Suppose that over two periods a stock price moves on a binomial tree

(i) Determine for what values of the riskless rate $r$ there is no arbitrage. From here on, fix $r=1 / 4$ and determine the equivalent martingale measure.

(ii) Find the time-zero price and replicating portfolio for a European put option with strike 15 and expiry $2 .$

(iii) Find the time-zero price and optimal exercise policy for an American put option with the same strike and expiry.

(iv) Deduce the corresponding (European and American) call option prices for the same strike and expiry.