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 holds because a violation would create:
- higher volatilityWhy not: Volatility is an input to option prices, not the consequence of the identity failing.
- a riskless arbitrage profit - correct
- greater counterparty riskWhy not: Counterparty risk concerns whether a party pays, not whether option prices are consistent with each other.
Arbitrage means making a risk-free profit with no net investment. Put-call parity says a call plus a bond and a put plus the asset pay exactly the same at expiry, so they must cost the same today. If one costs more, you can sell the dear one, buy the cheap one and lock in a sure gain. Traders do this until the gap closes. So a violation would create a riskless arbitrage profit, which is why parity holds.
A harder one
Put-call parity can be rearranged to synthesise a long underlying position from:
- a long call, a short put and a long riskless bond - correct
- a long put, a short call and a short riskless bondWhy not: That package replicates a short position in the underlying, the opposite of what is wanted.
- a long call, a long put and a long riskless bondWhy not: A call and a put together pay off on large moves in either direction, not the linear exposure the underlying has.
Put-call parity says Call + PV(X) = Put + S. Rearranged: S = Call - Put + PV(X). Read it as a recipe: buy a call, sell (write) a put, and buy a riskless bond that pays X at expiry. The call and the short put together pay S - X at expiry, whatever happens, and the bond adds X. The total is S, the same as owning the asset. So these three together copy the asset's payoff.
The hardest tier
An analyst observes market prices where S + p is greater than c + PV(X). What does this imply?
- An arbitrage exists: buy the stock-plus-put and sell the call-plus-bondWhy not: This is the reverse trade. It buys the dearer package and sells the cheaper one, which locks in a loss.
- No arbitrage exists: the gap just reflects illiquid option marketsWhy not: Illiquidity can explain why a gap survives, but the gap itself is a price relationship that should not exist.
- An arbitrage exists: sell the stock-plus-put and buy the call-plus-bond - correct
Put-call parity says S + p should equal c + PV(X), because the two packages pay the same at expiry. If S + p is greater, the left side is too expensive. Selling something dear and buying an identical payoff cheaply locks in a risk-free profit. So sell the share plus the put, and buy the call plus the bond. The difference is kept whatever the price does. That trading pushes prices back into line. Illiquidity might let a gap last, but the gap should not exist.
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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