Call options on bonds: Is the underlying the bond's clean or dirty price?

Are options on bonds written on the bond’s clean price or its dirty price?

The following question is from a Schweser quiz (Level 2 Module 38.3). Something is not right here. The question appears to either:

  1. assume that the option is written on the clean price of the bond, rather than the dirty price, i.e. the coupon payment will influence neither the bond’s price nor the call option’s price. (This would be strange since the clean price is not an actual market price, unlike the dirty price. I feel that if the underlying was the clean price, physical settlement of the contract would be difficult.)

  2. treat American call options (exercise at any time) like Bermudan options (exercise is allowed at specific times only, e.g. at the nodes, not between nodes).

Take for example the uppermost node of the last period (the one with int. rate = 8.56%) in the binomial tree below. The European/Bermudan-style call option allows exercising only at the end of the year, immediately after the $7 coupon has been paid, so their value is zero since the strike ($100) is more than the bond’s price ($98.56).

But an American call could be exercised 1 second before the end of the year, i.e. 1 second before it expires and 1 second before the $7 coupon is paid. So although at expiry the American option is worth 0, one second before it should be worth $5.56, no? Unless the option is written on the clean price? Why does Schweser appear only to allow exercising the option at the node rather than between nodes as is by definition allowed in American-style calls options?

Here’s the Schweser explanation, which does not consider the possibility of exercising just before the $7 coupon is paid:

In a nutshell: We can exercise American-style call options on stocks just before the stock goes ex-dividend. So why is there no discussion of exercising American-style call options on bonds just before the bond pays a coupon?

I think that the question’s moot; most (all?) options on bonds can be exercised only on coupon dates, and are exercised ex-coupon.

Here’s a presentation that illustrates this (see p. 3):

I, for one, wasn’t sure that that was the case, but suspected that it was.