Our model

Every price and every Greek on the site comes from one model, computed exactly. Here is the model, why we use it, and the conventions behind the numbers you see.

Black–Scholes–Merton

What we use

C = S\,e^{-qT}\,N(d_1) - K\,e^{-rT}\,N(d_2)

P = K\,e^{-rT}\,N(-d_2) - S\,e^{-qT}\,N(-d_1)

d_1 = \frac{\ln(S/K) + \left(r - q + \tfrac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}

d_2 = d_1 - \sigma\sqrt{T}

$S$: the stock's price now
$K$: the option's strike price
$T$: the time left to expiry, in years
$r$: the risk-free interest rate, continuously compounded
$q$: the stock's dividend yield, taken as paid continuously
$\sigma$: sigma, the volatility: how much the stock's returns swing in a year
$N(x)$: the chance that a standard normal value is below $x$
$\ln,\ e$: the natural logarithm, and its base $e$ , about 2.718

Fischer Black and Myron Scholes published the formula in 1973. The same year Robert Merton set it on a rigorous footing and extended it, among other things to stocks that pay dividends, which is the $q$ in the formulas. Scholes and Merton received the 1997 Nobel Prize in economics for this work; Black had died in 1995, and the prize is not awarded after death.

Bjerksund–Stensland

For comparison

\begin{aligned} C ={}& \alpha S^{\beta} - \alpha\,\phi(S, T, \beta, I, I) + \phi(S, T, 1, I, I) - \phi(S, T, 1, K, I) \\ & - K\,\phi(S, T, 0, I, I) + K\,\phi(S, T, 0, K, I) \end{aligned}

\begin{aligned} C ={}& \alpha S^{\beta} - \alpha\,\phi(S, T, \beta, I, I) \\ & + \phi(S, T, 1, I, I) - \phi(S, T, 1, K, I) \\ & - K\,\phi(S, T, 0, I, I) \\ & + K\,\phi(S, T, 0, K, I) \end{aligned}

The symbol $I$ stands for the trigger price: the model assumes the option is exercised the first time the stock reaches it. The symbol $\beta$ is set by the interest rate, the dividend yield and the volatility; $\alpha$ by the trigger, the strike and $\beta$ ; and $\phi$ is a helper built from the normal distribution, as $N$ is above. The formula holds while the stock is below $I$ ; at or above it, the call is worth its exercise value, $S - K$ , and a put follows from a call by the authors' own transformation.

Petter Bjerksund and Gunnar Stensland published this approximation in 1993 and refined it in 2002. It prices American options, which can be exercised before expiry, by assuming exercise at a flat trigger price. It is fast and close to the exact value, but it is an approximation.

Why we use Black–Scholes–Merton

It is a closed formula, so every Greek, up to the third order, is exact rather than estimated from small price changes.
It is the model textbooks, courses and broker education use to teach the Greeks, so what you learn here carries over.

Why your broker's numbers may differ a little

Some platforms price American options, for example with Bjerksund–Stensland. The difference shows mainly on deep in-the-money puts and on calls just before a dividend.
Different inputs: the interest rate, the dividend and the day count each platform uses.
The volatility: each platform takes its own implied volatility, often a different one for each strike and expiry.

The Greeks behave the same way in every case, and that is what we train.

Our conventions

Theta: Computed per year and shown per calendar day: the year's figure divided by 365.
Vega: Computed per 1.00 of volatility and shown per volatility point: a move from 20% to 21% changes the price by vega.
Rho: Computed per 1.00 of interest rate and shown per point, like vega.
Per share: Greeks are per share. A contract is 100 shares, so per contract is the per-share figure times 100 times the quantity; worked solutions show it beside the per-share figure, and the few questions asked per contract, where a per-share figure would round to zero, say so.
Time passing: Charm, veta and color measure what happens as time passes, not as time to expiry grows; some references state veta and color the other way around, with the opposite sign.
Day count: Calendar days, 365 to the year: $T$ is the days to expiry divided by 365. A typical one-day stock move is sized on 252 trading days.

Education only, not financial advice. Options involve substantial risk. Read the full disclaimer.