Black–Scholes

Black–Scholes–Merton values European calls and puts with a constant continuous dividend yield. All amounts use the same chosen currency; prices are per underlying unit, without a contract multiplier.

r and q are annual continuously compounded rates, entered in percent. σ is annual volatility in percent; time is in years. The software accepts positive S/K up to 10¹², r/q from −100% to 100%, σ from 0% to 500%, and time from 0 to 100 years.

Delta is price sensitivity to S; gamma is its second derivative. Vega and rho are per one percentage point. Theta is annual calendar-time decay divided by 365, not a trading-day convention. Details show d₁/d₂, annual theta, unit-rate Greeks and the put–call parity residual. At expiry the price is intrinsic value; at zero volatility it is the discounted deterministic payoff. Greeks are not shown in those edge cases.

IV mode uses market price instead of an input volatility, and solves within 0–500%. Model bounds are checked first. A rounded price at a bound, tiny premiums or flat prices can make IV indeterminate or sensitive to rounding; zero at the lower bound is a compatible convention, not proof of actual zero volatility. Very small values use scientific notation; unsupported numerical scales show an error. Constant volatility/rates and European exercise are model assumptions, not market forecasts.

Input edits update after a short pause; option and calculation changes update immediately. Enter updates immediately. Reset restores the example. Only raw inputs, option and mode are kept locally; results and open details are not saved. Reload restores inputs without calculating. Applying solved volatility explicitly updates the volatility input.