Options
Black-Scholes Option Pricing Calculator
Price European call and put options with the Black-Scholes-Merton model from spot, strike, days to expiry, rate, volatility and dividend yield.
The Black-Scholes model is a formula for the theoretical fair value of a European option from spot price, strike, time to expiry, volatility and interest rate.
Call theoretical price
$10.45
- Call price
- $10.45
- Put price
- $5.57
Black-Scholes-Merton with continuous dividend yield. European exercise.
Worked example
Price a 1-year at-the-money call: spot $100, strike $100, 5% risk-free rate, 20% volatility, no dividend.
- d1, d2
- computed from spot, strike, rate, volatility and time
- Call price
- $10.45
- Put price (same inputs)
- $5.57
The theoretical call is worth $10.45 and the put $5.57 — the $4.88 gap reflects the cost of carrying the underlying at a 5% rate for one year.
How this is calculated
The Black-Scholes-Merton price uses:
d1 = [ln(S/K) + (r − q + σ²/2)·T] ÷ (σ·√T)d2 = d1 − σ·√TCall = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2)Put = K·e^(−rT)·N(−d2) − S·e^(−qT)·N(−d1)
Time T is days ÷ 365, and rates and volatility are decimals. We use a high-precision normal CDF so the price is accurate to many decimal places.
When to use this calculator
Use this to get a theoretical fair value for a European-style option from spot, strike, time, volatility and rates — either to sanity-check a quoted market price or to see how value should change if one input moves.
It is a learning tool as much as a pricing tool. Varying one input at a time — a week less to expiry, a 5-point volatility bump — shows how each parameter feeds the price, which is the intuition behind the Greeks.
Remember its limits: the model assumes European exercise, constant volatility and no dividends unless adjusted. American-style equity options, especially near dividends, can diverge from the Black-Scholes value.
Pair it with the implied volatility calculator to close the loop: price an option at your volatility estimate here, compare against the market's quote, and the difference between your estimate and the implied figure is the disagreement you are trading.
Common mistakes
- Entering volatility as a whole percentage number (e.g. 20) instead of a decimal (0.20), inflating the price by 100x.
- Using calendar days instead of years for time to expiry without converting (days ÷ 365).
- Ignoring dividend yield for dividend-paying stocks, which understates call prices and overstates put prices.
Frequently asked questions
- What does the Black-Scholes model calculate?
- It gives the theoretical fair value of a European call or put from the spot price, strike, time to expiry, risk-free rate, volatility and dividend yield.
- What are its assumptions?
- It assumes European exercise, constant volatility and rates, lognormal returns and no transaction costs — so real prices can differ, especially for American options.
- Why is the put price different from the call price at the same strike?
- Put-call parity: call − put = spot × e^(−qT) − strike × e^(−rT). At S=K=$100, T=1, r=5%, sigma=20%, the $10.45 call and $5.57 put differ by exactly that discounted forward-minus-strike gap.
- What happens if volatility is entered as 20 instead of 0.20?
- The model treats sigma as a decimal (annualised), so entering 20 means 2000% volatility and the price will be nonsensically high — always enter volatility as a decimal or percentage per the field's label.
- Does this price American options correctly?
- No — Black-Scholes is a European-exercise model. American options (which allow early exercise) can be worth more, especially for puts or dividend-paying calls.
- How sensitive is the price to volatility?
- Very — see Vega on the Greeks calculator for the exact price change per 1 percentage-point change in volatility; it's usually one of the largest sensitivities for at-the-money options.