# American Vs European Options behavior with fixed strikes and varying expiration

Following is from page 10 of Fengler (2005), "The prices of American calls for the same strikes must be nondecreasing, Merton (1973), and in the absence of dividends, this property translates to European calls as well."
I have read in this document Handbook of Comp Fin that this doesn't translate to european options since there theta can change sign, so how does no dividends affect this nature of european options.

Thank you.

• For a European Call the Theta reverses sign only if the dividend rate q is higher that rhe risk free rate $r_f$. (So with $r_f\ge 0$ and $q=0$ you are safe. – noob2 Apr 21 '17 at 18:18
• @noob2, I missed that thought, thanks for clearing that up. I need to brush up my options basics. – quantfin_enthusiast Apr 21 '17 at 18:40
• ... actually I am not sure I am right. Checking... – noob2 Apr 21 '17 at 18:43
• @noob2. In Hull there are two cases where the theta of a long option position can be positive. 1. ITM European put on a non-dividend paying stock 2. ITM European call on a currency with a very high interest rate – quantfin_enthusiast Apr 21 '17 at 18:50
• @quantfin_enthusiast Consider the following thought experiment where a dividend-paying stock is expected to pay a very large dividend in exactly 6M. Two European options are written on that stock. They have exactly same strike but different expiries: $T_{\pm} = 6M \pm 1day$. Which one is the most expensive? Now suppose that instead these are American options, which one is the most expensive? – Quantuple Apr 21 '17 at 20:01