Price European options using Black-Scholes model. Visualize Greeks (delta, gamma, theta, vega) and payoff diagrams.
Enter an underlying price, strike, time to expiration, volatility, risk-free rate, and option type to calculate a theoretical Black-Scholes value for a European option. The tool also presents delta, gamma, theta, and vega, plus an expiration payoff diagram. A common source of confusion is comparing the current model value with the expiration payoff. Before expiration, remaining time and volatility can give an out-of-the-money option positive value even though its immediate exercise value is zero.
A concrete input and expected output from the current implementation.
Input
Option type: Call Underlying price: 100 Strike price: 100 Time to expiration: 1 year Volatility: 20% Risk-free rate: 0%
Expected output
Option price: 7.9656; delta: 0.53983; gamma: 0.019848; theta: -3.9695 per year, equivalent to about -0.01088 per calendar day when divided by 365; vega: 0.39695 per volatility percentage point.
With no dividends, d1 = 0.1 and d2 = -0.1. The call value is 100 x [Phi(0.1) - Phi(-0.1)] = 7.9656, and the stated Greeks follow from the same inputs.
For a non-dividend-paying underlying, the calculation forms d1 = [ln(S/K) + (r + sigma^2/2)T] / [sigma sqrt(T)] and d2 = d1 - sigma sqrt(T). The call value is S Phi(d1) - K exp(-rT) Phi(d2). The corresponding put uses K exp(-rT) Phi(-d2) - S Phi(-d1).
Delta measures first-order price sensitivity to the underlying, gamma measures how delta changes, theta measures sensitivity to elapsed time, and vega measures sensitivity to volatility. The payoff diagram instead shows the option's value at expiration across underlying prices. For a call that value is max(S - K, 0); for a put it is max(K - S, 0).
No. It is a theoretical present value based on the supplied price, strike, time, rate, and volatility. Actual option premiums also reflect bid-ask spreads, supply and demand, liquidity, dividends, and volatility patterns that a single Black-Scholes volatility input does not capture. Treat the result as a model benchmark.
Gamma and vega are the same for European calls and puts with identical inputs under standard Black-Scholes assumptions. Delta and theta generally differ, although the zero-rate, at-the-money example makes call and put theta equal. No, equal gamma and vega do not imply equal prices; the price difference is governed by put-call parity.
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