Black-Scholes Option Pricing Calculator

Price European call and put options, compute Greeks, and estimate implied volatility with the Black–Scholes–Merton model.

Black-Scholes Calculator

same currency as K

Use years (e.g. 0.5) or days (e.g. 182). The model internally uses years.

Continuously compounded equivalent is used internally.

Use 0 for non-dividend-paying assets.

Historical or implied volatility, annualized.

Leave empty if you only want theoretical prices and Greeks.

Theoretical Prices Black–Scholes–Merton

Call price C
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Put price P
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d₁
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d₂
–

Greeks (per 1 unit of underlying)

Call Greeks

Delta
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Gamma
–
Vega
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Theta (per day)
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Rho
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Put Greeks

Delta
–
Gamma
same as call
Vega
same as call
Theta (per day)
–
Rho
–

Vega is shown per 1% change in volatility. Theta is per calendar day.

Implied Volatility (σimp)

Enter a market price to compute implied volatility.

How the Black–Scholes option pricing model works

The Black–Scholes–Merton model gives a closed-form formula for the fair value of a European call or put option on an underlying asset whose price follows a geometric Brownian motion with constant volatility.

Call and put prices with continuous dividend yield q:

\( C = S e^{-qT} N(d_1) - K e^{-rT} N(d_2) \)
\( P = K e^{-rT} N(-d_2) - S e^{-qT} N(-d_1) \)

where

\( d_1 = \dfrac{\ln(S/K) + (r - q + \tfrac{1}{2}\sigma^2)T}{\sigma \sqrt{T}} \),   \( d_2 = d_1 - \sigma \sqrt{T} \)

S = spot price, K = strike, T = time to maturity in years, r = risk-free rate, q = dividend yield, σ = volatility, N(·) = standard normal CDF.

Greeks in the Black–Scholes model

This calculator also returns the main Greeks, which measure sensitivity of the option price:

  • Delta – change in option price for a 1-unit change in the underlying price.
  • Gamma – change in Delta for a 1-unit change in the underlying price.
  • Vega – change in option price for a 1 percentage point change in volatility.
  • Theta – time decay, i.e. change in option price per day as time passes.
  • Rho – change in option price for a 1 percentage point change in interest rates.

Key Greeks (per unit of underlying):

\( \phi(d_1) = \dfrac{1}{\sqrt{2\pi}} e^{-d_1^2/2} \)

Call Delta: \( \Delta_c = e^{-qT} N(d_1) \)
Put Delta: \( \Delta_p = -e^{-qT} N(-d_1) \)

Gamma: \( \Gamma = \dfrac{e^{-qT} \phi(d_1)}{S \sigma \sqrt{T}} \)

Vega (per 1% σ): \( \text{Vega} = 0.01 \times S e^{-qT} \phi(d_1) \sqrt{T} \)

Call Theta (per year): \( \Theta_c = -\dfrac{S e^{-qT} \phi(d_1) \sigma}{2\sqrt{T}} - r K e^{-rT} N(d_2) + q S e^{-qT} N(d_1) \)

Put Theta (per year): \( \Theta_p = -\dfrac{S e^{-qT} \phi(d_1) \sigma}{2\sqrt{T}} + r K e^{-rT} N(-d_2) - q S e^{-qT} N(-d_1) \)

The calculator converts Theta to a per-day value by dividing by 365.

Call Rho: \( \rho_c = 0.01 \times K T e^{-rT} N(d_2) \)
Put Rho: \( \rho_p = -0.01 \times K T e^{-rT} N(-d_2) \)

Implied volatility with Black–Scholes

Implied volatility is the volatility σ that makes the Black–Scholes price equal to the observed market price. There is no closed-form solution, so we solve for σ numerically.

This tool uses a robust root-finding algorithm with bounds (e.g. 0.1% to 500% annual volatility) and stops when the pricing error is very small or a maximum number of iterations is reached.

Model assumptions and limitations

  • Underlying follows a geometric Brownian motion with constant volatility and drift.
  • Markets are frictionless: no transaction costs, continuous trading, and no arbitrage.
  • Interest rate and dividend yield are constant and known.
  • European exercise only (no early exercise).

In practice, volatility is not constant and markets are not frictionless, so Black–Scholes prices are approximations. Traders often work with implied volatilities and volatility surfaces rather than raw prices.

Practical tips for using this Black–Scholes calculator

  • Use a risk-free rate that matches the option’s maturity (e.g. Treasury yield of similar tenor).
  • For equity index options, include the dividend yield of the index (or use the repo rate adjustment).
  • Check that time to maturity is expressed consistently (years vs days).
  • Use implied volatility from liquid options as input when valuing related structures.

Black–Scholes FAQ

What is the Black–Scholes model used for?

It is used to estimate the fair value of European call and put options on stocks, indices, FX and other assets. It provides a closed-form formula for the option price under specific assumptions about the underlying price process and market frictions.

Can I use this calculator for American options?

The formula is strictly valid for European options. For non-dividend-paying stocks, the Black–Scholes price is often a good approximation for American options. For dividend-paying stocks or deep in-the-money puts, early exercise can be optimal and the true American option value will be higher than the Black–Scholes price.

What units should I use for time and volatility?

Time to expiration T must be in years inside the formula. This tool lets you enter either years or days and converts days to years using 365 days. Volatility σ is annualized (e.g. 20% per year). If you have daily volatility, multiply by √252 to annualize.

Why is my implied volatility result “no solution found”?

This usually means the market price you entered is inconsistent with any volatility in the allowed range (e.g. below intrinsic value or extremely high). Check that the option type (call/put), strike, underlying price, and time to expiration are correct, and that the market price is realistic.