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1 Chapter 16 Option Valuation

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2 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

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

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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 yielddecreases

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5 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 = $18 Stock price = $20

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

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

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8 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 – 1 = 4.50 The value of the portfolio today is 4.5 e – =

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9 Valuing the Option The portfolio that is long 0.25 shares short 1 option is worth The value of the shares is (= ) The value of the option is therefore (= – )

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10 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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11 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.

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12 Black-Scholes Option Valuation X = Exercise price. = 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 Standard deviation of annualized cont. compounded rate of return on the stock

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13 Call Option Example S o = 100X = 95 r =.10T =.25 (quarter) =.50 = 0 d 1 = [ln(100/95)+(.10-0+( 5 2 /2))]/( /2 ) =.43 d 2 =.43 - (( /2 ) =.18

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14 Probabilities from Normal Dist. N (.43) =.6664 Table 17.2 d N(d) Interpolation

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

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16 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?

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17 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

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18 Put Option Valuation: Using Put-Call Parity 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

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19 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

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