[FabHK]

European and American calls cost the same on non-dividend paying stocks (on dividend paying stocks, it might make sense to exercise an American just before the ex-date).

Either way, as was pointed out, in reality BS is used as a deterministic one-to-one mapping between option prices and BS vols. Then, from market quotes (either as prices or as BS vols) a vol-surface is fitted (as a function of strike and expiry time), from which a stochastic process is fitted that correctly re-prices all these points (using a model such as "local vol" or "stochastic vol" or a combination of those two, or others), and then everything is priced of that.

[sokoloff]

American style options are inherently more valuable. Imagine you had options on a stock that experienced a sharp but possibly temporary move. As a holder of an American style option, you could benefit from that temporary move, making it more valuable.

[FabHK]

By put-call parity, C - P = S - df K, thus C = S - df K + P > S - K.

This is contingent only on the discount factor df being <= 1, and P >= 0, which is basically always the case. Thus, the value of the call exceeds the exercise value, making exercise never optimal.

Exercise for the reader: Understand why the same argument doesn't work for puts (or calls on dividend paying stocks).

[im3w1l]

The way the market is typically modeled, temporary moves are not a thing.

[sokoloff]

The way the market actually exists, temporary moves are definitely a thing.

[dahfizz]

> European and American calls cost the same on non-dividend paying stocks

All else being equal, I would prefer to buy an option contract I can exercise at any time vs one I can only exercise on a certain date. It doesn’t make intuitive sense they would be priced the same, can you please elaborate?

[Renevith]

The parent is assuming that you can always sell your option to someone else for its fair value. If that's the case, there would never be a time where it's optimal to exercise a call option, because the optionality will always make the option value higher that the value of owning the stock.

This is shown in the article: the curved lines representing the option value are always above the straight lines of the final option payoff (the value if exercised).

This is not necessarily true for put options or for call options if the stock pays dividends. In those cases the option value can be below the payoff line and early exercise would be better than selling the option.