# Does put-call parity hold for a compound option with underlying American option?

Say there is an American put option that expires $N$ months from today.

A call-on-put (CoP) option provides the owner the right to buy the American put option in exactly $M < N$ months (but no sooner). A corresponding put-on-put (PoP) option provides the owner the right to sell the American put option in exactly $M < N$ months (but no sooner).

So the underlying option is American and the compound on top (PoP or CoP) is of European style.

Now for usual single European options, the put-call parity holds and for single American options the put-call parity does not hold.

Since the compound option depends on an underlying American option, does this mean that the put-call parity does not hold for the compound option?

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