Black-Scholes Option Pricing Calculator
Estimate the fair value of European call and put options with the Black-Scholes model. Enter the spot price, strike price, time to expiry, risk-free rate, and volatility to get theoretical call and put prices.
Formula
Black-Scholes: C = S·N(d1) − K·e^(−rt)·N(d2)
- d1 = [ln(S/K) + (r + σ²/2)t] ÷ (σ√t), and d2 = d1 − σ√t.
- N(·) is the cumulative standard normal distribution; this tool uses a fast approximation of it.
- The call price is S·N(d1) − K·e^(−rt)·N(d2); the put price follows from put-call parity.
- Enter the risk-free rate and volatility as annual percentages, and time to expiry in years.
- The model assumes European exercise, no dividends, constant volatility and rate, and lognormal prices — real option prices can differ.
At-the-money 1-year call
Inputs
- Spot Price: 100
- Strike Price: 100
- Time to Expiry (years): 1
- Risk-Free Rate (%): 5
- Volatility (%): 20
With spot = strike = 100, 1 year to expiry, a 5% rate, and 20% volatility, the model prices the call near 10.45 and the put near 5.57.
Frequently asked questions
What is the Black-Scholes model?
It is a classic formula for pricing European options based on the spot price, strike, time to expiry, risk-free rate, and volatility.
What does volatility mean here?
It is the expected annual standard deviation of the underlying's returns. Higher volatility raises both call and put prices because larger price swings are more likely.
Does this handle dividends?
No. The basic model assumes no dividends. For dividend-paying stocks you would use an adjusted version, so treat these values as approximations.
What is put-call parity?
It is the relationship C − P = S − K·e^(−rt) that links call and put prices, which is how the put price is derived here.
Are American options priced the same?
Not exactly. Black-Scholes prices European options, which can only be exercised at expiry. American options, exercisable any time, can be worth slightly more.