What is the Black Scholes Model used for?

It shows the theoretical fair value of a European call or put option based on six input parameters: stock price, strike price, time to expiration, risk-free interest rate, volatility and dividend yield. Traders use it to calculate options prices and compare them on a common scale.

What are the differences between call and put option prices?

The value of a call option increases as the stock price rises above the strike price, while the value of a put option increases as the stock price falls below the strike price. As both options are linked by Put-Call Parity, this calculator computes both simultaneously using the same combination of inputs.

What are Option Greeks?

The Greeks measure how an option's price changes in response to individual influencing factors. Delta measures the reaction to stock prices, gamma is the reaction of delta, theta is time value loss, vega is volatility and rho is interest rates. They are indicators that reflect not only the price but also the risk of a position.

What is implied volatility?

Implied volatility is the volatility for which the Black-Scholes model gives an option value that equals the actual market price of the option. Since volatility is the only input to the model that cannot be directly observed, implied volatility provides a way to express market expectations about future price changes.

Why does the model use the risk-free rate?

The model calculates the option price by considering it as a hedged portfolio with a risk-free interest rate. This interest rate is used to discount the exercise price to its current value. Typically, the return on short-term government bonds is used. The impact on options with shorter maturities is small, while the impact on options with longer maturities is larger.

Black-Scholes Calculator

Option details

What do you want to do?

Price the option

Find implied volatility

Stock price (spot)

Strike price

Time to expiration

Volatility

Risk-free interest rate

Dividend yield

Option prices

Call option price

Put option price

Show the option Greeks

Delta, gamma, theta, vega, and rho for the call and the put.

Call option price

Put option price: $
Moneyness (spot / strike)

Call intrinsic value: $
Call time value: $

Put intrinsic value: $
Put time value: $

The option is at the money: the spot price equals the strike.

Show the value chart

Plot call and put value against the underlying price.

Value chart

The Black-Scholes calculator computes the price of European call options or put options and shows the price sensitivity to changes in market variables. By inputting the stock price, exercise price, time to expiration, volatility, risk-free interest rate, and dividend yield it will return the theoretical value of call options and put options as well as the option Greeks. It is also possible to compute implied volatility from the market traded option prices.

The role of the Black-Scholes model

This model was published by Fischer Black and Myron Scholes in 1973 and later extended by Robert Merton to include dividends. It provides a closed-form expression for the fair value of options that can only be exercised at expiration. The model assumes that the stock price follows a log-normal random walk with constant volatility, and calculates the price as the cost of a portfolio used to fully hedge the option. The resulting single formula converts six inputs into a single price.

Formula

For stocks that pay dividends continuously, call and put option values are as follows:

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)

The two intermediate terms are:

d_1 = \frac{\ln(S/K) + \left(r - q + \frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}}\qquad d_2 = d_1 - \sigma\sqrt{T}

N represents the cumulative distribution function of a standard normal distribution. The following table shows the individual input parameters and examples:

Symbol	Meaning	Example
S	Stock price (spot)	300
K	Strike price	250
T	Time to expiration, in years	1
r	Risk-free interest rate	3 percent
q	Dividend yield	0 percent
sigma	Volatility	15 percent

With these values, the value of the call option calculated by the model is approximately 58.82 and the put option is approximately 1.43. Since the current price of 300 is well above the strike price of 250, the call option is deep in-the-money with most of its value coming from intrinsic value and little time value.

To use this calculator, follow these steps:

Start with the Pricing Mode. Enter the stock price and strike price, set the time to expiration in days, weeks, months or years, then enter volatility, risk-free interest rate and dividend yield. The call option prices, put option prices and all of the greeks will be updated immediately as you make your inputs. If you switch to the implied volatility mode, you can perform a reverse calculation. Enter the market price of the option and select whether it is a call or put option, then the tool will calculate the volatility required to reproduce that price.

How to interpret the Greek letters:

The Greek letters measure how the option price changes when an input parameter varies. These are the values that most traders actually rely on in making their trading decisions.

Greek	Measures the change in price for	Sign
Delta	a 1 unit move in the stock price	+ for calls, - for puts
Gamma	delta itself, per 1 unit stock move	+ for both
Theta	one day of time passing	usually - for both
Vega	a 1 percent change in volatility	+ for both
Rho	a 1 percent change in interest rates	+ for calls, - for puts

Delta can be thought of as a rough estimate of the probability that an option will be "in the money" at expiration. Gamma reaches its maximum value near the "at-the-money" area and shows the rate of change in Delta. Theta represents the daily cost of owning an option, since its time value is continuously decaying. Vega shows why volatility is one of the most discussed inputs among traders while Rho is most important for options with long remaining lifetimes.

Intrinsic Value and Time Value

Any option price has two components: intrinsic value and time value. Intrinsic value is the amount an option would be worth if it were exercised at that moment in time. For call options, this is the difference between the underlying's current price and the strike price; for put options, it is the difference between the strike price and the underlying's current price. Intrinsic value has a floor of zero. Time value is what exceeds intrinsic value and represents a premium for the possibility that the option will become further "in the money" before expiration. Time value is highest near the "at-the-money" area, tapering off to zero at expiration.

Implied volatility

Volatility is the only input that cannot be directly observed in the Black-Scholes model. Implied volatility uses the model in reverse: given an actual traded option price, it calculates the rate of volatility that would produce that price. Traders use this to compare options on different stocks using a common metric and see if one option appears under- or over-priced relative to market expectations. This calculator determines implied volatility by searching for the volatility rate at which the model price matches the inputted price.

Assumptions and limitations

The Black-Scholes model is a model, so the prices calculated are estimates and not guaranteed values. It assumes that options are European style and can only be exercised at maturity, that volatility and risk-free interest rate are constant, that stock price follows lognormal without jumps, and there are no taxes or transaction costs. As these assumptions do not always hold in real markets, traders tend to focus on implied volatility rather than specific fixed values. For options where early exercise is possible, the binomial tree model or American option model is more suitable.

This calculator is for educational purposes only and does not constitute investment advice, actual market prices of options may differ from the values calculated here. Trading in options involves real risk of loss. Please consult a qualified professional before trading.

Frequently asked questions

What is the Black Scholes Model used for?: It shows the theoretical fair value of a European call or put option based on six input parameters: stock price, strike price, time to expiration, risk-free interest rate, volatility and dividend yield. Traders use it to calculate options prices and compare them on a common scale.
What are the differences between call and put option prices?: The value of a call option increases as the stock price rises above the strike price, while the value of a put option increases as the stock price falls below the strike price. As both options are linked by Put-Call Parity, this calculator computes both simultaneously using the same combination of inputs.
What are Option Greeks?: The Greeks measure how an option's price changes in response to individual influencing factors. Delta measures the reaction to stock prices, gamma is the reaction of delta, theta is time value loss, vega is volatility and rho is interest rates. They are indicators that reflect not only the price but also the risk of a position.
What is implied volatility?: Implied volatility is the volatility for which the Black-Scholes model gives an option value that equals the actual market price of the option. Since volatility is the only input to the model that cannot be directly observed, implied volatility provides a way to express market expectations about future price changes.
Why does the model use the risk-free rate?: The model calculates the option price by considering it as a hedged portfolio with a risk-free interest rate. This interest rate is used to discount the exercise price to its current value. Typically, the return on short-term government bonds is used. The impact on options with shorter maturities is small, while the impact on options with longer maturities is larger.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Investopedia: Black-Scholes Model
Definition, formula, inputs, and worked explanation.
Black, F. and Scholes, M. (1973). The Pricing of Options and Corporate Liabilities
The original Journal of Political Economy paper.
Corporate Finance Institute: Black-Scholes-Merton Model
Model assumptions, formula, and the Greeks.