Black-Scholes Option Price Calculator
The Black-Scholes model calculates the theoretical fair value of European call and put options. Published in 1973 by Fischer Black and Myron Scholes (with contributions from Robert Merton), it remains the foundational formula in derivatives pricing. The model inputs are the current stock price, the option's strike price, the continuously compounded risk-free interest rate, the time to expiration expressed in years, and the annualized volatility of the underlying asset. This calculator computes both the call price (the right to buy) and the put price (the right to sell) simultaneously, along with the intermediate d1 and d2 values. Use annualized figures for volatility and the risk-free rate; convert days to expiration by dividing by 365.
Black-Scholes formula
d1 = [ln(S/K) + (r + sigma^2/2) * T] / (sigma * sqrt(T))
d2 = d1 - sigma * sqrt(T)
Call = S * N(d1) - K * e^(-rT) * N(d2)
Put = K * e^(-rT) * N(-d2) - S * N(-d1)
Where N() is the cumulative standard normal distribution function, S is the stock price, K is the strike price, r is the risk-free rate (decimal), T is time to expiration in years, and sigma is volatility (decimal).
Understanding the Black-Scholes model
- The call price represents the right (not obligation) to buy the underlying at the strike price at expiration.
- The put price represents the right (not obligation) to sell the underlying at the strike price at expiration.
- d1 reflects the moneyness of the option adjusted for time and volatility; N(d1) is the option's delta for a call.
- d2 = d1 - sigma*sqrt(T) represents the risk-neutral probability that the option expires in the money.
- All else equal, higher volatility increases both call and put prices (vega effect), while longer time to expiration also increases option value.
Frequently asked questions
What is the Black-Scholes model?
The Black-Scholes model is a mathematical framework for pricing European-style options, published by Fischer Black and Myron Scholes in 1973. It assumes the underlying asset follows geometric Brownian motion with constant volatility and a constant risk-free rate.
What inputs does Black-Scholes require?
The model requires: current stock price (S), strike price (K), risk-free interest rate (r, annualized), time to expiration (T, in years), and implied or historical volatility (sigma, annualized). Dividends require the Merton extension.
Does Black-Scholes work for American options?
No. Black-Scholes prices European options only, which can only be exercised at expiration. American options, which can be exercised early, require binomial tree methods or the Bjerksund-Stensland approximation for accurate pricing.
What is d1 and d2 in the formula?
d1 = [ln(S/K) + (r + sigma^2/2) T] / (sigma sqrt(T)) and d2 = d1 - sigma sqrt(T). N(d1) and N(d2) are cumulative standard normal probabilities used to weight the stock price and the discounted strike.
What are the limitations of Black-Scholes?
The model assumes constant volatility (real markets show volatility smiles), log-normal returns (fat tails exist), continuous trading, no transaction costs, and no dividends. Despite limitations it remains the industry benchmark for European option pricing.
Official sources
- Black, F. and Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy, 81(3), 637-654. Available via JSTOR: jstor.org/stable/1831029.
- Options Clearing Corporation (OCC): theocc.com.
- CBOE Options Institute: cboe.com/education.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.