Black-Scholes Options Calculator
Calculate theoretical call and put option prices using the Black-Scholes model.
Enter stock price, strike, volatility, and more.
What Is the Black-Scholes Model?
The Black-Scholes model calculates the theoretical price of stock options. It was developed in 1973 by Fischer Black, Myron Scholes, and Robert Merton — and it won the Nobel Prize in Economics in 1997.
Think of it this way: if you could buy the right (but not the obligation) to purchase a stock at a specific price in the future, how much should that right be worth today? That is exactly what Black-Scholes tells you.
Quick Options Primer
- A Call Option gives you the right to buy a stock at a fixed price (the “strike price”) before a certain date. If the stock goes up, your call becomes more valuable.
- A Put Option gives you the right to sell a stock at a fixed price before a certain date. If the stock goes down, your put becomes more valuable.
The Black-Scholes Formulas
Call Price:
C = S × N(d₁) - K × e^(-rT) × N(d₂)
Put Price:
P = K × e^(-rT) × N(-d₂) - S × N(-d₁)
Where:
d₁ = [ln(S/K) + (r + σ²/2) × T] / (σ × √T)
d₂ = d₁ - σ × √T
| Symbol | Meaning |
|---|---|
| S | Current stock price |
| K | Strike price (the price you can buy/sell at) |
| T | Time to expiration (in years) |
| r | Risk-free interest rate (annualized) |
| σ (sigma) | Volatility of the stock (annualized standard deviation) |
| N(x) | Cumulative standard normal distribution (probability that a standard normal variable is less than x) |
| e | Euler’s number (≈ 2.71828) |
| ln | Natural logarithm |
Worked Example
Stock price (S) = $100 Strike price (K) = $105 Time to expiration (T) = 0.5 years (6 months) Risk-free rate (r) = 5% (0.05) Volatility (σ) = 20% (0.20)
Step 1: Calculate d₁: d₁ = [ln(100/105) + (0.05 + 0.04/2) × 0.5] / (0.20 × √0.5) d₁ = [-0.04879 + 0.035] / 0.14142 = -0.09725
Step 2: Calculate d₂: d₂ = -0.09725 - 0.14142 = -0.23867
Step 3: Look up N(d₁) and N(d₂): N(-0.09725) = 0.4613 N(-0.23867) = 0.4057
Step 4: Call price: C = 100 × 0.4613 - 105 × e^(-0.05×0.5) × 0.4057 C = 46.13 - 105 × 0.9753 × 0.4057 = 46.13 - 41.54 = $4.59
Step 5: Put price (using put-call parity): P = 4.59 - 100 + 105 × e^(-0.05×0.5) = 4.59 - 100 + 102.41 = $7.00
What Each Input Does to Option Price
| Input Increases | Call Price | Put Price |
|---|---|---|
| Stock price ↑ | Goes up | Goes down |
| Strike price ↑ | Goes down | Goes up |
| Time to expiry ↑ | Goes up | Goes up |
| Volatility ↑ | Goes up | Goes up |
| Risk-free rate ↑ | Goes up | Goes down |
Limitations
The Black-Scholes model assumes:
- The stock price follows a log-normal distribution
- No dividends are paid during the option’s life
- Markets are efficient (no arbitrage)
- Volatility and the risk-free rate are constant
- European-style options only (no early exercise)
Real markets violate all of these assumptions to some degree, which is why actual option prices differ from Black-Scholes prices. The model is still widely used as a starting point and benchmark.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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