Black-Scholes Calculator
The Black-Scholes calculator estimates the fair theoretical price of a European call or put option from the stock price, strike, time to expiry, volatility, and interest rate. It is the foundation of modern options pricing. This is a theoretical value for a European option under the model's assumptions (constant volatility, no dividends, log-normal returns). Real option prices differ because those assumptions rarely hold exactly.
Price an option
Adjust the assumptions. Results update in your browser only.
Theoretical call price
$10.45
For these assumptions, the model prices the call at $10.45 and the expiration break-even is $110.45.
Breakdown
- d1
- 0.35
- d2
- 0.15
- Break-even
- $110.45
- Intrinsic value
- $0.00
Payoff at expiration
Net payoff at expiration after subtracting the theoretical premium.
The payoff chart plots net profit or loss at expiration across stock prices around the strike, with markers for strike and break-even.
How the Black-Scholes calculator works
Black-Scholes estimates a theoretical option premium for European exercise with no dividends. The payoff diagram then subtracts that premium from expiration payoff so you can see break-even and net profit or loss.
The calculator converts the stock price, strike, time, rate, and volatility into d1 and d2, then applies the standard normal distribution to estimate the present value of a European call or put.
d1 = (ln(S/K) + (r + sigma^2/2) x T) / (sigma x sqrt(T))
d2 = d1 - sigma x sqrt(T)
Call = S x N(d1) - K x e^(-rT) x N(d2)
Put = K x e^(-rT) x N(-d2) - S x N(-d1)- S is stock price, K is strike price, T is years to expiry, r is the risk-free rate, and sigma is volatility.
- N(d1) and N(d2) are standard normal cumulative probabilities.
- The payoff diagram uses the model price as the premium paid and plots long call or long put payoff at expiration.
When to use it
Helpful for
- Benchmarking a European call or put against a clean theoretical model.
- Seeing how volatility, time, and rates affect option value.
- Estimating break-even and expiration payoff after paying the theoretical premium.
Can mislead when
- The option is American-style and early exercise matters.
- The underlying pays dividends during the option life.
- Volatility is changing quickly or the option has a wide bid-ask spread.
Common mistakes
- Using historical volatility when the market is pricing a very different implied volatility.
- Applying the standard no-dividend model to dividend-paying stocks without adjustment.
- Using it as an exact price for American options that can be exercised early.
- Ignoring bid-ask spreads, commissions, assignment rules, and liquidity.
Worked example
The default inputs use a 100 stock price, 100 strike, 1 year to expiry, 5% risk-free rate, and 20% volatility. The model gives d1 of 0.35, d2 of 0.15, a call price near 10.45, and a put price near 5.57.
| Input | Value |
|---|---|
| d1 | 0.35 |
| d2 | 0.15 |
| Call price | $10.45 |
| Put price | $5.57 |
| Call break-even | $110.45 |
Frequently asked questions
Black-Scholes gives a theoretical fair value as a benchmark. An option trading well below its Black-Scholes value can look cheap and one above it rich, but real prices differ because the model assumes constant volatility, no dividends, and European exercise, which rarely all hold.
Implied volatility is the volatility figure that makes the Black-Scholes price equal the option's actual market price. Traders often back it out from market prices to compare how expensive options are across strikes and dates.
Not exactly. The standard model prices European options, which can only be exercised at expiry. American options allow early exercise and are usually worth at least as much, so they need adjusted models.
It assumes constant volatility, no dividends, frictionless trading, and log-normally distributed returns. Real markets show changing volatility, dividends, and fat tails, so the model is a starting benchmark rather than a precise price.
Screen optionable stocks
Use the screener to compare liquidity, volatility, valuation, and fundamentals before modeling option scenarios.