Early execise of American Call on Non-Dividend paying stock.

Question

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 1

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.