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Black–Scholes calculator
This Black–Scholes calculator is a free option pricing calculator for European calls and puts. Enter stock price, strike, time to expiration, volatility, risk-free rate, and dividend yield to see theoretical prices, a full Greeks table (Delta, Gamma, Theta, Vega, Rho), and intrinsic vs time value. Use presets like Textbook ($400 / $350) or ATM · 30 days, or type your own inputs. Expand Calculation details for d₁, d₂, and time in years; export CSV or PDF when you are done. We do not load live quotes or implied volatility—you supply every assumption. Not the intrinsic value calculator (stock fair value via DCF/Graham) or a broker options chain. For education only—not investment advice.
What is the Black–Scholes model?
The Black–Scholes–Merton model estimates the theoretical price of a European option from spot price, strike, time to expiration, volatility, interest rates, and a continuous dividend yield. It is a teaching and risk benchmark—not a promise that an option will trade at that price.
New to calls and puts? Start with Investor.gov’s introduction to options. Exercise style (European vs American) matters for real listings—we cover that below.
Call and put prices
Theoretical European call and put values plus intrinsic and time value breakdowns update live.
Option Greeks table
Delta, Gamma, Theta, Vega, and Rho with conventions documented on this page (theta per day; vega per 1 vol point).
Example presets
Load Textbook, ATM · 30 days, or ITM call example to match common classroom numbers.
CSV/PDF export
Download inputs, prices, Greeks, and disclaimers for Excel or notes—calculations stay in your browser.
Pair with stock average calculator for share cost basis or P/E ratio calculator for equity valuation context—not option pricing.
How to use this Black–Scholes calculator
- Enter stock price (spot) and strike price.
- Set time to expiration in days (365-day year) or open More options to enter years.
- Add annualized volatility, risk-free rate, and dividend yield (all in percent).
- Optional: tap a preset chip above the calculator for a worked example.
- Read call and put prices, the Option Greeks table, and Calculation details for d₁ and d₂; export CSV or PDF if you need a record.
You enter volatility directly. This page does not back out implied volatility from a market quote—you plug in σ and get a model price.
Choosing volatility, rates, and dividend yield
Volatility (σ) is the hardest input. In practice, traders often start from implied volatility on a nearby strike (from a broker chain) or from historical volatility over a lookback window. For homework, instructors frequently use 15%–30% annualized. Higher σ raises both call and put time value (all else equal).
Risk-free rate (r) should match the currency and horizon of the option—many examples use a short-maturity Treasury yield as a teaching default. The model assumes r stays constant until expiration.
Dividend yield (q) is the continuous yield on the stock. A rough estimate is annual dividends per share ÷ spot price. Set q = 0 for indices or stocks with no dividend during the life of the option. Dividends lower call values and raise put values in this framework.
| Input | Calculator default | Teaching tip |
|---|---|---|
| Spot / strike | $100 / $100 | ATM starting point; move K for ITM/OTM |
| Time | 30 days | Switch to years in More options for LEAPS |
| Volatility σ | 25% | Match IV from chain or use 20% in textbook examples |
| Risk-free r | 5% | Short Treasury yield in same currency |
| Dividend yield q | 0% | Raise q before ex-div when modeling yield impact |
Defaults load on first visit; presets override them with published classroom numbers.
Calls, puts, and moneyness
A call is the right to buy the underlying at the strike; a put is the right to sell at the strike. This calculator prices European calls and puts exercisable at expiration only, using the same inputs for both sides so you can compare call vs put sensitivity.
Moneyness compares spot S to strike K (for calls):
| Term | Typical condition (calls) | Intuition |
|---|---|---|
| At-the-money (ATM) | S ≈ K | Often high time value; call delta often near 0.50 before rates/dividends |
| In-the-money (ITM) | S > K | Call has positive intrinsic max(S−K,0); put is out-of-the-money |
| Out-of-the-money (OTM) | S < K | Call intrinsic is zero; value is mostly time value and volatility |
Use the ITM call example preset to see intrinsic near $10 with additional time value on top. OTM options can still have positive model price because there is time left and volatility can move spot before expiration.
Black–Scholes formula
The model gives closed-form prices for European options: no binomial tree is required. You first compute d₁ and d₂ from inputs, then apply the standard normal cumulative distribution N(·) in the call and put equations below. The same structure underlies many desk and spreadsheet implementations—only conventions (day count, American exercise, vol surface) differ in production systems.
European call and put
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 + ½σ²)T] / (σ√T)
d₂ = d₁ − σ√T
Where S is spot, K strike, T time in years, r and q are continuous rates (decimals in the engine), σ is volatility, and N(·) is the cumulative standard normal distribution. Rates and yields are entered as percentages in the calculator UI.
Black–Scholes calculator with Greeks
Many searches ask for a Black–Scholes calculator with Greeks—sensitivities that describe how the model price moves when inputs change. This page shows Delta, Gamma, Theta, Vega, and Rho for calls and puts alongside prices.
Delta (Δ) — option price change per $1 move in the stock. Gamma (Γ) — how delta changes per $1 stock move. Theta (Θ) — time decay per calendar day here. Vega (ν) — change per 1 percentage-point move in volatility (20% → 21%). Rho (ρ) — change per 1 percentage-point move in the risk-free rate.
Broker platforms may use different day counts or American exercise, so treat these as model outputs aligned with the formulas on this page.
| Greek | What it measures | Typical sign (calls / puts) |
|---|---|---|
| Delta (Δ) | Price change per $1 move in the stock | 0 to 1 / −1 to 0 |
| Gamma (Γ) | Change in delta per $1 stock move | Positive for both (peaks near ATM) |
| Theta (Θ) | Time decay per calendar day | Usually negative (long options lose time value) |
| Vega (ν) | Change per +1 vol point (σ) | Positive for both (long vol) |
| Rho (ρ) | Change per +1 rate point (r) | Often positive calls / negative puts |
Compare broker Greeks only after aligning conventions—especially theta per calendar day versus per year and vega per vol point versus per 1% of volatility.
Assumptions and limitations
Black–Scholes is a simplifying model. It is excellent for teaching and for a first-pass price, but listed options can trade away from the formula when real markets violate these assumptions:
- European exercise only—no early exercise premium for American options.
- Constant volatility σ and constant rates (r, q) until expiration.
- Continuous trading, no jumps, and lognormal returns—no earnings gaps or halts.
- No bid–ask spread, commissions, borrow costs, or margin financing.
- No discrete dividends—only a continuous yield q (lumpy ex-div dates can matter for American calls).
- Unlimited liquidity to delta-hedge—academic frictionless markets.
Where live quotes often differ
Volatility smile/skew, event risk, borrow on hard-to-short names, and American early exercise (especially puts on dividend stocks) are common reasons broker marks do not match a single σ Black–Scholes run. Use this page to learn mechanics and test sensitivities—not to replicate a live chain without aligning every convention.
European vs American options
Exercise style changes whether early exercise is optimal. Most single-stock equity options in the U.S. are American. Some index products (for example many S&P 500 index options) are European, which is why academic examples and index-focused desks still lean on Black–Scholes.
| European | American | |
|---|---|---|
| Exercise | Only at expiration | Any time up to expiration |
| Early exercise | Not allowed | Can be optimal (e.g. deep ITM put, dividend call) |
| Typical U.S. listings | Some index options | Most equity options |
| This calculator | Yes — European only | Not modeled |
American options generally need a binomial or numerical model, or your broker’s pricer. For European index-style homework, this calculator matches the usual textbook setup.
When is Black–Scholes still used?
Despite known limitations, the model remains the default language for options education and for translating volatility into dollars. Common uses today:
- Coursework and interviews — closed-form calls/puts and Greeks with fixed inputs.
- Sensitivity analysis — see how price moves when σ, T, or r changes (Greeks table).
- Benchmarking — enter your own σ and compare the model price to a live quote (you supply volatility; we do not infer it from the quote).
- Index and European-style products — closer match when early exercise is not relevant.
- Risk intuition — delta hedging thought experiments (not live execution).
For stock valuation (DCF, Graham, margin of safety), use our intrinsic value calculator—not the option intrinsic value rows in this tool.
Black–Scholes in Excel
Use NORM.SDIST(x, TRUE) for N(d) in Excel or Google Sheets. Put spot S, strike K, time T (years), rates r and q (decimals), and volatility σ (decimal) in labeled cells. Compute d₁ and d₂ with LN and SQRT, then build call and put cells with EXP.
Example structure: d1 = (LN(S/K) + (r - q + 0.5*σ^2)*T) / (σ*SQRT(T)) and d2 = d1 - σ*SQRT(T). Export CSV from this calculator to paste your inputs into column A before wiring formulas.
There is no separate Excel file to download—use CSV export, the formulas above, or the in-app Calculation details panel.
Black–Scholes calculator example
Tap a preset above the calculator or enter the values below. Display rounds to two decimals for prices, same as the live tool.
| Scenario | Inputs | Result (approx.) |
|---|---|---|
| Textbook (preset) | S $400 · K $350 · 365 days · σ 20% · r 3% · q 1% | Call ≈ $65.67 · Put ≈ $9.30 |
| ATM · 30 days (preset) | S $100 · K $100 · 30 days · σ 25% · r 5% · q 0% | Call ≈ $3.06 · Put ≈ $2.65 |
| ITM call (preset) | S $110 · K $100 · 60 days · σ 22% · r 4% · q 0% | Call > intrinsic $10 (time value adds premium) |
Load Textbook ($400 / $350), ATM · 30 days, or ITM call example for live Greeks and CSV/PDF export.
Educational disclaimer
This Black–Scholes calculator is for learning and illustration. Outputs are theoretical European prices and Greeks from the inputs you provide—not live market quotes, tax advice, or a recommendation to trade.
- We do not fetch option chains, implied volatility surfaces, or Treasury curves.
- American exercise, discrete dividends, and volatility skew are not modeled.
- CSV/PDF export repeats the same illustrative math for your records.
Not investment, tax, or legal advice. Review definitions at Investor.gov — options and confirm any trade with a licensed professional and your broker’s disclosures.
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Frequently asked questions about the Black–Scholes calculator
What is a Black–Scholes calculator?
A Black–Scholes calculator is an option pricing calculator that applies the Black–Scholes–Merton model to estimate European call and put prices. You enter stock price, strike, time to expiration, volatility, risk-free rate, and dividend yield; the tool returns theoretical prices, option Greeks, and optional CSV/PDF export.
What is the Black–Scholes model used for?
It converts inputs (spot, strike, time, volatility, rates, dividends) into a theoretical option value and Greeks. Students use it for homework; investors use it to explore sensitivities and compare a model price to quotes—always as an estimate, not a trading signal.
Is Black–Scholes still relevant?
Yes for education and baseline pricing. Professionals add volatility surfaces, American exercise, and event risk, but Black–Scholes remains the standard reference for how spot, time, and vol flow into option value.
What is the Black–Scholes formula?
European call: C = S·e^(−qT)·N(d₁) − K·e^(−rT)·N(d₂). European put: P = K·e^(−rT)·N(−d₂) − S·e^(−qT)·N(−d₁), with d₁ and d₂ from spot, strike, time, rates, yield, and volatility. This page computes those values automatically when you enter inputs.
Can you do Black–Scholes in Excel?
Yes. In Excel or Google Sheets, define d₁ and d₂, use NORM.SDIST(x,TRUE) for N(d), and combine with EXP and LN. Or export CSV from this calculator and finish the spreadsheet from your saved inputs.
Does this calculator include Greeks?
Yes. You get a full Black–Scholes calculator with Greeks: Delta, Gamma, Theta (per calendar day), Vega (per 1 vol point), and Rho (per 1 rate point) for calls and puts.
What interest rate is used in Black–Scholes?
You choose a constant risk-free rate (annual percent) that matches your currency and horizon—often a short Treasury yield in teaching examples. The model holds that rate fixed until expiration.
European vs American options—what’s the difference?
European options can be exercised only at expiration. American options can be exercised earlier, which matters for many U.S. equity options and some dividend scenarios. This calculator prices European options only.
What volatility should I enter?
Use implied volatility from an options chain when comparing to a quote, or historical volatility / a class assumption (often 15%–30% annualized). Enter it as an annual percent; we do not download market data for you.
What is implied volatility?
Implied volatility is the σ that makes the model price match a market premium. Here you type σ to get a model price—we do not solve implied vol backward from a quote on this page.
How is time to expiration entered?
Enter days by default (we convert with a 365-day year). Open More options to enter years instead—for example long-dated LEAPS.
What units are Theta and Vega?
Theta is change in option value per calendar day. Vega is change in value for a 1 percentage-point move in volatility (e.g. 20% to 21%).
What is time value vs intrinsic value?
Intrinsic value is immediate exercise value: max(S−K,0) for calls, max(K−S,0) for puts. Time value is option price minus intrinsic—the extra premium for time and uncertainty. Both are shown in the results breakdown.
How is this different from the intrinsic value calculator?
This tool prices listed options (calls/puts) and shows option intrinsic/time value. Our intrinsic value calculator estimates stock fair value with Graham and DCF—not option premiums.
What is at-the-money (ATM)?
At-the-money means spot is near strike (S ≈ K). ATM options often carry more time value; call delta is often near 0.50 before rates and dividends shift the curve. Load the ATM · 30 days preset to try it.
Is this Black–Scholes calculator free?
Yes. Calculations, Greeks, presets, and CSV/PDF export are free with no sign-up. Inputs are processed in your browser—they are not sent to our servers.
Can I export results?
Yes. Click Export CSV or Export PDF after you calculate. Downloads include inputs, prices, Greeks, and the educational disclaimer.
Is this investment advice?
No. This page is for learning and scenario math only—not buy or sell recommendations. For definitions, see Investor.gov on options.