[leplen]

American options can be exercised at any time at the investor's discretion. This means the instrument has a maximum duration, but the actual duration is up to choice of the option holder, which you can't model with an equation.

[auc]

This is subtly incorrect.

An American option should be priced assuming that the option is optimally exercised, otherwise this would create a soft arbitrage opportunity. The difficulty is determining when the option is optimally exercised because it depends on several potentially unknown and difficult to model factors.

[l33t7332273]

Why does having variable duration mean an equation cannot model this type of option?

[hfhdjdks]

The point is not whether it can be modeled (it can), but whether you can get a closed form solution for it (you can't...or we don't know how for now)