Skip to content
All library documents

Put–Call Parity for Compound Options on American Options

Article Quant Q&A · Author: Liam Williams
Read the original

Summary

This exchange considers whether European-style compound calls and puts retain put–call parity when the underlying is an American option. The central lesson is that the answer depends on the contract’s definition and whether the underlying option can disappear before the compound option’s exercise date. A general parity argument applies when the underlying is a tradeable asset that remains available under the contract.

If the American option can be exercised early and thereby cease to exist, a compound contract tied to that specific option may be cancelled, breaking the assumptions behind parity. A different structure could define the underlying as an option whose American exercise period begins only when the compound is exercised; in that case, parity may hold. The exchange provides conceptual reasoning rather than a derivation or pricing example. Its conclusion therefore depends on the payoff terms, settlement rules, and treatment of early exercise in the actual contract.

Key ideas

  • Put–call parity depends on the compound contract’s payoff and survival conditions.
  • Parity may fail if early exercise removes the underlying American option before compound expiry.
  • A contract that activates the underlying option’s exercise window at compound exercise may preserve parity.
  • The specific legal and settlement terms determine which parity argument applies.

Tags

OptionsDerivatives pricing
Full text
# Does put-call parity hold for a compound option with underlying American option?


# 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?

## Answer by Alexey Kalmykov (score 3)

https://quant.stackexchange.com/a/3202

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.

## Answer by Brian B (score -1)

https://quant.stackexchange.com/a/3220

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.