.18.5714.20.5793

14. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Call Option Value C o = S o e -  T N(d 1 ) - Xe -rT N(d 2 ) C o = 100 X.6664 - 95 e -.10 X.25 X.5714 C o = 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?

15. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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) (1-.5714) - $100 (1-.6664) P = $6.35

16. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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 = 13.70 + 95 e -.10 X.25 - 100 P = 6.35

17. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. 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

18. McGraw-Hill/Irwin © 2004 The McGraw-Hill Companies, Inc., All Rights Reserved. Portfolio Insurance - Protecting Against Declines in Stock Value Buying Puts - Limitations