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Even though dividends are discrete, literature often makes the assumption of continuous dividends (mostly in the case of indices but the individual stocks as well).

The dividend yield denoted by q is often considered as an adjustment to the risk free rate (i.e. r-q).

My question is, what happens to American Call options if r-q < 0? Is it now possible to exercise before maturity so it can no longer be calculated as a European option? Logic says you can early exercise but I am not sure.

Some footnote: In discrete dividend case we know that we should only exercise American Calls before maturity if the excess value of the option is less than the dividend. Otherwise value of the American Option will always be greater than the exercise price. This is mainly due to r > 0, and in the rare case of r < 0 American Puts become equivalent to European Puts.

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I think that for any $q>0$ it becomes optimal to exercise an American call for a sufficiently high spot price $S$: if the spot increases enough, the dividend yield corresponds to sufficient cash dividend to render exercise optimal.

This would happen irrespective of the value of $r$ or the sign of $r-q$. What matters is that, for a given strike $K$, the price of a European call is of the order $$ C \sim S\ e^{-qT}-K\ e^{-rT} \text{ for large }S $$ as both cumulative Normals go to one. This can become negative for large enough $q$, even though $S>>K$. The holder of an American option would not allow for this intrinsic value to become negative, and therefore would exercise early.

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Plus, now I see it is kind of a silly question since I made the mistake of confusing the effect of the sign of $r$ to the options with the effect of $q$ as a dark brother of $r$ missing the step they do not go together (hence $K$ is unaffected by $q$). – berkorbay Aug 4 '14 at 12:41

The excercise should happen if the present value of the dividend yield minus risk free interest exceeds the value of the related put. As noted before this is only true if a position is deep in the money and volatility is low.
On single stocks this decision is usually done right before dividend ex-date.

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With 0% interest rates r-q is almost always < 0. Dividends are pretty much never continuous, so for an American option if the dividend you collect and interest you forgo collecting (or paying) being short the stock is larger than the value of the put (since when you punch a call you sell that call and buy 100 shares, thus selling the put). It's also worth noting that r is < 0 in the pricing model for many stocks now. These stocks are known as "Hard to Borrow" because you have to pay to be short them. If you want to know more about how people trade in hard to borrow stocks it would require a much lengthier explanation. Just know that if you're going to be paying 20 cents to carry short stock over the life time of the call and the put is only worth 5 cents, it's probably a punch.

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