Introduction
The Black-Scholes calculator helps you find the fair price of a call option or put option. It uses the Black-Scholes model, one of the most important formulas in finance. This model was created by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s. Traders, investors, and students use it every day to price stock options.
To get a result, you enter five key inputs: the stock price, the strike price, the risk-free interest rate, the volatility of the stock, and the time to expiration. You can also add a dividend yield if the stock pays dividends. The calculator then gives you the option price, the Option Greeks (Delta, Gamma, Theta, Vega, and Rho), and a full step-by-step solution so you can see exactly how the math works.
The Greeks tell you how the option price changes when market conditions shift. For example, Delta shows how much the price moves when the stock goes up by one dollar, and Theta shows how much value the option loses each day. A chart is also included so you can see how the option price changes across a range of stock prices.
This tool uses the Merton extension of the Black-Scholes formula, which accounts for stocks that pay a continuous dividend yield. All results update instantly as you change your inputs. Once you know the fair price, you can use our Options Profit Calculator to estimate the potential gain or loss on a trade.
How to Use Our Black-Scholes Calculator
Enter a few details about your option and the underlying stock below. The calculator will give you the fair price for both call and put options, the Option Greeks, a step-by-step solution, and a price chart.
Option Type: Choose Call if you want the right to buy the stock, or Put if you want the right to sell it. Your selection will be highlighted in the results.
Stock Price (S): Enter the current market price of the stock in US dollars. This is the price the stock trades at right now.
Strike Price (K): Enter the strike price of the option in US dollars. This is the price at which you can buy or sell the stock if you use the option.
Risk-Free Rate (r): Enter the annual risk-free interest rate as a percent. A common choice is the yield on a US Treasury bond that matches your option's time frame. Our Bond Yield Calculator can help you determine the current yield on government bonds.
Volatility (σ): Enter the annual volatility of the stock as a percent. You can use the implied volatility from the market or the stock's historical volatility. Historical volatility is based on the standard deviation of past returns.
Dividend Yield (q): Enter the stock's annual dividend yield as a percent. If the stock does not pay a dividend, leave this field blank and it will default to zero.
Time to Expiration (t): Enter how much time is left until the option expires, shown in years. For example, 6 months is 0.5. You can also pick an expiry date from the calendar, and the calculator will fill in this field for you.
Once all fields are filled in, click Calculate to see your results. Click Reset at any time to return all inputs to their default values.
What Is the Black-Scholes Model?
The Black-Scholes model is a math formula used to find the fair price of a stock option. An option is a contract that gives you the right to buy or sell a stock at a set price before a certain date. The model was created by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s. It changed how people trade and price options forever. For a broader look at option pricing and payoff scenarios, see our Options Calculator.
How It Works
The formula uses five key inputs to calculate an option's price:
- Stock Price (S) — the current price of the stock.
- Strike Price (K) — the price at which you can buy or sell the stock if you use the option.
- Time to Expiration (t) — how long until the option expires, measured in years.
- Risk-Free Rate (r) — the return you would earn on a safe investment like a government bond.
- Volatility (σ) — how much the stock price is expected to move up or down over time.
This calculator also includes a dividend yield (q) input, which accounts for stocks that pay dividends. You can find a stock's yield using our Dividend Yield Calculator, or estimate future dividend income to better understand the underlying asset. This feature is based on the Merton extension of the original model.
Calls and Puts
There are two types of options. A call option gives you the right to buy a stock at the strike price. A put option gives you the right to sell a stock at the strike price. This calculator prices both at the same time so you can compare them. After pricing, you can use the Stock Profit Calculator to evaluate the profit or loss on the underlying shares themselves.
What Are the Greeks?
The Greeks measure how sensitive an option's price is to changes in the inputs. Here is what each one tells you:
- Delta (Δ) — how much the option price changes when the stock price moves by $1.
- Gamma (Γ) — how fast delta itself changes as the stock price moves.
- Theta (Θ) — how much value the option loses each day as time passes. This is also called time decay.
- Vega (ν) — how much the option price changes when volatility goes up or down by 1%.
- Rho (ρ) — how much the option price changes when the risk-free interest rate shifts by 1%.
Key Assumptions
The Black-Scholes model assumes stock prices follow a normal distribution of returns, volatility stays constant, there are no transaction fees, and the option can only be exercised at expiration (European-style). It also relies on continuous compounding of the risk-free rate, a concept closely related to compound interest. Real markets do not always follow these rules, but the model remains the most widely used starting point for option pricing in finance. For a deeper look at how present-day cash flows relate to future values, try our Present Value Calculator or Investment Calculator.