Black Scholes Calculator

Price a European call or put option from the spot price, strike, time to expiry, risk-free rate, and volatility, and see delta, d1, and d2.

Quick Facts

Formula
Call = S·N(d1) − K·e^(−rT)·N(d2)
d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T); d2 = d1 − σ√T. Assumes a European option with no dividends.

Your Results

Calculated
Option price
-
Fair value per share
Delta
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Price change per $1 of underlying
d1
-
Standardized drift-adjusted moneyness
d2
-
N(d2) = risk-neutral chance ITM

Ready

Enter the option inputs and press Calculate.

What this calculator does

This tool applies the Black-Scholes-Merton model to price a European call or put option. You supply five inputs — the current spot price S, the strike price K, the time to expiration T in years, the annual risk-free rate r, and the annual volatility σ — and it returns the option's fair value along with delta, d1, and d2. The model was published by Fischer Black and Myron Scholes in 1973, with Robert Merton providing the underlying theory; it remains the standard closed-form benchmark for option pricing.

The formula

A European call is worth C = S·N(d1) − K·e−rT·N(d2), and by put-call parity a put is worth P = K·e−rT·N(−d2) − S·N(−d1). The two helper terms are:

  • d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T) — a standardized measure of how far the option is in or out of the money once drift is included.
  • d2 = d1 − σ√T — under the risk-neutral measure, N(d2) is the probability the call finishes in the money.

Here N() is the cumulative distribution function of the standard normal distribution, and e−rT discounts the strike back to today. This calculator evaluates N() with the Zelen & Severo rational approximation, which is accurate to about seven decimal places.

Interpreting the output

The option price is the model's fair value per share of the underlying. Delta — N(d1) for a call, N(d1) − 1 for a put — estimates how much the option price moves for a $1 change in the underlying, and roughly indicates the equivalent number of shares the option behaves like. Volatility has the largest effect on price: higher σ makes both calls and puts more valuable, because a wider range of outcomes increases the chance of a large favorable move while losses stay capped at the premium.

Assumptions and limits

  • Prices European options only (exercisable at expiry), and assumes the underlying pays no dividends.
  • Assumes constant volatility and a constant risk-free rate, lognormal price movements, and frictionless trading — real markets deviate from these.
  • Enter the rate and volatility as annual percentages (5 for 5%, 20 for 20%); the calculator converts them to decimals internally.
  • For high-stakes trading decisions, treat the result as a model estimate and confirm with market quotes or a licensed professional.

Frequently Asked Questions

What is the Black-Scholes formula?
For a European call, price = S·N(d1) − K·e^(−rT)·N(d2), where d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T) and d2 = d1 − σ√T. S is the spot price, K the strike, T the time to expiry in years, r the risk-free rate, σ the volatility, and N() the standard normal cumulative distribution. The put price follows from put-call parity: P = K·e^(−rT)·N(−d2) − S·N(−d1).
What inputs does the calculator need?
Five values plus the option type: the current spot price of the underlying, the strike price, the time to expiration in years, the annual risk-free interest rate, and the annual volatility (the standard deviation of returns). This calculator assumes a European option on a non-dividend-paying asset with a constant rate and volatility.
What are d1, d2, and delta?
d1 and d2 are standardized distances that feed the normal distribution term in the formula; N(d2) is the risk-neutral probability that a call finishes in the money. Delta is the option's sensitivity to a $1 move in the underlying: N(d1) for a call and N(d1) − 1 for a put.
Does this handle American options or dividends?
No. Standard Black-Scholes prices European options, which can only be exercised at expiry, and this tool assumes no dividends. American options and dividend-paying underlyings need adjustments such as the binomial model or a dividend-yield term, which are outside this calculator's scope.

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