INVESTMENTS | BODIE, KANE, MARCUS Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin CHAPTER 18 Option Valuation

INVESTMENTS | BODIE, KANE, MARCUS 18-2 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

INVESTMENTS | BODIE, KANE, MARCUS 18-3 Figure 18.1 Call Option Value before Expiration

INVESTMENTS | BODIE, KANE, MARCUS 18-4 Table 18.1 Determinants of Call Option Values

INVESTMENTS | BODIE, KANE, MARCUS 18-5 Restrictions on Option Value: Call Call value cannot be negative. The option payoff is zero at worst, and highly positive at best. Call value cannot exceed the stock value. Value of the call must be greater than the value of levered equity. Lower bound = adjusted intrinsic value: C > S 0 - PV (X) - PV (D) (D=dividend)

INVESTMENTS | BODIE, KANE, MARCUS 18-6 Figure 18.2 Range of Possible Call Option Values

INVESTMENTS | BODIE, KANE, MARCUS 18-7 Figure 18.3 Call Option Value as a Function of the Current Stock Price

INVESTMENTS | BODIE, KANE, MARCUS 18-8 Early Exercise: Calls The right to exercise an American call early is valueless as long as the stock pays no dividends until the option expires. The value of American and European calls is therefore identical. The call gains value as the stock price rises. Since the price can rise infinitely, the call is “worth more alive than dead.”

INVESTMENTS | BODIE, KANE, MARCUS 18-9 Early Exercise: Puts American puts are worth more than European puts, all else equal. The possibility of early exercise has value because: –The value of the stock cannot fall below zero. –Once the firm is bankrupt, it is optimal to exercise the American put immediately because of the time value of money.

INVESTMENTS | BODIE, KANE, MARCUS Figure 18.4 Put Option Values as a Function of the Current Stock Price

INVESTMENTS | BODIE, KANE, MARCUS Stock Price C 10 0 Call Option Value X = 110 Binomial Option Pricing: Text Example

INVESTMENTS | BODIE, KANE, MARCUS Alternative Portfolio Buy 1 share of stock at $100 Borrow $81.82 (10% Rate) Net outlay $18.18 Payoff Value of Stock Repay loan Net Payoff Payoff Structure is exactly 3 times the Call Binomial Option Pricing: Text Example

INVESTMENTS | BODIE, KANE, MARCUS C C = $18.18 C = $6.06 Binomial Option Pricing: Text Example

INVESTMENTS | BODIE, KANE, MARCUS Alternative Portfolio - one share of stock and 3 calls written (X = 110) Portfolio is perfectly hedged: Stock Value90120 Call Obligation0 -30 Net payoff90 90 Hence C = $81.82 or C = $6.06 Replication of Payoffs and Option Values

INVESTMENTS | BODIE, KANE, MARCUS Hedge Ratio In the example, the hedge ratio = 1 share to 3 calls or 1/3. Generally, the hedge ratio is:

INVESTMENTS | BODIE, KANE, MARCUS Assume that we can break the year into three intervals. For each interval the stock could increase by 20% or decrease by 10%. Assume the stock is initially selling at $100. Expanding to Consider Three Intervals

INVESTMENTS | BODIE, KANE, MARCUS S S + S + + S - S - - S + - S S S S Expanding to Consider Three Intervals

INVESTMENTS | BODIE, KANE, MARCUS Possible Outcomes with Three Intervals EventProbabilityFinal Stock Price 3 up1/8100 (1.20) 3 = $ up 1 down3/8100 (1.20) 2 (.90) = $ up 2 down3/8100 (1.20) (.90) 2 = $ down1/8100 (.90) 3 = $72.90

INVESTMENTS | BODIE, KANE, MARCUS 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 distribution will be less than d Black-Scholes Option Valuation

INVESTMENTS | BODIE, KANE, MARCUS X = Exercise price e = , the base of the natural log r = Risk-free interest rate (annualized, 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 the stock Black-Scholes Option Valuation

INVESTMENTS | BODIE, KANE, MARCUS Figure 18.6 A Standard Normal Curve

INVESTMENTS | BODIE, KANE, MARCUS S o = 100X = 95 r =.10T =.25 (quarter)  =.50 (50% per year) Thus: Example 18.1 Black-Scholes Valuation

INVESTMENTS | BODIE, KANE, MARCUS Using a table or the NORMDIST function in Excel, we find that N (.43) =.6664 and N (.18) = Therefore: C o = S o N(d 1 ) - Xe -rT N(d 2 ) C o = 100 X e -.10 X.25 X.5714 C o = $13.70 Probabilities from Normal Distribution

INVESTMENTS | BODIE, KANE, MARCUS Implied Volatility Implied volatility is volatility for the stock implied by the option price. Using Black-Scholes and the actual price of the option, solve for volatility. Is the implied volatility consistent with the stock? Call Option Value

INVESTMENTS | BODIE, KANE, MARCUS Black-Scholes Model with Dividends The Black Scholes call option formula applies to stocks that do not pay dividends. What if dividends ARE paid? One approach is to replace the stock price with a dividend adjusted stock price Replace S 0 with S 0 - PV (Dividends)

INVESTMENTS | BODIE, KANE, MARCUS Example 18.3 Black-Scholes Put Valuation P = Xe -rT [1-N(d 2 )] - S 0 [1-N(d 1 )] Using Example 18.2 data: S = 100, r =.10, X = 95, σ =.5, T =.25 We compute: $95e -10x.25 ( )-$100( ) = $6.35

INVESTMENTS | BODIE, KANE, MARCUS P = C + PV (X) - S o = C + Xe -rT - S o Using the example data P = e -.10 X P = $6.35 Put Option Valuation: Using Put-Call Parity

INVESTMENTS | BODIE, KANE, MARCUS 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

INVESTMENTS | BODIE, KANE, MARCUS Figure 18.9 Call Option Value and Hedge Ratio

INVESTMENTS | BODIE, KANE, MARCUS 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

INVESTMENTS | BODIE, KANE, MARCUS Figure Profit on a Protective Put Strategy

INVESTMENTS | BODIE, KANE, MARCUS Figure Hedge Ratios Change as the Stock Price Fluctuates

INVESTMENTS | BODIE, KANE, MARCUS Hedging 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.

INVESTMENTS | BODIE, KANE, MARCUS Hedging and Delta The appropriate hedge will depend on the delta. Delta is the change in the value of the option relative to the change in the value of the stock, or the slope of the option pricing curve. Delta = Change in the value of the option Change of the value of the stock

INVESTMENTS | BODIE, KANE, MARCUS Example 18.6 Speculating on Mispriced Options Implied volatility = 33% Investor’s estimate of true volatility = 35% Option maturity = 60 days Put price P = $4.495 Exercise price and stock price = $90 Risk-free rate = 4% Delta = -.453

INVESTMENTS | BODIE, KANE, MARCUS Table 18.3 Profit on a Hedged Put Portfolio

INVESTMENTS | BODIE, KANE, MARCUS Example 18.6 Conclusions As the stock price changes, so do the deltas used to calculate the hedge ratio. Gamma = sensitivity of the delta to the stock price. –Gamma is similar to bond convexity. –The hedge ratio will change with market conditions. –Rebalancing is necessary.

INVESTMENTS | BODIE, KANE, MARCUS Delta Neutral When you establish a position in stocks and options that is hedged with respect to fluctuations in the price of the underlying asset, your portfolio is said to be delta neutral. –The portfolio does not change value when the stock price fluctuates.

INVESTMENTS | BODIE, KANE, MARCUS Table 18.4 Profits on Delta-Neutral Options Portfolio

INVESTMENTS | BODIE, KANE, MARCUS Empirical Evidence on Option Pricing The Black-Scholes formula performs worst for options on stocks with high dividend payouts. The implied volatility of all options on a given stock with the same expiration date should be equal. –Empirical test show that implied volatility actually falls as exercise price increases. –This may be due to fears of a market crash.