I have been under the impression that European options trade at a discount to American options on a similar underlying. You should pay a little extra to have the privilege of exercising anytime. This seems to be true for XEO versus OEX options. However, when I checked DJX versus DIA, it seems that the DIA puts are slightly cheaper at the same strike. Of course, the underlying is not exactly the same, but close. Anybody have an explanation? NDX and QQQ are not as easy to compare since I think you have to multiply the QQQ bid and asks by 40 (?) to make a comparison.