Black-Scholes Calculator (call & put prices + Greeks)

Understanding the Black-Scholes option calculator — concept and principle

This calculator prices European call and put options with the Black–Scholes–Merton model and gives the main Greeks: delta, gamma, vega, theta and rho, plus d₁ and d₂. Use it for options courses, checking broker quotes, understanding how volatility and time affect option prices, and building intuition for hedging.

How it is calculated

• d₁ = [ln(S/K) + (r − q + σ²/2)T] ÷ (σ√T); d₂ = d₁ − σ√T.
• Call = S e^(−qT) N(d₁) − K e^(−rT) N(d₂).
• Put = K e^(−rT) N(−d₂) − S e^(−qT) N(−d₁).
• Delta: e^(−qT) N(d₁) for a call, e^(−qT)[N(d₁) − 1] for a put. Gamma = e^(−qT) N′(d₁) ÷ (Sσ√T). Vega = S√T N′(d₁) e^(−qT), shown per 1 point of volatility.
• Theta is shown per calendar day; rho per 1 point of interest rate.

S share price, K strike, T years to expiry, σ volatility, r risk-free rate, q dividend yield, N the standard normal distribution.

The principle

The model assumes the share price follows a random walk with constant volatility, so its logarithm at expiry is normally distributed. A portfolio of the option and a changing amount of shares can be made riskless at every instant, so it must earn the risk-free rate; this pins down the option's value without needing the share's expected return. N(d₂) is the risk-neutral probability that a call ends in the money, and N(d₁) is the hedge ratio. Put and call prices are linked by put-call parity.

Worked example

Default inputs: S = 100, K = 105, T = 0.5, σ = 30%, r = 4.5%, q = 0.

1. d₁ = [ln(100/105) + (0.045 + 0.045) × 0.5] ÷ (0.3 × √0.5) ≈ −0.0179; d₂ ≈ −0.2300.
2. N(d₁) ≈ 0.4929, N(d₂) ≈ 0.4090.
3. Call = 100 × 0.4929 − 105 × e^(−0.0225) × 0.4090 ≈ $7.29.
4. Put ≈ $9.96 (check: call − put = 100 − 105e^(−0.0225) ≈ −2.66).
5. Call delta 0.4929, put delta −0.5071, gamma 0.0188, vega 0.282, call theta −0.0284 per day.

How to read the result

The price is a fair value under the model's assumptions; market prices also reflect supply, demand and the market's implied volatility. Delta near 0.5 means the option is close to at the money. Vega shows how much the price changes when volatility moves one point, often the most important sensitivity for buyers. Negative theta is the daily cost of holding the option as time passes.

Common mistakes

• Entering volatility or rates as decimals when the fields expect percentages.
• Using calendar months for T instead of years.
• Applying the model to American options with early exercise value, such as puts on high-rate assets.
• Ignoring dividends, which lower call values.

Frequently asked questions

What does N(d₂) mean?

The risk-neutral probability that a call option finishes in the money.

What is implied volatility?

The volatility that makes the model price equal to the market price of the option.

Does Black–Scholes work for American options?

Not exactly. It ignores early exercise, which matters for some American options.

Sources: John C. Hull, Options, Futures, and Other Derivatives, 10th ed. (Pearson, 2018), ch. 15, 17 and 19