Put-Call Parity

Put call parity is a fundamental relationship in options pricing. It says that, for European options with the same strike and expiration, a call and a put must be priced consistently with the underlying asset and interest rates. If they are not, traders can build offsetting positions to lock in a profit, and that buying and selling pushes prices back into line. Hans Stoll described the relationship formally in 1969. It explains why calls and puts move together, how synthetic positions work and how to spot mispricing.

The formula#

For a non dividend paying stock:

C - P = S - K / (1 + r)^T

• C: call price
• P: put price
• S: current stock price
• K: strike price
• r: risk free interest rate
• T: time to expiration in years

Rearranged: C + K / (1 + r)^T = P + S. A call plus cash equal to the present value of the strike is worth the same as a put plus the stock.

With continuous compounding the discount factor becomes e^(minus rT), and with known dividends you subtract their present value from S.

Why it must hold#

Consider two portfolios:

• Portfolio A: one call plus cash that will grow to K at expiration.
• Portfolio B: one put plus one share.

Both portfolios are worth the same in every outcome, max(S, K). So they must cost the same today. If they did not, you could buy the cheap one, sell the expensive one and pocket the difference with no risk.

Conversions and reversals#

These are arbitrage trades used by market makers and professional firms. See Arbitrage Strategies.

Synthetic positions#

Rearranging parity shows how to build one position from others:

See Synthetic Positions.

What parity tells traders#

• Calls and puts share the same implied volatility at the same strike and expiry, in theory. If they did not, parity would be violated. See Implied Volatility (IV).
• Interest rates matter: higher rates make calls more valuable relative to puts.
• Dividends matter: expected dividends lower call values and raise put values.
• Hard to borrow stocks: when shorting is costly, puts can trade richer than parity suggests, because the reversal arbitrage requires shorting the stock.

American options#

For American options, early exercise turns parity into a range rather than an exact equation. The relationship still holds approximately and is widely used, but small deviations can persist. See American vs European Options.

Frequently asked questions#

What is put call parity?#

A relationship stating that a call minus a put with the same strike and expiry equals the stock price minus the present value of the strike, for European options.

Why is put call parity important?#

It keeps call and put prices consistent, explains synthetic positions and identifies arbitrage opportunities when prices get out of line.

Does put call parity work for American options?#

Only approximately. Early exercise means it becomes a range of prices rather than an exact equality.

Next, learn to draw any option strategy's profit and loss in Option Payoff Diagrams.

Sources#

• Wikipedia, Put call parity