013 - 13 European Option Pricing Answers to Questions and...

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97 Answers to Questions and Problems 1. What is binomial about the binomial model? In other words, how does the model get its name? The binomial model is binomial because it allows for two possible stock price movements. The stock can either rise by a certain amount or fall by a certain amount. No other stock price movement is possible. 2. If a stock price moves in a manner consistent with the binomial model, what is the chance that the stock price will be the same for two periods in a row? Explain. There is no chance. In every period, the stock price will either rise or fall. Therefore, in two adjacent peri- ods, the stock price cannot be the same. From this period to the next, the stock price must necessarily rise or fall. However, the stock price can later return to its present price. This depends on the up and down factors for the change in the stock price. 3. Assume a stock price is $120, and in the next year, it will either rise by 10 percent or fall by 20 percent. The risk-free interest rate is 6 percent. A call option on this stock has an exercise price of $130. What is the price of a call option that expires in one year? What is the chance that the stock price will rise? Our data are: Therefore, C 5 0.0556($120) 2 $5.03 5 $1.64. The probability of a stock price increase is: ( R 2 D )/( U 2 D ) 5 (1.06 2 0.8)/(1.1 2 0.8) 5 0.8667 N * 5 C u 2 C d ( U 2 D ) S 5 2 2 0 (1.1 2 0.8) 120 5 0.0556 B * 5 ( C u D 2 C d U ) [( U 2 D ) R ] 5 2(0.8) 2 0(1.1) (1.1 2 0.8) (1.06) 5 $5.03 R 5 1.06 DS 5 $96 US 5 $132 C d 5 $0 C u 5 $2 13 European Option Pricing
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98 CHAPTER 13 EUROPEAN OPTION PRICING 4. Based on the data in question 3, what would you hold to form a risk-free portfolio? Because C 5 N * S–B *, the portfolio of C–N * S 1 B * should be a riskless portfolio. 5. Based on the data in question 3, what will the price of the call option be if the option expires in two years and the stock price can move up 10 percent or down 20 percent in each year? Terminal stock prices in two periods are given as follows: UUS 5 $145.20, DDS 5 $76.80, and UDS 5 DUS 5 $105.60. The probabilities of these different terminal stock prices are: p uu 5 (0.8667) (0.8667) 5 0.7512; p ud 5 (0.8667) (0.1333) 5 0.1155; p du 5 (0.1333) (0.8667) 5 0.1155; and p dd 5 (0.1333) (0.1333) 5 0.0178. The call price at expiration equals the terminal stock price minus the exercise price of $100, or zero, whichever is larger. Therefore, we have C uu 5 $15.20, C dd 5 0, C du 5 C ud 5 0. We have already found that the probability of an increase is 0.8667, so the probability of a down movement is 0.1333. Because the option pays off only with two increases, we need consider only that path. Thus, the value of the call is: 6. Based on the data in question 3, what would the price of a call with one year to expiration be if the call has an exercise price of $135? Can you answer this question without making the full calculations? Explain. From question 3, we see that US 5 $132. This is not enough to bring the call into-the-money. Therefore, we know that the call must expire worthless, so its current price is zero.
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013 - 13 European Option Pricing Answers to Questions and...

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