Use put-call parity to link option prices with the underlying's price
The idea
Put-call parity says that a put plus the underlying equals a call plus the present value of the exercise price: p + S = c + X/(1+r)^T. Both sides are worth the same at expiry whatever the underlying does, so if they cost different amounts today the difference is a riskless profit. For European options on an underlying that pays no income, the relationship is therefore not a model but an accounting identity enforced by arbitrage, which is why it holds regardless of what anyone thinks volatility will be. Its practical use is construction: rearranged, it shows that any one of the four instruments can be synthesised from the other three.
Formulas
Put-call parity
c + X ÷ (1 + r)ᵗ = p + S₀
- c
- the price of a European call
- p
- the price of a European put, same strike and same expiry
- X
- the exercise price both options share
- r
- the risk-free rate
- t
- the years to expiry
- S₀
- the price of the underlying now
A call plus cash equal to the strike, and a put plus the share, pay the same in every state at expiry, so they cost the same now. European options only, and no dividends before expiry, or the two sides no longer match.
A practice question
Put-call parity relates the prices of a call, a put, the underlying and:
- a riskless bond with face value equal to the exercise price — correct
- a second option on a different assetWhy not: Parity is a statement about one underlying. Two assets have no reason to be linked by an identity.
- the dividend yield onlyWhy not: Dividends enter as an adjustment for a paying underlying; the fourth leg of the relation is a bond.
The bond is what carries the exercise price back to today, which is why the relationship depends on interest rates.
A harder one
A call costs $6, the underlying trades at $50, the exercise price is $50 and its present value is $48. The put is worth:
- $8Why not: Adds the gap between the strike and its present value to the call, rather than solving the identity.
- $6Why not: The call's price. Parity makes the put equal to the call only when the underlying equals the discounted strike.
- $4 — correct
Rearranging parity gives p = c + PV(X) - S, which is 6 plus 48 minus 50.
The hardest tier
If the observed put price is above what parity implies, the arbitrage is to:
- buy the put and the callWhy not: Buys the expensive leg. The overpriced side is what should be sold.
- sell the call and the bondWhy not: Sells the cheap side and leaves the overpriced put untouched.
- sell the put and the underlying, and buy the call and the bond — correct
Sell the expensive side and buy the cheap one; the two positions cancel at expiry leaving the difference.
In the app: 18 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
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