Based on P78 [[DerivativeFinancialProducts]].
Most popular Compound option is a call on a put. (I think vanilla Puts are the overall most popular option in Eq and FX, and embedded Calls are the most popular bond options.) Given a lot of buyers are interested in puts, there’s a natural demand for calls-on-puts.
In a Compound option there are 2 layers of fees (i.e. premiums) and 2 expiry dates —
The front fee is what you pay up front to receive the front option. If you exercise front option, you do so by buying the back option, paying the back fee as a 2nd premium to the dealer. Therefore the back fee is a conditional fee.
At end of front option’s lifespan, the back option’s protection period would start. (Remember in this case the back option is a protective put.)
It’s quite common that at end of the front window, the back option has become unnecessary, or back fee has become too expensive given the now reduced risk.