McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved Option Valuation Chapter 21

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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 )

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved Allowable Range for Call Call Value S0S0 PV (X) + PV (D) Upper bound = S 0 Lower Bound = S 0 - PV (X) - PV (D)

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved Stock Price C 75 0 Call Option Value X = 125 Binomial Option Pricing: Text Example

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved C C = $53.70 C = $26.85 Binomial Option Pricing: Text Example

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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).

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved Generalizing the Two-State Approach

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved S S + S + + S - S - - S + - S S S S Expanding to Consider Three Intervals

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved X = Exercise price 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 Black-Scholes Option Valuation

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved N (.43) =.6664 Table 17.2 d N(d) Interpolation Probabilities from Normal Dist

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved N (.18) =.5714 Table 17.2 d N(d) Probabilities from Normal Dist.

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved Put Value Using Black-Scholes P = Xe -rT [1-N(d 2 )] - S 0 [1-N(d 1 )] Using the sample call data S = 100 r =.10 X = 95 g =.5 T =.25 95e -10x.25 ( )-100( ) = 6.35

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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)

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved 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.

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved Hedging and Delta The appropriate hedge will depend on the delta. Recall the delta is the change in the value of the option relative to the change in the value of the stock. Delta = Change in the value of the option Change of the value of the stock

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved Mispriced Option: Text Example Implied volatility = 33% Investor believes volatility should = 35% Option maturity = 60 days Put price P = $4.495 Exercise price and stock price = $90 Risk-free rate r = 4% Delta = -.453

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved Hedged Put Portfolio Cost to establish the hedged position 1000 put options at $4.495 / option$ 4, shares at $90 / share 40,770 Total outlay 45,265

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved Profit Position on Hedged Put Portfolio Value of put option as function of stock price: implied vol. = 35% Stock Price Put Price $5.254 $4.785 $4.347 Profit (loss) for each put (.148) Value of and profit on hedged portfolio Stock Price Value of 1,000 puts $ 5,254 $ 4,785 $ 4,347 Value of 453 shares 40,317 40,770 41,223 Total45,571 45,555 45,570 Profit

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved Home Assignment Required: problems 2, 7 (3 rd ed). problems 2, 7 (5 th ed). an additional problem (see next slide) closely follow financial news!

McGraw-Hill/Irwin Copyright © 2001 by The McGraw-Hill Companies, Inc. All rights reserved Home Assignment Assume that each year there are only 2 possible states of the economy: "up" and "down". If the economy goes "up" an "ABC" share's price, which is today $100, increases by 20%, and if the economy goes "down" it decreases by 30%. What should be the price of a 2 years call option on ABC's share with an exercise price of $105, if the riskless rate of return is 8% per year?