7. In each period of time, a certain stock either goes down 1 with probability 0.41, remains the same
with probability 0.20, or goes up 1 with probability 0.39. Assume that the changes in successive time
periods are independent. Approximate the probability that, after 700 time periods, the stock will be
up more than 10 from where it started.
8. Consider two European put options, both of which have expiration time T. Suppose the exercise prices
of the two puts are $K_1$ and $K_2$ respectively. Prove that
$K_1 - K_2 \ge P_1 - P_2 \ge 0$,
where $P_i$ is the price of the put with strike price $K_i$, $i = 1, 2$ and $K_1 \ge K_2$.
9. Three European put options on a stock have the same expiration date and strike prices of $55, $60,
and $65. The market prices are $3, $5, and $8, respectively. Explain how a butterfly spread can be
created. Construct a table showing the profit from the strategy. For what range of stock prices would
the butterfly spread lead to a loss?
