2. Gamma: why delta will not stay still

Derivatives 4: The Greeks

The previous module ended with a supposedly neutral position that stopped being neutral as soon as the market moved. That is the observation that gives the second Greek its point. Delta is not a constant of the option: it is an instant snapshot that expires with the underlying's first move.

Gamma is the rate at which delta changes

Gamma measures how much delta changes for every euro the underlying moves. If delta is speed, gamma is acceleration.

On our one-month 100-strike call, with the share at 100 and an assumed implied of 35%, gamma is around 0.04 a share. Translated: for every euro the underlying rises, that call's delta rises about four hundredths.

• With the share at 100, delta 0.52.
• With the share at 101, delta around 0.56.
• With the share at 105, delta around 0.70.

Notice the detail in the third point. The linear estimate (0.52 plus five times 0.04) would give 0.72, and the real value is somewhat below, close to 0.70. The reason is that gamma itself shrinks as the price moves away from the strike. Gamma too is a snapshot, and it too expires.

Why it is greatest at the money and close to expiry

Gamma is not spread evenly. It concentrates in two zones, and both answer to the same idea: gamma is high where the uncertainty about whether the option will end up worth anything is resolved quickly.

• Close to the strike. A call with a strike of 130 and one with a strike of 70 have stable deltas, close to 0 and to 1 respectively: a euro up or down does not change the verdict. The 100-strike one lives on the border, and there every euro reshuffles the forecast.
• Close to expiry. An at-the-money call with six months to run has a lazy delta: there is half a year for everything to change, so a euro barely moves it. The same call with three days to run has an extremely jumpy delta, capable of going from 0.30 to 0.80 on a 2% move, because the time to put things right has run out. In the limit, on expiry day delta can only be 0 or 1, and it jumps from one to the other as it crosses the strike.

This second zone is the one that surprises people. A long horizon gives the option a lot of value but little gamma; a short horizon gives little value and a great deal of gamma. They are different things and it is best not to confuse them.

Gamma in your favor and gamma against you

Whoever buys options, calls or puts, has positive gamma. Whoever sells them, negative gamma. And that difference of sign defines two opposite experiences of the same market.

With gamma in your favor, the position improves on its own. If the underlying rises, your delta becomes more positive and you gain faster; if it falls, your delta becomes more negative and you gain on the way down. The hedge adjusts in your direction without your doing anything, and when you rebalance you end up selling after rises and buying after falls.

With gamma against you, the mirror image happens and it is unpleasant. Every move leaves you worse positioned than you were, and to get back to neutral you have to buy after it has risen and sell after it has fallen. Here is the mechanical explanation of why hedges fail precisely when they are most needed: it is neither bad luck nor panic, it is the sign of gamma multiplying the size of the move.

And there is a nuance of size worth taking on board: gamma's effect grows with the square of the move, not in proportion. A 10% jump does not do twice the damage of a 5% one, but close to four times. That is why somebody with gamma against them can survive whole months of normal moves without breaking sweat and be ruined in a single session.

Worked case: the hedge that breaks on its own

A trader sells 10 contracts of our one-month 100-strike call and hedges the delta by buying shares. All the figures follow the previous module's assumptions.

• Delta of the short position: minus 0.52 a share, that is, minus 520.
• He buys 520 shares at 100 euros. Total delta: zero. Position hedged.
• First session. The share rises to 105. The call's delta rises to 0.70, so his short position becomes minus 700. With the 520 shares, his net delta is minus 180: he is short in a rising market. To get back to zero he buys 180 shares at 105.
• Second session. The share returns to 100. The call's delta goes back to 0.52, the short position returns to minus 520, and with 700 shares he is left with a delta of plus 180. To get back to zero he sells 180 shares at 100.

The underlying has ended exactly where it started. The trader has bought 180 shares at 105 and sold them at 100: 900 euros of loss on the rebalancing. In exchange, he has collected the passage of time for two days, around 70 euros a day on our assumptions, that is, around 140 euros. The balance of the round trip is roughly 760 euros against him.

Nobody got the direction wrong. The bill was decided not by the price's destination but by its route. And that is the module's lesson: a delta-hedged position is not a riskless position, it is a position that has swapped the risk of direction for the risk of the route.

What to take away

Delta neutrality is a point-in-time thing and it has to be renewed. The more gamma a position has, the faster it spoils and the more often it has to be readjusted. With gamma in your favor, every readjustment tends to add; with gamma against you, to subtract. The other half of the trade-off is missing: if having gamma in your favor were free, everybody would have it. What is paid for it is the business of the next module.