Brealey. Myers. Allen Chapter 21 Solution - CHAPTER 21...

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171 CHAPTER 21 Valuing Options Answers to Practice Questions 1. a. 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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172 b. 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
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173 2. a. The diagram on the left below displays the possible stock prices. The diagram on the right displays the corresponding call option values. u – 1 = (¥6000/¥5000) – 1 = 0.200 d – 1 = (¥4167/¥5000) – 1 = –0.167 Let p equal the probability of a rise in the stock price. Then, if investors are risk-neutral: (p × 0.20) + (1 – p) × (–0.167) = 0.03 p = 0.537 The value of the call is: The possible stock prices and the corresponding put option values are shown in the diagrams below: The value of the put is: b. Let X equal the break-even exercise price. Then the following must be true: X – ¥5000 = [(p)(¥0) + (1 – p)(X – ¥4167)]/1.03 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: ¥5,680.21 At any higher exercise price, the put should be exercised immediately. 5000 6000 4167 260.68 500 0 260.68 1.03 0) (0.463 500) (0.537 = × + × 599.20 1.03 1333) (0.463 0) (0.537 = × + × 5000 6000 4167 599.20 0 1333
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174 c. Again, let X equal the break-even exercise price. Then the following must be true: X – ¥5000 = [(p)(¥0) + (1 – p)(X – ¥4167)]/1.14 Solving for X, we find that the break-even exercise price is: ¥5,569.69 At any higher exercise price, the put should be exercised immediately. d. The diagram on the left below displays the possible stock prices. The diagram on the right displays the corresponding call option values. The value of the call is: 3. a. The future stock prices of Moria 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. 100 80 125 60 105 75 48 131.25 84 ? 0 0 0 32.42 51.25 4 5000 6000 4167 166.73 319.79 0 166.73 1.03 0) (0.463 319.79) (0.537 = × + ×
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175 If stock price decreases to $80 in month 6, then the call is worthless. If stock price increases to $125, then, if it is exercised at that time, it has a value of ($125 – $80) = $45. 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 Moria Mining are: With dividend Ex-dividend 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 $80 in month 6, then the call is worthless. If stock price increases to $125, then, if it is exercised at that time, it has a value of ($125 – $80) = $45. 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: 27.41 1.10 0) (0.33 5) (0.67 $ $ 4 $ = × + × 100 80 125 51.2 80 125 64 100 ? 0 0 0
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