The Greeks, without the calculus

The Greeks are four questions about the same position, each asking what happens to its value when one thing moves and everything else stands still.

You do not need the calculus to use them. You need to know which one is currently your problem.

Delta — how much like the underlying is this?

Delta is the change in the option's value for a one-point move in the underlying. A call runs from 0 (hopelessly out of the money) to 1 (so deep in the money it behaves like stock). A put runs from 0 to −1.

The useful reading is as equivalent position size. A 0.4-delta call on 100 shares behaves, for small moves, like owning 40 shares. Sum the deltas across a book and you have your net directional exposure in a single number — which is exactly what a trading desk's risk system reports.

Gamma — how fast is delta changing?

Gamma is the rate of change of delta. High gamma means your directional exposure is unstable: the position becomes rapidly more long as the price rises and more short as it falls.

Gamma is largest at the money and close to expiry. This is why the last day of a contract's life is the dangerous one — a position that was comfortably hedged in the morning can be badly wrong by the afternoon without the underlying doing anything dramatic.

Long options are long gamma, which is pleasant: the position gets longer as it wins. Short options are short gamma, which is the opposite, and it is the mechanism by which sold options produce sudden large losses.

Theta — what does waiting cost?

Theta is the value lost per day from time passing alone. It is negative when you are long options and positive when you are short.

Theta and gamma are two sides of one trade. Being long gamma — profiting from movement — is paid for with negative theta. Being short gamma earns theta and sells you the risk. There is no position that is long gamma and long theta; if you think you have found one, check the trade again.

Vega — what if the market changes its mind about risk?

Vega is the change in value for a one-point move in implied volatility. It is not actually a Greek letter, which tells you something about how the vocabulary grew.

Vega matters more than most people expect. You can be exactly right about direction and still lose, because implied volatility collapsed after the event you were positioned for. This is routine around earnings: the move happens, and the option holder loses anyway.

A straddle is nearly pure vega at the money — flat delta, and its value is almost entirely a view on volatility. Drag the volatility slider and watch the pre-expiry line lift and fall while the expiry line stays exactly where it is. That difference is what you are actually trading.

Which one kills you

In roughly this order:

1. Gamma, because it turns a manageable position into an unmanageable one without warning, and it is worst exactly when you have least time to react.
2. Vega, because it moves against you at the same moment as everything else — volatility rises when markets fall, so a hedge gets more expensive precisely when you want it.
3. Theta, which is at least honest. It takes a predictable amount every day and never surprises anyone.
4. Delta, which is the risk everyone thinks about and the one most easily hedged.

The calculator reports all four aggregated across a whole position, which is the only level at which they are useful — individual leg Greeks rarely tell you what you want to know.