Consider an americal Call option on an underlying paying dividends. Then it is often argued that it is only optimal to exercise right before the dividend is paid out, otherwise one will not exercise.
Now what if the dividend is continuous -can one then always see exercise?
Furthermore, is a reasonable assumption that the above strategy is possible? I have little experience with how things actually works, but is it, in practice, known beforehand when lump sum dividends are paid out?
When dividends are continuous, they are essentially negative interest rates, so you should price options w.r.t. new interest rate $\hat r := r-d$ where $r$ is the original interest rate and $d$ is the continuous dividend yield. If $\hat r>0$ then the price of the call is still a submartingale, so early exercise is not (strongly) optimal, however in a more realistic setting $\hat r<0$ so that pricing of the call becomes as intricate as that of the put: at each moment of time some strikes are early exercise and some are not.
In practice, it is often known how often does the company pays/decides on the dividends. They always anounce the dividend amount and date in advance, before that you have to price the dividends using some model.
in the continuous case, you can regard the dividend rate as the interest on a foreign bank account if we invest it so the number of shares grows at the rate $d.$ So we can think it as a call option on a foreign exchange rates. Now calls and puts are the same thing in foreign exchange just by changing viewpoint. So the pricing is just as hard for calls as for puts and exercise can happen any time depending on the relative interest rates.
