How this calculator works
Put-call parity links the prices of a European call, a European put, the underlying asset and a zero-coupon bond with a face value equal to the strike. A portfolio of a call plus cash equal to the present value of the strike produces the same payoff at expiry as a put plus one share, so the two must cost the same today. Rearranging gives C - P = S - PV(K). If market prices violate the relationship, an arbitrageur could buy the cheap combination and sell the expensive one for a riskless profit, ignoring transaction costs. The check is only valid for European options on a non-dividend-paying asset, or with adjustments for dividends, and assumes a constant risk-free rate.
Formula
C - P = S - K e^(-rT). Solving for a price: C = P + S - K e^(-rT) and P = C - S + K e^(-rT). The present value of the strike is K e^(-rT), and the parity difference is (C - P) - (S - PV(K)). A value of zero means the prices satisfy parity.
Worked example
With a spot price of $100, a $100 strike, a one-year expiry and a 4.5% risk-free rate, the present value of the strike is $95.62. If the put costs $5, parity implies a call of 5 + 100 - 95.62 = $9.38. Conversely a call of $9.38 with a put of $5 leaves a parity difference of zero.
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Frequently asked questions
Does parity hold for American options?
Only as an inequality. Early exercise means an American call can be worth more than the European value, so the exact equality does not hold. It does hold for European options.
How do dividends change the result?
A dividend paid before expiry lowers the stock price that the holder receives, so the stock term becomes S minus the present value of the dividends, or S e^(-qT) for a continuous yield.
What does a non-zero difference mean?
Either the prices are stale or from different times, or there is an arbitrage opportunity before transaction costs, taxes and financing. In practice these costs usually close the gap.
Why use the forward price?
The implied forward price S e^(rT) shows what the stock must be worth forward for parity to hold. The present value of the strike is the forward price discounted back at the risk-free rate.