Black Scholes Option Pricer
Estimate European call and put values with the Black Scholes model and review five option sensitivities.
Earlier editorial review
- Reviewed
- 2026-09-25
- Methodology
- 1.1
Sources
How it works
Enter positive spot and strike values, calendar days until expiration, annual volatility as a percentage, and an annual continuously compounded rate as a percentage. A negative finite rate is accepted.
The calculator reports call and put values for a dividend free European option using the same assumptions. It also reports delta, gamma, theta, vega, rho, and a numerical put call parity check.
Change one assumption at a time to inspect model sensitivity. Selecting a ticker attempts to fill the spot value through a Yahoo Finance quote request. Check the value shown; quote availability and timing may vary.
Values are expressed per one underlying unit and in the same price unit as the spot and strike inputs. The calculator does not apply a contract multiplier. Results depend on the entered assumptions.
Step by step
- Enter the spot, strike, days, volatility, and rate.
- Review both option values, the Greeks, and the parity check.
- Change one input and compare how the model outputs respond.
Worked example
Enter spot 100, strike 100, 365 calendar days, annual volatility 20%, and annual continuously compounded rate 5%. The dividend free European model returns a call value of about 10.4506 and a put value of about 5.5735, per one underlying unit.
Call delta is about 0.6368. Put call parity differs by less than 0.000001 in this calculation. These values are covered by the calculation tests.
When to use this calculator
Use this calculator to compare theoretical European option values under explicit assumptions or to inspect how the model responds when one input changes.
For a market comparison, use an option quote with matching expiration, strike, exercise style, and dividend treatment. The model omits several features that can affect traded prices.
Common input mistakes
Enter 20 for 20% volatility, not 0.20. Use calendar days to expiration and keep the rate and volatility annualized.
Do not read the parity check as evidence of executable arbitrage. Confirm the option style, expiration, strike, quote timing, and units before comparing with a market price.
Formula
C = S × N(d₁) − K × e^(−rT) × N(d₂)
P = K × e^(−rT) × N(−d₂) − S × N(−d₁)
d₁ = [ln(S) − ln(K) + (r + σ² / 2) × T] / (σ × √T)
d₂ = d₁ − σ × √T
N(x) = cumulative standard normal distribution
T is calendar days divided by 365. Volatility σ and continuously compounded rate r are annual percentages converted to decimals. The model assumes European exercise, no dividends, constant volatility, and no transaction costs.
Assumptions and limitations
This is a dividend free European option model. It does not model early exercise, dividends, changing volatility, volatility skew, jumps, liquidity, bid and ask spreads, commissions, contract multipliers, or taxes. Extreme inputs outside the supported numerical range can produce an unavailable result. Market prices can differ from theoretical values. This calculation is educational and is not investment advice.
Key terms
- Black Scholes model
- A mathematical model for estimating European call and put values from spot, strike, time, annual volatility, and an annual continuously compounded rate, with no dividends assumed.
- Option Greeks
- Sensitivities that describe how model values respond to changes in spot, time, volatility, or rate under the stated assumptions.
- Put call parity
- A relationship between European call and put values, spot, and discounted strike under matching assumptions. The check is mathematical and does not establish executable arbitrage.
Related tools
Use Option Breakeven to calculate the price where an option's value at expiration equals its cost. Use Options Greeks to explore how model sensitivities change across inputs.
Stock Chart shows historical prices for an underlying. A model estimate here is not a market quote and does not establish that an option is mispriced.
FAQ
What does this Black Scholes calculator estimate?
The model estimates European call and put values from spot, strike, time, annual volatility, and a continuously compounded annual rate. It assumes no dividends. The result is a theoretical value under those assumptions, not a live quote or execution price.
What units does each output use?
Delta and gamma describe sensitivity to the underlying price. Theta is the estimated value change per calendar day closer to expiration. Vega and rho show sensitivity to a one percentage point change in volatility and rate. Prices and delta are per one underlying unit. No contract multiplier is applied.
What does the put call parity check mean?
For this model, call plus discounted strike should equal put plus spot. The displayed check measures numerical consistency between the calculated prices. It does not show that a real trade is available at those values or establish an arbitrage opportunity.
Can I use this result for an American option?
No. This calculation assumes European exercise only at expiration. An American option can be exercised earlier, so its value may differ. Dividends can also affect exercise decisions and are not included here.
Should volatility be entered as a percentage or decimal?
Use an annual volatility expressed as a percentage, such as 20 for 20%. Implied volatility from an option chain is an input inferred from market prices, while historical volatility summarizes past price movement. The calculator does not determine which value is appropriate.
Related tools
Option Breakeven calculates expiration thresholds, Options Greeks compares model sensitivities, and Stock Chart displays historical underlying prices.