Using put-call parity, derive the theoretical European call price: C = P + S − K ÷ (1 + r)^t, linking call, put, underlying, and strike.
Input Data
Results
At a glance:Put-call parity states that a call equals a put plus the spot minus the present value of the strike: C = P + S − K/(1+r)^t for European options.
Formula
callPrice = putPrice + spotPrice − strikePrice / (1 + ratePct%)^years
$$$C + \\dfrac{K}{(1+r)^t} = P + S$$$$$$C = P + S - \\dfrac{K}{(1+r)^t}$$$$$$5 + 100 - \\dfrac{100}{1.05^1} \\approx 9.76$$$How to Use
- Enter the put price, spot, and strike.
- Enter the risk-free rate and years to expiry.
- Read the theoretical call price.
FAQ
What is put-call parity and why does it hold?
Put-call parity is a no-arbitrage relation in option pricing: a long call plus cash equals a long put plus the underlying stock, because both portfolios pay the same at expiry (max(S−K, 0) plus cash vs max(K−S, 0) plus stock). With the same strike and expiry, their values must be equal today, otherwise an arbitrage profit exists. The formula is C + K·e^(−rT) = P + S.
How do I read the theoretical call price result?
If the two sides are unequal, the difference signals a mispricing. A positive gap means buying the cheaper side and selling the dearer side would, in theory, lock in a risk-free profit at expiry. In real markets such gaps are tiny and vanish quickly after costs, so treat the result as a mispricing signal, not a trade to chase.
Why must the options be European and same strike/expiry?
European options can only be exercised at expiry, so the payoff at expiry is exact and the parity holds cleanly. American options may be exercised early, and different strikes or expiries break the one-to-one payoff match, so the basic formula no longer applies and needs adjustments.
Does this apply to Hong Kong-listed options?
Hong Kong's exchange-traded stock options are American style, so pure European put-call parity does not hold exactly; a looser inequality applies. Index options and the underlying market still reflect the same no-arbitrage logic. Use this calculator as a teaching model; for real pricing, account for early-exercise and local market conventions.
What is the relationship between put-call parity and the implied volatility smile?
They are two different lenses on option pricing. Put-call parity links the prices of calls and puts of the same strike/expiry (a 'cross-price' no-arbitrage relation), ensuring both sides stay in line; the implied-volatility smile describes how implied vol differs across strikes for the same expiry, reflecting the market's view of tail risk. The parity must hold even when a smile exists — if the call and put implied vols for the same strike diverge, the parity breaks and an arbitrage appears, which is exactly how traders spot mispricings. In short: parity is a hard arbitrage constraint between call and put prices; the smile is a soft market expectation expressed in vol. Together they describe options: the smile must not violate the parity. This calculator checks the parity; a real smile needs the vol surface from market data.
References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.