Second-order Greeks: When the letters themselves move

First-order Greeks say how an option's price moves. Second-order ones say how the Greeks themselves move.

It is the difference between speed and acceleration. Delta is a speed. Gamma is its acceleration.

Gamma: how much does delta change when the share moves one euro?

Back to our option on a share at one hundred euros, with a delta of zero point five. Say a gamma of zero point zero three.

The share gains one euro: delta goes to zero point five three. The option follows the move a little more.

The share gains ten euros: delta climbs towards zero point eight. The more the share rises, the more the option follows it.

With these rounded numbers, a ten euro rise brings in about six euros fifty. A ten euro fall costs about three euros fifty.

That is convexity. Whoever buys the option gains faster than they lose. Whoever sells it lives the opposite.

This gift has a price. High gamma goes with high theta: you pay every day for this curvature.

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Gamma is strongest when the share is close to the strike price, and expiry is near.

That is why option sellers dread the last days. A small move can then tip everything over.

Second letter: vanna. Delta moving when volatility moves.

If expected agitation rises, an option far from the strike has more chance of reaching it. Its delta changes, without the share moving.

Third letter: volga. It is the vega of vega: how sensitivity to volatility changes when volatility changes.

It measures convexity to agitation. It matters most when volatility jumps suddenly, in the middle of a crisis.

Fourth letter: charm. Delta eroding with time, even if nothing else moves.

An option far from the strike sees its delta slide towards zero, day after day. An option already winning sees its own slide towards one.

A trader perfectly hedged on Friday may no longer be on Monday. Just because time has passed.

And then there are the letters almost nobody uses: speed, zomma, color.

Speed: gamma changing when the share moves. Zomma: when volatility moves. Color: when time passes.

These are derivatives of derivatives. Useful to a few market makers, useless to almost everyone else.

What to remember fits in one image: an option is not a straight line. It is a curve.

The first order gives the slope. The second says how the slope turns. And that is often where surprises hide.

Selling an option means selling curvature. While all stays calm, it is income. When things speed up, it is a debt.