As an options trader trading American style options, should one be considered with the probability of an option expiring in the money? Or simply the probability of the option touching the In the money point? Because as soon as it touches the ITM, the trader can close the position.What is the need to worry about the probability of the option 'expiring' ITM as opposed to touching ITM?
Conceptually, yes, you need to worry about it. As a practical matter, it's less likely to be exercised until expiry or shortly prior.
The way to think about paying a European option is: [Odds of paying out] = [odds that strike is in the money at expiry]
Whereas the American option can be thought of as: [Odds of paying out] = [odds that strike price is in the money at expiry] + ( [odds that strike price is in the money prior to expiry] * [odds that other party will exercise early] ).
This is just a heuristic, not a formal financial tool. But the point is that you need to consider the odds that it will go into the money early, for how long (maybe over multiple periods), and how likely the counterparty is to exercise early.
Important considerations for whether they will exercise early are the strategy of the other side (long, straddle, quick turnaround), the length of time the option is in the money early, and the anticipated future movement.
A quick buck strategy might exercise immediately before the stock turns around. But that could leave further gains on the table, so it's usually best to wait unless the expectation is that the stock will quickly reverse its movement. This sort of counter-market strategy is generally unlikely from someone who bought the option at a certain strike, and is equivalent to betting against their original purchase of the option. So most of these people will wait because they expect the possibility of a bigger payoff.
A long strategy is usually in no hurry to exercise, and in fact they would prefer to wait until the end to hold the time value of the option (the choice to get out of the option, if it goes back to being unprofitable). So it usually makes little sense for these people to exercise early. The same goes for a straddle, if someone is buying an option for insurance or to economically exit a position.
So you're really just concerned that people will exercise early and forgo the time value of the American option. That may include people who really want to close a position, take their money, and move on. In some cases, it may include people who have become overextended or need liquidity, so they close positions. But for the most part, it's less likely to happen until the expiration approaches because it leaves potential value on the table. The time value of an option dwindles at the end because the implicit option becomes less likely, especially if the option is fairly deep in the money (the implicit option is then fairly deep out of the money).
So early exercise becomes more meaningful concern as the expiration approaches. Otherwise, it's usually less worrisome but more than a nonzero proposition.