# Black-Scholes Calculator (Option Pricing)

URL: https://speedycalculator.com/calculators/finance/black-scholes/
Last updated: 2026-10-25

> The Black-Scholes calculator prices European call and put options from the stock price, strike, time to expiry, volatility, risk-free rate and dividend yield, and gives the option Greeks delta, gamma, vega and theta.

## Inputs

- Current stock price
- Strike price
- Time to maturity (years, days)
- Volatility (annual) (%)
- Risk-free interest rate (%)
- Dividend yield (%)

## Results

- d₁
- d₂
- Call option price
- Put option price
- Call delta
- Put delta
- Gamma
- Vega (per 1% volatility)
- Call theta (per day)

## 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√T
```

## Explanation

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

FieldWhat to enterCurrent stock priceAmount in your currencyStrike priceAmount in your currencyTime to maturityUnits: years, daysVolatility (annual)In %Risk-free interest rateIn %Dividend yieldIn %

### 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 priceCall option price505.80e-5750.221005.6512523.1215046.7220096.34

## Worked 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.

## FAQ

**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).


## Sources

- [Black, F. & Scholes, M. (1973) – The pricing of options and corporate liabilities](https://doi.org/10.1086/260062)