# pricing american calls on non dividend paying stocks

It is never optimal to exercise an american call option early if it is written on a stock that doesn't pay dividends, yet when pricing such an option, using a binomial model, we check whether or not it is optimal to exercise at each node.

I find it strange that it is never optimal to exercise early yet we take into account in its price the payoff from exercising early.

Does anyone know a good argument to explain this? Consider the following example of a call option

it is clearly optimal to exercise early everywhere, so it implicit in the question that the stock underlying the option pays dividends?

• Who is this "we", kemosabe? Oct 22, 2015 at 15:53

The argument that the American and European call are worth the same is model independent. So it holds for the binomial model. So there is no need to check to see if the early exercise occurs because it won't.

Of course, if you have written general purpose code, it is much easier to test for early exercise and always have the test fail than to try and deal with special cases.

• so the expected present value of not exercising the call option at any node is always greater than the payoff of exercising at that node? Oct 23, 2015 at 2:42
• yes that is true Oct 23, 2015 at 2:58
• Doesn't my example contradict that? is there a proof of that statement Oct 23, 2015 at 3:01
• your spreadsheet doesn't look right to me. You have absolute moves but p is the same everywhere. It's a very standard model-free result see eg chapter 2 of my book concepts or my answer here: quant.stackexchange.com/questions/3486/… Oct 23, 2015 at 3:14

It's been proven mathematically that it's never optimal to exercise an American call without dividend. If your spreadsheet shows otherwise, it has to be wrong.

It's a bad idea to drop off the check for early exercising just because you know it'll never happen, because your code will break for anything else, say a dividend-paying American call.

Furthermore, it's a good practice to keep the code for model validation.

Even though it is not optimal to exercise early, you still have that right, thus the price must reflect it

• why must the price reflect something that is never used?.. Oct 23, 2015 at 2:05
• "Not optimal" does not mean never used. I don't know all the practical reasons why you might exercise early, but it's there if you want to. This early exercise feature is what makes American style options at least as valuable as similar European options. Oct 23, 2015 at 2:26

IMHO the code checks for early exercise because it is a general purpose code. That "if" statement will never be true for a non-dividend paying stock. It does not harm and it is good to make your code general.