Black-Scholes Calculator — Option Pricing Model
The Black-Scholes model gives the theoretical fair value of a European call or put option. Enter the stock price, strike, time to expiry, risk-free rate and implied volatility to price the option and see its key Greeks.
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Black-Scholes theoretical fair value of the European call
- 1
Time to expiry (years)
T = 90 ÷ 365 = 0.2466 - 2
d₁
[ln(100 ÷ 100) + (0.05 + 0.2² ÷ 2) × 0.2466] ÷ (0.2 × √0.2466) = 0.1738Measures how far the stock price is above the strike, adjusted for drift and volatility. - 3
d₂
0.1738 − 0.2 × 0.4966 = 0.0745 - 4
N(d₁) — delta probability weight
N(0.1738) = 0.569 - 5
N(d₂) — risk-neutral prob. in-the-money
N(0.0745) = 0.5297 - 6
Call price
100 × 0.569 − 100 × 0.9877 × 0.5297 = 4.5790
How does this calculator work?
Black-Scholes call price: C = S·N(d₁) − K·e^(−rT)·N(d₂) where d₁ = [ln(S/K)+(r+σ²/2)T]/(σ√T) and d₂ = d₁−σ√T. N(d₂) is the risk-neutral probability of expiring in-the-money. The model assumes European exercise, constant volatility, no dividends and no transaction costs.
Formula
How this is calculated
The Black-Scholes-Merton (BSM) model, published in 1973, prices European-style options (exercisable only at expiry) under a set of simplifying assumptions: the stock follows a log-normal random walk with constant volatility σ, there are no dividends, no transaction costs, and a constant risk-free rate r. Under these conditions the call price is C = S·N(d₁) − K·e^(−rT)·N(d₂) and the put price follows by put-call parity: P = C − S + K·e^(−rT).
The two intermediate values d₁ and d₂ determine the probability weights. N(d₂) is the risk-neutral probability that the option expires in-the-money; the distribution curve below shows d₁ and d₂ on the standard normal and shades the N(d₂) area. The Greeks — Delta (sensitivity to stock price), Gamma (rate of change of Delta), Vega (sensitivity to volatility), and Theta (time decay per day) — are derived analytically from the same formula and are shown in the stats grid.
The BSM model is a benchmark, not gospel. Real markets have volatility smiles (implied vol differs by strike), jumps in stock prices, and early-exercise value for American options. This calculator prices European options only. The normal CDF uses the Abramowitz & Stegun polynomial approximation with error < 7.5×10⁻⁸.
Frequently asked questions
A call gives the right to buy the stock at the strike price; its value rises as the stock rises. A put gives the right to sell at the strike; its value rises as the stock falls. Both are priced simultaneously by Black-Scholes and linked by put-call parity: C − P = S − K·e^(−rT).
You can use historical volatility (standard deviation of daily log-returns, annualised by multiplying by √252) or implied volatility (the σ at which the BSM price equals the current market price, back-solved from traded option quotes). Implied volatility is forward-looking and reflects market expectations; historical volatility looks backward.
The original BSM formula assumes no cash dividends. A common adjustment for discrete dividends is to subtract the present value of expected dividends from the current stock price S before entering it here. For continuous dividend yield q, replace S with S·e^(−qT) in the formula — this adjustment is not built in, so make the substitution manually.
TG we-Calculate Editorial Team. (2026). Black-Scholes Calculator — Option Pricing Model [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/black-scholes-calculator
TG we-Calculate Editorial Team. "Black-Scholes Calculator — Option Pricing Model." TG we-Calculate. 2026. https://we-calculate.com/calculator/black-scholes-calculator.
TG we-Calculate Editorial Team, "Black-Scholes Calculator — Option Pricing Model," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/black-scholes-calculator
@misc{wecalculate_black_scholes_calculator, title = {Black-Scholes Calculator — Option Pricing Model}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/black-scholes-calculator}}, year = {2026}, note = {TG we-Calculate} }
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