The Black-Scholes-Merton model (1973) earned a Nobel Prize and remains the standard for pricing options. It assumes constant volatility and rates and log-normal prices – real markets show a volatility “smile”, so treat results as a fair-value benchmark.
How to use the Black-Scholes Calculator
- Current stock price
- Strike price
- Time to maturity
- Volatility (annual)
- Risk-free interest rate
- Dividend yield
Results update as you type, and every field accepts the units shown in its menu.
How it works
Enter the current stock price, strike price, time to maturity, volatility, risk-free rate and dividend yield.
Tips
- Use annualised volatility and interest rates as decimals per year.
- Remember that one standard option contract usually covers 100 shares.
- Options lose time value as expiry approaches – re-run the numbers as dates change.
- Derivatives can lose more than you expect; understand the maximum loss before trading.
Common mistakes to avoid
- Mixing days and years for time to expiry.
- Ignoring commissions and the bid–ask spread.
- Forgetting dividends when pricing options on dividend-paying stocks.
Inputs explained
| Field | What to enter |
|---|---|
| Current stock price | Amount in your currency |
| Strike price | Amount in your currency |
| Time to maturity | Units: years, days |
| Volatility (annual) | In % |
| Risk-free interest rate | In % |
| Dividend yield | In % |
What each result means
- d₁ = (ln([Current stock price]÷[Strike price]) + ([Risk-free interest rate]÷100 – [Dividend yield]÷100 + ([Volatility (annual)]÷100)^2÷2)×[Time to maturity])÷([Volatility (annual)]÷100×sqrt([Time to maturity]))
- d₂ = [d₁] – [Volatility (annual)]÷100×sqrt([Time to maturity])
- Call option price = bs([Current stock price], [Strike price], [Time to maturity], [Risk-free interest rate]÷100, [Dividend yield]÷100, [Volatility (annual)]÷100, 0)
- Put option price = bs([Current stock price], [Strike price], [Time to maturity], [Risk-free interest rate]÷100, [Dividend yield]÷100, [Volatility (annual)]÷100, 1)
- Call delta = exp(-[Dividend yield]÷100×[Time to maturity])×ncdf([d₁])
- Put delta = exp(-[Dividend yield]÷100×[Time to maturity])×(ncdf([d₁]) – 1)
- Gamma = exp(-[Dividend yield]÷100×[Time to maturity])×npdf([d₁])÷([Current stock price]×[Volatility (annual)]÷100×sqrt([Time to maturity]))
- Vega (per 1% volatility) = [Current stock price]×exp(-[Dividend yield]÷100×[Time to maturity])×npdf([d₁])×sqrt([Time to maturity])÷100
Example values
How the main result changes when Current stock price changes and everything else stays at the example values above:
| Current stock price | Call option price |
|---|---|
| 50 | 5.80e-5 |
| 75 | 0.22 |
| 100 | 5.65 |
| 125 | 23.12 |
| 150 | 46.72 |
| 200 | 96.34 |
Formula
Call = S e^(−qT) N(d₁) − K e^(−rT) N(d₂); Put = K e^(−rT) N(−d₂) − S e^(−qT) N(−d₁)d₁ = [ln(S/K) + (r − q + σ²/2)T] ÷ (σ√T); d₂ = d₁ − σ√TWorked example
Inputs — Current stock price: $100; Strike price: $105; Time to maturity: 0.5 years; Volatility (annual): 25 %; Risk-free interest rate: 4.5 %; Dividend yield: 1 %.
Results — d₁ = -0.0886; d₂ = -0.2654; Call option price = $5.65; Put option price = $8.81; Call delta = 0.4624; Put delta = -0.5326; Gamma = 0.02237; Vega (per 1% volatility) = 0.2796; Call theta (per day) = -0.0229.
Frequently asked questions
What does the Black-Scholes model calculate?
The theoretical price of European options.
Which volatility should I use?
Implied volatility from the market, or historical volatility as an estimate.
Does it work for American options?
Approximately for calls on non-dividend stocks; American puts need other models.
What formula does the Black-Scholes Calculator use?
Call = S e^(−qT) N(d₁) − K e^(−rT) N(d₂); Put = K e^(−rT) N(−d₂) − S e^(−qT) N(−d₁).