McGraw-Hill/Irwin © 2007 The McGraw-Hill Companies, Inc., All Rights Reserved. Option Valuation CHAPTER 15

15-2 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

15-3 Figure 15-1 Call Option Value Before Expiration

15-4 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

15-5 Binomial Option Pricing: Text Example Stock Price C 75 0 Call Option Value X = 125

15-6 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

15-7 Binomial Option Pricing: Text Example C C = $53.70 C = $26.85

15-8 Another View of Replication of Payoffs and Option Values Alternative Portfolio - one share of stock and 2 calls written (X = 125) Alternative Portfolio - one share of stock and 2 calls written (X = 125) Portfolio is perfectly hedged Portfolio is perfectly hedged Stock Value50200 Call Obligation Net payoff50 50 Hence C = or C = 26.85

15-9 Generalizing the Two-State Approach

15-10 Generalizing the Two-State Approach C CUCU CDCD C U U C U D C D D

15-11 Figure 15-2 Probability Distributions

15-12 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.

15-13 Black-Scholes Option Valuation X = Exercise price.  = Annual dividend yield of underlying stock e = , 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

15-14 Figure 15-3 A Standard Normal Curve

15-15 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 =.43 d 2 =.43 - ((  5 .25) 1/2 d 2 =.43 - ((  5 .25) 1/2 =.18 =.18

15-16 Probabilities from Normal Dist. N (.43) =.6664 Table 17.2 d N(d) Interpolation Interpolation

15-17 Probabilities from Normal Dist. N (.18) =.5714 Table 17.2 d N(d)

15-18 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?

15-19 Figure 15-4 Implied Volatility of the S&P 500 (VIX Index)

15-20 Put-Call Parity Relationship S T X Payoff for Call Owned 0S T - X Payoff for Put Written-( X -S T ) 0 Total Payoff S T - X S T - X

15-21 Figure 15-5 The Payoff Pattern of a Long Call – Short Put Position

15-22 Arbitrage & Put Call Parity Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal. Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal. C - P = S 0 - X / (1 + r f ) T If the prices are not equal arbitrage will be possible If the prices are not equal arbitrage will be possible

15-23 Put Call Parity - Disequilibrium Example Stock Price = 110 Call Price = 17 Put Price = 5 Risk Free = 5% Maturity =.5 yr Exercise (X) = 105 C - P > S 0 - X / (1 + r f ) T > (105 e (-.05 x.5) ) 9 > > 7.59 Since the leveraged equity is less expensive; Since the leveraged equity is less expensive; acquire the low cost alternative and sell the high cost alternative acquire the low cost alternative and sell the high cost alternative

15-24 Example 15-3 Arbitrage Strategy

15-25 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

15-26 Put Option Valuation: Using Put-Call Parity P = C + PV (X) - S o = C + Xe -rT - 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

15-27 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 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

15-28 Figure 15-6 Call Option Value and Hedge Ratio

15-29 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

15-30 Figure 15-7 Profit on a Protective Put Strategy

15-31 Figure 15-8 Hedge-Ratios Change as the Stock Price Fluctuates

15-32 Empirical Tests of Black-Scholes Option Pricing Implied volatility varies with exercise price –Lower exercise price leads to higher option pricing –If the model was complete implied variability should be the same for different exercise prices

15-33 Figure 15-9 Implied Volatility as a Function of Exercise Price