Black-Scholes & Greeks Calculator
Price European calls and puts and see all five main Greeks. Supports a continuous dividend yield (the Merton extension).
Option value vs. stock price
How to use
- Enter the stock price, strike, days to expiration, implied volatility, risk-free rate, and dividend yield.
- Results update as you type. The table shows call and put values and Greeks per share.
- Use the chart to see how each option's value changes with the stock price, holding everything else constant.
Formulas explained
N() is the standard normal CDF and n() is its density. T is in years (days/365).
Worked example Hypothetical
Hypothetical textbook case: S = $100, K = $100, 1 year, r = 5%, σ = 20%, no dividend.
- Call ≈ $10.45, put ≈ $5.57. Put-call parity check: 10.45 − 5.57 = 100 − 100e^(−0.05) = 4.88. ✓
- Call delta 0.637, gamma 0.0188, vega $0.375 per vol point, call theta ≈ −$0.0176/day, call rho ≈ $0.532 per 1% rate.
FAQ
Is theta per calendar day or trading day?
Per calendar day (annual theta ÷ 365). Some platforms divide by 252 trading days, which gives a larger number.
Why use dividend yield?
Expected dividends lower call values and raise put values. The Merton model treats dividends as a continuous yield q.
Does this work for American options?
It is exact for European options. American options can be worth more when early exercise is optimal, mainly deep ITM puts or calls before a dividend.
Why is vega shown per 1%?
The raw derivative is per 1.00 (100 vol points). Dividing by 100 gives the dollar change for a one-point IV move, which is how brokers show it.
Black-Scholes calculator explained for beginners
What this tool does
It estimates the theoretical "fair" price of a call and a put from six inputs, and shows the Greeks, which describe how that price changes when something moves. It is a model, not a live quote.
What each input means
- Stock price
- The price the stock trades at right now. Use the last price from your broker or a quote site.
- Strike
- The price at which the option lets you buy (call) or sell (put).
- Days to expiration
- Calendar days until the option expires. Count from today to the expiration date on the contract.
- Implied volatility (IV)
- How much the options market expects the stock to move over a year, written as a percent. 30% means the market is pricing roughly a 30% one-standard-deviation move over a year. Your broker shows IV for each option; higher IV means pricier options.
- Risk-free rate
- The interest rate you could earn on cash with essentially no risk, such as a short-term US Treasury bill. It has a small effect on option prices.
- Dividend yield
- The stock's yearly dividend as a percent of its price. Dividends slightly lower call values and raise put values. Use 0 if the stock pays none.
How to read the results
The call and put prices are per share; multiply by 100 for one contract. Delta is roughly how much the option moves for a $1 stock move. Gamma is how fast delta changes. Theta is how much value the option loses per day from time passing. Vega is how much it gains or loses per 1 point of IV. Rho is the effect of a 1 point change in interest rates.
A short worked example (hypothetical)
Stock $100, strike $100, 30 days, IV 30%, rate 4%, no dividend. The model gives a call near $3.59 and a put near $3.26. The call delta is about 0.53, so a $1 rise in the stock would add roughly $0.53 per share ($53 per contract), all else equal.
Common mistakes
- Typing 0.30 instead of 30 for IV.
- Treating the model price as what you will actually get. Real quotes include a bid/ask spread and IV skew.
- Using it for American options on dividend stocks without remembering early exercise can matter.
- Forgetting that theta and vega here are per share, not per contract.