# 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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Put-call parity is a model free relationship, i.e. it makes no assumptions regarding the underlying. The underlying can be any trade-able asset. So it should hold in your case.

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Agree with Alexey, and would add that this principle is as of the result of the absence of arbitrage assumption, which has to hold regardless of what the dynamics of the underlying is. –  Peter Irojah Apr 8 '12 at 22:56