QUESTION $4$ Consider a EUROPEAN PUT option and an AMERICAN PUT option. Each option is on one share of Smart R Us Inc. Each option has an EXERCISE PRICE OF $$120.$ The two options have one period remaining until expiration. The current price of Smart R Us is $$100.$ By the end of the period the share price will either increase by $20\%$ or decrease by $20\%.$ The stock will not pay dividends. The riskless interest rate over the period is $10\%.$

a$)$ Calculate the current binomial price of the EUROPEAN put option.

b$)$ Suppose that the current market price of the EUROPEAN put option is $$5\mathrm{.}00.$ Is there a profitable arbitrage? If yes, design the arbitrage, show that it is riskless, and calculate the profits on expiration day.

c$)$ Calculate the current binomial price of the AMERICAN put option.