Call Option Calculator (Black-Scholes)

The Black-Scholes model prices European-style call and put options on a non-dividend-paying stock. It assumes the stock follows geometric Brownian motion with constant volatility and a constant risk-free rate.

The formulas:

d1 = [ln(S/K) + (r + sigma^2 / 2) * T] / (sigma * sqrt(T)) d2 = d1 - sigma * sqrt(T)

Call price C = S * N(d1) - K * exp(-rT) * N(d2) Put price P = K * exp(-rT) * N(-d2) - S * N(-d1)

where N is the standard normal cumulative distribution function.

Put-call parity always holds: C - P = S - K * exp(-r*T). Useful as a sanity check.

The Greeks measure how the option price changes with each input:

Delta is the first derivative with respect to S — how much the option price moves per dollar change in the stock. Call delta is N(d1), between 0 and 1. Put delta is N(d1) - 1, between -1 and 0.

Gamma is the second derivative with respect to S. It is the same for calls and puts and largest for at-the-money options near expiry. High gamma means delta changes fast — the option is very sensitive to stock moves.

Theta is the time decay (per day in this calculator). Negative for both calls and puts on long positions — options lose value as time passes.

Vega is the sensitivity to volatility (per 1 percentage point change). Always positive for long positions.

Rho is the sensitivity to interest rate (per 1 percentage point change). Positive for calls, negative for puts.

The chart shows call and put prices across stock prices from 0.5K to 1.5K, with the strike marked. The hockey-stick shape near expiry is the intrinsic value; the smooth curve at longer expiries shows time value.

Limitations: Black-Scholes assumes constant volatility, no dividends, European exercise (only at expiry, not American-style early exercise), and lognormal stock returns. Real markets violate all of these, which is why traders use the model as a benchmark and adjust empirically (volatility smile, dividend corrections, binomial trees for American options).