The Greeks · quants.md

An option’s price depends on the underlying price, time, volatility and rates. The Greeks are its partial derivatives with respect to each. They let you treat a complicated position as a sum of simple exposures.

Delta

\Delta = \frac{\partial V}{\partial S}

How much the option moves per unit move in the underlying. Under Black–Scholes, a call’s delta is $N(d_1)$ , between 0 and 1; a put’s is $N(d_1) - 1$ , between $-1$ and 0. At the money, delta is about 0.5. Delta is also the hedge ratio: short $\Delta$ shares per long call to be locally directionless.

Gamma

\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{N'(d_1)}{S\sigma\sqrt{T}}

How fast delta changes. Gamma is the same for calls and puts, largest at the money, and explodes as expiry approaches. Long options are long gamma: you gain from big moves in either direction, because your delta grows in the direction the market moves.

Theta

\Theta = \frac{\partial V}{\partial t}

Time decay. Long options lose value as time passes, all else equal. Theta is the rent you pay for gamma.

The central trade-off

For a delta-hedged option, the Black–Scholes PDE rearranges to

\Theta + \tfrac{1}{2}\Gamma\,\sigma^2 S^2 \approx 0

so the daily P&L of a hedged long option is about

\tfrac{1}{2}\Gamma S^2\big(\text{realised move}^2 - \sigma_{\text{implied}}^2\,\Delta t\big)

You make money if the stock moves more than implied volatility priced in, and lose theta if it moves less. Buying an option and hedging it is a bet on realised volatility versus implied volatility.

Vega

\mathcal{V} = \frac{\partial V}{\partial \sigma} = S\,N'(d_1)\sqrt{T}

Sensitivity to implied volatility, usually quoted per one vol point. Vega is largest at the money and for long-dated options. Long options are long vega.

Rho

\rho = \frac{\partial V}{\partial r}

Sensitivity to interest rates. Small for short-dated equity options; relevant for long-dated ones and in high-rate environments.

Signs at a glance

	Call (long)	Put (long)
Delta	+	−
Gamma	+	+
Theta	− (usually)	− (usually)
Vega	+	+
Rho	+	−

Greeks of a book

Greeks add across positions (scaled by quantity), which is what makes them useful: a desk can hold thousands of options and summarise its risk in a handful of numbers. A typical market-maker stays close to delta-neutral, manages gamma and vega limits, and earns the spread. See all five update live in the calculator.