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In a previous question this question came up.

In my mind, if I'm holding an option at time t, then there are possible future price paths where at t+k the option will be ITM but at T the option will be out of the money. Thus, I'd expect the value of an American call at time t to be higher than the European call. Apperently this is not the case, I found this reference that explains it in math, but LocalVolatility says there is a simple economic reason that I am missing. What is it ?

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In your question you consider a model with no dividends or repo rates. Put-call parity in this setting is

$$ C_0 - P_0 = S_0 - K e^{-r T} $$

Which implies that

$$ C_0 \geq \max \left\{ S_0 - K e^{-r T}, 0 \right\} $$

since $P_0 \geq 0$. When $r \geq 0$, then

$$ S_0 - K e^{-r T} \geq S_0 - K $$

where the right-hand side is the intrinsic value. Thus, the value of a European call option is always at least as high as the intrinsic value. Exercising the American option would only pay the intrinsic value. Thus, there is no advantage in exercising early. The corresponding right is worthless and the American call price is the same as the European call price.

Note that this result assumes that there are no dividends or repo rates and interest rates are non-negative. In the more general case it only holds when the discounted forward is not smaller than the spot. i.e. $F_t(T) e^{-r (T -t)} \geq S_t$.

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    $\begingroup$ Probably worth explaining why introducing repo/discrete divs changes the result since then the discounted forward value is lower than spot. $\endgroup$ – Ezy Jan 29 at 1:32
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Although it might seem that due to its flexibility, the American call option must be worth more than the European call, now we will look at why the call option is never optimal to exercise before expiry and hence why they both have the same value.

Suppose instead of exercising the American call option at time T1( which would give you a payoff of S-K), you sell the stocks short at T1( +S(T1) ) and buy them back at expiry by either exercising your option (price: K) or at the market price S whichever is lower. Hence they have the same value.

To summarize,

If you exercise the American call option at t1, profit = S(t1) - K

Second choice:

but now, you short your stocks at t1 : +S(t1) and at maturity, you decide to close the short either by exercising your contract(-K) or by buying the stocks in market (S(T)) whichever is lower. and

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  • $\begingroup$ I really like this trading argument! Could you please formulate it a bit clearer? I can imagine it gets more attention with some nice TeX... Anyways: nice illustration. $\endgroup$ – Richard Feb 4 at 9:19

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