In section 11.1 (Hull, 9th global), it says and I quote " Now consider the effect of the expiration date. Both put and call American options become more valuable (or at least do not decrease in value) as the time to expiration increases. Consider two American options that differ only as far as the expiration date is concerned. The owner of the long-life option has all the exercise opportunities open to the owner of the short-life option—and more. The long-life option must therefore always be worth at least as much as the short-life option. Although European put and call options usually become more valuable as the time to expiration increases (see Figure 11.1e, f), this is not always the case. Consider two European call options on a stock: one with an expiration date in 1 month, the other with an expiration date in 2 months. Suppose that a very large dividend is expected in 6 weeks. The dividend will cause the stock price to decline, so that the short-life option could be worth more than the long-life option " I think each paragraph looks right. However, if we read them together, why cannot we apply the reasoning for American options to European Options? Thank you.
General comment: an American option can always exercise early if it is beneficial to the option holder to do so. Specific: This part of the text you quote is simply highlighting the impact that holding a European call option on a dividend paying stock can have. Specifically, the 1 month call option expires before the ex dividend date and so will not reflect a dividend within its price. By contrast, the 2 month option will be priced to reflect the large dividend (which will reduce the price of the forward, all else equal). If large enough this effect could overwhelm the added juice / time value embedded within the longer dated option to such an extent that the 2 month call trades below the shorter-life option. [I've never seen this in practice but I accept that it is theoretically possible.]
