# Black-Scholes and Greeks Calculator

> Free Black Scholes calculator. Enter stock price, strike, days to expiry, rate, volatility and dividend yield to get call and put prices and the Greeks.

Source: https://learn.tradelabsai.com/tools/black-scholes-calculator/  
Track: Calculators · Level: Beginner · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "Black-Scholes and Greeks Calculator", https://learn.tradelabsai.com/tools/black-scholes-calculator/

The Black Scholes model, published by Fischer Black and Myron Scholes in 1973 and extended by Robert Merton, gives a theoretical price for European style options. It remains the foundation of options pricing and the language traders use to quote volatility. This calculator returns call and put prices along with the main Greeks: delta, gamma, vega and theta. It includes a continuous dividend yield, so it also suits index options. Enter volatility as an annual percentage, such as 20 for 20%.

## Calculator

*Interactive calculator: use it at https://learn.tradelabsai.com/tools/black-scholes-calculator/*

## How it works

```
d1 = [ln(S / K) + (r - q + σ² / 2) × T] / (σ × √T)
d2 = d1 - σ × √T
Call = S × e^(-qT) × N(d1) - K × e^(-rT) × N(d2)
Put = K × e^(-rT) × N(-d2) - S × e^(-qT) × N(-d1)
```

N is the cumulative normal distribution and T is time in years (days divided by 365). Vega is shown per one point of volatility and theta per calendar day. See [Black-Scholes Model](https://learn.tradelabsai.com/options/black-scholes-model/) and [The Option Greeks Explained](https://learn.tradelabsai.com/options/the-option-greeks-explained/).

**Example: An at the money one year option**
With the stock at $100, a $100 strike, one year to expiry, a 4% rate, 20% volatility and no dividends, the call is worth about $9.93 and the put about $6.00. The call's delta is about 0.62, so it gains about 62 cents for a $1 rise in the stock. Put call parity checks out: call minus put, about $3.92, equals the stock price minus the present value of the strike, $100 minus $96.08. Raising volatility to 25% increases both prices by roughly the vega, about 38 cents per volatility point, times 5. See [Put-Call Parity](https://learn.tradelabsai.com/options/put-call-parity/) and [Vega](https://learn.tradelabsai.com/options/vega/).

## The Greeks at a glance

| Greek | Measures | Lesson |
|---|---|---|
| Delta | Change in option price for a $1 move in the underlying | [Delta](https://learn.tradelabsai.com/options/delta/) |
| Gamma | Change in delta for a $1 move | [Gamma](https://learn.tradelabsai.com/options/gamma/) |
| Vega | Change in price for a 1 point change in volatility | [Vega](https://learn.tradelabsai.com/options/vega/) |
| Theta | Change in price for one day passing | [Theta](https://learn.tradelabsai.com/options/theta/) |

## Assumptions behind the model

| Assumption | Reality |
|---|---|
| Constant volatility | Volatility changes and differs by strike, the volatility smile. See [Volatility Smile and Skew](https://learn.tradelabsai.com/volatility/volatility-smile-and-skew/) |
| Lognormal prices, no jumps | Markets gap and have fat tails. See [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/) |
| European exercise | American options can be exercised early. See [Binomial Option Pricing Calculator](https://learn.tradelabsai.com/tools/binomial-calculator/) |
| Continuous trading, no costs | Hedging has costs and happens in steps |
| Constant interest rate | Rates change, but matter little for short options |

Despite these limits, traders use Black Scholes as a common language: they quote options by implied volatility and adjust for the smile. See [Implied Volatility Calculator](https://learn.tradelabsai.com/tools/implied-volatility-calculator/).

## Tips

1. **Use implied volatility** from the market for pricing, not just historical volatility. See [Historical and Realized Volatility](https://learn.tradelabsai.com/volatility/historical-volatility/).
2. **For American calls on non dividend stocks,** Black Scholes is accurate, since early exercise is not optimal.
3. **For American puts or dividend paying stocks,** use a binomial model.
4. **Days to expiry:** use calendar days divided by 365 for consistency with most platforms.
5. **Check against market prices;** large differences usually mean your volatility input is off.

## Trying scenarios

Change the days to expiry from 365 to 30 and watch time value shrink and theta grow. Move the strike above and below the stock price to see delta move toward 1 or 0. Raise volatility and notice that both calls and puts become more expensive. These experiments build the intuition behind the Greeks faster than formulas alone.

## Frequently asked questions

### What does the Black Scholes model calculate?

The theoretical price of European style options from the underlying price, strike, time to expiry, interest rate, volatility and dividends.

### What volatility should I use?

Usually the option's implied volatility from market prices, or a forecast of future volatility if you are estimating fair value.

### Does Black Scholes work for American options?

It works well for American calls on stocks without dividends; for American puts and dividend paying stocks, binomial models are more accurate.

Next, work backwards from a price to volatility with the [Implied Volatility Calculator](https://learn.tradelabsai.com/tools/implied-volatility-calculator/).

## Continue learning

- Next lesson: [Implied Volatility Calculator](https://learn.tradelabsai.com/tools/implied-volatility-calculator/)
- Previous lesson: [Option Payoff Calculator](https://learn.tradelabsai.com/tools/option-payoff-calculator/)
- Related: [Option Payoff Calculator](https://learn.tradelabsai.com/tools/option-payoff-calculator/): Free option payoff calculator. Choose a call or put, long or short, enter strike and premium, and see profit or loss at expiry, break even and max risk.
- Related: [Black-Scholes Model](https://learn.tradelabsai.com/options/black-scholes-model/): The Black Scholes model prices European options from five inputs. Learn the formula, its assumptions, a step by step example and where the model breaks down.
- Related: [The Option Greeks Explained](https://learn.tradelabsai.com/options/the-option-greeks-explained/): The option Greeks measure how an option's price responds to price, time, volatility and rates. Learn what each Greek means and how traders use them together.
- Related: [Implied Volatility Calculator](https://learn.tradelabsai.com/tools/implied-volatility-calculator/): Free implied volatility calculator. Enter an option's market price, strike, expiry and rate to find the volatility the market is pricing in, plus the expected move.
- Related: [Put-Call Parity](https://learn.tradelabsai.com/options/put-call-parity/): Put call parity links the prices of calls, puts, the underlying and interest rates. Learn the formula, a worked example, arbitrage logic and synthetic positions.
- Related: [Binomial Option Pricing Calculator](https://learn.tradelabsai.com/tools/binomial-calculator/): Free binomial option pricing calculator using a Cox Ross Rubinstein tree. Price American or European calls and puts and see the early exercise premium.