Why sensitivities

An option's price depends on the spot, volatility, time, rates and dividends. The Greeks are the partial derivatives with respect to each, so they say how the price will move and how to hedge that movement. A book of options is managed almost entirely through its Greeks.

First order

For a call under Black-Scholes, with \(n(\cdot)\) the standard normal density:

$$ \Delta = e^{-qT} N(d_1), \qquad \Gamma = \frac{e^{-qT} n(d_1)}{S \sigma \sqrt{T}}, \qquad \mathcal{V} = S e^{-qT} n(d_1) \sqrt{T} $$ $$ \Theta = -\frac{S e^{-qT} n(d_1)\sigma}{2\sqrt{T}} - rKe^{-rT}N(d_2) + qSe^{-qT}N(d_1), \qquad \rho = K T e^{-rT} N(d_2) $$

• Delta is the hedge ratio: the number of shares that offsets one option. It runs from 0 to 1 for calls and from −1 to 0 for puts.
• Gamma is the same for calls and puts and peaks at the money near expiry. High Gamma means Delta changes quickly, so the hedge must be rebalanced often.
• Vega is also shared by calls and puts. Long options are long volatility.
• Theta is usually negative: the option loses value as time passes. Theta and Gamma are linked through the pricing equation, so a position long Gamma pays for it in Theta.
• Rho matters most for long-dated options. Calls gain from higher rates, puts lose.

To see Theta at work, watch time decay on the Black-Scholes page: the calendar runs to expiry with everything else held still, the value curve sinks onto the payoff, and Theta is drawn as the slope of the value at today's spot.

Second order

Vanna, Vomma, Charm, Speed, Zomma and Color describe how the first-order Greeks themselves move. Traders of volatility products watch Vanna and Vomma because they show how the hedge changes when volatility shifts. Charm and Color matter over weekends, when time passes with no trading.

Units

The raw derivatives are per unit of the variable: per one full point of volatility, per one full unit of rate, per year. Practitioners quote Vega per one volatility point (divide by 100), Rho per one percent of rate (divide by 100) and Theta per calendar day (divide by 365). The tables show both.

Numerical Greeks

When no formula exists, bump an input, reprice, and take the difference. The central difference

$$ \Delta \approx \frac{V(S + h) - V(S - h)}{2h} $$

is accurate to order \(h^2\). Trees and grids also offer the Greeks directly from their nodes, which is both faster and less noisy than bumping. Monte Carlo Delta by bumping is noisy unless the same random numbers are reused for both prices, which is why the simulator uses pathwise estimates where it can.