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.