Black–Scholes Calculator
Price a European call or put option with the Black–Scholes–Merton model from spot, strike, time, rate, volatility and dividend yield — with the option delta and the d₁, d₂ terms.
How it works
d₁ = (ln(S ÷ K) + (r − q + σ² ÷ 2)·T) ÷ (σ·√T) · d₂ = d₁ − σ·√T · Call = S·e^(−qT)·Φ(d₁) − K·e^(−rT)·Φ(d₂) · Put = K·e^(−rT)·Φ(−d₂) − S·e^(−qT)·Φ(−d₁) · Φ = standard normal CDF
The Black–Scholes–Merton model gives a closed-form fair value for a European option — one exercisable only at expiry — from six inputs: the spot price, the strike, the time to expiry, the risk-free rate, the volatility and the dividend yield. The two intermediate terms d₁ and d₂ measure, in standard-deviation units, how far in-the-money the option is expected to finish; Φ turns them into probabilities under the risk-neutral distribution. Volatility is the input the market cannot observe directly and the one the price is most sensitive to — higher volatility widens the range of possible outcomes and makes both calls and puts more valuable. Delta, the first derivative of the price with respect to the spot, tells you how many shares hedge one option and approximates the option’s sensitivity to a small move in the underlying.
Worked example
For a call with spot S = $100, strike K = $105, T = 0.5 years, r = 5%, σ = 20% and no dividends: σ√T = 0.2 × 0.7071 = 0.1414, d₁ = (ln(100 ÷ 105) + (0.05 + 0.02) × 0.5) ÷ 0.1414 = −0.0975, and d₂ = −0.0975 − 0.1414 = −0.2389. With Φ(d₁) = 0.4612 and Φ(d₂) = 0.4056, the call is 100 × 0.4612 − 105 × e^(−0.025) × 0.4056 ≈ $4.58, with a delta of about 0.46. The matching put is worth about $6.99.
Sources: Black–Scholes model — Wikipedia
Frequently asked questions
Does Black–Scholes work for American options?
Not exactly. The model prices European options, which can only be exercised at expiry. American options, exercisable any time before expiry, are worth at least as much and are usually valued with a binomial tree or numerical methods. For non-dividend-paying stocks an American call is worth the same as a European one, so Black–Scholes still applies to that case.
What is delta and why does it matter?
Delta is how much the option price changes for a $1 change in the underlying — roughly 0.46 here means the call gains about $0.46 if the stock rises $1. It is also the hedge ratio: holding 0.46 shares short against one call neutralizes small moves. Call deltas run from 0 to 1 (adjusted down by dividends), put deltas from −1 to 0.
Which input matters most?
Volatility. Unlike the spot, strike, rate and time, it cannot be read off a screen and must be estimated, and the option value is highly sensitive to it. In practice traders often invert the model — feeding in the market price to back out the "implied volatility" the market is pricing in — rather than using it to find the price.
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Educational estimate, not investment, tax or financial advice. Option prices and returns are never guaranteed and past performance does not predict the future. Confirm figures with a licensed advisor.