Chapter 16 Option Valuation

Option Values Intrinsic value - payoff 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 2

Time Value of Options: Call Value of Call Intrinsic Value Time value X Stock Price 3

Factors Influencing Option Values: Calls Factor Effect on value Stock price increases Exercise price decreases Volatility of stock price increases Time to expiration increases Interest rate increases Dividend yield decreases 4

A Simple Binomial Model A stock price is currently $20 In three months it will be either $22 or $18 Stock Price = $22 Stock price = $20 Stock Price = $18

A Call Option A 3-month call option on the stock has a strike price of 21. Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $0

Setting Up a Riskless Portfolio Consider the Portfolio: long D shares short 1 call option Portfolio is riskless when 22D – 1 = 18D or D = 0.25 22D – 1 18D

Valuing the Portfolio (Risk-Free Rate is 12%) The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22´0.25 – 1 = 4.50 The value of the portfolio today is 4.5e – 0.12´0.25 = 4.3670

Valuing the Option The portfolio that is long 0.25 shares short 1 option is worth 4.367 The value of the shares is 5.000 (= 0.25´20 ) The value of the option is therefore 0.633 (= 5.000 – 4.367 )

Example: Suppose the stock now sells at $100, and the price will either double to $200 or fall in half to $50 by the year-end. A call option on the stock might specify an exercise price of $125 and a time to expiration of one year. The interest rate is 8%. What is the option price today?

Black-Scholes Option Valuation Co = Soe-dTN(d1) - Xe-rTN(d2) d1 = [ln(So/X) + (r – d + s2/2)T] / (s T1/2) d2 = d1 - (s T1/2) where Co = Current call option value. So = Current stock price N(d) = probability that a random draw from a normal dist. will be less than d. 9

Black-Scholes Option Valuation X = Exercise price. d = Annual dividend yield of underlying stock e = 2.71828, 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 s = Standard deviation of annualized cont. compounded rate of return on the stock 10

Call Option Example So = 100 X = 95 r = .10 T = .25 (quarter) s = .50 d = 0 d1 = [ln(100/95)+(.10-0+(.5 2/2))]/(.5 .251/2) = .43 d2 = .43 - ((.5)( .251/2) = .18 11

Probabilities from Normal Dist. Table 17.2 d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700 12

Probabilities from Normal Dist. Table 17.2 d N(d) .16 .5636 .18 .5714 .20 .5793 13

Call Option Value Co = Soe-dTN(d1) - Xe-rTN(d2) Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 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? 14

Put Option Value: Black-Scholes P=Xe-rT [1-N(d2)] - S0e-dT [1-N(d1)] Using the sample data P = $95e(-.10X.25)(1-.5714) - $100 (1-.6664) P = $6.35

Put Option Valuation: Using Put-Call Parity P = C + PV (X) - So = C + Xe-rT - So Using the example data C = 13.70 X = 95 S = 100 r = .10 T = .25 P = 13.70 + 95 e -.10 X .25 - 100 P = 6.35 15

Exercise in class The stock price of Ajax Inc. is currently $105. The stock price a year from now will be either $130 or $90 with equal probabilities. The interest rate at which investors can borrow is 10%. Using the binomial OPM, the value of a call option with an exercise price of $110 and an expiration date one year from now should be worth __________ today. A) $11.60 B) $15.00 C) $20.00 D) $40.00 The stock price of Bravo Corp. is currently $100. The stock price a year from now will be either $160 or $60 with equal probabilities. The interest rate at which investors invest in riskless assets at is 6%. Using the binomial OPM, the value of a put option with an exercise price of $135 and an expiration date one year from now should be worth __________ today. A) $34.09 B) $37.50 C) $38.21 D) $45.45 Answer: A Difficulty: Hard Answer: C Difficulty: Hard