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Chapter 16 Option Valuation

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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

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**Time Value of Options: Call**

Value of Call Intrinsic Value Time value X Stock Price 3

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**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

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**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

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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

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**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

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**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 e – 0.12´0.25 =

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**Valuing the Option The portfolio that is**

long shares short 1 option is worth 4.367 The value of the shares is (= 0.25´20 ) The value of the option is therefore (= – )

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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?

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**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

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**Black-Scholes Option Valuation**

X = Exercise price. d = Annual dividend yield of underlying stock e = , 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

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**Call Option Example So = 100 X = 95 r = .10 T = .25 (quarter)**

s = d = 0 d1 = [ln(100/95)+(.10-0+(.5 2/2))]/( /2) = .43 d2 = ((.5)( .251/2) = .18 11

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**Probabilities from Normal Dist.**

Table 17.2 d N(d) Interpolation 12

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**Probabilities from Normal Dist.**

Table 17.2 d N(d) 13

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**Call Option Value Co = Soe-dTN(d1) - Xe-rTN(d2)**

Co = 100 X 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

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**Put Option Value: Black-Scholes**

P=Xe-rT [1-N(d2)] - S0e-dT [1-N(d1)] Using the sample data P = $95e(-.10X.25)( ) - $100 ( ) P = $6.35

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**Put Option Valuation: Using Put-Call Parity**

P = C + PV (X) - So = C + Xe-rT - So Using the example data C = X = 95 S = 100 r = .10 T = .25 P = e -.10 X P = 15

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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

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