Put-Call Parity Calculator
The formula
What the parity-implied price tells you
Put-call parity is the one exact identity in listed options: a call plus a discounted strike equals a put plus the stock. Break any leg out of it and you get synthetic stocks, conversions/reversals, and the market’s effective borrow rate.
Calculation: C + K·e^(−rT) = P + S − PV(div); any leg is solvable from the other three. Enter your inputs and every substitution is shown below the result.
Reading the result
A persistent gap bigger than spread + fees is not free money — it is usually hard-to-borrow stock, American exercise on the put, or a dividend assumption mismatch. Trade the parity only when you can fund or borrow at the implied rate.
Rules of thumb: Index parity gaps within 1–3 bps of funding; single-stock put-call IV differences of a few vol points are borrow, not mispricing
Limitations and best practice
The identity holds exactly only for European options; American deep-ITM puts carry early-exercise value that looks like a parity violation. Dividends shift the whole line: leave PV(div) at zero for non-payers, and use futures pricing for indices where carry is the dividend.
This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Frequently asked questions
What is a synthetic long stock?
Why is my put IV above call IV?
Can I arbitrage the parity gap?
Does parity need volatility?
Source & methodology
Stoll (J. Finance 1969); standard no-arbitrage identity.
Put-Call Parity: the number, the method and the caveats
This page was written for one job: turning your own inputs into a defensible figure, with the working shown. Searchers land on a Put-Call Parity page for three reasons — to plan, to compare options, and to check that somebody else’s arithmetic is honest. This tool serves all three, and everything after the calculator explains the model instead of hiding it.
Every investing metric in Investing & Markets is a bet about the future priced in today’s language. Put-Call Parity translates that bet: C + K·e^(−rT) = P + S − PV(div); any leg is solvable from the other three. The translation is mechanical and reliable; the judgement is deciding whether the inputs you feed it are honest forecasts or wishful numbers.
A calculation like this one is worth roughly as much as a careful sentence: it forces several vague considerations into one comparable quantity. Once Put-Call Parity is on the table, the argument can stop being about impressions.
Every figure in the prose is generated by the engine rather than written beside it — the worked example resolves to fair put price = $-105.40 /share on the default inputs, and the sensitivity section ranks the inputs that move the result — Underlying price first, at ≈20.4 % of the headline for a ten-percent error either way in that input — which is the thread the rest of this guide pulls on.
Inside the Put-Call Parity calculation: formula and every input
Here is the complete model this tool runs — C + K·e^(−rT) = P + S − PV(div); any leg is solvable from the other three — with each input explained below. If a competitor’s page gives you a different answer from the same inputs, this is the text that settles which of you is right.
Underlying price ($) (example value 215) is a plain measured quantity — the unit shown in the label is the unit the formula expects. If your source quotes a figure before and after some deduction, decide which one the formula wants and be consistent for every field.
Strike ($) (example value 108) is a measured number in the unit named by the label; the formula does not convert for you. Replace the example with the figure from your own records — the tool recomputes instantly and the printed steps follow your numbers.
Time to expiry (years) (example value 1) is a quantity rather than a category — precision here is cheap and worth having. Use the value as of today, not as of the last time you looked — time fields drift silently.
Risk-free rate (%) (example value 5) is a quantity rather than a category — precision here is cheap and worth having. Check the convention first: if the source says “6.5 % per year”, type 6.5 and let the formula do the rest.
Observed call price ($) (example value 4) is a measured number in the unit named by the label; the formula does not convert for you. If your source quotes a figure before and after some deduction, decide which one the formula wants and be consistent for every field.
Solve for (example value Fair put) is a branching choice rather than a figure you measure; the per-option effects are computed below. The example value is there to demonstrate the mechanics, not to describe your situation — overwrite it.
Observed put price ($) (example value 5) is a measured number in the unit named by the label; the formula does not convert for you. If your source quotes a figure before and after some deduction, decide which one the formula wants and be consistent for every field.
PV of dividends before expiry ($) (example value 0) is a measured number in the unit named by the label; the formula does not convert for you. This input feeds the formula directly; a defensible figure here is worth more than false precision anywhere else.
What each “Solve for” choice actually does: Fair put → $-105.40 /share · Fair call → $114.40 /share · Fair forward → $219.89 /share These figures are computed from this engine, not estimated — switch the dropdown above and you will land on exactly these numbers.
The default Put-Call Parity calculation, written out in full
The tool prints its own working; this is the same arithmetic on the example values, reproduced so you can follow it by hand. Starting figures: Underlying price ($), Strike ($), Time to expiry (years), Risk-free rate (%). The endpoint of that chain is $-105.40 /share of fair put price, sitting in the “Parity gap of -$110 per share (-$11,040 per contract) before costs” band.
- PV(K) = $108·e^(−4.5%×0.50) = $106
- Parity: C + PV(K) = P + S − PV(div) → fair put price = -$105
- Observed minus theoretical = -$110 per share
PV of strike — $105.60. If your inputs change, this is usually the first companion figure to move.
Synthetic put (C+PV(K)−S) — $-105.40 /share. This is the number experienced users quote when the headline alone would mislead.
Synthetic call (P+S−PV(K)) — $114.40 /share. Read next to the headline, this is the figure that shows the shape of the trade-off.
Conversion arbitrage edge — $-11,040.29 /contract. Compare it against the main result of $-105.40 /share before drawing conclusions.
Keep this worked example open while you enter your own figures: matching the shape of the substitution is the fastest way to spot a unit or convention mistake.
The Put-Call Parity corridor: your result across a realistic band
Single-point answers invite false confidence, so the same engine was run across a band around the input that matters most here. Each row is a full recalculation of Put-Call Parity, not an interpolation.
| Underlying price setting | Fair put price |
|---|---|
| 80 % of the example value | $-62.40 /share |
| 90 % of the example value | $-83.90 /share |
| example inputs | $-105.40 /share |
| 110 % of the example value | $-126.90 /share |
| 120 % of the example value | $-148.40 /share |
Across 5 settings of Underlying price, the result spans $-148.40 /share to $-62.40 /share. A decision that only works at one row of this table is a decision waiting for an input to move. Reproduce it in the form above by editing Underlying price alone; the substitution steps will follow your figures exactly.
The levers behind Put-Call Parity, ranked by effect
Generic advice says “it depends on your inputs”; this section says exactly how much. Each field was perturbed by ±10 % in isolation and the fair put price recomputed, so the ranking below is measured rather than asserted. For Put-Call Parity the ranking is measured rather than guessed, and it belongs to these defaults: shift the starting point and the order can shift with it.
- Underlying price: ten percent up gives $-126.90 /share, ten percent down gives $-83.90 /share — ≈20.4 % of the fair put price either way
- Nudge Strike by ±10 % and Put-Call Parity lands between $-115.96 /share and $-94.84 /share (sensitivity ≈10.0 %)
- Nudge Observed call price by ±10 % and Put-Call Parity lands between $-105.80 /share and $-105.00 /share (sensitivity ≈0.4 %)
- Nudge Time to expiry by ±10 % and Put-Call Parity lands between $-105.64 /share and $-105.17 /share (sensitivity ≈0.2 %)
- Risk-free rate: ten percent up gives $-105.64 /share, ten percent down gives $-105.17 /share — ≈0.2 % of the fair put price either way
Underlying price is the dominant lever here: ten percent either way moves the result by about 20.4 %, from $-126.90 /share to $-83.90 /share. Measure that input from a document rather than from memory, and treat Strike as the second priority (≈10.0 % per ten percent). The remaining fields can sit at safe defaults without materially changing the answer — knowing which is which is what separates a careful calculation from an anxious one.
Reading the result: bands, zones and thresholds
A number on its own is trivia; a number against a range is intelligence. On the example inputs the band reads “Parity gap of -$110 per share (-$11,040 per contract) before costs”; your own figures place you on a different rung of the same ladder. The sensitivity list shows how far a modest input error moves you along that ladder: for Underlying price, ±10 % covers $-126.90 /share to $-83.90 /share. The practical reading rules: read the band as a starting hypothesis, then test it: change the two dominant inputs by ten percent and see whether the verdict survives. Verdicts that survive stress are the ones worth acting on.
Related measures answer related questions, and the distinctions matter: Put-Call Parity is the quantity, Fair put price is this tool’s reading of it, and the decision that follows belongs to neither of them separately.
Common mistakes with Put-Call Parity (and the fix)
The identity holds exactly only for European options; American deep-ITM puts carry early-exercise value that looks like a parity violation. Dividends shift the whole line: leave PV(div) at zero for non-payers, and use futures pricing for indices where carry is the dividend. This calculator gives an estimate based on standard formulas; it is not professional financial, medical, legal or engineering advice. Verify anything you plan to act on with a qualified professional and your own records.
Copying a single headline number without checking the companion outputs. The tool reports PV of strike, Synthetic put (C+PV(K)−S) and Synthetic call (P+S−PV(K)) alongside the main result precisely because Put-Call Parity in isolation misleads — the secondary columns are where outliers and trade-offs show up.
Attention is also misallocated between fields. Having measured this engine, we can say where it belongs: Underlying price first, everything else second.
A final trap is presentation: reporting Put-Call Parity to more decimal places than the inputs support. The tool prints what the arithmetic gives; you should quote only what your least precise field justifies.
A quick check for any Put-Call Parity figure
- Compare the output against one published benchmark or a colleague’s independent estimate.
- Confirm the example values are plausible for your own situation before replacing them.
- Re-verify Underlying price against a source document — ten percent of error there moves the headline by ≈20.4 %.
- Recompute once with the dominant input deliberately set low, and see whether the decision changes.
- State, in one sentence, which convention each rate, unit or period in the form is using.
- Confirm the answer against one independent method or published benchmark for Put-Call Parity.
Five steps from raw inputs to a figure you can defend
Using a calculator well sounds trivial, but the difference between a decision-grade figure and a throwaway one lives entirely in the routine. The routine for Put-Call Parity takes a minute to learn and applies to every input above.
Step 1 — gather before you type. Put-Call Parity depends on underlying price ($), strike ($), time to expiry (years), risk-free rate (%), observed call price ($), solve for and the remaining fields listed above. Collect them from a source document — a statement, a report, an invoice, a log — rather than from memory, because recalled inputs are the single largest cause of wrong results. The input you reach for first is usually the one you measured longest ago.
Step 2 — enter the conservative case first. Resist the flattering scenario. Type the inputs you would defend in front of a sceptical reader, read the Fair put price ($-105.40 /share for the example values), and note the verdict band. Numbers are cheap to produce and expensive to retract; the extra minute is the cheapest insurance available.
Step 3 — play with the inputs on purpose. Ten percent up, ten percent down, one field at a time. Within a few minutes you will know this calculation better than most people who quote its results. One decimal place more than your data supports is one decimal place of fiction.
Step 4 — run the scenario trio. Expected, pessimistic, optimistic. The gap between pessimistic and expected is the number to plan against; the optimistic result is useful only as a ceiling. If the verdict changes with a ten-percent nudge, report the nudge alongside the number.
Step 5 — record the context, not just the number. Note today’s date and the inputs alongside the $-105.40 /share output. Six months from now the context is what turns this figure into a trend. A figure nobody can reproduce is a figure nobody will act on twice.
From calculation to decision: where Put-Call Parity earns its keep
Most Options & Derivatives disagreements are arithmetic disagreements in disguise. When Fair put price of $-105.40 /share sits on the table with its working shown, the conversation stops being about who sounds more confident and starts being about which input is wrong. Half of a good calculation is knowing which input you would hate to be wrong about.
The audience for this page is people who need one trustworthy number before their next decision. In professional Options & Derivatives practice the pattern is always the same — screen with Put-Call Parity, then invest slow analysis only in whatever survives the screen. If the answer changes your behaviour, it deserves a second run with different assumptions.
Re-running this page costs nothing and catches the slow drifts that never announce themselves. In Investing & Markets the expensive mistakes are rarely sudden; they are small changes in Underlying price ($) that nobody recalculated for a year. Comparisons are cheap: run the identical calculation for the alternative and look at the gap.
Put-Call Parity — frequently asked questions
What affects Put-Call Parity the most?
The short answer: Underlying price. The longer answer is the sensitivity table on this page, where every numeric input was probed by ten percent and the fair put price recalculated — Underlying price moved it furthest, by roughly 20.4 % ($-126.90 /share to $-83.90 /share).
Which matters more: strike or underlying price?
For Put-Call Parity, Underlying price wins the head-to-head (≈20.4 % against ≈10.0 % per ten-percent change). Knowing which input dominates is the difference between careful work and merely slow work.
How much does the “Solve for” choice change the answer?
The spread is not cosmetic: this choice alone takes Put-Call Parity from $-105.40 /share to $219.89 /share with every other input frozen, which is more influence than most of the numeric fields on this page can claim.
What is a typical Put-Call Parity?
On the worked example above Put-Call Parity comes out at $-105.40 /share. Treat that as one point on a distribution rather than a norm: the bands, the companion outputs and the sensitivity ranking tell you how wide the plausible range really is.
How do I calculate Put-Call Parity by hand?
Use the stated model — C + K·e^(−rT) = P + S − PV(div); any leg is solvable from the other three — and substitute in the order the tool prints (it starts from PV(K) = $108·e^(−4.5%×0.50) = $106). On the example inputs the hand route lands on $-105.40 /share, exactly what the calculator shows. A spreadsheet reproduces it in two or three cells; doing it once by hand is what makes the automated answer trustworthy.
What happens to Put-Call Parity if underlying price changes by 10 %?
Try it in the form above and the printed substitution will follow you; the pre-computed answer for the defaults is $-126.90 /share to $-83.90 /share across the ten-percent probe, i.e. ≈20.4 % sensitivity on the fair put price.
Is the Put-Call Parity result of $-105.40 /share good or bad?
The tool answers “what”, not “whether”: $-105.40 /share carries the verdict “Parity gap of -$110 per share (-$11,040 per contract) before costs” for the default case, which is a starting interpretation rather than a verdict on your situation. Re-run with your own figures and compare the bands.
Is Put-Call Parity accurate?
The arithmetic is exact: C + K·e^(−rT) = P + S − PV(div); any leg is solvable from the other three is evaluated with full floating-point precision and every substitution is printed in the worked example. What limits accuracy is input quality and the model’s assumptions, so treat the $-105.40 /share result as the correct answer to the question you typed — and check the assumptions section before acting on it.
How do I calculate Put-Call Parity manually?
Write down the inputs, then follow the printed steps in order — C + K·e^(−rT) = P + S − PV(div); any leg is solvable from the other three. A spreadsheet replicates it with two or three cells. The manual route is worth doing once to build intuition; after that the calculator saves time and eliminates transcription slips.
What is a synthetic long stock?
Call minus put at the same strike/expiry: parity says that combination behaves exactly like owning shares funded by the PV of the strike.
Why is my put IV above call IV?
Skew (real demand for protection) plus borrow — parity itself is satisfied; it is the flat-vol assumption of BS that fails, not parity.
Can this tool see the numbers I type?
It cannot. There is no submission step: the calculation happens in your browser and the result disappears when you close the page. Treat the fields above as a scratchpad rather than a form.
What the labels on this page mean
- PV of strike
- $105.60 for the Put-Call Parity worked case; it sits next to the main result because it carries context the headline leaves out.
- Synthetic put (C+PV(K)−S)
- $-105.40 /share on the example inputs — a companion figure printed beside the headline of Put-Call Parity so that no result is read in isolation.
- Synthetic call (P+S−PV(K))
- $114.40 /share for the Put-Call Parity worked case; it sits next to the main result because it carries context the headline leaves out.
- Fair put price
- The primary output of this tool: $-105.40 /share for the default inputs, computed exactly from C + K·e^(−rT) = P + S − PV(div); any leg is solvable from the other three.
- Worked example
- The calculation written out in full so that nothing on this page has to be taken on trust.
- Verdict band
- A named zone on the range of plausible results; the example inputs land in “Parity gap of -$110 per share (-$11,040 per contract) before costs”, and your inputs may not.
- Sensitivity (±10 %)
- How far the Put-Call Parity moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
Where the Put-Call Parity model comes from
There is no black box on this page — the engine evaluates C + K·e^(−rT) = P + S − PV(div); any leg is solvable from the other three. Source of record: Stoll (J. Finance 1969); standard no-arbitrage identity. Where a professional version of this calculation differs, it differs in inputs and jurisdiction-specific coefficients, not in the algebra — which is why re-checking your inputs beats hunting for a second calculator.
Nothing here replaces a qualified professional, and nothing here is approximate either — the arithmetic is exact, and the honesty is in the inputs you supply. The design constraint was set by usefulness rather than by comfort: an estimate you can interrogate beats an answer you have to trust.
What to do with your Put-Call Parity next
Re-run the calculation with your own figures, stress the two most sensitive inputs by ±10 %, and only then act on the result.
This page was written for people who need one trustworthy number before their next decision, and the promise is that nothing above required trust. Bookmark the tool, re-run it whenever an input moves by more than a few percent, and keep the date with the number.