THE NUMORIX GUIDE
How to use the Black-Scholes Calculator
Last reviewed September 14, 2026
What this calculator does
The engine uses the European Black-Scholes equations C = S N(d1) - K e^(-rT) N(d2) and P = K e^(-rT) N(-d2) - S N(-d1), with d1 = [ln(S/K) + (r + sigma^2/2)T] / (sigma sqrt(T)) and d2 = d1 - sigma sqrt(T).
Formula and method
The engine uses the European Black-Scholes equations C = S N(d1) - K e^(-rT) N(d2) and P = K e^(-rT) N(-d2) - S N(-d1), with d1 = [ln(S/K) + (r + sigma^2/2)T] / (sigma sqrt(T)) and d2 = d1 - sigma sqrt(T). S is stock price, K is strike, T is years, r is the risk-free decimal rate, and sigma is volatility as a decimal.
Variables and inputs
Enter Stock Price, Strike Price, Time to Expiry in years, Risk-Free Rate as a percentage, and Volatility as a percentage. The UI displays call and put prices plus d1 and d2. Rates and volatility are divided by 100 inside the engine.
Worked example
For S = $100, K = $100, T = 1, r = 5%, and volatility = 20%, d1 = (0 + (0.05 + 0.20^2/2)) / 0.20 = 0.35 and d2 = 0.15. With the engine's hardcoded normal-CDF approximation, the displayed call is about $11.91 and put about $7.03.
How to interpret the result
The call and put values are theoretical premiums under the same inputs. A higher stock price generally raises a call value, while more time and volatility usually increase both option values because they enlarge the range of possible outcomes.
Common mistakes to avoid
Enter volatility as an annual percentage such as 20, not a variance or 0.20 in the UI. Use years for time to expiry and match the strike and stock prices to the same share. Do not omit dividend yield from a real pricing comparison when the underlying pays dividends.
Assumptions and limitations
The route assumes European exercise, lognormal prices, constant volatility, a constant continuously compounded risk-free rate, no dividends, and frictionless trading. The normal-CDF helper is a hardcoded approximation and the engine does not model early exercise, bid-ask spreads, discrete dividends, liquidity, or changing volatility; zero time or volatility can also make the formula undefined.