The put-call parity calculator works with an identity rather than a model. Put-call parity says that a European call and a European put on the same underlying, with the same strike and the same expiry, are locked together by a relationship that holds regardless of what anyone thinks the underlying will do. Buy the call and sell the put, and you have manufactured a forward commitment to buy at the strike. That equivalence is what fixes the relationship, and it is why the identity needs no volatility input, no distributional assumption, and no view.
At Arb Digital we build free calculators that make the assumptions visible instead of burying them, and this one is a good example of why that matters. The identity is exact under its conditions and approximate the moment any of them slips. This page takes every price, rate and parameter from you, publishes no market data, and recommends nothing. It solves the algebra and reports what remains unexplained.
What This Put-Call Parity Calculator Does
You choose which leg to solve for — the call, the put, the spot price, or the discounted strike — and supply the rest. The calculator computes the continuous discount factor from the risk-free rate and the time to expiry, applies it to the strike, subtracts any present value of dividends from the spot, and rearranges the identity to isolate your chosen unknown. It also reports the parity gap implied by the numbers you entered, which is the amount by which the two sides of the identity fail to match before any solving takes place.
The synthetic long stock cost in the grid is the right-hand side of the identity: spot minus the present value of dividends. That figure is what buying the call and selling the put, then holding cash to cover the discounted strike, actually costs you. Seeing it as a number rather than a symbol tends to make the identity click.
How to Use It
- Pick the leg you want to solve for. The selected quantity is derived; the others are treated as given.
- Enter both option prices, the spot and the strike. The strike must be the same for both legs and the expiries must match. Parity across different strikes or different expiries is a different relationship entirely.
- Enter the risk-free rate and the time to expiry in years. The rate should be the continuously compounded rate for a horizon matching the expiry, in the same currency as the option.
- Enter the present value of dividends, if any. Discount each expected cash dividend with an ex-date before expiry back to today and enter the total. Leave it at zero for an underlying that pays nothing.
- Read the solved leg and the parity gap. The gap tells you how far the numbers you entered are from satisfying the identity.
The Formula — How Put-Call Parity Is Calculated
The identity is: call price minus put price equals spot price, minus the present value of dividends, minus the strike discounted back to today. Writing the discount factor as e raised to minus the rate times the time, the calculator rearranges that one equation four ways. The put is the call, minus the spot, plus the dividend present value, plus the discounted strike. The call is the put, plus the spot, minus the dividends, minus the discounted strike. The spot is the call, minus the put, plus the dividends, plus the discounted strike. And the discounted strike is the put, minus the call, plus the spot, minus the dividends — from which the calculator also reports the strike that discounted value implies.
The reasoning behind it is a replication argument, not a statistical one. Two portfolios that pay identically at expiry in every state of the world must cost the same today, or a riskless profit exists. A long call plus a short put at the same strike pays exactly the spot price minus the strike at expiry, whatever the spot turns out to be — which is the payoff of a forward at that strike. General educational material on options mechanics is published by the Options Industry Council, whose Options Pricing overview covers the inputs that drive option values, and the wider theoretical framing sits in the derivatives material published by the CFA Institute professional learning library.
A Worked Example, Verified Both Ways
Take a spot of 100, a strike of 100, a risk-free rate of five per cent, half a year to expiry, and no dividends. The discount factor is e to the minus 0.025, which is 0.975310. The discounted strike is therefore 97.5310. If the call trades at 8.00, the identity gives a put of 8.00 minus 100 plus 97.5310, which is 5.5310.
Now run it back the other way as a check. Starting from a put of 5.5310, the call is 5.5310 plus 100 minus 97.5310, which returns 8.0000 exactly. Solving for spot from a call of 8.00 and a put of 5.5310 gives 8.00 minus 5.5310 plus 97.5310, which is 100.0000. And solving for the discounted strike gives 5.5310 minus 8.00 plus 100, which is 97.5310 — implying a strike of 100.0000 once you undo the discounting. The identity is consistent in all four directions, and the parity gap on those figures is zero.
Change one thing and watch the structure work. Raise the rate to eight per cent with everything else unchanged, and the discounted strike falls to 96.0789. The same 8.00 call now implies a put of 4.0789. Higher rates make the deferred obligation to pay the strike cheaper in present-value terms, which raises calls relative to puts. Nothing about the underlying's prospects has changed; only the financing has.
The Assumptions That Have to Hold
This calculator computes the European, frictionless form of the identity, and every word of that matters. European exercise means the options can only be exercised at expiry. An American put can be exercised early — and rationally will be, in some deep in-the-money and high-rate situations — so American options satisfy only an inequality, a band, rather than an exact equality. If your quotes are for American-style contracts, a residual gap is expected rather than anomalous.
Dividends must be modelled if the underlying pays them. This page handles them as a known present value of discrete cash dividends subtracted from the spot, which is the standard treatment for a single stock. An index with a continuous dividend yield is usually handled by discounting the spot at that yield instead; entering the equivalent present value here gets you to the same place. Uncertain or unannounced dividends leave a real residual that no amount of algebra removes.
Frictionless borrowing and lending at the stated rate is the assumption that fails first in practice. The identity implicitly assumes you can borrow and lend freely at one rate, short the underlying without cost or restriction, trade in any size without moving prices, and pay no commissions, no bid-offer spread and no financing charge. Every one of those is false to some degree. Hard-to-borrow stock in particular carries a borrow fee that behaves exactly like an unmodelled dividend and can open a persistent apparent gap that nobody can capture.
What an Apparent Violation Usually Means
If your inputs produce a non-zero parity gap, the overwhelmingly likely explanation is a problem in the data or in the assumptions, not a free trade sitting unnoticed. In rough order of frequency: the call and put quotes were captured at different moments, so the underlying moved between them. The prices are mid quotes, but you would buy at the offer and sell at the bid, and the spread across four legs typically exceeds the gap. The contracts are American rather than European. A dividend with an ex-date before expiry has not been entered. The underlying is expensive or impossible to borrow. The interest rate used does not match the option's currency, tenor or compounding convention. Or one of the options is thinly traded and the last quote is stale.
This is not a counsel of despair about market efficiency; it is a statement about measurement. Genuine parity violations in liquid listed markets are small, brief and consumed by participants with far lower transaction costs than a screen quote implies. Treat the gap as a diagnostic on your own inputs — that is the use it actually has. Nothing on this page is a recommendation to buy, sell or arrange any position.
Where Parity Fits Alongside a Pricing Model
Parity and option pricing models do different jobs, and it is worth being clear on the boundary. A model such as Black-Scholes produces an absolute value for each leg from a volatility assumption; parity produces only a relationship between the two legs, with no volatility input at all. Our Black-Scholes calculator prices a European call or put and returns the Greeks, and its outputs will satisfy this identity automatically when fed consistent inputs — which makes parity a useful sanity check on any model implementation.
Parity is also the reason synthetic positions exist. A long call plus a short put replicates long exposure to the underlying; the reverse replicates a short. That construction is why the identity underpins conversion and reversal trades in market-making, and why it constrains implied volatilities: a European call and put at the same strike and expiry must carry the same implied volatility, because if they did not, parity would be violated. If you are working through the wider toolkit, the present value calculator handles the discounting on its own, the compound interest calculator covers compounding conventions, and the portfolio beta calculator aggregates directional exposure across holdings.
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Talk to Arb Digital All Free ToolsCommon Mistakes to Avoid
- Applying parity to American options. Early exercise turns the equality into an inequality. Expect a residual and do not read it as an anomaly.
- Forgetting dividends. An unmodelled dividend shifts the identity by its present value and looks exactly like a mispricing.
- Using different strikes or expiries. The identity is defined for one strike and one expiry. Anything else is a different relationship.
- Mixing compounding conventions. This page discounts continuously. Feeding it an annually compounded rate produces a discount factor that is slightly wrong at every tenor.
- Reading a gap as an opportunity. Bid-offer spreads across four legs, borrow costs and non-simultaneous quotes account for nearly every apparent violation.
Related Free Tools From Arb Digital
For pricing rather than relationships, the Black-Scholes calculator values a European call or put and returns the Greeks. For the discounting arithmetic on its own, use the present value calculator or the future value calculator. For portfolio-level risk, the portfolio beta calculator aggregates betas by weight and the Sharpe ratio calculator gives return per unit of volatility. Browse the free tools hub for the rest.
Frequently Asked Questions
Put-call parity is an identity linking the prices of a European call and a European put with the same underlying, strike and expiry. It states that the call price minus the put price equals the spot price, less the present value of any dividends, less the strike discounted back to today.
Not as an exact equality. American options can be exercised before expiry, so the relationship becomes an inequality that bounds the prices within a band rather than fixing them. A residual gap on American contracts is expected, not an anomaly.
Cash dividends with an ex-date before expiry are discounted back to today and subtracted from the spot price. This calculator takes that present value as an input. An index paying a continuous yield is usually handled by discounting the spot at that yield instead.
Almost always because of the data or the assumptions rather than a mispricing. Common causes are quotes captured at different moments, mid prices instead of tradeable bids and offers, American-style contracts, an unmodelled dividend, a borrow cost on the underlying, or a rate that does not match the option's currency and tenor.
In practice, very rarely. Executing the identity requires four simultaneous trades, each crossing a spread, plus financing and borrow costs. Genuine violations in liquid listed markets are small and short-lived. Treat a gap as a diagnostic on your inputs.
No. That is what distinguishes it from a pricing model. Parity is a no-arbitrage relationship between two option prices and needs no view on volatility or on the distribution of the underlying. A model such as Black-Scholes requires volatility to produce an absolute price for each leg.
Because the strike is paid at expiry, and a higher rate makes that deferred payment cheaper in present-value terms. The discounted strike falls, which widens the call price minus put price difference required by the identity.
This tool performs arithmetic on figures you enter and is for general education only. It is not investment, tax or financial advice, it recommends no trade, security or strategy, and it publishes no market data. Options carry substantial risk of loss. Speak to a licensed professional before making any decision.
