Let us consider an American call option with strike price K and the time to maturity be T. Assume that the underlying stock does not pay any dividend. Let the price of this call option is C$^a$ today (t = 0). Now, suppose that at some intermediate time t ($<$T), I decide to exercise my call option. Hence the profit is:
P1 = S(t) - K - C$^a$
I could then earn the interest on this profit and hence at maturity i will have:
P2 = P1*e$^{r(T-t)}$ = (S(t) - K - C$^a$)e$^{r(T-t)}$
Instead, I could have waited and exercised it at maturity. My profit would then be:
P3 = S(T) - K + Ke$^{rT}$ - C$^a$
I write this because i could have kept $K in the bank at t = 0 and earned a risk-free interest on it till maturity time T.
So here is my question: Merton (in 1973) said that it an American call on a non-dividend paying stock should not be exercised before expiration. I am just trying to figure out why it is true. Because there might be a possibility that P2 > P3.
P.S: I am not contesting that what Merton said is wrong. I totally respect him and am sure what he is saying is correct. But I am not able to see it mathematically. Any help will be appreciated!.
Thank You.
Answer
You compare apples and oranges here. You can't possibly compare the profit generated involving S(t) on one side and S(T) on the other side. at time t you do not know what the stock will be worth at time T. Merton made the statement in the context of deciding whether
- to exercise the call option at any time before expiration
OR
- to simply sell the call option in the market
and came to the conclusion that it is sub-optimal to exercise the option before expiration, but in light of the fact that he meant a comparison between exercise vs. selling the option, not between exercising and waiting till expiration.
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