I guess this a pretty easy question to answer, but I'm not able to get the intuition despite reading the concept a couple of times. So, the Greek Theta is almost always negative, except for when an American Put is deep in the money, the American Puts price may increase as the time to maturity nears and it may also be positive for the case of an American Call on a dividend paying stock.
However, wouldn't this just that it would be optimal to hold the American Put and the American Call on a dividend paying stock up to maturity rather than exercise them early as their values are actually increasing with the passage of time? But in reality we would only exercise an American Put or an American Call on a dividend paying stock prior to maturity, so my intuition is clearly wrong. Please could explain where my logic is failing?