7.5. Consider a stock that can appreciate by 50 percent or depreciate by 50 percent per period. Three periods from now, a stock with an initial value of

$32 per share can be worth (1) $108—three up moves; (2) $36—two up moves, one down move;

(3) $12—one up move, two down moves; or

(4) $4—three down moves. Three periods from now, a derivative is worth $78 in case (1), $4 in case (2), and $0 otherwise. If the risk-free rate is 10 percent throughout these three periods, describe a portfolio of the stock and a risk-free bond that tracks the payoff of the derivative and requires no future cash outlays. Then, fill in the question marks in the accompanying exhibit, which illustrates the paths of the stock and the derivative. Hint: You need to work backward. Use the risk-neutral valuation method to check your work.