Put-call parity
C - P = S_0 - PV(K)A call-minus-put position replicates a forward exposure when contract terms match.
Foundations
Put–Call Parity
A no-arbitrage relation linking European calls, puts, spot, and discount factors.
Mental model
Two portfolios with the same European payoff must have the same no-arbitrage value.
Put–call parity ties together European call and put prices with the same strike and maturity.
In its simplest form, C − P = S₀ − K·DF, where DF is the discount factor to maturity.
Explore the mechanism
Which portfolio of a call, put, underlying, and cash produces the same terminal payoff?
This graphic shows put–call parity: the payoff of a call-plus-bond portfolio and a put-plus-stock portfolio. Use the controls to see how today’s prices line up with the parity condition C − P = S₀ − K·DF.
Parity gap: 0.045C − P = 3.000 · S₀ − K·DF = 2.955
Numbers
Discount factor DF = e-rT ≈ 0.9704
C − P = 3.0000 · S₀ − K·DF = 2.9554
Parity gap (C − P − S₀ + K·DF) ≈ 0.0446 (near parity)
Implied P* from parity ≈ 5.0446 · implied C* ≈ 7.9554
Interpretation
The blue and green lines show call + bond and put + stockpayoffs. They coincide for all S_T, which is why their prices today must satisfy put–call parity.
When C − P matches S₀ − K·DF, the parity gap is essentially zero and neither side is obviously rich. If C − P is too high, the call side is rich relative to the put and spot; if it is too low, the put side is rich. In frictionless markets, traders could lock in arbitrage by buying the cheap portfolio and shorting the rich one until the gap closes.
UNDERSTAND
What to notice
If observed prices violate parity beyond frictions, a static arbitrage exists between option and underlying portfolios.
The quantitative structure
Notation
C- European call value
P- European put value
S_0- spot price
PV(K)- present value of the strike
This model assumes
- Call and put are European, share strike and maturity, and use consistent dividend and financing treatment.
A small example
Checking matched option quotes
Spot is 100, the present value of the strike is 95, and the call is worth 12.
- Parity requires C - P = 100 - 95 = 5.
- Solve 12 - P = 5, giving P = 7.
ResultA put price materially different from 7 would require explanation through conventions, costs, or inconsistent quotes.
Where it appears
- checking European option quotes
- constructing synthetic forwards
- diagnosing funding or dividend inconsistencies
Where the intuition breaks
What this view leaves out
- The textbook identity assumes matched strikes and maturities and consistent treatment of dividends, financing, exercise style, and transaction costs.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Put–Call Parity
Put–Call ParityYOU ARE HERE
Built onArbitrage and No-ArbitragePut–call parity is a no-arbitrage identity linking calls, puts, the underlying and discounting.
ValidatesBlack–Scholes Option PricingBlack–Scholes call and put prices should satisfy put–call parity when inputs and conventions are consistent.
ValidatesBinomial Option Pricing (CRR)A consistent binomial implementation should preserve put–call parity for comparable European contracts.
Built on · Arbitrage and No-Arbitrage
Put–call parity is a no-arbitrage identity linking calls, puts, the underlying and discounting.
Explore concept →