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The above picture shows the payoff at expiry(in gold) and at current time T+0(in blue) for a bull call spread.

I am trying to understand American options and to know if it has any significant advantage over the European options in terms of payoff.

Looking at the payoff, assuming volatility and other factors are constant, if price rises to $120 then we would be earning close to 5 at T+0, but if that was at expiration, we would be earning approximately the max profit potential of the spread which is somewhat above 12.5. This is the case for European options.

Now my question is, assuming both options that make up the spread are American options, and I exercise the long call and my short call owner(the buyer) exercises at the exact same time(at T+0), is my payoff still going to be a profit of 5(blue payoff) or above 12.5(gold payoff) at T+0?

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In general American options behave similar to European options but face another arbitrage boundary because of early exercise.

For example in the case where the underlying price rises to $S = \\\$120$ the value of your long call $C(K)$ with strike $K = 100$ has a lower limit. $C(K) \geq S - K = 120 - 100 = 20$. Otherwise you could buy the option and immediately realize the profit.

The problem is that your short call will never be exercised as it has no intrinsic value at the strike. But if the counterparty exercises you will realize the profit of about $ \\\$ 12.50$.

Note that the blue line does not apply in general to your case as this is the price of the option(s) under Black-Scholes model which does not apply to American options.

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