Hong Life CalculatorsUnderstanding the put-call parity calculator — concept and principle
Put-call parity is the no-arbitrage relationship between European call and put prices on the same share with the same strike and expiry. This calculator checks whether market prices satisfy it, finds the put price implied by the call (and the reverse) and shows which side to buy if there is a gap.
How it is calculated
c + K·e^(−rT) = p + S·e^(−qT)
- c, p: call and put prices; S: share price; K: strike; r: continuous risk-free rate; T: time to expiry in years; q: continuous dividend yield.
- Parity gap = (p + S·e^(−qT)) − (c + K·e^(−rT)).
- Implied put = c + K·e^(−rT) − S·e^(−qT); implied call = p + S·e^(−qT) − K·e^(−rT).
The principle
A call plus a zero-coupon bond paying K at expiry is worth max(ST, K) at expiry: if the share ends above K you exercise the call, otherwise you keep the bond's K. A put plus the share pays exactly the same. Two portfolios with identical payoffs must cost the same today, otherwise you could buy the cheaper one, sell the dearer one and lock in the difference with no risk. That is why, given one option price, the other is fixed.
Worked example
Default inputs (textbook case): S = 31, K = 30, r = 10%, T = 0.25 years, no dividend, call 3, put 2.25.
- Present value of the strike = 30 × e^(−0.025) ≈ 29.26 USD.
- Cost of call + bond = 3 + 29.26 = 32.26 USD; cost of put + share = 2.25 + 31 = 33.25 USD.
- Parity gap = 33.25 − 32.26 ≈ 0.99 USD.
- Implied put = 3 + 29.26 − 31 ≈ 1.26 USD; implied call = 2.25 + 31 − 29.26 ≈ 3.99 USD.
- Arbitrage direction: buy call + bond, sell put + share.
| Share price now | Parity gap: (put + share) − (call + bond) |
|---|---|
| 7.75 | -$22.26 |
| 15.5 | -$14.51 |
| 23.2 | -$6.76 |
| 31 | $0.99 |
| 38.8 | $8.74 |
| 46.5 | $16.49 |
| 62 | $31.99 |
How to read the result
A gap near zero means prices are consistent. A positive gap means the put-plus-share side is expensive relative to the call-plus-bond side. In practice bid-ask spreads, commissions, borrowing costs and early exercise of American options keep small gaps alive, so most apparent gaps are not tradable profits.
Common mistakes
- Applying parity to American options, where it holds only as inequalities.
- Using an annual simple rate instead of a continuous rate.
- Forgetting dividends, which lower the share term.
- Ignoring transaction costs when judging arbitrage.
Frequently asked questions
What is the put-call parity formula?
c + K·e^(−rT) = p + S·e^(−qT) for European options on the same underlying, strike and expiry.
Does put-call parity hold for American options?
Not exactly. It becomes a pair of inequalities because of the possibility of early exercise.
How do dividends affect parity?
They reduce the present value of the share term, so calls are cheaper and puts dearer.
Sources: Hull, J. C., Options, Futures, and Other Derivatives (put-call parity) · Stoll, H. R. (1969). The Relationship Between Put and Call Option Prices. Journal of Finance, 24(5).