Work with put-call-forward parity
The idea
Put-call-forward parity replaces the spot underlying in the standard relation with a forward contract: a long call plus the present value of the exercise price equals a long put plus the present value of the forward price. It is the same no-arbitrage statement, expressed in instruments that may be cheaper or more practical to trade than the underlying itself. Rearranging it shows that a synthetic forward can be built from a long call and a short put at the same strike, which is how option desks manufacture forward exposure without touching the cash market.
Formulas
Put-call-forward parity
c + X ÷ (1 + r)ᵗ = p + F₀ ÷ (1 + r)ᵗ
- c
- the price of a European call
- p
- the price of a European put, same strike and expiry
- X
- the exercise price
- r
- the risk-free rate
- t
- the years to expiry
- F₀
- the forward price agreed today for delivery at expiry
Put-call parity with the underlying replaced by a forward on it, which is what to reach for when the underlying cannot be held — an index, a commodity, a currency.
A practice question
A long call and a short put at the same exercise price and expiry together replicate what position?
- A long forward on the underlying — correct
- A short forward on the underlyingWhy not: That is the opposite combination, a short call with a long put.
- A risk-free bondWhy not: A bond has no exposure to the underlying, whereas this pair moves one for one with it.
Above the strike the call pays; below it the short put loses. The combined payoff line is the underlying less the strike at every price, which is a forward.
A harder one
What does put-call-forward parity substitute for the spot underlying in the standard parity relation?
- A risk-free bondWhy not: The bond is already in the relation, representing the discounted exercise price.
- The present value of the forward price — correct
- The option premiumWhy not: A premium is a price, not a claim on the underlying, and cannot stand in for it.
Holding a forward and the present value of its price is economically equivalent to holding the underlying, so the relation can be restated in forward terms.
The hardest tier
Why is put-call-forward parity useful even when the spot underlying is freely traded?
- Because it removes the need to discount the exercise priceWhy not: Discounting is still required; the substitution is on the underlying side of the relation.
- Because forwards are exempt from marginWhy not: Futures require margin and bilateral forwards often require collateral, so no exemption applies.
- Because forward exposure can be cheaper — correct
It can also be operationally simpler to obtain than the physical underlying. Commodities that must be stored, or indexes that would require buying hundreds of constituents, make the forward the practical instrument; the arbitrage logic is unchanged and only the cheapest way to express it differs.
In the app: 12 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
Take the free diagnostic →