Merton's no early exercise theorem

[JPW]

Hi,

there is theorem by Merton (sometimes called "no-early-exercise theorem") which says that an American Call has the same value as an European, because early exercise is never optimal (as far as I know, it can be proved by using the submartingale property).

Does this theoretical observation really hold in reality? Do American and European Calls in reality really have the same price?

I mean, if I hold an American option whose maturity is one year and after 2 months the stock is up 600%, I can not think that it can be a dominant strategy not to exercise the option but wait until it's expiration instead?!

Many thanks for your answers.

 

[JPAlonso]

The presence of dividends may make it optimal to exercise early.

If the expected value of the holding period between the dividend payment and the expiration is less than the value of the dividend itself, it makes sense to exercise the option just before the dividend date.

The fact that the stock is up 600% by itself does not make it optimal to exercise (according to Black-Scholes anyway) because it is assumed that we have no information about trends (the stock is equally likely to go up or down according to the model).

It would be interesting to see if it holds in practice!

edit: I guess that if the stock went up 600% percent, the intrinsic value would be so high that the premium would be negligible anyway, so maybe the problem takes care of itself.

 

[darth]

I mean, if I hold an American option whose maturity is one year and after 2 months the stock is up 600%, I can not think that it can be a dominant strategy not to exercise the option but wait until it's expiration instead?!

Many thanks for your answers.

The above does not mean that you should hold the option . It just means that selling the option will get you a higher profit than exercising it early. Also for stocks without dividends :
a) An american call option is worth the same as a European call option.
b) An american put option may be worth more than a European put option

You can refer to my favourite book R.L Macdonalds for more details on abovementioned.

 

[GoIllini]

It's true 98% of the time in this low-rates environment, but 2% of the time weird stuff happens and the American feature makes a difference. What happens if you own an in-the-money call on a stock that up and decides to pay a 5% dividend two weeks before expiry?

But yes, most people when they buy a call option are not thinking about these possibilities months and months out. They are thinking about the underlying stock's volatility and whether they are getting cheap or expensive insurance. They are thinking about the underlying's technicals and fundamentals.

I agree with the notion that it's probably better for a retail investor or even an institutional investor who's not in the vol business to sell rather than exercise, but you do need to know when to sell.

 

[financeguy]

just to add, in practice, we do not offer american options at the same rate as european options - we'll charge for the extra optionality you get in the ability to exercise early whether it should be the case in theory or not.

 

[quotes]

For European Options Call and Put, we have this equation:
C-P=S-K/exp(r(T-t))
Thus
C-P>S-K
C>S-K

Since the European Call C has value greater than its intrinsic value S-K, the more valuable American Call is larger than S-K as well.
Thus you can always sell an American Call on the market at a price greater than its intrinsic value.