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Black-Scholes Option Pricing Calculator

Calculate European call and put option values, along with Delta, Gamma, Theta, Vega, and Rho.

Formula Verified by Dr. Elizabeth Vance, PhD

Quick actions: Save inputs, view saved runs, print, share, or export results

Saved Runs

Inputs

Contract Specs

Stock Price ($)$

Strike Price ($)$

Time to Expiration (Years)

Market Assumptions

Risk-Free Rate (%)%

Annual Volatility (%)%

Results

Option Value

Call Option Price

$10.9310

Put Option Price

$3.8174

Call Option Greeks

Call Delta (Δ)0.6935

Call Theta (θ)-0.0242

Call Rho (ρ)0.2921

Put Option Greeks

Put Delta (Δ)-0.3065

Put Theta (θ)-0.0128

Put Rho (ρ)-0.1723

Shared Greeks

Gamma (Γ)0.0199

Vega (ν)0.2482

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Option values fluctuate with volatility and time decay. Save this page to recalculate contract prices and risk Greeks when market parameters shift.

Formula

Black-Scholes Call Price formula

C = S₀ N(d₁) − K e^(−rT) N(d₂)

Uses cumulative standard normal distributions to weight stock price and discounted strike price.

Step by step

How the calculation works

1Enter current stock price and option strike price.
2Define the time to expiration in fractional years (e.g. 0.5 for 6 months).
3Specify risk-free interest rate and annualized asset volatility.
4Review the theoretical call/put prices and active risk Greeks.

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How to understand and use this calculator

The Black-Scholes-Merton Model

The Black-Scholes-Merton model is a landmark mathematical formula used to estimate the fair market value of European-style options. Published in 1973, the model assumes that stock prices follow a geometric Brownian motion with constant volatility and a lognormal distribution.

The formula takes five primary inputs: the underlying asset price, the strike price of the option, the time to expiration, the risk-free interest rate, and the asset's volatility. It assumes no transaction costs, constant interest rates, and no dividend payments (though standard extensions adjust for dividends).

The Mathematical Formulas for Call and Put Options

The Black-Scholes formulas for European call option value $C$ and put option value $P$ are:

$$C = S_0 N(d_1) - K e^{-r T} N(d_2)$$

$$P = K e^{-r T} N(-d_2) - S_0 N(-d_1)$$

Where:

$$d_1 = \frac{\ln(S_0 / K) + (r + \sigma^2 / 2) T}{\sigma \sqrt{T}}$$

$$d_2 = d_1 - \sigma \sqrt{T}$$

And:

* $S_0$ is the current spot price of the underlying asset.

* $K$ is the option strike price.

* $r$ is the risk-free interest rate (annualized).

* $T$ is the time to expiration in years.

* $\sigma$ is the annualized volatility of the asset returns.

* $N(x)$ is the cumulative standard normal distribution function.

Understanding the Option Greeks

Option Greeks measure the sensitivity of the option's price to changes in the underlying model parameters. They are essential tools for hedging and risk management:

Delta ($\Delta$): The rate of change of the option price with respect to changes in the underlying stock price. Call Delta ranges from 0 to 1, while Put Delta ranges from -1 to 0.

Gamma ($\Gamma$): The rate of change of Delta with respect to changes in the underlying stock price. Gamma is highest for at-the-money options.

Vega ($\nu$): The sensitivity of the option price to changes in the volatility of the underlying asset. Vega is identical for both call and put options.

Theta ($\theta$): The sensitivity of the option price to the passage of time (time decay). Theta is typically negative, reflecting that options lose value as expiration approaches.

Rho ($\rho$): The sensitivity of the option price to changes in the risk-free interest rate.

Related concepts

Delta Hedging

A risk management strategy that offsets the directional exposure of an option position.

Implied Volatility

The asset volatility implied by the market price of an option.

Time Decay

The reduction in option value caused by the passage of time.

Frequently Asked Questions

What is the cumulative normal distribution function N(x)?+

N(x) represents the probability that a random variable from a standard normal distribution is less than or equal to x. In the call formula, N(d2) is the risk-adjusted probability that the option will expire in-the-money.

Why is volatility so important in option pricing?+

Volatility measures the size of price movements in the underlying stock. Higher volatility increases the probability that the stock price will exceed the strike price, making both call and put options more valuable.

What is the risk-free rate in Black-Scholes?+

The risk-free rate is the theoretical return on an investment with zero risk, typically represented by government bonds (e.g., US Treasury bills) matching the term of the option.

What is the difference between historical and implied volatility?+

Historical volatility measures the actual price fluctuations of a stock in the past. Implied volatility is the volatility back-calculated from current market option prices, reflecting the market's future expectations.

What does delta tell us about option probability?+

While not mathematically exact, traders often use Delta as a rough proxy for the probability that the option will expire in-the-money. For example, a 0.30 Delta option has roughly a 30% chance of expiring in-the-money.

How does time decay impact calls vs. puts?+

Theta is negative for both calls and puts, meaning that as days pass, the value of the option decays. Time decay accelerates as the option approaches its expiration date.

What are European-style options?+

European options can only be exercised at the exact expiration date. American options can be exercised at any time before expiration. The Black-Scholes formula is designed specifically for European options.

What is Gamma?+

Gamma measures the acceleration of Delta. A high Gamma means that Delta is highly sensitive to stock price changes, making the option position riskier or more reactive.

How does a dividend impact option pricing?+

Dividends reduce the underlying stock price on the ex-dividend date. This makes call options less valuable and put options more valuable.

Why is Vega identical for calls and puts?+

Both calls and puts benefit from increased volatility because they have limited downside risk but unlimited upside potential. Thus, a rise in volatility increases the price of both call and put contracts by the exact same Vega factor.

How to understand and use this calculator

The Black-Scholes-Merton Model

The Mathematical Formulas for Call and Put Options

The Black-Scholes formulas for European call option value $C$ and put option value $P$ are:

$$C = S_0 N(d_1) - K e^{-r T} N(d_2)$$

$$P = K e^{-r T} N(-d_2) - S_0 N(-d_1)$$

Where:

$$d_1 = \frac{\ln(S_0 / K) + (r + \sigma^2 / 2) T}{\sigma \sqrt{T}}$$

$$d_2 = d_1 - \sigma \sqrt{T}$$

And:

* $S_0$ is the current spot price of the underlying asset.

* $K$ is the option strike price.

* $r$ is the risk-free interest rate (annualized).

* $T$ is the time to expiration in years.

* $\sigma$ is the annualized volatility of the asset returns.

* $N(x)$ is the cumulative standard normal distribution function.

Understanding the Option Greeks

Option Greeks measure the sensitivity of the option's price to changes in the underlying model parameters. They are essential tools for hedging and risk management:

Delta ($\Delta$): The rate of change of the option price with respect to changes in the underlying stock price. Call Delta ranges from 0 to 1, while Put Delta ranges from -1 to 0.

Gamma ($\Gamma$): The rate of change of Delta with respect to changes in the underlying stock price. Gamma is highest for at-the-money options.

Vega ($\nu$): The sensitivity of the option price to changes in the volatility of the underlying asset. Vega is identical for both call and put options.

Theta ($\theta$): The sensitivity of the option price to the passage of time (time decay). Theta is typically negative, reflecting that options lose value as expiration approaches.

Rho ($\rho$): The sensitivity of the option price to changes in the risk-free interest rate.

Related concepts

Delta Hedging

A risk management strategy that offsets the directional exposure of an option position.

Implied Volatility

The asset volatility implied by the market price of an option.

Time Decay

The reduction in option value caused by the passage of time.

Frequently Asked Questions

What is the cumulative normal distribution function N(x)?+

Why is volatility so important in option pricing?+

What is the risk-free rate in Black-Scholes?+

The risk-free rate is the theoretical return on an investment with zero risk, typically represented by government bonds (e.g., US Treasury bills) matching the term of the option.

What is the difference between historical and implied volatility?+

What does delta tell us about option probability?+

How does time decay impact calls vs. puts?+

Theta is negative for both calls and puts, meaning that as days pass, the value of the option decays. Time decay accelerates as the option approaches its expiration date.

What are European-style options?+

What is Gamma?+

Gamma measures the acceleration of Delta. A high Gamma means that Delta is highly sensitive to stock price changes, making the option position riskier or more reactive.

How does a dividend impact option pricing?+

Dividends reduce the underlying stock price on the ex-dividend date. This makes call options less valuable and put options more valuable.

Why is Vega identical for calls and puts?+

Black-Scholes Option Pricing Calculator

Inputs

Contract Specs

Market Assumptions

Results

Option Value

Call Option Greeks

Put Option Greeks

Shared Greeks

Bookmark this Derivatives Tool

Black-Scholes Call Price formula

How the calculation works

Learn more about this calculation

ROI Calculation: Formula, Annualized Return, and Examples

Understanding Calculator Formulas: How DTC Turns Inputs into Results

Black-Scholes Option Pricing Calculator

Inputs

Contract Specs

Market Assumptions

Results

Option Value

Call Option Greeks

Put Option Greeks

Shared Greeks

Bookmark this Derivatives Tool

Key output metrics

Call vs. Put Option Value

Greeks Comparison

How to understand and use this calculator

The Black-Scholes-Merton Model

The Mathematical Formulas for Call and Put Options

Understanding the Option Greeks

Related concepts

Frequently Asked Questions

Black-Scholes Call Price formula

How the calculation works

Learn more about this calculation

ROI Calculation: Formula, Annualized Return, and Examples

Understanding Calculator Formulas: How DTC Turns Inputs into Results

Key output metrics

Call vs. Put Option Value

Greeks Comparison

How to understand and use this calculator

The Black-Scholes-Merton Model

The Mathematical Formulas for Call and Put Options

Understanding the Option Greeks

Related concepts

Frequently Asked Questions