Put-Call Parity: Wetin E Mean With Real Numbers

Put-call parity na fixed arithmetic wey tie call, matching put, underlying share, and cash loan together for one price. For European options wey get the same strike and expiration date, call price minus put price equal the discounted difference between the forward price of the stock and that strike. If you apply the relation to a real chain, e go give you the share price wey the options imply, and that one rarely match the price for the tape. The gap between both na where dividend and financing rate dey.

Wetín put-call parity really dey talk

If you write the relation out, na C - P = S - PV(K) - PV(D). C na call price and P na put price for the same strike and expiration. S na share price today. PV(K) na the strike discounted back with the financing rate, while PV(D) na the present value of dividends wey the share go pay before expiration. Call options carry the right to buy at the strike, while put options carry the right to sell at that price.

The algebra get one package behind am. Buy the call and sell the put at one strike, and for expiration day you go end up owning the share at that strike, no matter wetin price do: if price pass the strike, you exercise; if e fall below am, dem assign the share to you. The pair na synthetic long share position wey you buy on credit. You pay for the share at expiration instead of today, and you no collect dividend during that period. Parity na just the pricing of those two adjustments.

Imagine say share dey $100, one-year call get $100 strike and quote of $9, while matching put quote at $5, with no dividend between now and expiration. The pair cost $4. Add the strike back, and the options imply $104 for a share wey dey quote at $100. That $4 na the interest on deferred purchase price for one year, at roughly four percent. Those figures na illustration, chosen because the arithmetic simple.

The share price wey options imply

When you rearrange the relation, e go print its own share price: call price minus put price plus the strike. Do that subtraction for every strike at one expiration, and the answers go dey within pennies of each other. That number na the synthetic stock price. The distance between am and the close for the tape na the basis, or carry, of the synthetic. Interest up to expiration dey push the basis up. Dividends before expiration dey pull am down.

Wetín dey inside the parity gap

Na four things, and na only four:

• Interest up to expiration. To pay the strike later instead of now na loan for the life of the contract, and the pair charges for am.
• Dividends before expiration. Synthetic holder no collect any of them, so every dividend wey go ex before contract expire dey come out of the gap.
• Borrow cost. Whoever take the other side and short the share go pay stock loan fee, and that fee enter the same gap.
• Early exercise value for American contracts, wey turn the equality into a band.

The same relation across expirations

Interest dey accrue every day and dividends dey land on specific dates, so basis na curve, no be one number. An option chain carry another readout for that curve. Parity tie call and put at one strike to one forward, so pricer wey use the chain’s own rate and dividend go return the same implied volatility for both legs. Any daylight between the legs mean daylight between the assumptions.

The panel dey pinned to the past, so the numbers for this page no dey move: AAPL monthly expirations from July 2026 reach January 2027, read from the June 16, 2026 session, using the strike nearest the close for each one.

SELECT
    expiry_date,
    days_out,
    concat('$', toString(argMin(strike, atm_gap)))                        AS atm_strike_used,
    round(100 * argMin(call_iv, atm_gap), 1)                              AS call_iv_pct,
    round(100 * argMin(put_iv, atm_gap), 1)                               AS put_iv_pct,
    round(100 * (argMin(call_iv, atm_gap) - argMin(put_iv, atm_gap)), 1)  AS iv_gap_pct
FROM
(
    SELECT
        expiry_date,
        days_out,
        strike,
        call_iv,
        put_iv,
        abs(strike / spot_close - 1) AS atm_gap
    FROM
    (
        SELECT
            toString(expiration_date)                                    AS expiry_date,
            max(dateDiff('day', toDate(date), toDate(expiration_date)))  AS days_out,
            round(toFloat64(strike_price), 2)                            AS strike,
            maxIf(toFloat64(implied_volatility), leg = 'C')              AS call_iv,
            maxIf(toFloat64(implied_volatility), leg = 'P')              AS put_iv,
            max(toFloat64(underlying_close))                             AS spot_close
        FROM
        (
            SELECT
                date,
                expiration_date,
                strike_price,
                implied_volatility,
                underlying_close,
                upper(substring(ticker, length(ticker) - 8, 1)) AS leg
            FROM global_markets.options_greeks
            WHERE underlying_symbol = 'AAPL'
              AND date = '2026-06-16'
              AND expiration_date BETWEEN '2026-07-01' AND '2027-01-31'
              AND toDayOfWeek(toDate(expiration_date)) = 5
              AND toDayOfMonth(toDate(expiration_date)) BETWEEN 15 AND 21
              AND iv_converged = 1
              AND volume > 0
              AND abs(toFloat64(strike_price) / toFloat64(underlying_close) - 1) < 0.03
        )
        GROUP BY expiry_date, strike
        HAVING countIf(leg = 'C') > 0
           AND countIf(leg = 'P') > 0
    )
)
GROUP BY expiry_date, days_out
ORDER BY expiry_date

At 2026-07-17, 31 days out, the $300 strike carry call at 21.9 percent against put at 20.5 percent, with gap of 1.4 points. At 2027-01-15, 213 days out, the gap na 0.3 points. The two legs dey track each other down the whole calendar.

How to extract implied dividend or implied borrow rate

For implied dividend, take two expirations wey dey on both sides of company’s ex-dividend date. Compute basis for each one, then subtract. The extra interest across the extra days small and you fit calculate am from the current bill rate. Wetin remain na the cash wey the chain expect to go ex between the two dates. Compare am with the declared schedule for our ex-dividend dates and options page, and the chain’s own dividend assumption go show clearly.

For implied borrow rate, run the process backward. Fix dividend at the declared amount and financing at the bill rate. Compute the basis wey both produce, then compare am with the basis wey the chain quote. For share wey easy to borrow, both go land on top each other. For hard-to-borrow share, quoted basis go sit below the computed one, and the annualized shortfall na the borrow fee wey the options dey charge.

The first thing na to know which two prices to pull from the screen, and our option chain guide explain the layout.

Where American options break the equality

American options fit exercise on any day up to expiration. European options fit exercise only at expiration, and strict parity equality na for European case. For American contracts, e relax into a band. The band wider for whichever side get early exercise right wey carry value. American call only make sense to exercise early just before dividend wey bigger than its remaining time value. Na why early call exercise dey cluster on the day before ex-dividend date.

The same volatility test dey run across strikes at one expiration, and na there the band show how wide e be.

SELECT
    strike,
    round(100 * call_iv, 1)             AS call_iv_pct,
    round(100 * put_iv, 1)              AS put_iv_pct,
    round(100 * (call_iv - put_iv), 1)  AS iv_gap_pct
FROM
(
    SELECT
        round(toFloat64(strike_price), 2)                AS strike,
        maxIf(toFloat64(implied_volatility), leg = 'C')  AS call_iv,
        maxIf(toFloat64(implied_volatility), leg = 'P')  AS put_iv
    FROM
    (
        SELECT
            strike_price,
            implied_volatility,
            upper(substring(ticker, length(ticker) - 8, 1)) AS leg
        FROM global_markets.options_greeks
        WHERE underlying_symbol = 'AAPL'
          AND date = '2026-06-16'
          AND expiration_date = '2026-09-18'
          AND iv_converged = 1
          AND volume > 0
          AND abs(toFloat64(strike_price) / toFloat64(underlying_close) - 1) < 0.15
          AND toUInt32(toFloat64(strike_price)) % 5 = 0
    )
    GROUP BY strike
    HAVING countIf(leg = 'C') > 0
       AND countIf(leg = 'P') > 0
)
ORDER BY strike

The two lines dey run together across the ladder. At 255 strike, call imply 30.5 percent and put imply 29.3 percent, with gap of 1.2 points. Where gap open at the far strikes, possible reasons include stale closing print on one side, thin market at that strike, early exercise premium wey the model no carry, or dividend assumption wey differ from the one the chain dey price.

Conversions and reversals, the flow wey enforce am

Conversion na long shares, short call, and long put at one strike. Reversal na the opposite package. Both packages stay flat against share price at expiration. Wetin remain na financing and dividend terms. Market makers put them on when the pair stray far enough to cover their costs, and that flow pull quotes back toward the line. The edge belong to desks wey get cheap financing and stock loan book behind them, and this page no be guide for trading am.

How parity differ from put/call ratio

People dey confuse the two. Put-call parity na pricing identity between one call and one put at one strike, and e be arithmetic. Put/call ratio count volume or open interest across the whole chain, and people read am as sentiment gauge. Parity tell you wetin the pair must cost against the share. The ratio tell you wetin people trade.

FAQ

Wetin be the put-call parity formula?

For European options wey get the same strike and expiration, call price minus put price equal share price, minus present value of the strike, minus present value of dividends wey dem pay before expiration. Rearranged: call price minus put price plus the strike give the share price wey options imply.

Put-call parity dey hold for American options?

No be strict equality. Early exercise rights turn am into a band around the European relation.

How I fit calculate implied dividend from option chain?

Take call and put at one strike and expiration. Compute call minus put plus the strike, then subtract the share price. That gap na interest up to expiration minus expected dividends. Calculate the interest from the bill rate across the same number of days, and wetin remain na the dividend wey the chain dey price.

Why call and put at the same strike get the same implied volatility?

Parity tie both prices together without volatility term inside the link. So model wey price one from a given forward and discount rate go price the other with the same inputs.

Every panel for here come with the SQL underneath am. If you wan run the same ladder for another name or farther expiration, ask for am in plain English on the Strasmore terminal.