The Greeks

Delta, gamma, theta, and vega: the partial derivatives that tell a trader how an option moves.

The idea

The Greeks are the partial derivatives of an option's price, and they measure how that price responds when the market moves. Black–Scholes prices an option as a function $V(S, t, \sigma)$ of the stock price, the time, and the volatility, and a partial derivative of $V$ records its response to one input with the others held fixed.

Four Greeks carry most of the work:

$\Delta = \frac{\partial V}{\partial S}, \qquad \Gamma = \frac{\partial^{2} V}{\partial S^{2}}, \qquad \theta = \frac{\partial V}{\partial t}, \qquad \nu = \frac{\partial V}{\partial \sigma}$

Delta is the change in value per dollar of stock; gamma is the rate of change of delta itself; theta is the change per unit of time; vega is the change per unit of volatility.

Because the Greeks are derivatives, we use them as a local approximation. A move $\Delta S$ in the stock changes the option's value by roughly

$\Delta V \approx \Delta \cdot \Delta S + \tfrac{1}{2}\Gamma\,(\Delta S)^{2},$

with the linear term supplied by delta, the quadratic correction by gamma, and a further term added for each other input that moved.

The figure shows this geometrically: the value curve $V(S)$ is convex, delta is the slope of its tangent at the current price $S_{0}$, and over a large move $\Delta S$ the curve pulls away from that tangent by the gamma correction $\tfrac{1}{2}\Gamma(\Delta S)^{2}$ — which is why the tangent alone cannot be trusted far from $S_{0}$.

Ways to work on it

• Walkthrough. How delta, gamma, theta, and vega measure an option's sensitivity to price, time, and volatility.
• Practice. Use the Greeks to size hedges and estimate an option's change in value.
• Hardest. Estimate a value change from all the Greeks at once, then build a delta-neutral position.

Not sure where to start? Take the ten-question placement test.