Black-Scholes Calculator
Enter six numbers and get the theoretical price of the call and the put together, with every step of the calculation on screen — d₁, d₂, N(d₁), N(d₂) and the five Greeks. Nothing to submit; the page is already showing a worked example.
Call option
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Per contract (×100): —
Put option
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Per contract (×100): —
Black-Scholes-Merton, European exercise — see the working · Checked against the original paper's worked case
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- Call intrinsic value
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- Call time value
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- Put intrinsic value
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- Put time value
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The working
Every intermediate value the model computed on the way to those two prices. It updates as you type. This is here because "what number came out" and "where did that number come from" are the same question for most people who look up this formula.
Step 1 — d₁ and d₂
d₁ = [ ln(S / K) + ( r − q + σ² / 2 ) × T ] ÷ ( σ × √T )
d₂ = d₁ − σ × √T
- T (years)
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- √T
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- σ × √T
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- d₁
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- d₂
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- N(d₁)
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- N(d₂)
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- N(−d₁)
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- N(−d₂)
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Step 2 — discount and combine
C = S × e−qT × N(d₁) − K × e−rT × N(d₂)
P = K × e−rT × N(−d₂) − S × e−qT × N(−d₁)
- S × e−qT
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- K × e−rT
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- S × e^(−qT) × N(d₁)
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- K × e^(−rT) × N(d₂)
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- K × e^(−rT) × N(−d₂)
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- S × e^(−qT) × N(−d₁)
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Step 3 — what those numbers mean
- Chance the call finishes in the money
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- Chance the put finishes in the money
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- Breakeven if you pay the model price for the call
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- Breakeven if you pay the model price for the put
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Those two percentages are N(d₂) and N(−d₂) — risk-neutral probabilities, which is not the same thing as the real-world chance of the stock getting there. They are the probabilities the model prices with, under an assumption that every asset drifts at the risk-free rate. Read them as an internal quantity of the model, not as a forecast.
The Greeks — how the price moves when something changes
Sensitivities of the two prices above, calculated as exact derivatives rather than by nudging the inputs. Collapsed by default because most visitors want the price; open it if you want the risk profile.
| Greek | Call option | Put option | What it tells you |
|---|---|---|---|
| Delta (Δ) | — | — | Dollars the option gains per $1 the stock rises. Also roughly how many shares one option behaves like. |
| Gamma (Γ) | — | — | How much delta itself changes per $1 move. High gamma means the position's character changes quickly. |
| Theta (Θ) | — | — | Dollars lost per calendar day, all else equal. Negative for anything you own. |
| Vega (ν) | — | — | Dollars gained per one percentage point rise in volatility. Identical for the call and the put. |
| Rho (ρ) | — | — | Dollars gained per one percentage point rise in the risk-free rate. Usually the smallest of the five. |
All figures are per share, so multiply by 100 for one contract. Theta is per calendar day; vega and rho are per one percentage point, which is the unit you can actually use — the textbook convention of per 1.00 would make them a hundred times larger.
Put-call parity — check us with a pocket calculator
For any correct pair of European prices, the call minus the put must equal the discounted spot minus the discounted strike. Both sides are shown so you can verify the two numbers above against each other without trusting us. If they ever disagree, something on this page is wrong and we would like to hear about it.
- C − P
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- S × e−qT − K × e−rT
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You have a theoretical price
The next question is almost always what you actually make if you buy it — which is a different calculation, on a different page, and your numbers travel with you.
A model price tells you what the option should cost. It does not tell you what you make if you buy it at that price. The payoff calculators take it from here — the price above is carried across as the premium, along with your spot and strike, so you land on the breakeven and the profit or loss without retyping anything.
How to use it
Enter the spot and the strike
Both straight off your broker's screen. The strike is the exercise price written into the contract; the spot is what the share trades at right now.
Enter the days until expiry
Calendar days — the number on the contract, not a fraction of a year. The page converts it and shows you the year figure the formula uses, so nothing is happening out of sight.
Enter a volatility
This is the one input with no obvious right answer, and the one that moves the price most. If your chain quotes an implied volatility, use it. If not, the guide below explains what to reach for instead.
Read both prices, and the working
The call and the put are priced from the same inputs at the same time. Underneath them, every intermediate value the model produced is on screen, so you can check any step by hand.
The Black-Scholes formula
Published by Fischer Black and Myron Scholes in 1973, and extended by Robert Merton the same year to handle a stock paying a continuous dividend yield. It is the version used here, which is why there is a dividend field. Scholes and Merton received the Nobel Memorial Prize in Economic Sciences for it in 1997; Black had died in 1995.
C = S × e^(−qT) × N(d₁) − K × e^(−rT) × N(d₂)
P = K × e^(−rT) × N(−d₂) − S × e^(−qT) × N(−d₁)
d₁ = [ ln(S / K) + ( r − q + σ² / 2 ) × T ] ÷ ( σ × √T )
d₂ = d₁ − σ × √T
S is the stock price, K the strike, T the time to expiry in years, r the risk-free rate, q the dividend yield and σ the volatility — the last three as decimals, so 4% enters the formula as 0.04. N(·) is the cumulative standard normal distribution: N(x) is the probability that a standard normal random variable comes in below x.
The default example, worked through
A $105 call and a $105 put on a $100 stock, 45 days out, 25% volatility, a 4% rate and no dividend. T is 45 ÷ 365 = 0.1233 years and σ√T is 0.0878. ln(100 ÷ 105) is −0.0488, and adding (0.04 + 0.03125) × 0.1233 gives −0.0400, so d₁ = −0.0400 ÷ 0.0878 = −0.4557 and d₂ = −0.5435. From there N(d₁) = 0.3243 and N(d₂) = 0.2934. The strike discounts to 104.4835, so the call is 100 × 0.3243 − 104.4835 × 0.2934 = $1.77, and the put is 104.4835 × 0.7066 − 100 × 0.6757 = $6.26. One contract of each costs $177 and $626.
Where to get each input
The spot, the strike and the days are simply read off the contract, and getting them wrong is not a real risk. The other three are where the answer is actually decided, and they are worth a paragraph each. Ranked by how much they move the price: volatility first by a wide margin, then dividend yield on a high-yielding stock, then the rate, which for a short-dated option barely matters at all.
Volatility — the input that decides the answer
This is the annualised standard deviation of the stock's returns, expressed as a percent, and it is the only input the market does not hand you. If your option chain quotes an implied volatility, use that: it is the market's own number, backed out of the price people are actually paying. If it does not, a historical volatility over a period similar to your holding period is the usual substitute — many large-cap US stocks sit in the 20-35% range and individual small caps are frequently double that. Be aware of the circularity: implied volatility is derived from option prices, so pricing an option with it tells you whether your assumptions match the market's, not whether the market is right.
Risk-free rate
The annualised yield on a government security maturing around the same time as the option — a Treasury bill for a short-dated contract. Continuously compounded in the formula, though at current levels the difference from the quoted annual yield is a rounding error against everything else. For an option expiring in weeks, moving this input by a full percentage point typically changes the price by a cent or two; rho, in the Greeks table, tells you exactly how much for your inputs.
Dividend yield
The annual dividend as a percentage of the share price, treated as paid continuously rather than in quarterly lumps — a simplification that is close enough for most equities and materially wrong for a stock about to pay a large special dividend. Dividends accrue to the shareholder rather than the option holder, so a yield lowers every call price and raises every put price. Leave it at zero for a non-payer.
What the model assumes
Every one of these is an assumption you are accepting when you read the two prices above. They are stated because a number whose assumptions you cannot see is a number you should not rely on:
- European exercise only. The model prices an option that can only be exercised at expiry. Almost all US single-stock options are American and can be exercised any day — see the section below on what that changes.
- Lognormal returns and constant volatility. The stock is assumed to drift smoothly with a volatility that never changes. Real markets gap, and volatility is neither constant across strikes nor across time — that is the volatility smile, also below.
- Continuous, frictionless trading. No commissions, no bid-ask spread, no borrowing limits, and the ability to trade continuously. Your fill will differ from the model price partly for these reasons alone.
- A known, constant risk-free rate and dividend yield. Both are single numbers here, held fixed to expiry, and the dividend is treated as a continuous yield rather than as discrete payments on known dates.
- ACT/365 day count. Days are converted to years by dividing by 365. Some desks use 252 trading days; on a 45-day option the difference is worth well under a cent at typical volatilities, but it is a choice and this is the one made here.
- Nothing about whether the trade is a good idea. A theoretical price is one input to a decision. It is not a recommendation, a forecast, or a claim that the market's price is wrong.
American options — what this calculator cannot do
Black-Scholes prices European options, which can only be exercised on the expiry date. Nearly every US single-stock option is American and can be exercised on any day up to expiry, and that extra right cannot be worth less than nothing — so an American option is worth at least as much as its European twin. In practice the gap is often negligible: for a call on a stock paying no dividend, early exercise is never optimal, and the two values are identical. The gap opens up for puts, which can be worth exercising early when they are deep in the money, and for calls on a stock about to pay a substantial dividend. Pricing those properly needs a binomial or trinomial tree rather than this formula. We would rather tell you that than quietly hand you a European number with an American label on it.
Why the market price is not the model price
If you price a real contract here and compare it to the quote, the two will usually differ, and the model is not necessarily the one that is wrong. The largest single reason is the volatility smile: Black-Scholes assumes one volatility for every strike, but the market consistently prices out-of-the-money puts at a higher implied volatility than at-the-money options, because it charges more for crash protection than a lognormal distribution says it should. Feed a single volatility into the formula and you will price options away from the money differently from the market by construction. Add the bid-ask spread, discrete dividends, early-exercise value on American contracts, and interest rates that are not quite the ones you entered, and a difference of a few percent is unremarkable.
Worked examples
The same 30-day option at 25% volatility and a 4% rate, with the stock in four different places — and what happens to the split between intrinsic and time value.
- Deep in the money: $120 stock, $100 strike. The call is worth $20.34, of which $20.00 is intrinsic value and only $0.34 is time value. There is very little optionality left to pay for — the model puts the chance of finishing in the money at 99.5% — so the contract behaves almost exactly like the stock. The matching put is worth a cent.
- At the money: $100 stock, $100 strike. The call is $3.02 and the put is $2.69, and every cent of both is time value: intrinsic value is zero for each. This is where time value peaks and where theta bites hardest, because the outcome is genuinely uncertain and the option is pure optionality.
- Out of the money: $80 stock, $100 strike. The call rounds to $0.00 — a 25% move in 30 days is far enough out on the distribution that the model gives it a chance of roughly 0.1%. The put is worth $19.67, slightly less than its $20 intrinsic value, because you receive the strike a month from now rather than today.
- One day left: $100 stock, $105 strike. The call is worth essentially nothing, because the stock has one day to travel 5% and volatility of 25% a year is about 1.3% a day. This is what the model means by time value decaying to zero — at expiry, the price collapses onto the payoff, which is where the payoff calculators take over.
Questions about Black-Scholes
Including the three that come up most: what to put in the volatility field, why the model price differs from the quote, and what the model does when the option is American.
The inputs
What volatility should I enter?
If your option chain shows an implied volatility for the contract, use that — it is the market's own estimate, backed out of the price people are paying. If it does not, use the historical volatility of the stock over a window similar to your holding period. Many large-cap US stocks sit somewhere in the 20-35% range; individual small caps are often well above that. It is the input that moves the price most, so if you are unsure, try a range rather than a single number.
If I use implied volatility, am I not just getting back the price I started with?
Essentially yes, and that is worth understanding. Implied volatility is defined as the volatility that makes the model reproduce the market price, so feeding it back in returns roughly the quote. That makes the exercise a consistency check rather than a valuation. The model becomes informative when you supply your own volatility estimate and compare the result to the quote — the difference is the market's volatility view versus yours.
Why does this ask for days when the formula uses years?
Because the number you have is days. Your contract expires on a date, not in 0.1233 years, and most calculators make you do that division yourself before they will help. This page does it and shows you the year figure next to the field, so nothing is hidden — you can still check the arithmetic.
Calendar days or trading days?
Calendar days, divided by 365. Some desks prefer trading days over 252, which is a defensible alternative that treats volatility as accruing only when the market is open. The two conventions give very similar prices — well under a cent apart on a 45-day option at typical volatilities — but they are not identical, and this page uses ACT/365 throughout.
Which risk-free rate should I use?
The annualised yield on a government security maturing near the option's expiry — a short-dated Treasury bill for a contract expiring in weeks. For most equity options the choice barely matters: rho in the Greeks table shows how much the price moves per percentage point, and for short-dated contracts it is usually a cent or two.
How do I handle dividends?
Enter the annual dividend as a percentage of the current share price. The model treats it as paid continuously rather than in quarterly lumps, which is a reasonable approximation for a steady payer and a poor one for a stock about to go ex-dividend on a large special payment. Dividends go to shareholders, not option holders, so a yield lowers call prices and raises put prices.
Can I use this for index or currency options?
The mathematics carries over — an index option uses the index's dividend yield in the q field, and a currency option uses the foreign interest rate there. The wording on this page assumes equity options, and index options are usually European and cash-settled, which actually fits Black-Scholes better than single-stock options do. Check the contract's multiplier yourself; the per-contract figures here assume 100.
Reading the result
Why do you show the call and the put together?
Because they cost nothing extra to compute — they come from the same d₁ and d₂ — and because seeing both is genuinely informative. The pair is also what makes the put-call parity check below the result possible, which is the easiest way to confirm the two numbers are internally consistent.
Is the price per share or per contract?
The headline price is per share, which is how options are quoted. The line underneath shows the same figure multiplied by 100 for one standard US equity contract. If you are pricing an adjusted contract after a split or merger, the multiplier may not be 100 and the per-contract figure will be wrong.
What is the difference between intrinsic and time value?
Intrinsic value is what the option would be worth if it expired right now — the stock price minus the strike for a call, floored at zero. Time value is everything else: what you are paying for the possibility that the stock moves further in your favour before expiry. At expiry time value is zero and only intrinsic value remains, which is exactly what the payoff calculators work with.
Why is the time value showing as negative?
That is real, not a bug, and it only happens on a deep in-the-money European option — most often a call on a stock with a meaningful dividend yield, or a deep in-the-money put. The holder of a European contract cannot exercise early to collect those dividends or to get the strike cash sooner, so the option can be worth slightly less than its intrinsic value. An American option, which can be exercised early, never trades below intrinsic value for exactly that reason.
Is N(d₂) really the probability the option expires in the money?
It is the risk-neutral probability, which is a specific and narrower thing. The model works in a world where every asset is assumed to drift at the risk-free rate, and N(d₂) is the chance of finishing in the money in that world. The real-world probability depends on the stock's actual expected return, which the model deliberately never asks for. Treat N(d₂) as an internal quantity of the pricing calculation, not as a forecast you can bet on.
What is the put-call parity line for?
It is a check you can run on us. For European options the call price minus the put price must equal the discounted stock price minus the discounted strike — an identity that follows from arbitrage, not from the model. Both sides are printed so you can confirm the two prices agree with each other. If they ever do not, something on this page is broken and the contact form is at the bottom.
The breakeven figure — is that a Black-Scholes output?
No, and it is labelled carefully for that reason. It is simply the strike plus the model price for the call, or the strike minus it for the put: what the stock would have to reach by expiry for you to break even if you paid exactly the theoretical price. It is there as the bridge to the payoff calculators, which handle profit and loss properly.
The Greeks
What does delta actually tell me?
How many dollars the option gains for each $1 the stock rises, per share. A delta of 0.32 means the option moves about 32 cents on a $1 move, and that one contract behaves roughly like 32 shares. It also drifts as the stock moves, which is what gamma measures.
Is theta per day or per year?
Per calendar day on this page, because that is the only version anyone can use. The formula produces an annual figure, which we divide by 365. It is negative for any option you own — time passing costs you money — and it accelerates as expiry approaches for an at-the-money contract.
Why are vega and gamma identical for the call and the put?
Because put-call parity ties the two prices together with terms that do not depend on volatility, so differentiating with respect to volatility gives the same answer for both. The same argument applies to gamma. Delta, theta and rho do differ, since the parity relationship does involve the stock price, time and the interest rate.
What units are vega and rho in?
Per one percentage point, per share. So a vega of 0.126 means the option gains about 12.6 cents if volatility goes from 25% to 26%. The textbook convention states them per 1.00 — that is, per 100 percentage points — which produces numbers a hundred times larger and correspondingly harder to use.
Are the Greeks approximated?
No. They are the exact analytic derivatives of the pricing formula, not finite differences. The site's test suite checks each of them against a numerical derivative of the price function as an independent verification, and against the sign and range each one must obey.
Limits of the model
Does this work for American options?
Not exactly, and we would rather say so. Black-Scholes prices European exercise only. An American option is worth at least as much, because early exercise is an extra right. For a call on a stock paying no dividend the values are identical, since exercising early is never optimal. The difference matters for puts and for calls on dividend payers, and pricing those needs a binomial tree instead.
Why does my broker's price differ from this?
Usually the volatility smile. The model assumes one volatility for every strike, while the market prices out-of-the-money puts at a higher implied volatility than at-the-money options — it charges more for crash protection than a lognormal distribution implies. Add the bid-ask spread, discrete dividend dates, early-exercise value on American contracts, and a rate that is not quite the one you entered, and a few percent of difference is normal.
When is the model most likely to be wrong?
Around earnings and other scheduled events, where the stock's distribution is not remotely lognormal and a single volatility number cannot describe it. Also for very short-dated contracts, deep out-of-the-money strikes where the smile is steepest, illiquid names with wide spreads, and any stock about to pay a large special dividend. It is most trustworthy for liquid, at-the-money, medium-dated options on a stock with no unusual events pending.
How accurate is the arithmetic itself?
The normal distribution function is computed with the Abramowitz and Stegun rational approximation, whose error is bounded at 7.5×10⁻⁸ across the whole range — around eleven orders of magnitude below the cent the prices are rounded to. The engine is tested against the published worked example from the standard textbook treatment and against put-call parity on every case. Model error dwarfs arithmetic error here by an enormous margin.
Is anything I type stored?
No. The calculation runs entirely in your browser; nothing you enter is sent anywhere or saved. The Copy link button encodes your inputs into a URL so you can share or bookmark a calculation, and that URL only exists wherever you paste it.
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Checked against published examples
A calculator that only agrees with itself proves nothing. The engine behind this page is unit-tested on every build, and the checks are the ones you could run yourself:
- The standard textbook worked example — a call with the stock at 42, strike 40, six months to expiry, a 10% rate and 20% volatility — publishes d₁ = 0.7693, d₂ = 0.6278, a call of 4.76 and a put of 0.81. This engine returns 0.7693, 0.6278, 4.7594 and 0.8086. Enter those inputs above and check.
- Put-call parity is asserted on every test case, including the degenerate ones: C − P must equal S × e^(−qT) − K × e^(−rT) to within 10⁻⁹. It is also displayed live beneath the result, so the check is available to you and not only to us.
- All five Greeks are compared against numerical derivatives of the price function — bump an input, difference the prices, and confirm the analytic formula agrees. An independent calculation of the same quantity is stronger evidence than a formula checked against itself.
- The boundaries are tested rather than avoided: zero days to expiry must return exactly the intrinsic value, zero volatility must return the discounted forward payoff, and a thirty-year option must not overflow. A missing volatility must be refused outright rather than quietly treated as zero — an early version of this engine did exactly that, and the test now prevents it.
Black & Scholes (1973), Journal of Political Economy 81(3) — the original paper, with Merton's dividend extension published the same year in the Bell Journal of Economics and Management Science. Neither author nor journal has any connection with this site.
Published examples are used as arithmetic checks only. Agreeing with a textbook proves the formula is implemented correctly; it says nothing about whether the model's assumptions hold for the contract you are pricing, which is the section above.
Method and limitations
Every figure on this page comes from the Black-Scholes-Merton formulas shown above, evaluated in your browser with a continuous dividend yield, ACT/365 day count, and the Abramowitz and Stegun approximation to the normal distribution. There is no market data feed, no server call and no stored state — the page knows only the six numbers you typed.
Built by CalcStocks. It has not been reviewed by a licensed derivatives professional, and we would rather say so than imply an endorsement that does not exist. A theoretical price is one input to a decision and is not investment advice. The model prices European exercise only and rests on assumptions — constant volatility, lognormal returns, frictionless trading — that are approximations rather than facts about markets.