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CH21

# Fundamentals of Corporate Finance + Standard & Poor's Educational Version of Market Insight

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CHAPTER 21 Valuing Options Answers to Practice Questions 1. a. 188 844 . 0 / 1 ; 185 . 1 5 . 0 24 . 0 = = = = u d e u \$45 \$53.33 \$37.98 \$63.19 \$45.01 \$45.01 \$32.06 887 . 0 / 1 ; 127 . 1 25 . 0 24 . 0 = = = = u d e u \$45 \$50.72 \$57.16 \$64.41 \$35.41 \$50.70 \$39.92 \$44.98 \$39.91 \$72.60 \$57.14 \$57.14 \$44.97 \$44.97 \$35.40 \$31.41 \$35.40 \$27.86

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b. 189 809 . 0 / 1 , 236 . 1 5 . 0 3 . 0 = = = = u d e u \$45 \$55.62 \$36.41 \$68.75 \$45.00 \$45.00 \$29.46 \$45 \$52.29 \$60.76 \$70.60 \$33.36 \$52.32 \$38.75 \$45.02 \$38.77 \$82.04 \$60.79 \$60.79 \$45.05 \$45.05 \$33.38 \$28.72 \$33.38 \$24.73 861 . 0 / 1 ; 162 . 1 25 . 0 3 . 0 = = = = u d e u
a. Let p equal the probability of a rise in the stock price. Then, if investors are risk- neutral: (p × 0.15) + (1 - p) × (-0.13) = 0.10 p = 0.821 The possible stock prices next period are: \$60 × 1.15 = \$69.00 \$60 × 0.87 = \$52.20 Let X equal the break-even exercise price. Then the following must be true: X – 60 = (p)(\$0) + [(1 – p)(X – 52.20)]/1.10 That is, the value of the put if exercised immediately equals the value of the put if it is held to next period. Solving for X, we find that the break- even exercise price is \$61.52. b. If the interest rate is increased, the value of the put option decreases. If there is an increase in: The change in the put option price is: Stock price (P) Negative Exercise price (EX) Positive Interest rate (r f ) Negative Time to expiration (t) Positive Volatility of stock price ( σ ) Positive Consider the following base case assumptions: P = 100, EX = 100, r f = 5%, t = 1, σ = 50% Then, using the Black-Scholes model, the value of the put is \$16.98 The base case value along with values computed for various changes in the assumed values of the variables are shown in the table below: Black-Scholes put value: Base case 16.98 P = 120 11.04 EX = 120 29.03 r f = 10% 14.63 t = 2 21.94 σ = 100% 35.04 190

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a. The future stock prices of Matterhorn Mining are: With dividend Ex-dividend Let p equal the probability of a rise in the stock price. Then, if investors are risk-neutral: (p × 0.25) + (1 - p) × (-0.20) = 0.10 p = 0.67 Now, calculate the expected value of the call in month 6. If stock price decreases to SFr80 in month 6, then the call is worthless. If stock price increases to SFr125, then, if it is exercised at that time, it has a value of (125 – 80) = SFr45. If the call is not exercised, then its value is: Therefore, it is preferable to exercise the call. The value of the call in month 0 is: b. The future stock prices of Matterhorn Mining are: With dividend Ex-dividend 191 100 80 125 60 105 75 48 131.25 84 ? 0 0 0 32.42 51.25 4 SFr27.41 1.10 0) (0.33 5) (0.67 4 = × + × 100 80 125 51.2 80 125 64 100 SFr32.42 1.10 4) (0.33 51.25) (0.67 = × + ×
Let p equal the probability of a rise in the price of the stock. Then, if investors are risk-neutral: (p × 0.25) + (1 - p) × (-0.20) = 0.10 p = 0.67 Now, calculate the expected value of the call in month 6. If stock price decreases to SFr80 in month 6, then the call is worthless. If stock price increases to SFr125, then, if it is exercised at that time, it has a value of (125 – 80) = SFr45. If the call is not exercised, then its value is: Therefore, it is preferable to exercise the call. The value of the call in month 0 is: a. The possible prices of Buffelhead stock and the associated call option values (shown in parentheses) are: Let p equal the probability of a rise in the stock price. Then, if investors are risk-neutral: p (1.00) + (1 - p)(-0.50) = 0.10 p = 0.4 192 220 (?) 110 (?) 440 (?) 55 (0) 220 (55) 880 (715) ? 0 0 0

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