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
What does put-call-forward parity substitute for the spot underlying?
- A forward contract - correct
- A second optionWhy not: Adding another option would change the payoff being replicated rather than restate the same one.
- A risk-free bondWhy not: A bond alone has no exposure to the underlying. A bond with face value equal to the forward price is used together with the forward contract to rebuild the underlying's payoff.
Put-call parity says: call + risk-free bond = put + underlying. Put-call-forward parity swaps the underlying for a long forward plus a risk-free bond whose face value is the forward price, because that pair pays the same at expiry. Underlying ≈ Forward + F0 / (1 + r)^T. Example: forward price 105, rate 5%, one year. The bond is worth 105 / 1.05 = 100, the same as the spot price. It is the same no-arbitrage idea, using instruments that can be easier to trade.
A harder one
What does put-call-forward parity substitute for the spot underlying in the standard parity relation?
- The discounted exercise priceWhy not: That term is already in the relation, paired with the call. The substitution is made on the underlying side.
- 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.
Standard put-call parity says: c + X / (1 + r)^T = p + S, where c is the call price, p the put price, X the exercise price and S the spot price. Instead of owning the asset (S), you can hold a forward plus a risk-free bond that grows to the forward price. That gives the same payoff at expiry. So S is replaced by the present value (today's worth) of the forward price, F0 / (1 + r)^T. Example: F0 = 105, r = 5%, T = 1 gives 105 / 1.05 = 100.
The hardest tier
Why is put-call-forward parity described as the same statement as ordinary put-call parity?
- Because both apply only to European options on one expiryWhy not: True of both, and it is a shared condition rather than the reason they are the same statement.
- Because the discounted forward price equals the spot price - correct
- Because both assume the options are at the moneyWhy not: Parity holds for any common strike, so moneyness plays no part.
Ordinary put-call parity is: Call + PV(Strike) = Put + Spot price. With no income or storage costs, the forward price is the spot price grown at the risk-free rate (the rate on a safe investment): Forward = Spot x (1 + r)^T. So Forward / (1 + r)^T = Spot. Swapping the spot price for the discounted forward price therefore changes nothing. For example, spot $100, r = 5%, one-year forward $105: 105 / 1.05 = $100. If the asset pays income or has carrying costs, the forward price already reflects them, so the two forms still agree.
In the app: 12 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
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