QUESTION $5$

You own a EUROPEAN CALL option and an AMERICAN CALL option, each on one share of We Know Options Inc., and each with an EXERCISE PRICE OF $$90.$ The current share price is $$100,$ and the two call options have one period until expiration. 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 $4\%.$

A$)$ Calculate the current binomial price of the EUROPEAN call option.

B$)$ Suppose that the current market price of the EUROPEAN call option is $$25\mathrm{.}00.$

Is there a profitable arbitrage? If yes, design the arbitrage, show that it is riskless, and calculate the profits on expiration day.

Now, suppose that the stock went ex$-$dividend by the amount of $$20$ an instant before the current share price became $$100.$

C$)$ Calculate the current binomial price of the AMERICAN call option immediately

after the stock went ex$-$dividend and its price became $$100.$

D$)$ Calculate the current binomial price of the AMERICAN call option immediately

before the stock paid $$20$ dividends.