Black-Scholes and Greeks 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#

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 and The Option Greeks Explained.

The Greeks at a glance#

Assumptions behind the model#

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.

Tips#

1. Use implied volatility from the market for pricing, not just historical volatility. See Historical and Realized 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.