# 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?

## 2 Answers

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.

• 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

It is important to think about what the contract really is. If you have a call-on-american-option where the underlying is the "usual" american option that could be exercised away, then put-call parity will not hold because the contract will presumably be cancelled in cases where the underlying has disappeared before the tenor of the compound.

On the other hand, perhaps your contract is specified as an option-on-hybrid-option, where the underlying is an option whose american exercise window begins when the compound is exercised. In this case, as with all assets that don't disappear, put-call parity will hold.

• Sorry can you elaborate how the contract can be cancelled? And how the underlying american option can "disappear" (assuming market liquidity)? – Alexey Kalmykov Apr 10 '12 at 15:31
• Call Put parity always hold in a frictionless market. In case of compound options Call-On-Put + PV(K) = Put + Put-on-put . If the parity does not hold anywhere it is presence of arbitrage opportunity and in that case under efficient market hypothesis the market quickly adjusts the price and arbitrage vanishes. – ash Nov 26 '12 at 1:47
• As I say, without specifying the contract terms better, the question is unanswerable. Put-call parity always holds in a frictionless market, true, but only if you assume that the asset will exist at option expiration. Consider the case of a takeover to see how this might end up being ambiguous if the contract has not specified what happens in these outcomes. Of more immediate concern is that if the underlying American option has been exercised, what does the contract say about the payoff? Depending on those terms, put call parity either will or will not hold. – Brian B Nov 26 '12 at 16:52