Problem 5 Suppose a stock, which pays no dividends, sells for $10 today. Next period, it will either move to $7 or $14. You do not know the probabilities of these two outcomes. Riskless zero coupon bonds, paying $1.10 in one period, cost $1.00 today.
1. What price would an at-the-money call sell for today?
2. If you wished to synthetically manufacture the at-the-money call option, how many bonds would you buy? How many shares of stock?
3. If the call sold for $3.00, how would you capture arbitrage profits?
4. Now, consider what the stock might do in the second period. If it moves to $14 in the first period, it can either move up to $18 or down to $11 in the second period. If it moves to $7 in the first period, it can either move up to $11 or down to $4 in the second period. Assuming that riskless zero-coupon bonds, paying $1.10 in the second period, cost $1.00 at the end of the first period, what price would an at-the-money call sell for today?