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Why is early exercise generally not recommended for an in-the-money option?
The crucial insight is that the alternative to early exercise of an American call is not necessarily to hold it to expiry, but to sell it. And selling it, at its value, is always better than exercising it.  Note that this holds only for options on assets that don't pay dividends. Here's the proof, using Put-Call-Parity. We know that at expiry T, we have (using a Call and a Put both struck at K): C(T) - P(T) = S(T) - K (if this is not clear to you, consider the case where S is less than, equal to, or greater than K at maturity, and go through each of them.) If the stock S doesn't pay any dividends (and there is no cost of carry etc.), we can replicate both sides now at time 0; we just buy one call, sell one put (that gives us the left hand side), buy the stock, and borrow money so that at time T we have to repay K (that gives us the right hand side). That means that now, we only need to borrow df * K, where df is the discount factor, and is less than one (assuming the good old pre-2009 world where interest rates are positive). Thus: C(0) - P(0) = S(0) - df * K.  Rearranging gives: C(0) = S(0) - df * K + P(0).  That's the value of the call, if we sell it (or hold it). However, if we exercise, we only get: C_ex = S(0) - K Now, we see that C(0) > C_ex, because we subtract less (df*K < K), and add P(0).