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# Answer to Question #12661 in Other Management for John

Question #12661
4. The current price of a stock is \$22, and at the end of one year its price will be either \$27 or \$17. The annual risk-free rate is 6.0%, based on daily compounding. A 1-year call option on the stock, with an exercise price of \$22, is available. Based on the binominal model, what is the option&#039;s value? a. \$2.43 b. \$2.70 c. \$2.99 d. \$3.29 e. \$3.62
1
2012-08-09T10:01:04-0400
Answer is c (\$ 2,99)The stock&rsquo;s range of payoffs in one year is \$27 - \$17 = \$10. At expiration, the option will be worth \$27 - \$22 = \$5 in case the stock price is \$27, and zero if the stock price is \$17.

The range of payoffs for the stock option is \$5 &ndash; 0 = \$5. We equalize the range to find the number of shares of stock:

Option range / Stock range = \$5/\$10 = 0.5

With 0.5 shares, the stock&rsquo;s payoff will be either \$13.5 or \$8.5. The portfolio&rsquo;s payoff will be \$13.5 - \$5 = \$8.5, or \$8.5 &ndash; 0 = \$8.5.

The present value of \$8.5 at the daily compounded risk-free rate is: PV = \$8.5 / (1+ (0.06/365))365 = \$8.005.

The option price is the current value of the stock in the portfolio minus the PV of the payoff: V = 0.5(\$22) - \$8.005 = \$3.00.

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