Put-Call Parity Calculator

Check whether call and put prices satisfy put-call parity (C + K·e−rT = P + S), see the theoretical fair value implied for the other option, and measure the size of any arbitrage gap.

Quick Facts

Parity formula
C + K·e^(−rT) = P + S
Holds for European-style options on the same underlying, strike, and expiration, assuming no dividends.
If the two sides don't match
The gap is a theoretical arbitrage
A mispriced call or put relative to the other can, in theory, be locked in as a riskless profit before costs.

Your Results

Calculated
Theoretical put price
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Implied by the call: C − S + PV(K)
Theoretical call price
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Implied by the put: P + S − PV(K)
Parity gap
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(C + PV(K)) − (P + S); 0 means parity holds
Arbitrage signal
-
Reading of the parity gap

Ready

Enter the stock price, strike, risk-free rate, time to expiration, and both option prices, then press Calculate.

How the Put-Call Parity Calculator works

Put-call parity is a no-arbitrage relationship between the price of a European call, a European put, the underlying stock, and a risk-free bond — all sharing the same strike price and expiration date. If the market prices of the call and put ever drift away from this relationship, a riskless profit becomes theoretically available. This calculator checks the relationship against your numbers and shows you the theoretical fair value implied for each side.

The formula

For a non-dividend-paying underlying, put-call parity states:

C + K·e−rT = P + S

where C is the call price, P is the put price, S is the current stock price, K is the shared strike price, r is the annual risk-free rate (continuously compounded), and T is time to expiration in years. The term K·e−rT is the present value of the strike — what you'd need to invest today at the risk-free rate to have exactly K in cash at expiration.

Rearranging gives the two theoretical prices this calculator reports: P = C − S + K·e−rT (the put implied by the call) and C = P + S − K·e−rT (the call implied by the put). The parity gap is simply (C + K·e−rT) − (P + S); a value of zero means the two sides balance exactly.

Worked example

With a $100 stock, a $100 strike, a 5% risk-free rate, and 0.5 years to expiration, the present value of the strike is K·e−rT = 100 × e−0.025 ≈ $97.53. If the call trades at $8.00, parity implies a theoretical put price of 8.00 − 100 + 97.53 = $5.53. If the put actually trades at $5.53, the two sides of the equation match almost exactly and there is no meaningful mispricing to exploit.

Where the arbitrage comes from

If the left side (call + present value of the strike) is worth more than the right side (put + stock), the call is relatively overpriced: selling the call, buying the put, buying the stock, and borrowing the present value of the strike locks in the difference as a riskless profit at expiration, before transaction costs. If the right side is worth more, the reverse trade — buying the call, selling the put, shorting the stock, and investing the present value of the strike at the risk-free rate — captures the gap instead. In liquid markets these gaps are usually tiny and close quickly because professional traders act on them.

Assumptions and limits

  • European exercise: the formula is exact for European options, which can only be exercised at expiration. American options can be exercised early, so their prices only satisfy parity as an inequality, not an exact equality.
  • No dividends: this calculator assumes the underlying pays no dividends before expiration. A known dividend yield q shifts the relationship to C + K·e−rT = P + S·e−qT, which lowers the effective stock price term.
  • Frictionless markets: the model ignores bid-ask spreads, commissions, margin costs, and borrowing constraints on shorting stock — all of which eat into any real-world arbitrage before it is fully captured.

Frequently Asked Questions

What is put-call parity?
Put-call parity is the relationship C + K·e^(−rT) = P + S, which must hold for a European call and put that share the same underlying stock, strike price, and expiration date (assuming no dividends). It says that a call plus a risk-free bond worth K at expiration has the same payoff, and therefore the same price today, as a put plus the stock itself.
What does it mean if the calculated prices don't match?
If (call price + present value of the strike) does not equal (put price + stock price), the two sides are mispriced relative to each other. In theory a trader could sell the overpriced side, buy the underpriced side, and finance the position at the risk-free rate to lock in the gap as a riskless profit — this is the classic conversion/reversal arbitrage. In real markets, transaction costs and bid-ask spreads usually absorb small gaps.
Does put-call parity apply to American options?
Not exactly. American options can be exercised before expiration, which adds early-exercise value that the parity equation does not price. For American options, parity holds only as a pair of inequalities (a range), not the exact equality used here — this calculator's formula is the European case.
What if the stock pays dividends?
Expected dividends before expiration reduce the stock's forward value, which shifts the parity relationship to C + K·e^(−rT) = P + S·e^(−qT), where q is the continuous dividend yield. This calculator assumes a non-dividend-paying underlying; for dividend-paying stocks, treat the reported gap as an approximation that overstates any true mispricing.

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