📊 Black–Scholes Calculator
Price European call and put options given spot, strike, rates, yield, volatility, and time.
Call Price
10.4506
Put Price
5.5735
d1 / d2
0.3500 / 0.1500
How to Use This Calculator
Enter the current underlying price S, the strike price K, the annual risk‑free rate r, the continuous dividend yield q (use 0 if none), the annualized volatility σ, and the time to expiration in years. The tool computes European option prices using the Black–Scholes–Merton model with continuous compounding. You can approximate months as fractions of a year (e.g., 6 months = 0.5). Volatility and rates should be provided in percent; the calculator will handle conversion to decimals internally. Results update instantly as you adjust inputs, returning the theoretical call and put values together with the intermediate d1 and d2 terms for diagnostics.
Formula
C = S e^(-qT) N(d₁) − K e^(-rT) N(d₂), P = K e^(-rT) N(−d₂) − S e^(-qT) N(−d₁)
d₁ = [ln(S/K) + (r − q + ½σ²)T] / (σ√T), d₂ = d₁ − σ√T
Where: N(·) is the standard normal CDF; r is the continuously compounded risk‑free rate; q is the continuous dividend yield; σ is volatility; T is time to expiry in years.
About Black–Scholes Calculator
This calculator implements the canonical Black–Scholes–Merton model for pricing European call and put options with continuous dividend yield. It is intended for analysts, traders, students, and anyone who wants a clean, reproducible way to benchmark option prices under the classic assumptions: log‑normal price dynamics, frictionless markets, constant rates and volatility, and no arbitrage. While real markets often deviate from these assumptions, the model remains a foundational reference for thinking about the trade‑off between time, volatility, and moneyness. By exposing both prices and the d1/d2 terms, the tool helps you understand how inputs map to probabilities in the risk‑neutral measure and provides a quick sense of sensitivity to key parameters like volatility and time.
Frequently Asked Questions
Does this support American options?
No. The Black–Scholes–Merton formula applies to European options, which can be exercised only at expiration. For American options without dividends, European and American call prices coincide; for puts, early exercise can have value. If you require early‑exercise features, binomial trees or finite difference methods are more appropriate. This tool does include a continuous dividend yield parameter, which is common for equity indices and can materially impact the call/put relationship.
What volatility should I enter?
Enter the annualized implied volatility in percent if you’re pricing from market quotes, or historical volatility if you’re running what‑if scenarios. Keep in mind that implied volatilities vary by strike and maturity (the “vol surface”), while Black–Scholes assumes a single constant σ. For quick sanity checks, this simplification is fine; for production‑grade valuation, use a model calibrated to the surface.