1. a. Is it ever optimal to exercise an American Call option early if the underlying stock has zero dividends?

b. Is it ever optimal to exercise an American Put option early if the underlying stock has zero dividends? If ever, when?

c. In the case of American Call options on stock with dividends, explain how an increase in the volatility of the stock return affects the probability of early exercise.

d. In the case of an American put option, what happens to the probability of early exercise if the dividend is increased? The ex-dividend date is during the lifetime of the option and there are no announcement effects. We are looking purely at the size of the dividend.