Put–Call Parity

A no-arbitrage relation linking European calls, puts, spot, and discount factors.

Explanation

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.

If observed prices violate parity beyond frictions, a static arbitrage exists between option and underlying portfolios.

optionsparityno-arbitrage

Interactive visualisation

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

Spot today S₀: 100.00Strike K: 100.00r (annual rate): 3.00%T (years): 1.00Call price C: 8.00Put price P: 5.00

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.