European option parity and arbitrage check
Put-Call Parity Calculator
Solve for a European call, put, stock price or strike, or compare market prices against put-call parity with continuous dividend yield.
Rate sensitivity
| Risk-free rate | PV of strike | Prepaid forward | Implied call | Implied put | Observed gap |
|---|
A displayed edge is a mathematical snapshot, not guaranteed arbitrage. Bid/ask spreads, execution timing, stock borrow, taxes, dividends and early exercise can remove it.
What the put-call parity calculator solves
Put-call parity links European calls and puts with the same underlying, strike and expiration. This calculator can solve for the call premium, put premium, stock price or strike. In check mode it compares the two sides of the relationship and estimates the gross and after-fee discrepancy across the selected contracts.
Put-call parity formula
- C is the European call premium per share.
- P is the European put premium per share.
- S is the current stock price and K is the common strike.
- r is the continuously compounded risk-free rate, q the continuous dividend yield and T time in years.
The term K × e−rT is the present value of the strike. The term S × e−qT is the prepaid forward value of the stock. With no dividends, set q to zero.
Verified example
For S = US$100, K = US$100, a US$8.50 call, a US$6.10 put, 180 days, a 5% continuously compounded risk-free rate and a 1% dividend yield, the present value of the strike is about US$97.56 and the prepaid forward value is about US$99.51. The call-plus-bond side is about US$0.46 per share more expensive than the put-plus-prepaid-forward side. Across one 100-share contract, the gross discrepancy is about US$45.63 and approximately US$40.63 remains after US$5 of estimated fees.
How to interpret a parity difference
- If C + PV(K) is greater, the call-plus-bond side is relatively expensive.
- If P + prepaid forward is greater, the put-plus-stock side is relatively expensive.
- A zero or very small difference means observed prices are close to theoretical parity.
- For a non-dividend stock, the textbook conversion combines long stock, long put, short call and borrowing; the reverse conversion reverses those positions. With dividends, the exact replication uses a prepaid forward or dividend-adjusted stock position.
Limits and assumptions
This equality is designed for European options. American options can be exercised early and generally follow bounds rather than exact equality, especially around dividends. The model uses continuous compounding and a continuous dividend yield; it does not model discrete dividend dates, hard-to-borrow costs, margin, bid/ask spreads or asynchronous quotes.
Prices are limited to 1,000,000,000 with at most six decimal places, days to 36,500, rates from −99% to 1,000%, dividend yield from 0% to 1,000%, and total contract shares to 1,000,000,000. Currency selection changes formatting only and performs no foreign-exchange conversion. Results are educational estimates, not investment, tax or legal advice.
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